0.999... = 1
While looking into using html for academic docs, I typed this one up (based on Thomas Park's PubCSS formatting of ACM SIG Proceedings articles).
As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ?
Anyway, please feel free to point out any errors in the attached note.
(For some reason, internal links in the pdf don't seem to work, not sure why; they're find in the original html.)
EDIT: Incidentally noticed that one of the reference links had the wrong url. Fixed.
As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ?
Anyway, please feel free to point out any errors in the attached note.
(For some reason, internal links in the pdf don't seem to work, not sure why; they're find in the original html.)
EDIT: Incidentally noticed that one of the reference links had the wrong url. Fixed.
Comments (611)
Either way, I think the attached proof is valid, but others are invited to point out errors.
Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same.
Looks fine to me!
I think there's a common source of confusion related to the one you pointed out, 1 isn't in the infinite series, but the infinite series converges to its supremum, which is 1. People confuse maximum and supremum a lot.
This argument isn’t actually valid, because it could arguably be the case (if not for other, valid proofs that 0.999... = 1) that 0.999... is the very last number before 1, so there is nothing between them even though they’re not (this hypothetical person would argue) the exact same.
I think the main issue is whether someone is familiar with the notation conventions within mathematics.
Both the rationals and the reals are densely ordered.
For any two different numbers, there's a third between them.
I suppose they might say that 0.999... isn't a number.
Indeed, if that argument was taken to be valid in mathematics, I could also argue that since there is no natural number between 9 and 10 to be found, that means that 9=10.
Besides a notation like 1.000 isn't a mathematical notation if it doesn't differ from 1. If there is no difference the preferred and obliged notation is 1 not 1.000.
1.000 is a valid notation in applied sciences like physics and chemistry as in applied sciences there is something like numbers of significance. 1.000 in physics or chemistry refers to any number in the collection [0.9995,1,0005>, while in mathematics 1.000 just means 1.
And it's a fairly large area; might scare some away.
It would be valid when restricted to the real numbers as originally intended, I think. There does not exist an x between the limit of that sequence and 1; it would simultaneously have to be greater than every sequence element of 0.9 0.99, 0.999... and less than 1, but since that sequence is monotonic, the "least" such strictly positive real number strictly between the sequence limit and 1 doesn't exist as you can always find one less than it. It's another way of reading the epsilon N proof.
The naturals aren't densely ordered like the rationals and the reals. ;)
You can say The limit = 1 but not .999...= 1
{0.9, 0.09, 0.009, ... }
is thus the limit of the series
{0.9, 0.9 + 0.09, 0.9 + 0.09 + 0.009, ...}
or in other words the limit of the series
{0.9, 0.99, 0.999, ... }
which you can clearly see converges to 1.
“0.999...” is of course also equal to the sum of the infinite series
{0.9, 0.09, 0.009, ... }
because that’s just what decimal notation means. And since the limit of the series of partial sums of that infinite series is 1, that means the total sum of that infinite series represented by 0.999... is also 1, so 0.999... = 1.
depends on how accurate the system needs to be. NASA does have room for error but it has extremely less room for error than chevrolet does. 0.999 might be good for some systems but not good enough for other systems.
Everything can be quantified including the personality of a person or people. I'm not sure everything should be quantified though. There is an ancient book that talks about that in regards to whether a nation should have a census. I'm not actually against modern censuses.
0.999... is not equal to 0.999
0.999... minus 0.999 equals 0.001
true but how does that negate what i said. Are you familiar with scientific notation. Something to consider is Pie is a never ending set of integers after the decimal. For NASA the number of those integers would need to be much greater than for Chevrolet engineers. Perhaps i wasn't clear enough the first time.
.999 and 1 are not the same thing. Is that what the OP was saying? That would be incorrect. In some systems you could claim it is close enough while in other systems they would not be close enough. Everything including personalities can have systems analysis and design principles applied to them.
typically when "0.999... = 1" is brought up, you'll find a flurry of objections, you just watch. :)
Either way, I think the attached proof is valid, but others are invited to point out errors.
Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same.
___________________________
Perhaps if the OP was written with more clarity you could prove otherwise but 0.999 is different from 0.999999 and also different from 1.0 or 1. Would you like to rewrite the OP so that my above statement is invalid?
9/9 = 9 × (1/9) = 9 × (0.111...)
1/9 doesn't equal exactly 0.111 nor 0.111111. But for some systems it would be close enough.
I'm confused what sort of system would this sort of concept be applicable too. I would agree that for some systems 0.111 would be close enough and for NASA (they have less room for error), 0.1111111111111111111111111111111111111111111 might be required.
A NASA engineer may invoke a handful of mathematical theorems and formulae out of physics, involving ?, differentiation and integration (which are calculus related to limits), only to find that x + 7m was what they were looking for. For example, there are all kinds of rules of differentiation, many of which are, or can be, proven by limits. Also, no manner of discussion and voting can somehow make ?2 mysteriously become 1.414.
Years ago I vaguely recall having done error analysis in physics experiments. Tedious. Maybe that's more along the lines of what you're thinking of?
What's this ancient (science/mathematics) book you're referring to anyways?
Quoting christian2017
A sensible person would just use 1/9. Loss-less. Unless or until they needed to write it out differently anyway.
Looks like the confuzzlement mentioned earlier.
There are infinitudes of numbers. Therefore there aren't numbers? Hmm...
lets not worry about that ancient book because i was replying to someone else.
Yes 1/9 is fine. Like i said it depends on the system as to how many integers after the decimal will be used for a variable or component of an equation.
Yes limits as for example in calculus make things alot more simple if a person took high school calculus. I'm familiar with the greek method of exhaustion as well.
I don't have a problem with what you are saying however am i correct that you could summarize what your OP is stating with: "some systems require more precision than others in terms of how many integers are used after the decimal"? I wouldn't be surprised if you have a less clear but at the same time more professional way of saying what i just said. Most of the people on here including me are amateur arm chair quarterback mathematicians.
0.111 does not equal 1/9. It’s close, and you’re saying it might be close enough for some purposes, but for others you might need more 1s. But so long as you have finitely many 1s, it won’t equal exactly 1/9.
But 0.111... (with that “...”, that’s very important) equals EXACTLY 1/9, by definition. It has infinitely many 1s. That’s what the “...” means: “keep repeating the preceding pattern forever.”
0.111 x 9 = 0.999, which is not 1.
0.111... x 9 = 1/9 x 9 = 0.999... = 1, exactly.
i agree. I'm used to the line that goes over the 0.9999.... in which case you wouldn't need the .... . ______________________________________ i don't type alot of math equations so i didn't equate .... with that line that goes over the integers that come after the decimal. You win. No sarcasm intended.
lol. this is the bullshit we do on here. this is so we drink less alcohol.
Depends on how one looks at it, the collection of any of those three are all infinite. I don't see a difference in the applied logic, no matter how densely ordered the collection of number is, the logic applied remains the same.
the ... means the limit is taken. IE, .999... = 1
Try this: .999... is not a number because it has a indefinite extension. A number is an object and an object cannot have an indefinite extension. So discussing the status of .999... under the presumption that it is a number, is misleading oneself by starting with a false premise.
If a person wanted to say "1" they would say 1. If a person wanted to say ".999..." they would say .999.... The two symbols have a distinct meaning and it is not a case of using different symbols to refer to the same thing. The one refers to an object, the other something indefinite. Why try to argue that they each refer to the same thing? Since it is so obvious that the two symbols have a different meaning, it requires accepting that false premise just to start the discussion. Then as soon as you accept the false premise there is nothing to discuss anymore, because the falsity has been validated.
But it has to do with the way language is being used. 'The total sum' is not the same thing as 'equals'.
The total sum cannot be explicitly demonstrated. It is really a concept that cannot be substituted with 'equals'.
Quoting fdrake
Yes but 'limit' is not the same as 'equals'.
0.999... is equal to the limit of the sequence {0.9, 0.99, 0.999, ...} is equal to 1.
Actually it is, that's why they use the equals sign. It's the entire essence of calculus.
And 1 - 0.999... = 0.000... = 0 (sanity check)
True. But they don't mean .999... = 1. They mean the Limit of the infinite sum = 1. There's a difference.
The sum being 9/10 + 9/100 + 9/1000 + ...
That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which is 1.
But the problem is what does .999... mean? How many 9s are we talking about? An infinity of them, of course. But what is an infinity 'of' something?
I don't think so. .999... describes the series. The limit of it is infinitely far away.
Well there's a 9 in place 1, a 9 in place 2, and 9 in place 3, and so on...
There's a 9 in place 20, a 9 in place 4 billion... apparently, there's a 9 in all places n where n is a number.
So, how many numbers are there?
What is [math]\frac 13[/math] in decimal?
Then you don't know what the symbols mean and should read the OP's article!
There are always people that disbelieve in it. It isn't so surprising, since it involves infinity, limits, convergence and monotonicity. What I find especially frustrating about it is that people can be shown a formal proof of it with sources for everything and still refuse to read it and ask exploratory questions.
I have no idea!!! I suspect there is 'an infinity of 3s' but what does that mean? That's the crux of the biscuit.
Quoting InPitzotl
I don't know because I don't know if 'how many' applies to infinity. At the beginning of the theory of limits mathematicians were careful to say that we should say a series 'tends towards' a limit. It is a conservative statement.
Quoting fdrake
But does anybody know? Intuitively yes, we can see that the limit is 1. But limit is not the same as equals. The argument is subtle. What is being said is 'After an infinity of 9s'. That is what I am suspicious about. I'm not sure what 'an infinity of' means. Or if it is a coherent statement.
Yes, many people do. Those who've taken the time to study it.
0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1.
0.999... is not an element of the sequence {0.9,0.99,0.999,...}, it is the limit of that sequence, which is 1.
.999... could not be the limit. To write the limit you'd have to have .999999999999999999999999999999999999999999999999999 - an infinity of 9s. And we can't write that, whatever it means.
.999... is a symbol for 'an infinity of' 9s. But what does that mean?
Which is exactly why you write 0.999"...". It is the limit.
Yes, I understand what you are saying. But if infinity is not a number how can you have an infinity "of"?
It's [math]0.\overline{3}[/math].
In one respect it is shorthand.
The sequence {0.9,0.99,0.999,..} is the sequence of partial sums [math]s(n)=\sum_{i=1}^{n}9 \times 10^{-i}[/math]. IE {0.9, 0.99,0.999...}={s(1),s(2),s(3),...}. 0.999... is the limit of that sequence. It is equal to the limit of that sequence. Which is 1. 0.999... is equal to1.
In another respect, and you will not like this even more because the math is more advanced, the cardinality of the set of sequence elements {0.9,0.99,0.999,...} is aleph-null, the smallest infinity. That transfinite number is not a real number.
If you don't understand these issues, you should read through @jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you.
Infinitesimals have never really been understood rigorously. Have you heard of Berkeley's "Ghosts of departed quantities"? Below 'Fluxions' means infinitesimals.
Ghosts of departed quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities (Boyer 1991), Berkeley wrote:
It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?[6]
Edwards describes this as the most memorable point of the book (Edwards 1994). Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions. (Katz & Sherry 2012)
Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).
It has nothing to do with infinitesimals.
I have read through it. These are mathematical expressions and as such they are symbols. They represent infinity. But mathematicians were aware of these issues when formulating the calculus and they cautioned against saying 'equals'. They said we should say 'Tends towards the limit'
That only shows you didn't understand it. @jorndoe defines what a limit is in it! It doesn't even need infinity in the definition.
The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1.
Note that in the article cited in the op they don't write
[math]\sum_{i=1}^{\infty}[/math] etc = x
They write
lim [math]\sum_{i=1}^{\infty}[/math] etc = x
These are two different concepts.
How do you know it's infinity and not, say, an octillion?
Quoting EnPassant
So if you want to be conservative, just say what you're talking about... "the number of counting numbers".
Nice. But it still begs the question: what does it mean to say x = 0.999...?
It means an infinity of 9s but what can that mean when infinity is not a number?
You have to say x = lim 9/10 + 9/100 + 9/1000 + ...
And we are back to square 1. (Uh, I mean square 0.999...)
Quoting InPitzotl
Because I know it is not any nameable number.
[math]1 = \sum_{i=1}^{\infty} 9 \times 10^{-i} = \lim_{n \rightarrow \infty} s(n) =0.999...[/math]
Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 =
It should be written
lim [math]\sum[/math]
not simply [math]\sum[/math]
What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.
Is x=y ?
It depends on how you read these expressions. I'll grant you that 0.999... can be identical to the limit of the series if that's how you interpret it. But if you do you interpret it as a limit not as equals.
You can say 0.999...= 1 if by that you mean the limit of 0.999...
What is the difference between
[math]\sum[/math] = x and
lim [math]\sum[/math] = x ?
Need to figure out that sexy equation mode.
You answered only half of the question... the half about your knowing that it's not any nameable number. What about the other half... how do you know it's infinite?
ETA: Probably obsolete now... once you accept that it's infinite, we could then enumerate as statements the meaning of each finite decimal expansion, and agree that we have no such infinite statement; then we can define the infinite statement to mean the limit (after possibly a quibble that we're defining the finite expansion's meaning anyway).
That is exactly how it is meant. That is what 0.999... means.
The infinite sum notation [math]\sum_{n=1}^{\infty}[/math] just means [math]\lim_{n\rightarrow \infty} \sum_{i=1}^{n}[/math], that [math]\lim[/math] means; for a sequence [math]\forall \epsilon >0 \exists N : \forall n > N |s(n)-L| < \epsilon [/math], the value [math]L[/math] is the limit. [math]0.999...[/math] is the limit of the sequence of partial sums [math]s(n)[/math], which is equal to 1.
Wrote a guide here. It's essentially LateX if you're familiar with it, though without lots of the standard packages.
[math]<3[/math]
:up:
to .
I just left it out for brevity. I'm sure you know what i mean.
Quoting fdrake
Ok, I'll accept that. But what we are talking about here is subtle and the " = " sign in calculus can be misleading:
[math]\sum 1/x[/math] is a literal sum.
lim [math]\sum 1/x[/math] is not a sum. It is the limit towards which the sum (over the range) converges.
That is the difference.
The first post on this math forum explains how to us the math tags.
https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts
Great. That means you accept [math]0.999...=1[/math]!
Only according to a strict interpretation of the ' = ' sign: 1 is not the sum. It is the limit of the sum. So 0.999... = 1 does not mean it is literally 1. It means 1 is the limit.
Quoting EnPassant
You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1.
Yes of course. I have not said it is not the limit. I said 0.999... is not equal to 1 if we are talking about a literal sum. If we are talking about a limit, yes, the limit is 1. I keep saying a sum and a limit are not the same thing.
https://en.wikipedia.org/wiki/Limit_(mathematics)#:~:text=In%20mathematics%2C%20a%20limit%20is,)%20%22approaches%22%20some%20value.&text=The%20concept%20of%20a%20limit,direct%20limit%20in%20category%20theory.
Yes, 'partial' sums. That means the sums are finite. Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a limit. As more terms are added indefinitely, the limit is approached more closely. That is what calculus is saying.
What I am saying is that a literal infinite sum is probably an incoherent concept.
It does though. It defines the sum of an infinite series as the limit that the partial sums approach.
Whether or not .999...qualifies as a numeral is a matter of interpretation. What I meant, as you seem to have difficulty in understanding, is that it does not signify a number. Whether or not .999... is a numeral is a semantic issue involving the meaning or definition of "numeral" being applied, and is irrelevant to the point that I was making, that .999... does not signify a number. If you'd like to discuss this point, please do not just create a distraction like that.
A numeral is a special type of sign. To know whether .999... qualifies as a numeral would require a definition which dictates the criteria for being a numeral.
But as I said, that issue is just a distraction. What matters to the present discussion is that .999... does not represent a number. Nor does .111... represent a number, and that's the problem with the op. Whether or not these could still be numerals, which do not represent numbers, is a matter of one's interpretation of "numeral", and this is not relevant.
The rest of your post doesn't seem to make any sense.
Hm. So present the object 1; that object to which "1" refers. Then your point will be made.
But you cannot. Numbers are not objects.
"1" does not refer to anything.
Quoting tim wood
:rofl:
[quote=Berkeley]Ghosts of departed Quantities[/quote]
goes along with (contemporary) calculus. They do occasionally come up as matters of convenience or tradition (e.g. in notation), but not of necessity.
Ghosts couldn't comprise a "boundary" between 0.999... and 1.
[sub]
23, 24, 25, 26, 27, 28
[/sub]
What a silly thing to say. .999, eighteen, XVI, and .999... all represent numbers.
In other words, the quantity 1 = the quantity 0.999...
The only difference is in the expression.
The difficulty with accepting the truth of 1 = 0.999...is in no small part due to the fact that the one's place in 0.999... is vacant while that in 1 is occupied by the digit 1. Quite obviously there's a discrepancy here - how can 1.000... be equal to 0.999...?
But seeing it that way is to ignore the infinite 9's that follow the decimal point in 0.999... Infinity should never be ignored is the main lesson here.
I understand what you are saying but a literal infinite sum is not considered in calculus. If partial sums are added they approach the limit. If more terms are added it gets closer to the limit and so on. This is how people like Cauchy formulated calculus. They don't consider literal infinite sums. And this is the question I am raising: is the concept of an infinite sum coherent?
When infinitesimals were invented/discovered it was asserted that, they are so small, no matter how many of them are added together, you end up with another infinitesimal. This is why Berkeley balked at these 'Ghosts of departed quantities.' They were so small you couldn't do anything with them.
As I see it, the conclusion that was being reached was that an infinite sum of zeros add to 1. And this did not make sense. So infinitesimals were invented. This is why I have reservations about literal infinite sums.
I'd disagree since I can express the outcome of the limit of 1 divided by x with x approaching zero, but I cannot simply divide by zero. Further more depending whether I approach zero with x from infinity or from minus infinity provides two vastly different outcomes. Saying that the limit of something equals the equal sign is effectively saying that minus infinity = infinity in the provided example. Hence in mathematics we say that 1 divided by zero is unsolvable rather than saying that it's both infinity and minus infinity.
I take my definition of "number" from OED: "an arithmetical value representing a particular quantity and used in counting and making calculations". Notice specifically the criteria "particular quantity". This rules out the possibility that .999... is a number.
Show me a definition of "number" which allows that .999... is a number.
Kummer did not believe in real numbers; "God made the integers and all the rest is the work of man". This is a bit extreme as real numbers - whatever they are - are cool. Without them we would not have calculus.
I think the system of real numbers allows that "number" remain undefined, indefinite, and this is why "the real numbers" is not a fixed system. Rigorous defining of "number" has been withdrawn for the sake of producing the real numbers.
...and there's Meta's problem.
Family Resemblance.
"Numbers are symbols that are only useful and meaningful when applied to real world situations."
That's true for physics. That's even true for the so-called applied math. But this is not true for math in general. There exist huge math theories that have no applications at all, and math people often tend to be proud of such math that can not be used for any practical needs. Most (not all) of tensor-related math is a good example.
When I was a schoolboy, and we learned infinite (both periodic and non-periodic) decimal fractions in the class, there was a clear phrase in the textbook we used, that 0.999... is taken as another representation for 1. Surely the situation may be different in other countries, but I'm still a bit surprised anyway.
Ah, I think I understand. This is a language barrier. In the language spoken by the mathematics community, .999... represents the same particular quantity that 1 does.
Yes exactly, calculus works in practice. You can sum [math]10^{1000000000000000000000000}[/math] terms and that's fine because it is a finite sum.
But what is [math]10^{\infty}[/math]?
It may well be that the infinite sum is 1 but mathematicians were suspicious about such a concept because infinity is not a number. This is why calculus is formulated in terms of limits, not infinite sums.
In geometry the length of the line - in this example [math]\sqrt{2}[/math] - is exact. But the decimal expansion representing it is not, unless we go to an infinite number of places.
Sounds like mathematical poetry.
No, you're not getting the point. There's no need for any "infinite sum", schoolchildren don't learn such complicated math that early. I don't really remember what school year it was, may be 6th (that is, 12-13 years old children were learning this). The phrase "0.(9) is another representation of 1" came without any explanations in that textbook; only several years later, I understood why it is true.
Quoting EnPassant
And so what, actually? What do you think an "infinite decimal fraction" is? Well, by definition, it is exactly that famous Limit. For this particular case, you can either take the limit of the sequence 0.9, 0.99, 0.999, ..., or you also can work in terms of so called series (see https://en.wikipedia.org/wiki/Series_%28mathematics%29) and consider the series of 9/(10^n), that is, 9/10 + 9/100 + 9/1000 +..., the result will be exactly the same. Mathematicians are good with both. Furthermore, I'd say mathematicians are "suspicious" (well, this is the right word) about infinite decimal fractions as such. Math is not done on decimal fractions; physics is, but physics is different. For a mathematician, rational number is a fraction n/m (where n is integer, m is natural). The problem is that there exist sequences of rationals that obviously have limit, but the limit can not be represented as n/m, so in order to "close up" the set of numbers, we need more numbers. That's how "real" numbers appear. Infinite decimal fractions are just one simple case of these sequences and series.
Exactly. They enjoy what they do just like poets enjoy writing their poems.
But this is what is being asserted: 1/10 + 1/100 + ...taken to an infinite sum of terms. Let S be this literal infinite sum. What is S? Is it 1 or [math]\infty[/math]? That is what the problem is, we don't know what a literal infinity is because finite algebra does not apply to literal infinities.
Earlier on someone wrote a very convincing 'proof':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1
All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1.
But what if S is infinite? That is, what if an infinite sum of terms is infinite?
Then we have [math]9\infty = 9[/math]
[math]\infty = 1[/math]
That's the thing, we don't really know what S is because you can't apply finite arithmetic to infinite sums.
Is strong encryption a real world situation?
It is not a proof at all, there's no theorem here. This is standard and well-known technique to convert a periodic decimal fraction back to rational form. Yes, for 0.(9) it works as expected.
Quoting EnPassant
It is not. For any decimal fraction, the limit of both the corresponding sequence and the corresponding series is always finite, which is obvious enough: it is always between 0 and 1, there can be no infinity here.
Furthermore, the equality 9S = 9 has the obvious solution S = 1, and it doesn't have any other solutions. In most math theories, infinity is not considered as a number so we can't multiply 9 and infinity; however, if we add infinity to the domain (which is rarely done in math, but is still possible), then 9 times infinity will be again infinity, not 9. So infinity is not a solution for the equation, even if we agree to use infinity as a number.
BTW, when it comes to limits, including the limits of "infinite sequences", we don't need the infinity as such. If you recall the definition of the notion of limit, the symbol "infinity" (I can't figure out how do you make them appear here) is not used in that definition at all. In this context, the word "infinity" simply means that we can take as many members of a sequence as we want or need, and no one will stop us from getting more of them. Actually, the word "infinity" in math is not as complicated and scary as you can expect.
This doesn't exist, because there's no limit for the sequence 10, 100, 1000, ... (That is, the sequence {10^n}). If you take, e.g., 1.1 instead of 10 (or actually any number between 1 and 2, but strictly lesser than 2), the limit will exist, so we'll be able to say what it is. Actually, as far as I remember, the limit will be the number of 1. There will even be a finite sum of the series (like 1.1 + 1.1*1.1 + 1.1*1.1*1.1 + ...).
This standard "proof" is of course bullpucky. It's true, but not actually a proof at this level. Why? Well, as you yourself have pointed out, the field axioms for the real numbers say that if [math]x[/math] and [math]y[/math] are real numbers, then so is [math]x + y[/math]. By induction we may show that any finite sum is defined. Infinite sums are not defined at all.
To define infinite sums, we do the following:
* We accept the axiom of infinity in ZF set theory, which says that there is an infinite set that models the Peano axioms. We call that set [math]\mathbb N[/math]. The axiom of infinity is a very powerful assumption that allows us to get higher mathematics off the ground. However, the axiom of infinity is manifestly false in the physical world. It's precisely at this point that mathematics diverges from physics. It doesn't matter how useful math is for physics. One must realize that no mathematical truth can have ontological significance in the physical world. The fact that .999... = 1 is in the end no more meaningful than asking why the knight moves as it does in chess. It's just a consequence of the rules of a formal game.
* Having modeled [math]\mathbb N[/math] within set theory, we use equivalence relations to build up the sets [math]\mathbb Z[/math] and [math]\mathbb Q[/math] of integers and rationals, respectively.
* Using Dedekind cuts (or any of a number of other constructions) we define the real numbers [math]\mathbb R[/math] and show that they are a complete, totally ordered, Archimedean field. Such a field is categorigal, meaning that any other such field is isomorphic to the one we've constructed. Complete in this context means that the reals have the least upper bound property, which says in effect that there are no "holes" in the real numbers. This is the property that characterizes the reals and distinguishes them from the other famous densely ordered set, the rationals.
* Having rigorously defined the real numbers and shown that they are essentially unique, we then define the limit of a sequence of real numbers via the usual epsilon definition.
* Having defined the limit of a sequence, we then define (as you have pointed out) the sum of an infinite series as the limit of the sequence of partial sums.
* Having done all that, we then prove a theorem that says that if we have a convergent sequence, we can multiply each of its terms by a constant, and the resulting sequence converges to that constant times the sum of the original sequence.
That last theorem is the ONLY WAY to justify [math]10 \times .999... = 9.999...[/math]
So using that proof requires mathematical principles and reasoning far more sophisticated than the mere fact that .999... = 1. This "proof" is at best a heuristic for beginners. I wouldn't object to it if it were presented this way. But every time someone writes [math]10 \times .999... = 9.999...[/math], they are implicitly invoking the theorem on term-by-term multiplication of a convergent infinite series by a constant; and they are leading students into confusion.
As you have noted, addition in the real numbers is only defined for finite sums. To define infinite sums requires a whole lot of technical machinery based ultimately on the axiom of infinity. We must in fact make a powerful conceptual leap, one that contradicts everything we know about the real world, to get a satisfactory theory of the real numbers.
Same remarks for the 1 = 3 x 1/3 = 3 x .333... = .999... proof. A heuristic for beginners, but hopelessly bogus as an actual mathematical proof. Not because it's wrong, but rather because it is secretly invoking mathematical principles that are deeper and more sophisticated than the fact claiming to be proved.
Then it wouldn't be representable by a repeating decimal. Only series that converge (not diverge to infinity) can be represented by repeating decimals. So you'll never have this problem with a proof of that sort.
You need to read back a few pages to see what I'm saying. It is like this-
It is being asserted that 1/10 + 1/100 + ...taken to an infinity of terms is 1.
If we take a finite number of terms they converge to a limit. But a limit is not a sum. It is what a finite sum converges to. We can't know what an infinite sum is. It may well be 1 but it may also be infinity. We don't know because you can't apply finite arithmetic to infinity. You can't jump to an infinite sum and assume it is 1.
No. Calculus is formulated in terms of finite sums and limits. You can't jump to infinity and expect the rules of finite arithmetic to apply. Jumping from the finite to the infinite is an infinite distance and we don't know what happens there.
The limit of the series of finite sums represented by 1/1 + 1/2 + 1/4 + 1/8 ... is 2. You do agree that you can calculate a limit, right? We can know that the limit of that series is not infinity, right? What do you think "the limit is 2" means, if not "this will never add up to more than 2, no matter how many terms you add"? And given that, we know that it will never, ever add up to infinity.
By a finite number of terms.
Quoting Pfhorrest
Yes, you are correct but what is an infinite sum of terms? What happens when you sum an infinity of terms? Calculus does not account for this.
The limit of the series of partial sums. By definition.
Quoting Wikipedia
You need to provide the link. As a working definition that may well be useful but there are still problems with infinite sums.
It can be defined using 10-adics:
[math]\underset{n\to+\infty}{lim}{|10|^n_{10}}=0[/math]
The word "Wikipedia" at the bottom of the quote is a clickable link to the article in question.
They don't say the infinite sum, ie the sum of all terms.
Quoting Wikipedia
It hardly matters. An infinite sum is undefined because nobody has ever computed it. The problem is that while it is ok to apply the logic to finite quantities, nobody knows what an infinite sum is.
Actual mathematicians do.
Emphasis mine.
An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it.
Quoting Wikipedia
...that's a term of art. It means to increase without bound. You're choking on mathematical language that you think represents some ideal thing that it just doesn't represent.
ETA: Quite honestly, this whole discussion of infinite sums reminds me a lot of the maths video from Look Around You:
...if I'm to understand correctly, the quibble you have is about what the "actual" infinite sum "actually" adds up to. Ironically, your quibble includes the notion that infinity is not a number. So I have no idea what you're talking about. Mathematicians define infinite sums differently.
See next answer.
Quoting InPitzotl
That still begs the question what is an infinite sum if nobody has ever computed it? You can't jump from the finite to the infinite and expect finite rules to apply. And it is questionable that it has been defined. All that has been rigorously defined is a limit.
That begs the question of what the heck you mean when you're talking about an infinite sum. Mathematicians regularly compute what they mean by it. It's the thing you're talking about that's nonsense, not the thing mathematicians are talking about.
Quoting EnPassant
Case in point... what are you talking about?
Quoting EnPassant
No, it's factual that it has been defined. Definitions aren't handed to us from an abstract guy giving out tablets in some Platonic/Pythagorean plane of existence. They're established by people... in this case, it's technical definitions given by mathematicians. They define it. You question that they define it, but that doesn't erase the fact that they, indeed, defined it.
Quoting EnPassant
The infinite sum itself has been defined to be the limit... by mathematicians... who are the both the speakers of and designers of the language of math.
They compute limits which are not the same as sums. A limit is what a finite sum converges to.
I'm aware of that. But it has not been explicitly defined. It has been defined in terms of limits which are limits of finite sums.
The sum of an infinite series is defined as the limit of the sequence of partial sums of the series.
I don't understand your objection. You are entirely right that there is no such thing as an infinite sum until we define it. And how do we define it? We first define the limit of a sequence, then we define the sum of an infinite series as the limit of sequence of the partial sums. To me this seems rather clever. We have something that at first makes no sense, then we MAKE it make sense with a clever definition. What exactly are you objecting to?
Given that the sum is by definition the limit, then by definition it's the same. You keep tripping over this same point.
Quoting EnPassant
Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit.
Quoting EnPassant
Right after the citation @Pfhorrest gave:
Quoting wikipedia
Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time). However, if you postulate it this way - something like "when we see the infinity sign, we can't know anything" - then (despite you thereby postulate math mostly doesn't exist) perhaps no one can argue. This just means you don't believe in math (again, despite that, actually, math doesn't need to be believed in). With precisely the same effect you can deny to believe anything at all, e.g., me or other people on this forum, postulating we simply don't exist. Or, you can postulate that the Earth is flat. It is impossible to disprove postulates, because any proof must be based on postulates, too, and if you postulate something, then every postulate that contradicts becomes false in your own universe.
By the way, if you state that infinity is something that can't be researched and/or known of, then these damn infinite decimal fractions will disappear in fear. They are based on the assumption we can work with infinities.
Quoting EnPassant
Definitely we can, and we always do. Actually, this is all higher math (in contrast to elementary math) is all about. Correctness of such a "jump" is shown and proved centuries ago. I can tell you even more. There's beautiful piece of math named "Functional analysis" (https://en.wikipedia.org/wiki/Functional_analysis), which works with spaces that have infinite number of dimensions (I believe this should impress more than just an infinity in a single miserable dimension), and, BTW, this piece of math has a lot of practical applications. May be I surprise you if I say that Pythagorean theorem perfectly works in some of these (3D? 4D? 5D? infinite-D!) spaces.
If we return to the topic, 0.999... doesn't need any higher math or any magic to be 1. It is simply a practical fact which, as I mentioned before, I was told about in elementary school.
Sorry Banno, but in logic definitions are prerequisite. Family resemblance might suffice as a description of meaning in common vernacular, but mathematics is logic.
Quoting InPitzotl
If that's the case, then why have two distinct representations for one and the same thing?
Quoting tim wood
I believe shenanigans is an apt word for a description of "real numbers". Modern mathematics contains a lot of sophistry, of which some is used for deception. Mathematics is loaded with tricks which the mathemgjicians have designed for the purpose of hiding contradictions.
Quoting tim wood
A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle. These items were designed as ideals, but the irrationality of the ratios demonstrates that this effort was a failure. Space cannot be represented as distinct dimensions, as the irrationality of these two dimensional figures demonstrates. One dimension is incommensurable with another, whether you represent the relationship between them as a curved line or as a right angle.
I'd point out that 2 + 2 = 4, but we've previously determined that you don't even believe that.
What do you mean "If that's the case, then"? There seems to be an implicit assumption that every thing should have exactly one name. Who exactly is making that assumption? It's not me, and it can't be you... does it say "Metaphysician Undercover" on your birth certificate? (And isn't 1 also equal to the fractions 1/1, 2/2, 3/3, and so on anyway?)
Any reason whatsoever for there being two distinct representations for the same thing would do. What's it to you that there are two of them? Is there supposed to be an objection here?
Quoting jorndoe
So 1/9 is a number, even for Meta, but 0.111... is not? And this despite their being equal?
Is this Meta's claim?
It looks to go beyond this. Not only is 0.111... not a number, but there's no such thing as squares, because dimensions are incommensurable (@tim wood asked the question I was thinking before I got to it... and that was his response). Circles aren't real, so maybe trigonometry is a lie. Looks to me like Meta's a strange sort of Pythagorean?
No, we know what a limit is. But as I keep saying, a limit is not a literal infinite sum. Weierstrass did not formulate calculus in terms of literal infinities. He formulated it in terms of finite sums converging to a limit. I'm not arguing that limits are not as they are defined. I'm saying a literal infinite sum is qualitatively different to a finite sum. The definition of a limit only makes sense in terms of finite sums converging to a limit. This is how the limit was rigorously defined by Weierstrass, Cauchy...That definition says that the finite sum can get ever closer to the limit. But you are jumping from the finite to the infinite and saying finite calculations still apply to infinite sums.
The limit of 1/2 + 1/4 + 1/8... is 1. But what is the infinite sum, as a literal infinity? We cannot assume it is 1 just because finite arithmetic points in that direction.
Not "all" higher math is all about. Lots of topology topics, for example, don't revolve about infinities.
Quoting Andrey Stolyarov
True, but frequently it concerns spaces of functions that are not so elaborate, like complex valued functions of the form
[math]S=\{{{f}_{s}}(t)={{s}^{2}}{{t}^{2}}+itSin(s+t):\text{ }s\in [0,1],\text{ }t:0\to 1\}[/math]
, where the space extends over values of s (uncountably infinite, but one dimensional). One then considers a sigma-algebra of subsets of these functions and defines a kind of measure of these sets. Then, frequently, a functional, which takes a function to a real or complex number. For instance,
[math]\mathbb{F}[{{f}_{s}}]=Su{{p}_{t}}\left| {{f}_{s}}(t) \right|[/math]
And so on. Just a simple example. Not everything in abstract math is infinite dimensional, though infinity comes into play.
Oh, indeed. Meta also cannot calculate an object's velocity, either, since for him the notion is absurd.
I pointed this out long ago. It is important to note that we are talking fractured ceramics here, not cranial Faraday cages. Hence the exercise becomes one of identifying and tracing the crack.
Seems you have a short memory. What we previously determined is that I do not believe that 2+2 is the same thing as 4. Remember? You argued that 2+2 is identical to 4, ignoring the difference between equivalent and identical.
Quoting InPitzotl
The question was why does the same thing have two names. There was no implicit assumption that the same thing ought not have the same name, but an implicit assumption that if the same thing does have two distinct names, there is a reason for it having two distinct names.
I don't believe that ".999..." and "1" refer to the exact same thing. So I'm asking you, who apparently does believe this, why does mathematics, as a single unified discipline, have these two distinct symbols to refer to the exact same thing. If you could answer this for me, then you might help me to believe what you believe. Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings.
You might argue as others have, that it is a difference which does not make a difference. But in acknowledging that it is a difference you accept the fact that they do not refer to the exact same thing. So I warn you that this would be a self-refuting argument.
Quoting Banno
Did I say that I agree that 1/9 is a number? Check my definition of number, "particular quantity". How could 1/9 ever be construed as a particular quantity? A fraction is not a number.
Quoting tim wood
"Triple-barreled" now. Looks like I'm moving up in the world.
Gorgeous!
Divide the pie amongst the nine of us - but none for Meta, since a ninth is not a quantity of pie!
A ninth of that particular pie is a particular quantity. A ninth, or 1/9, is not a particular quantity. Are you capable of understanding this?
One is a particular quantity, and therefore a number; 1/9 is not. This is because a fraction must be a fraction of something in order that it signify a particular quantity, 1/9 of this, or 1/9 of that. But 1/9 on its own does not signify any particular quantity.
Right, that's why one is called a number, it's a value representing a particular quantity (as per the definition I offered), the quantity of 1. On the other hand, 1/9 does not represent any particular quantity unless it is qualified with 1/9 of 9, or 1/9 of 90, etc..
No, I'm not following. It seems that on your account we can only use the word "quantity" to apply to discreet individuals: one pie, two pies, and so on; the word cannot be applied to partial individuals: half a pie, a quarter of a pie.
Is that your contention?
Because that seems wrong. We do talk of half a pie as being a quantity of pie.
No that's not my contention. One is a quantity, two is a quantity, three is a quantity and so is four, etc.. 1/9 is not a quantity because it is a fraction, and for it to refer to a particular quantity it must be specified what it is a fraction of. 1/9 of 9 is a different quantity from 1/9 of 18, which is a different quantity from 1/9 of 27, etc.. So 1/9 on its own does not refer to any particular quantity.
Quoting Banno
I'm not denying that people talk that way, just like they say .999... is a number. I'm saying these people are wrong. Half of this pie is a different quantity from half of that pie. So half a pie is not a particular quantity at all. Even though people might talk as if it is a particular quantity of pie, we'd be fools to believe them. If you bought half a small pizza would you complain because you expected to get the same quantity as half a large?
I invited anyone to provide a better definition of "number", one which would provide for these "non-particular quantities" to be called numbers, but none has been provided. So I still believe that concepts such as "real numbers" operate without an acting definition of "number", providing for all sorts of tomfoolery.
Five is a quantity, six is a quantity, and Heavens to Betsy, I do believe there are more! :lol:
So instead of arguing "there are two names for a thing therefore there are two things", which is a red herring, you're just arguing "there are two names for a thing and there's no reason for it having a second name therefore there are two things", which is just a red herring with weasel (obviously if a thing has two names, there's a reason it has two names... it was named twice; and obviously that doesn't count... so, the weasel is in what constitutes "a reason"). Adding a weasel to a red herring is still not an argument, though I suppose the weasel would love the snack.
Quoting Metaphysician Undercover
Okay.
Quoting Metaphysician Undercover
But that would be silly because your premise that two names must refer to two things is a red herring. Also, it's a bit fishy:
Quoting Metaphysician Undercover
If you cannot agree that a fraction is a number, how are you even qualified to talk about the meaning of .999... in the first place?
Quoting Metaphysician Undercover
I detect some language loaded to the brim with irrelevancies.
...why does this sound like the hook of a con to me? My "belief" isn't relevant here (except insofar as I'm part of the math community which, technically, I am, but it's just a tiny part)... the terms here are terms of art in the math community. As mentioned before, the math community defines and uses these terms. And the way they use it, .999...=1. The definitions therefore are matters of fact. If you have any issues, it's with the proofs. But you're not pointing those out... you're just rattling about nonsense of two names having to refer to two things... it's your core broken intuition, and just propping it up with loaded language isn't going to fix what's broken here.
If we can't agree that 1/9 of a pie is a particular quantity of pie, then we can't have the conversation you want. But it's irrelevant anyway.
On the off chance that someone else is curious, yes, there's a reason that the decimal system representation of numbers gives two names for the same numbers, and it's not unusual for various systems to do so. On the off chance MU replies to this with a rebuttal, it's irrelevant... your entire two names is two things argument is dubious, so there's literally nothing to argue against including this.
Quoting Metaphysician Undercover
But isn't one of this pie a different quantity from one of that pie?
Quoting Metaphysician Undercover
We agree on seeing tomfoolery, we just disagree on where we see it.
But equally, if you bought 1 small pizza would you complain because you expected to get the same quantity as 1 large?
None of what you have said makes your contention clear.
You say 1/9 of 9 is a different quantity from 1/9 of 18; Is 1/9 of three yet another quantity? But surely you must say that ? is not a quantity...
But this all still leaves hanging why you think 3 is a quantity but ? isn't...
If you are going to use the word "quantity" in a way that is so at odds with how everyone else uses it, you might need to put some more effort into explaining why.
No, I believe the two symbols have different meaning, and I've given the reasons why I believe that. You claim that the symbols refer to the same thing so I want to know the reasons why you believe this.
As indicated in the op, it is not the case that the same thing is named twice. It's very clear that "1/9X9" does not say the same thing as "1". So your claim that the same thing was named twice is false.
Quoting InPitzotl
As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. It seems you do not know what a definition is.
Quoting InPitzotl
As I explained to Banno, it's very clear that "1/9 of a pie" does not indicate a particular quantity of pie, because pies vary in size. If your inability to accept this fact rules you out of this conversation then so be it.
Quoting InPitzotl
No, why would you think that? One of anything is the same quantity as one of anything else. It is one, which is a quantity. We are talking about quantity in an absolute, abstract sense now. But if we are talking about a quantity of pie, then clearly one large pie is a different quantity than one small pie.
The lesson you ought to take from this is that 1/9, as a fraction does not refer to any quantity in any sense whatsoever, because it needs to be qualified. In order to have any meaning whatsoever, we need to indicate the thing which is to be thus divided. To talk about a division without any thing divided, is to talk about a useful tool, which is doing nothing. And the tool which divides quantities is not itself a quantity.
Quoting Banno
Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem.
Quoting Banno
Let's start with this definition. A "quantity" is something which can be measured. The simple act of counting, 1,2,3,4,5,etc., when there are no objects being counted, is an act of measuring imaginary things. These imaginary things are called "numbers". So a numeral represents an imaginary quantity, which is called a number. A quantity is something which can be measured and in this case the measurement is counting. Now look at "1/3". It represents a ratio, which is a specific relationship between two distinct quantities, or numbers. A relationship between two numbers is not the same thing as a number, therefore we ought not try to represent it as a number.
The relationship between two numbers (indicating determinate measurable quantities), is not necessarily a measurable quantity itself. When it is not, there's a word for this "incommensurable". Why create the illusion that incommensurable things are actually not incommensurable, and insist that this illusion is truth.
It's just that i think about extremely trivial things, which is not a common trait. But the important things are already over thought so why not?
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
You're all over the place here. You have a definition of number that refers to a value (read the newer version of OED; cf to definition 1b of your revision). 1 and .999... being equivalent means they refer to the same value. And don't think I didn't catch that suddenly "refer to" changed to "are"; nevertheless, it's common language to use forms of "to be" to represent equivalence under equality. If .999... represents the same "particular quantity" that 1 does, they refer to the same value, which is what it means to say that they are the same thing.
Quoting Metaphysician Undercover
Your "therefore" is thwarted by the definition of a number. Equivalence under equality means that the left hand side has the same value as the right hand side. Your OED definition of number is that of a value. Therefore, equivalence in this context means referring to the same number, since it's the same value. And you're complaining about tomfoolery?
Quoting Metaphysician Undercover
...
Quoting Metaphysician Undercover
Because pies vary in size?
Quoting Metaphysician Undercover
Apparently not. One pie is the same as one pie even if they are different sizes, but one ninth of a pie is not the same as one ninth of a pie because they are different sizes. I know special pleading when I see it. Again, you're all over the place.
Quoting Metaphysician Undercover
Uhm... but...:
Quoting Metaphysician Undercover
...yet:
Quoting Metaphysician Undercover
...and:
Quoting Metaphysician Undercover
What conversation pray tell are you even talking about? How can .999... have a second meaning if .9 means 9/10 and 9/10 is allegedly a problem? And how come you can't be honest about what you're inviting me to do? The problem isn't that you're missing that conversation about why there are numbers that have two representations in the decimal system... the problem is that you don't believe decimals are possible because you have a quixotic quest against fractions, and yet you present to claim that you believe .999... has a meaning at all. I'm not the problem here, MU; I can easily have that conversation with someone who isn't so wrapped up in your fictional world of fraction-denial. I just can't have this conversation with you because you can't face the fact that there's a thing to discuss.
But again, it's irrelevant, because your two-names-means-two-things premise is still as dubious as it ever was.
1/9 is a proportional relationship. In geometry if the side of the square is [math]1[/math] the diagonal is [math]\sqrt{2}[/math] and the proportion is [math]1:\sqrt{2}[/math]
I missed this post. Yes, the definition of the sum is the same as the limit. But I am talking about an actual sum. An explicit infinite sum that you can write down. This of course is not possible because it requires an infinity of calculations. That's the point I'm making. I understand sums and limits but that is not really how the question can be answered. It can only be answered by an explicit infinite sum. What makes me suspicious is the paradoxes that exist at infinity.
I think your problem lies with the distinction between pure and applied maths rather than a distinction between 1 and 1/9. 1, 1/9, -1/9 , 0.9, 0.999i are all numbers in the realm of pure maths.
Then when you start applying them to pies you enter the realm of applied maths which is a different realm. 1 pie and 1/9 of pie take on meaning but not so much -1/9 pie or 0,99i pies.
I realize that, but there's no "actual sum" to speak of outside of this definition. In principle I could give an intuitive argument for why .999...=1 using the idea that each digit in the decimal is dialing in on the "address" of the number it refers to. In such an argument I could say that if you have an infinite number of 9's after the string .999, then the resulting string dials in on the address of 1 itself. But in using this argument and applying it to a repeated decimal, I would in effect be using the limit definition.
Quoting EnPassant
I understand that as well... but analogously, divergent infinite sums can't use the definition above; only convergent ones. But that's precisely why we would apply this definition to infinite sums for convergent infinite sums (and in certain cases we can apply a definition to divergent sums, but with different definitions).
But I think in the bigger picture this thing boils away, because in the state of affairs that we're in, the "actual sum" for a convergent infinite sum has been defined, by this definition. In essence, the infinite sized string .999... is like a word, and we have assigned a formulaic definition for all words matching this pattern, including that one... and by that definition, .999...=1. We could say then that the address .999... has been assigned in such a way that the number assigned to it is 1.
Or phrased another way, I'm not sure you can actually say what it is you're disagreeing with meaningfully. To do so you would have to reify .999...'s definition without using the provided one, and claim that whatever reified thing you came up with "has a problem". But what problem? If the problem is it hasn't been defined, then it's a bit vacuous. If the problem is we don't know what it is, then you're presuming it has a meaning... in what sense does it have a meaning? What meaning did you assign to it? That's the problem I'm raising here... you cannot talk about this reified "actual sum" unless you can talk about it, and I'm not sure you've convinced me there's a thing to talk about.
Would be kind of tedious for physicists and cosmologists to have to check whether their results had exceeded "the largest number". :D
Just have to remember that ? isn't a real number, can't be shuffled into arithmetic calculations (+-×/) just like that.
And there are any number of ways to write 1.
0.5 + 0.5 = 2 - 1 = 1 × 1 × 1 = [math] \frac{\sqrt{2}}{\sqrt{2}}[/math] = [math]1^5[/math] = [math]1 + \sum_{n \in \mathbb{N}_+} (1 - 1)^n[/math] = ...
No numerical difference.
And we can reason about [math]\sum_{n \in \mathbb{N}_+} 9(\frac{1}{10})^n[/math] and such if we're careful. (y)
Why would any of this be a problem anyways...?
(I didn't see the formalities implying a contradiction here in the thread.)
What I'm saying is very simple. Suppose you had a kind of God calculator that would print out the actual addition of 9/10 + 9/100... to an infinity of terms, what would that be, 1 or infinity? That's what I mean by the actual sum.
To show how quirky infinite sums are consider the following (this is not meant to answer anything, it is just to illustrate how strange things become at infinity)
Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity
Now sum 1/2 + 1/4 + 1/8...in view of the above theorem. No term in this series is zero, they are all positive quantities. So we are summing an infinity of positive quantities, some of the bigger than others...
Again, I'm not trying to answer anything here but it is worth contemplating.
Only in the special case you describe of adding the same thing to itself forever. Diminishing quantities act differently. Otherwise Achilles could never pass the tortoise, or even get started running.
One of those two series you gave diverges; it does not have a limit. The other converges: it has a limit. The second one never gets anywhere close to infinity no matter how long you run it. It would only ever even get up to 1 if you ran it forever, with your “God-calculator”.
This is exactly what limits are for. Only a series without a limit sums to infinity. A series with a limit sums to that limit.
This becomes false as soon as k = x.
1/2 + 1/4 + 1/8 = 0.875
1/3 + 1/3 + 1/3 = 1
I know, I made a mistake. Let me rethink how to formulate it...
You say it in your notes:
Yes, we know the answer, we know how it works.
The sorry fact is, that we cannot either describe or simply cannot understand infinity as clearly as we would want. Or infinitesimal and it's relation to numbers.
But that begs the question: you say it don't sum to infinity because it don't sum to infinity. That is the very thing that is being questioned. I know the limit is 1. But that limit is defined by finite arithmetic. I am asking what really happens at infinity. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.
Here's what I read is going on. You want to talk about an "actual sum" in a meaningful sense, outside of the provided definition. You intuit that it means something, but I'm not convinced it actually does.
To convince me, however, you metaphorically appeal to the God calculator, and sprinkle in "actual" as adjectives. But that's not convincing for me. I read both the metaphor and the adjective as just reifying.
God didn't give us addition on tablets; we invented it. The "base" definition of addition works recursively down to base cases, so that's fine for finite numbers of terms. But you cannot reduce an infinite recursion down to base cases. So there's no a priori definition of infinite sums.
Quoting EnPassant
Quoting EnPassant
It could be infinity; but it doesn't have to be infinity. You have to define what you mean by infinite sums first before you even get to say this sum is infinity. But let's grant that theorem; it works at least for one definition:
Quoting EnPassant
...then this still does not follow. There are infinite sequences of terms in the range (0,1) such that for any such term x, there's only a finite number of terms greater than or equal to that x. In fact, 9/10, 9/100, 9/1000, ... is such a sequence. So even if you're going to try to apply some variant of a squeeze theorem to prove that this sum is infinite, you just can't do it... because there is no term in this series for which you're adding it or any larger number an infinite number of times.
I don't insist it doesn't sum to 1. You may well be right. I'm saying we don't know because we can never have an actual infinite sum. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.
What is the 'last' term in the sequence 1/2, 1/4, 1/8...? No need to answer, it is a rhetorical question. But surely all terms - the whole infinity of them - are positive and > 0. Right? Now sum an infinity of positive quantities...
That’s where you’re wrong. Just flat wrong. You tried to show that true and I showed it false in just three terms.
The same positive quantity added to itself infinitely many times is infinity, sure. But not every series is like that. No convergent series is like that. A series like that can have no limit. Any series with a limit is unlike that. A series with a limit sums to that limit. That’s what the limit is.
That's insufficient to use your theorem, as I explained in my previous reply.
1/x + 1/x + 1/x+...+1/x
Now let x go to infinity-
[math]\sum_{i=1}^{\infty} 1/x = \infty[/math]
dubious. Take x=10^9. I happen to know off the top of my head the left hand side goes below 1/10^9 at term 30 (because I work with computers). So in binary, the sum on the left is 0.111...11 with 30 1's. Take that sum and divide it by 1/10^9, you get a finite number... call that number's ceiling y. At term y, the sum on the right equals the sum of the left before term 30, and you're just adding smaller and smaller terms on the left. In fact, by the time you reach term 10^9 on the right, the right sum becomes 1; and the left sum by that term is simply 0.111...11 with 10^9 1's in binary, which is less than 1. After that, every term you add is going to be less than 10^9 on the left, and equal to 10^9 on the right.
This is false soon as the number of terms is greater than or equal to x, after which point the bottom sum is greater than 1 and the top sum is still less than 1.
x is not static. I'm saying if there are the same number of terms in each. Now increase x indefinitely with the same number of terms top and bottom.
...and you'll find the inequality always breaks down for some number of terms, and all terms after that. In fact, you can cheat... whatever positive integral x you specify, it will break down at the xth term.
x is a power of 2.
1/2 + 1/4 + 1/8+....+1/1024 >
1/1024 + 1/1024 + 1/1024+....+1/1024 for the same number of terms.
Now go to infinity with x. You are still adding an infinity of positive terms.
So that's 10 terms. What happens at the 1024th term?
Your left sum is 0.111....11 with 1024 1 bits in binary. Your right sum is 1. Is 0.111...11 with 1024 1-bits greater than 1?
The sum of terms 1/x is infinity if 1/x > 0.
What are you talking about?
At the 1024th term on the left, we're adding 1024 terms... in binary point, 0.1, 0.01, 0.001, ... , 0.00...001 (with 1 in the 2^-1024th place). That sum is 0.111...11 (with 1024 1's).
At the 1024th term on the right, we're adding 1024 terms, each of which is 1/1024... that is by definition of multiplication equal to 1024*(1/1024), which is 1024/1024=1.
After the 1024th term, we're adding numbers on the left much smaller than 1/1024; in fact, they're smaller than 1/2^1024. And on the right, for each of these, we're just adding 1/1024.
What is [math]1/2^c[/math] added to itself infinitely?
[math]1/2^c[/math] added [math]2^c[/math] times is [math]1[/math]
Now add another [math]2^c[/math] terms[math] = 1[/math]
1 + 1 + ... = [math]\infty[/math]
It doesn't matter. An infinite sum of equal infinitesimals must be infinite.
Everyone here does that. No, what I'm curious about is the apparent absence of humility. Given that others have thought about these issues - many others, over centuries - and given that your way of thinking is so at odds with the way these others have approached the topic, I wonder at the absence of self-correction.
Your latex is garbled... let me generalize and math this for you. For any positive integral x, no matter how large:
[math]\displaystyle\sum_{i=1}^{x}{2^i} < \sum_{i=1}^{x}{x} = 1[/math]
...intuitively, you can see this by "argument from binary". The left sum is always:
0.111...11 with x 1-bits. That's always less than 1.
...and for each y>x:
[math]\displaystyle 2^y < 1/x[/math]
Quoting EnPassant
There is no infinite sum of equals on the left side. For any positive x, no matter how small, there are only a finite number of terms greater than x in that infinite sum. Quick proof...
1. Pick your x.
2. Write 1/x in binary
3. Count the digits; call the number of digits plus one n.
4. 2^n is greater than your 1/x.
5. 2^-n is less than your x.
6. All terms after the nth term are less than 2^-n.
(eta: corrections)
1/2 + 1/4 + 1/8+....+1/1024 > 1/1024 + 1/1024 + 1/1024+....+1/1024
LHS has the same number of terms as RHS
Now let the number of terms run to infinity and the sum on RHS is infinite at infinity.
The next inequality would be-
1/2 + 1/4 + 1/8+....+1/2048 > 1/2048 + 1/2048 + 1/2048+....+1/2048
You keep handwaving through the same argument's flaw.
You have an inequality that's true for term 10:
0.1111111111b > 1024+1024+1024+1024+1024+1024+1024+1024+1024+1024
That's all fine and dandy. But it doesn't hold at term 1024:
0.11111111......111b ? 1024+1024+1024+1024+1024+1024+1024....1024
(with 1024 1-bits on the left, and 1024 1024-terms on the right).
In fact, at that term:
0.11111111......111b < 1024+1024+1024+1024+1024+1024+1024....1024
...and after that term, you're adding values less than 2^-1024 on the left, which is << 1/1024... but for each such term, you're adding 1/1024 on the right.
So that it works at term 10 is irrelevant, because the inequality fails at term 1024 and for all terms after it. You can't go from 10 into infinity without passing 1024.
I'm not up to speed on binary. I don't think you understand what I'm saying.
Are all of the terms in 1/2, 1/4, 1/8...positive and > 0? Yes.
For all c 1/2^c is positive and > 0.
Let c run to infinity and sum. Now you have an infinite sum of positive quantities > 0 and that's infinite.
There's no point in saying calculus says otherwise because calculus does not deal explicitly in infinite sums. You need the God calculator for that.
All we have as evidence for the impossibility od three being divided by nine is your insistence. Further, I am certain that dividing none by three is a reasonably straight forward activity.
Contemplate this; I have three dozen pies. I divide them amongst the nine of us.
I guess your response will be to the effect that there are 36 pies, not three dozens, and hence that this is not an example of 3 divided by 9, but of 36 divided by 9.
You have done this several times in this thread; taking a common way of speaking and arguing that it is incorrect. You did the same in the conversation where you denied that one could calculate the velocity of an object at a given time, and indeed you did much the same thing in the discussion of the Tractatus.
@Pfhorrest, @ InPitzotl, @A Seagull: The discussion here is not mathematics, which Meta plainly has misunderstood. I said before that the interest here is in identifying the origin and progress of the crack. It seems to me to lead to Aristotle, and to a curiously inert language, in which "number", "velocity", "fact" and other terms are arbitrarily and unilateral forbidden their usual use. Meta is of interest because of his inability to see that ? is just another number, that velocity is calculable and that facts change over time. The lesson is that one can show someone the solution to their philosophical issues, but they may not see that solution.
Personally I think mathematics is not really about numbers. Mathematics is more about harmonies and proportion. Numbers are 'markers' in the symphony of proportion and relation. The real music of mathematics is beyond numbers. Just a thought...
I'm going to take a stab at your confusion then. Here is the full form of the inequality:
1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024 > 1024+1024+1024+1024+1024+1024+1024+1024+1024+1024
There are ten terms here. The sum on the left as it turns out is 1023/1024. The sum on the right is 10/1024. 1023/1024 > 10/1024, as you said.
I can't write the full form of 1024 terms without basically flooding the channel... 1024 terms itself is large enough, but the value of 2^1024 itself is huge:
179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216
But we can write the sums in power notation. The sum of 1024 terms of 1/2+1/4+...+1/2^1024 = ((2^1024)-1)/2^1024. That is less than 1 (just barely... by 1 divided by that huge number above, but less is less). But 1024*(1/1024) is equal to 1. So:
1/2+1/4+1/8+...+1/2^1023+1/2^1024 < 1024+1024+...+1024
...because:
((2^1024)-1)/2^1024 < 1
Quoting EnPassant
Of course you don't, because you keep replying to me. But that's not what the problem is. The problem is that you don't understand what you're saying.
Quoting EnPassant
Dubious. Your argument was based on this theorem:
Quoting EnPassant
There is no such term 1/x that is added to itself infinitely often in 1/2+1/4+1/8+...; nor is there a "squeeze term" such that in that sequence there are terms >=1/x added to themselves infinitely often. For this reason you cannot apply Theorem 1. If you disagree, name the number; but I already gave you a generic refutation... for any number you name, I can tell you how many finite terms there are in the sequence >=1/x, and you cannot name a positive number such that there are an infinite number. Theorem 1 requires something that's not there... therefore, you cannot apply it.
Hang on - you can't have it both ways; you can't both say that we know how it works and yet we don't understand it.
If we know the answer, what exactly is missing?
Hence mathematics sets out a multiplicity of patterns, of ways of speaking that do not lead to contradiction.
And so it should not be a surprise that some of these patterns, these ways of speaking, can be used for doing things like economics and physics.
I'm not talking about the 1024th term. I'm saying-
1/2+1/4+1/8+...+ 1/2^c to k terms - whatever the value of k
>1/2^c + 1/2^c + 1/2^c to k terms. There are the same number of terms in each series.
I remember all too well, which is why I'm not joining in with the rest of the gang arguing with you. Nice job trolling them all though. Your objection isn't to the theory of convergent infinite series of real numbers. Your objection starts with 2 + 2 = 4. You seem to agree.
I surely didn't ignore the difference between equivalent and identical. On the contrary I wrote many posts carefully explaining the difference; and showing through actual mathematical proof, as well as other methods, that 2 + 2 and 4 designate the same mathematical object. One object, two names.
You do agree that a single object or thing may have more than one name. Earlier you claimed this as some kind of problem. Your position seems absurd on its face.
Okay, I think I got it (incidentally, c=k here, right?), but the same objection applies. You still can't apply theorem 1, because you still can't name an x for which you have an infinite number of terms of the value 1/x such that 0<1/x<1. Every x you name is finite; therefore, every term in your sequence is finite. You don't have an infinite number of 1/x for any 0<1/x<1, so you can't apply Theorem 1.
Or think of it this way. Note that every time you add a term, you change all of the terms. We go from 1/2, to 1/4+1/4, to 1/8+1/8+1/8, and so on. But note also that we can actually sum these partial terms too... 1/2=1/2, 1/4+1/4=1/2, 1/8+1/8+1/8=3/8, and so on.
The general form here is:
[math]\displaystyle\sum_{i=1}^{k}{\frac{1}{2^k}} = \frac{k}{2^k}[/math]
But when you generalize this, you're talking about what that form becomes, so you really mean:
[math]\displaystyle\lim_{k\to\infty}{\sum_{i=1}^{k}{\frac{1}{2^k}}} = \lim_{k\to\infty}{\frac{k}{2^k}}[/math]
But your theorem 1 just says:
[math]\displaystyle\lim_{\to\infty}{\sum_{i=1}^{k}{\frac{1}{x}}}=\infty[/math] where [math]0<\frac{1}{x}<1[/math]
...so doesn't apply.
I do not argue against the fact that mathematicians believe that .999..., and 1 refer to the same value. The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim.
Quoting InPitzotl
My argument, if you've read what I posted, is that .999... does not represent a particular quantity. I suggest that you come back when you've got an argument to make. The fact that mathematicians believe that .999... has the same value as 1 is just evidence that they are wrong, it's not an argument that .999... refers to a particular quantity.
Quoting InPitzotl
Sorry, I have no idea of what you're talking about here. I never said .9, or 9/10 is a problem. I said these do not represent any particular quantity, and ought not be considered as numbers. It is the belief that they are numbers which is what I consider to be a problem.
Quoting Pfhorrest
I believe that zero is a very complex idea with numerous different meanings, some inconsistent with each other, as exemplified by imaginary numbers.
Quoting A Seagull
I agree that these are all considered by mathematicians, to be numbers. What I haven't seen yet is a definition of "number" which validates this belief. It is possible that a person, or even a whole group of people, believe that a certain thing is such and such a type of thing, but when a clear definition of that type of thing is made, it turns out that the thing is actually not that type of thing. Take Pluto for example. Everyone believed it was a planet, until a clear definition of "planet was made, then the people realized that Pluto actually wasn't a planet. The same might be the case with some of these things which people believe are numbers. Until a clear definition of "number" is produced we will not know if this is the case. According to the definition I proposed, some of these are not numbers.
Quoting Banno
I wonder where you get your idea of correct from. That everyone does it, doesn't make it correct, read my example above. You support mob rule?
Quoting Banno
If you knew your example was so bad, why present it? Clearly "three dozen" does not represent a quantity of three, just like "four score" does not represent a quantity of four, and "twenty six" does not represent a quantity of twenty. Sometimes I wonder Banno, how you can go so far as to conceive such bad arguments. It must take strenuous effort to make your arguments so bad.
Quoting Banno
I've explained very thoroughly why 1/2 is not a number. I have yet to see a counter argument, only your extremely bad example which premises that "three dozen" represents a quantity of three. That false premise disqualifies the argument as unsound.
Quoting fishfry
I didn't claim this is a problem, that was Pitzotl''s misinterpretation. I said that if the same thing has two distinct names, there is a reason for that.
:grin: A trite reply, as was expected.
Quoting Metaphysician Undercover
You missed the point of the example, as is your habit.
Quoting Metaphysician Undercover
Well, no; what you did was explain how you use the word "number" in a rather eccentric fashion. You told us nothing about numbers.
So that's an end to this discussion, I think. There's nothing here but Meta's queer usage.
Yes, you got it. The point I'm making is that 1/k is positive and > 0. Even as you go to infinity 1/x can't be zero. So, you are summing an infinity of positive terms > 0 which is infinity, right? As the number of terms taken increases 1/x decreases but never becomes zero.
These are constructed by two limits, the left part describes some sequence with the surreal number as its lower limit, and the right number describes some sequence with the surreal number as its upper limit, and that further this mode of representation is powerful enough to produce all the other sorts of numbers... except imaginary numbers.
Now the Wolfram discussion on Limits makes the point that a limit is said to exist if the limit approached from below is the same as the limit approached from above; the same construction as the surreals.
An example fo the mathematics that disappears when one denies the argument in the OP.
There's something extraordinary in the creativity of mathematics. Consider the imaginaries; we all know you can't take the root of a negative number; but despite that if we call the root of -1 "i" we can have even more fun with numbers...
Please, please, don't start calling this trash "theorems". And stay away from LaTeX, it's not like playing with a shovel and pail in a sandpile. What you don't know can hurt you. I beg you, leave this disaster zone and return to the relatively safe comforts of philosophical musing. :groan:
But I note that your OED definition talks about values referring to the same particular quantity. And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value.
Until you do, there's nothing to argue against. You have no point to make, just a problematic claim. And by Hitchen's razor, I can dismiss that without argument.
Your problem is your problem though, not mine.
Quoting Metaphysician Undercover
No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language.
You can do whatever you wish, but I'm under no obligation to take you seriously, especially at your word.
Quoting Metaphysician Undercover
That's not an argument, it's a claim.
And I told you why that's inconsistent with the views you presented. It's still there in the post. To help you out, I repeated it at the top. But the barrier between us (and also you and many others) goes far deeper than this. You're trying to have a conversation without speaking the language. That's made even worse by your refusal to even consider speaking it, which is made even worse by your having unfalsifiable "opinions" on how the language should even work. All of this is a grand recipe for having pointless arguments, but nobody is interested in having pointless arguments with you. We have to clear this barrier before it's even possible to have a conversation with you.
Quoting Metaphysician Undercover
Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you.
That's true for all finite x. But you need it to be true for an infinite x. To see the problem, here's a "troll proof" that infinity is finite. 1. 1 is finite. 2. For all x where x is finite, x+1 is finite. 3. By 1 and 2, and infinite recursion, infinity is finite, QED.
Think of this "troll proof" analogous to your conjecture. Your inequality holds for all finite x's, no matter how big the x is. But also, no matter how big the x is, you only have a finite number of terms. But to apply Theorem 1, you need two things: (a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1.
To write the difference between 0.999... and 1, write a zero, a decimal point, and then infinitely many zeroes, and then "when you finish", write the "final" 1.
Problem is, you will never finish, because the zeroes are infinite, so the difference between 0.999... and 1 is 0.000... forever. In other words, just 0.
And if the difference between two things is 0, there is no difference between them; they are the same.
There was no point to your example, as is your habit. You started from a false premise and tried to make something out of it.
Quoting Banno
I took my definition of "number" straight from the first entry in my OED. I'm still waiting for an alternative definition, one which allows that 1/2 signifies a number. Your response to the definition was very lame: "Family Resemblance".
Quoting InPitzotl
I don't deny that in some cases different symbols represent the same quantity. The op does not provide one of those instances.
Quoting InPitzotl
I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity.
Quoting InPitzotl
I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. To represent a particular value they need to be qualified.
Quoting InPitzotl
I haven't seen that definition. care to provide it?
Quoting InPitzotl
The argument is very clear in my discussion with Banno. You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.
Ok, but can't this be also said for 0.999...? Adding terms and then saying 'at infinity'. You can't have (b) at any finite number of the terms 9/10, 9/100...but ya gotta say 'at infinity' sometime if you assert that the literal infinite sum is 1.
They don't just believe that 0.999... = 1. They've proved it.
There really is nothing to discuss here. The disagreements are flat Earther stuff.
MU, you're pretending here to be making an argument about .999... = 1:
Quoting Metaphysician Undercover
A, therefore B, where A is .999... = 1, and B is some rambling about equivalence. But here we don't merely have equivalence, we have equality. Because we have equality, they do represent the same value. I've never heard of someone so far gone as to commit an amphiboly by changing the word. But in this post, and here?:
Quoting Metaphysician Undercover
...you're still not talking about the relevant point. You're still not advancing any reason why you think equality represents different values, or why you think the same value representing a particular quality can actually wind up representing different qualities. Nor are you even trying to make this point; you're just, instead, playing hide-the-ball.
Quoting Metaphysician Undercover
Quoting Wikipedia
That definition was already discussed in the thread. And that definition is used in the pdf provided by the op in section 1. By that definition, .999... = 1 exactly.
Quoting Metaphysician Undercover
In other words, 65 pennies, a dime and a quarter is not worth a dollar because pennies are 1/100th of a dollar and that's not a particular quantity of money. I mean, sure, some pennies are smaller than other pennies slightly; but some dollar coins are also smaller than other dollar coins. But apparently the pennies being smaller implies that pennies aren't a particular value, whereas the dollar coins being smaller does not indicate such a thing. Such is the tomfoolery I've heard from you so far. That's a garbage argument that can be ignored just on its merits.
Quoting Metaphysician Undercover
Yes, but it's all gibberish nonsense.
Quoting Metaphysician Undercover
Sure they do. 1/2 represents one half. As you said, one of anything represents a particular quantity. The quantity that half represents is very clear... that is the multiplicative inverse of two. It takes two halves to make the quantity one.
Quoting Metaphysician Undercover
What argument?
Your "premise" isn't a premise... it's a pointless language game. It shows you cannot speak the language (or at the very least, refuse to). You invented some niche and uninteresting alternate meaning for "particular quantity" that mathematics speakers do not use. The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. So you're not really advancing a "view" of quantities, you're promoting a language that's uninteresting. Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem is.
.999... is an infinite string of 9's. There's no problem with that per se.
Quoting EnPassant
We don't have to say "at infinity"... it's an infinite string of 9's. We add one 9 for all finite numbers, there are an infinite number of finite numbers, therefore there are an infinite number of 9's. But I think you misunderstand what the problem is...
In the infinite string .999...., I can say that the first digit after the decimal is 9. That's how I construct that infinite string. It's done homogeneously... it's always there at all steps.
In contrast, you cannot say what finite value x is when you have your infinite case, or that 0 < 1/x < 1 is true in that case. Your x changes every time you increment even by 1. What x is is inhomogeneous; it's diferent every time.
9 is the n-th digit in my string at all steps in the construction of it starting at n, going on indefinitely. x is a different value at every step in your construction. When I extend my string to infinity, 9 is still the first and nth digit. When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1.
Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? If we're talking about an infinite sum the same applies: by that way of looking at it you would not have 0 < 1/2^c < 1
Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1
But the assertion being made is that 1/10 + 1/100+... can be taken to an infinity of terms and summed to 1. If we take a finite number of terms it won't be 1. That's what I mean by an actual or literal sum. You have to go the whole way.
Such as the vicissitudes of these things. Ramanujan summed the natural numbers and got -1/12.
Calculus is a way of reasoning but there are other ways of reasoning that arrive at different results. Calculus is fine as long as we are talking about finite sums tending towards a limit. But when you go to infinity things get sticky. This is one of the reasons calculus was formulated in terms of finite sums going to the limit, so that the paradoxes at infinity would not interfere.
That doesn’t sum to 1, that sums to 1/9.
You do know how to calculate a limit, don’t you?
No.
Series 1
Step 1: 1/2
Step 2: 1/2+1/4
Step 3: 1/2+1/4+1/8
...
Step 10: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
Step 11: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024+1/2048
...
Step with all finite numbers: 1/2+1/4+1/8+1/16+...
Series 2
Step 1: 1/2
Step 2: 1/4+1/4
Step 3: 1/8+1/8+1/8
...
Step 10: 1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024
...
Step with all finite numbers: ? + ? + ? + ? + ? + ...
In series 1, at the "step with all finite numbers", the first term is 1/2. 1/2 is finite, and 0 < 1/2 < 1. In series 2, what is the first term at the "step with all finite numbers"? It's just that "step with all finite numbers" that has an infinite number of terms. Can you name the finite number x such that ? is 1/x? Can you say that 0 < ? < 1? Can you even say what ? is?
Quoting EnPassant
Sorry, what is c here and how does that relate to 0.999...?
Quoting EnPassant
Yes, by using a different definition for a divergent infinite sum. I toyed with that here:
Quoting InPitzotl
...since ...999=-1 in 10-adics.
It's the same idea 0.999.. = 9(1/10 + 1/100 + ...)
What seems to be happening here is that 1/x = 0 at infinity.
Then you have the absurd(???) conclusion that
1/infinity = 0
1 = 0 x infinity
So 0 + 0 + ... = 1 after an infinity of terms.
There was an Indian mathematician in the Middle Ages who asserted this (I forget his name)
But most mathematicians probably would not accept this.
But there is also a geometric way to "prove" this.
Take the x,y axis and mark off the unit length from 0 to 1.
This unit represents an infinite string of points all lined up in a straight line.
What is the width of each point? Zero. They are dimensionless.
But they add up to 1 unit width. How do da?
zero width + zero width + ... = extension???
Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. Go figure...
So you're talking about 0.111...? Then @Pfhorrest's post applies:
Quoting Pfhorrest
Quoting EnPassant
There is no "at infinity" here though. Every term here is a finite number; there's just an infinite number of finite numbers. Think of it intuitively this way... imagine the set you're trying to picture... it has "at infinity" in it, and maybe some other things. Remove every infinite-th step from this; we only care about finite steps. But we do want all of the finite steps. Now you still have an infinite set, but it only has finite terms in it. That is the thing we're describing.
Quoting EnPassant
If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined.
Quoting EnPassant
But you could do the same thing with a segment of length 2, 50, 0, and -7. So that infinite sum could also add up to 2, 50, 0, or -7, or any other value. This is what undefined refers to.
Quoting EnPassant
No, they aren't. The unit position is not defined in terms of infinite additions of 0. That would be useless, since infinite additions of 0 is undefined.
That was a typo, I meant to say 9/10 + 9/100 + ...
Quoting InPitzotl
Yes, because it can't be defined in terms of calculus but the question remains, what is it?
Quoting InPitzotl
Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit radius can be 1 inch or 1 light year.
It's undefined! :wink:
Quoting EnPassant
No, it's not arbitrary. It's just infinitely non-specific. That sum genuinely is sometimes 1 inch, sometimes a light year, sometimes 0, sometimes negative. So it's undefined. You can't reduce the sum to 1 inch if it could be negative or a light year. So it's useless to ponder whether it "truly" is 1 inch or "truly" is a light here, because your infinite sum doesn't give you the information to distinguish any length from any other.
Sometimes you can get that information elsewhere. But from just this, you just can't say.
Say 0 + 0 + 0 + ... = 50 units. Simplify-
(0 + 0 + 0 + ...)/50 = 1
But 0/50 = 0. It's just a matter of reducing to the lowest terms and the same logic obtains.
Okay, so that sum is 50 units.
Quoting EnPassant
You can only say that if you're literally talking about that 50 unit thing, because:
Quoting EnPassant
Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. To recap, that sum is undefined.
But you still have 0 + 0 + ... = something
If you do. It can also be 0. It can also not be anything. It can also be anything.
But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit width.
You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever.
The reason the unit circle is a standard in geometry is because what applies to the unit applies to a circle of radius 10 or 100 or 1000. It is the internal geometric logic of the thing that matters, not the arbitrary measurement. It is like the difference between inches and centimeters. 1 inch = 2.5 cm. This has more to do with convention than anything fundamental.
Not talking about the unit circle... just unity on the number line, and the idiom "going in circles" which means to retrace your paths over and over.
Well, @Metaphysician Undercover hasn't addressed the two proofs from the OP. All he has done is to assert that 1/9, and other fractions, are not numbers. His argument is an appeal to the authority of the OED.
Incidentally, and to my great amusement, the OED definition of fraction is "...numerical quantity that is not a whole number...", contradicting Meta's assertion that fractions are nether numbers nor quantities.
Hang on... isn't the flat earth two dimensional?:
Quoting Metaphysician Undercover
I mentioned before - I don't know if you noticed it - that this thread is not about mathematics so much as about the psychology of crackpots, of which Meta is certainly one.
i've been engaging with Meta for years - his posts are like a broken tooth that one keeps probing with one's tongue.Perhaps there is something to be gained here, not by treating Meta's posts seriously, but by looking at how he avoids confronting the truth.
For example he provides a restricted definition that suits his purposes, and when challenged he demands 'I invited anyone to provide a better definition of "number"'; a "have you stoped beating your wife" response.
He engages in pedantry for rhetorical purposes, as when he avoided @Pfhorrest's point concerning cones, and when he dithers on fractions being quantities, and pretending to @fishfry that his argument pivots on the distinction between equivalent and identical.
When met with a refutation he will deny it and simply repeat the refuted argument.
Then there is the paranoia, as in his reply to you that "Modern mathematics contains a lot of sophistry, of which some is used for deception".
Again, there is the outstanding point that he fails to directly address the two arguments presented in the OP. I think this is in order to avoid rigour.
And there is an extraordinary lack of self-awareness, of just how far he has deviated from orthodoxy, and how much of mathematics he must reject. Of course, for him this might be seen as proof of his intellectual courage, his rugged individuality.
Elsewhere his lack of comprehension has derailed whole threads - repeatedly, in the case of discussions of Wittgenstein on rules and on private language.
But is he evil? I don't think so, although you rightly question his motivation. I would suggest that it's not malevolence; but that he has found that adopting this approach brings attention. He is, probably only semi-consciously, espousing extremes in order to get a reaction.
And it works.
Not so well. Can't think why.
They think they've proved it. Staring from a false premise does not make a sound proof. But if one doesn't recognize the falsity of the premise...
Quoting InPitzotl
"1" represents a value which is a quantity. No one has demonstrated how 1/9 represents any particular quantity, because as I've explained, it does not represent a quantity. So how do you claim they are equal or equivalent?
Quoting InPitzotl
I see no such definition. Perhaps you can produce that definition so that we can determine whether 1/9, and .111... represent numbers. According to the definition of "number" which I provided they do not represent numbers. Where's your definition of "number"?
Quoting InPitzotl
Sorry, I have no idea of what you're talking about again. I wish you could make a greater effort to make clear what you want to say.
Quoting InPitzotl
I don't see how you can say that one half represents any particular quantity, when one half of two is different from one half of four which is different from one half of eight, etc.. The quantity represented by "one half" is clearly, and completely dependent on the context. So how could "one half" on its own, represent any particular quantity.
So, you claim that there is a quantity called one half. That's nonsense. There is no such thing as one half, unless it is a half of something, and that "something", which is required of necessity for the existence of the half, sets the value for the quantity which the half is a half of. If you really think that there is such a thing as a half which is not a half of something, and is an actual quantity all on its own, then show it to me, give me an example. Show me a half which is not a half of something. It's impossible, because "half" is by definition half of something. I'm sure you must really understand this though, that there is no such thing as a half which is not a half of something, and that one half cannot have any quantitative value whatsoever unless it is stated what it is a half of. Are you just playing dumb?
And no, it does not take two halves to make one, that's a falsity. Two halves are made by dividing one. Two equal things together, are two, not one. And one is a unity which is not necessarily made from two halves. I'm shocked that you are unfamiliar with these fundamental principles.
Quoting InPitzotl
Ha ha, that's ridiculous. 1/9 can be any quantity you want, depending on the size of the whole which is being divided nine ways. 1/9 can be one if the whole is nine, it can be two if the whole is eighteen, it can be three if the whole is 27, it can be four if the whole is thirty six, and so on and so forth. The idea that 1/9 itself, is a particular quantity is utter lunacy. And if mathematics speakers really use the term in this way, then I'd have to say that they really do not know what they're talking about. Don't you agree? How can something (1/9), which can be absolutely any quantity whatsoever, be said to be a particular quantity? And how can you not see the ridiculousness of the claim that it is a particular quantity?
Quoting InPitzotl
I've already demonstrated this, numerous times already now. It's utter nonsense to insist that 1/9 is any particular quantity, when it's very clear that it can be any quantity whatsoever. And please don't suggest as Banno did, that 1/9 really means one divided in nine equal parts, because that's something impossible.
Quoting Banno
I'm still waiting for a better definition of "number". "Family Resemblance" doesn't suffice in logic.
Quoting Banno
Actually, this doesn't contradict my claim at all. It just shows that the definitions of mathematicians contradict themselves. That's the problem with mathematics, which I will not cease to demonstrate, it's loaded with contradictions.
Quoting Banno
Let's just say that there is something to be gained from looking at how people, in general, avoid confronting the truth, and not single out any individuals here. Can we leave it at that? Or are you so absolutely certain that what you claim is the truth, and what I claim is false, that you would single out me as the one who is so certain?
Quoting Banno
The op clearly deals with "a matter of representing numbers". If it is the case that some of the symbols used in the op do not actually represent numbers, then we have a false representation. Therefore we require a definition of "number" to determine whether or not there is such a false representation. I provided a definition of "number". According to this definition of "number" we have a false representation in the op.
You are not willing to accept the truth of this, so you reject my definition. Now we have no definition of "number", and no way to resolve the question of whether the op gives us a false representation. Therefore I implore you to provide a better definition of "number", so that we can truly see whether there is a false representation or not, resolve this issue to everyone's satisfaction, and get on to something less trivial.
Quoting Banno
There is no point to addressing the arguments themselves until we determine whether or not there is a false premise. Is this a matter of representing numbers or not. I think it's very clearly not such a matter. You disagree. Where's your argument? What is your criteria for "a matter of representing numbers", which makes you so strongly believe that this is actually a matter of representing numbers?
I see what is expressed in the op as a matter of dividing magnitudes. And, it has been demonstrated numerous times, over and over again throughout history, that some theoretical divisions cannot be represented in number. Claiming to have a numerical representation of what cannot be represented in number, is clearly a false claim. Don't you agree?
:lol:
Redacted. Move on to Tim's question.
@Metaphysician Undercover, perhaps in answer to Tim's question you might set out where the flaw is in this calculation - regardless of wether the items involved are numbers or not, where in your view does this go wrong?
Wrong. I have. Also, Banno has:
Quoting Banno
...using your own dictionary. And your dictionary has, as demonstrated by Banno. Your problem is that you don't understand the language; that's confounded by the fact that you think you do.
FTFY. A ninth is the specific particular quantity corresponding to dividing one into nine equal units. That's why your same dictionary that you quoted the definition of a number in says that a fraction is a number.
Quoting Metaphysician Undercover
By proof, such as the one given in the OP.
Quoting Metaphysician Undercover
We're using your OED definition.
Quoting Metaphysician Undercover
Wrong. According to that definition, they are numbers. You just don't understand that definition... see above.
Quoting Metaphysician Undercover
That's quite interesting. What I was saying here is a direct analog of your points about fractions and pie applied to money according to my best assessment of what gibberish you're trying to push. So if you yourself don't understand this, maybe you should heed the advice you're trying to give me.
Quoting Metaphysician Undercover
It's only ridiculous to you, because you don't speak the language.
Quoting Metaphysician Undercover
We've already addressed this... you're saying nothing about fractions that doesn't also apply to counting numbers. To have a point you must special plead it.
Quoting Metaphysician Undercover
What are you talking about? A whole pie is one pie, not nine pies, eighteen pies, or twenty seven pies. You mean groups. Taking a particular quantity of equal sized groups is just multiplication. If I were at a farmer's market and they had a carton of a dozen eggs, I might could barter getting one half of a dozen. He'll give me six eggs. Or maybe I need more... maybe I need two dozens. He'll give me 24 eggs. Even your precious one dozen is twelve eggs. You're choking on multiplication.
Quoting Metaphysician Undercover
Nope. I would say you had some severe misunderstanding of math.
Quoting Metaphysician Undercover
1/9 is only a ninth of 1. But you can take 1 of anything, including groups.
Quoting Metaphysician Undercover
Because I understand how it makes sense, because I understand it.
It's a slow day, so I dug out both the Concise and the Shorter OED.
Neither contains Meta's definition.
How is that to be explain'd?
The first definition, given in both with slightly differing wording, is
Followed by
and then...
and then on to other related uses.
I found it on some random russian vocabulary site: (https://slovar-vocab.com/english/fundamental-vocab/number-6810737.html) (ETA: Unhiding this link). An expanded version can be found here.
SO what do we decide - did Meta lie, or was he misled?
The former has no attribution. The latter says "powered by Oxford" (OUP). I personally was granting that maybe he had one of those ancient analog thingies made of trees; can't quite trace it further than that (though TBH I didn't try too hard past those two).
That's what I was using, so that's no excuse.
Lexico claims "All definitions and translations are written by Oxford lexicographers". So we might give him the benefit of our doubt and conclude that he was misled rather than malicious.
I only really know enough about maths to enjoy the explanations given here, not to provide any of my own, but I do know about the psychology of crackpots (we prefer to use the term 'nutjobs' nowadays, in these more enlightened times, unless the patient is allergic to nuts, in which case 'fruitcake' is fine).
What goes on here, I think, is that a person develops a fear of that which they do not understand. I think it's born of the extent to which we are no longer in control of our livelihoods (but that's another story entirely). The point is that philosophy-talk - the grammatical form of the arguments that philosophy uses - acts like heroin, in that it supplies a way of distancing oneself from that uncertainty and complexity. Rather than words being used, as they really are, to fumble about in the dark trying to get other people to act in ways we'd like, they turn into containers to bind stuff to, to pin it down and appear to stop it from being so ineffable.
Dictionary definitions are the drug dealers here. Supplying a few thousand such strong-boxes in which we can lock uncertain aspects of the world we experience. Numbers are no longer a complicated product of our minds, slightly intangible in places, occasionally contradictory if taken out of context, with some odd consequences we can quite get our heads around (like i). They now become what the dictionary says they are (usually ignoring the second and third definitions) - tamed and chained.
It's a weird flipping of our rational capacity from being that which tries to make sense of an already existent world to that which creates that world according to its rules, and thereby regains control over it. Instead of trying to work out some way of modelling what we experience, we simply claim to only experience that which we have modelled. A good proportion of the arguments here could be summed up as "It seems that way to me, therefore it must be that way". Of course, one acting this way needs a filler, something to explain all that which is beyond their current imagination - hence almost everyone using this drug is also religious - God of the Gaps.
Anyway, thanks to the mathematicians here who do patiently point out the flaws in these 'solipsistic' arguments, they do make interesting reading for an interested non-mathematician.
Cheers.
I suggested elsewhere that the motive might have been a search for recognition; perhaps this is transference on my part.
That link between the avoidance of ambiguity, political conservatism and religion is cogent.
Yeah, I think that's true. I see it as part of the whole uncertainty issue. I mentioned this briefly on the Lazerowitz thread. Socially, the rise of science as powerful force, I think, results in a backlash rise in alternative 'expert' fields where they are immune to being demonstrably wrong. I think the same thing happens on a smaller scale here, and again is facilitated by the grammatical structure of philosophy-talk. Faced with a field in which it appears one can be demonstrably shown to be wrong, recognition is harder to come by and more fragile when attained. A simpler tactic is to set up an alternate set of rules and , regardless of their utility, raise oneself as an expert in those. One cannot be demonstrably wrong, one is instantly the world's foremost expert and one did not even have to leave one's armchair. Of, corse many see this as a hollow victory because the very public rejection of such rules is sufficient to pour cold water on any feelings of grandeur.
Many people are immune, or resistant, to updating their beliefs in the face of rejection by their peers. This is usually a good thing because it's how we get innovation and resistance to oppression etc. But again here philosophy-talk lets us down, it gives the impression that we can do this with language too - that we can reject the 'oppressive, conservative' use of terms to refer to A and insist they refer to B - ignoring the fact that language is a social endeavour, agreement is the substance of it, not a side-effect.
So yeah, I think you're right, people do maintain these very private structures as a kind of 'cheat mode' for the progression to recognition, and I think the means by which they do it is to mistake language for the kind of thing where innovation is bold and entrepreneurial and so become immune to the rest of their language community responding as if they were mad.
The argument, in its cleanest, most direct form, is given in the OP:
Quoting jorndoe
Meta did not directly address this, or any other such proof. Instead he went to an irrelevance, his claim that 1/9 is not a number.
Even if 1/9 were not considered a number, the proof would stand.
I suppose that for him it's not the mathematical process that is at fault so much as the picture of how mathematics works - for Meta, mathematicians have failed to understand their topic, and so their proofs are irrelevant. His task becomes not the rejection of the proof, but the correction of the mathematician's picture.
Hence, the mathematicians who engaged with him could never succeed in showing him how he went wrong; for Meta their very arguments are based on false premises. They may have thought that they were in a discourse about mathematics, but they were in a discourse about Meta's certainty.
Yes, that's exactly it. Proving a theorem by the rules of mathematics is seen as irrelevant because the game is not to accept the rules and try to understand where they lead, the game is to take one's current understanding and construct rules which make it right.
It's obviously a triggered response at some point in their life. Children cannot develop at all if they were born with such an attitude, no understanding of anything would come about. It must therefore be something these people decide at some point in their life - "That's it, no more understanding, no more modelling, from now on the world has to change to fit what I already understand!".
What I can't decide is which came first. Whether the need to respond this way promotes that kind of philosophy, or whether that kind of philosophy entices people into responding that way. If there was no form of discourse in which one could appear to argue about what a number "really is", would they invent one to meet the need, or would they be out of alternatives and have to fumble by with only half understanding like the rest of us?
Which premise is false?
And don't you think that the professional mathematicians know more about maths than you do? I don't know what your area of expertise is but I'm pretty certain that it's something else.
Perhaps philosophy allows someone like Meta to hide what he is doing. If this were a thread about Wittgenstein, only those who had read Wittgenstein and the associated literature would recognise the misrepresentation in which Meat engages - as indeed did happen in the threads involving Sam (of blessed memory) and others.
It's the topic - mathematics is clear cut, and so it's harder to hide.
When it's so close to the usual definitions, yeah.
Quoting tim wood
I've already given my proof, based in my definition of "number". It's in my posts directed at Banno. Don't listen to Banno here because Banno's form of discussion is to pay no attention to what the other person says.
We can talk about making a theoretical division which is impossible to do, such as dividing one in nine equal parts, which is impossible to do. In reality it is contradictory to divide one into any parts, because then you are saying that it is not one, but however many parts you are dividing it into. Instead of recognizing that division of some quantities is impossible, some mathemagicians have proposed a new system of "numbers", which allows that impossible divisions can be represented as numbers.
The problem is that now they have so-called "numbers" which are outside the criteria of the definition of "number" (as presented by me), yet the mathemagicians provide no clear new definition of "number" which allows that these representations such as .111..., are actually representative of numbers. They give them a name "real numbers", so that they can refer to them, but the concept of "number" is just left vague, undefined, and full of inconsistencies.
Since I have provided a very clear definition of "number", and according to this definition many representations which are classed as real numbers are not actually numbers, so we can conclude that these real numbers are not actually numbers, based on that definition. If you want to demonstrate that my logic is unsound, I suggest you show me that my definition is false, by producing the true definition of "number", the one which allows that all real numbers are actually numbers. Otherwise we can look for a better name for things like 1/9, one which better represents what they are, such as "relations between quantities", or something like that .
Quoting Banno
I've already laid that out for you:
Quoting Metaphysician Undercover
The op uses an expression which represents an impossible division, 1/9, a ratio which cannot be expressed in numbers. There are different forms of divisions which cannot be expressed in numbers, some are called "irrational numbers". Since they are not numbers it was a mistake to start calling them numbers. But this is the bad influence which common vernacular has over logic, it inclines us to replace the rigorous logical definitions which are required for sound logic, with family resemblance. (It's similar to a number, so let's just called it a number, and then we can have a bunch of different types of numbers).
Yes, I think that's true. I'd go as far as to say that some philosophical discourse is deliberately engineered to serve this function.
To slice a pizza into equal slices, try a pizza cutter. Ten paragraphs of nonsense gibberish can be refuted with one kitchen appliance.
To most people, cutting one thing into multiple pieces is a triviality. Where's the contradiction you're describing? Cut one pizza into four equal slices, and you have four slices, each of which is one fourth of the pizza. What you're saying translates that there's a contradiction here because I'm saying that there are four slices and one pizza. How's that a contradiction? You would have to change the concept of "contradiction" to something that can actually be done, and the concept of "impossible" to something you can actually do, to make this argument stick, because we damned well can slice a pizza into four slices.
We can also cut each of those four slices in half, making eight slices. One of the four slices we had before is the same quantity of pizza as two of the eight slices we produce; i.e., it's a "particular quantity". It all works, MU. There's no contradictions except in your fictitious world where you can't use pizza cutters. In reality, we have pizza cutters and we can and do slice pizza.
Sure, yet we do know some things at least, and can reason to some extent if careful.
Don't just ? × ? - ? - 7 + ? / ? + 3 / ?, for one.
The amount of naturals isn't a natural, for another.
Maybe ? could be said to be a quantity that's not a number.
Quoting jgill
(y) (I'd hit "Like", but this will have to do)
Actually, that's a good lot of philosophy right there. ;)
Saw something fly by about adding zeros, but:
[math]\sum_{n \in \mathbb{N}} 0 \ = \ 0[/math]
This is a ramification of the contents of the thread: 0.999...= 1 started by @jorndoe. Thanks jorndoe
First off, the simplest infinity that we know of is the natural numbers: N = {1, 2, 3,...}. The infinite set N is generated by the simple iteration of adding one to the preceding number as so: 1; hen 1 + 1 = 2; then 2 + 1 = 3; and so on and so forth.
What I'm particularly concerned about is the ratio between consecutive elements in the set N. The ratios look like below:
1) 1; there is no ratio here as there is no natural number that precedes 1
2) 2; the ratio is 1 : 2 = 0.5
3) 3; the ratio is 2 : 3 = 0.666...
4) 4; the ratio is 3 : 4 = 0.75
5) 5; the ratio is 4 : 5 = 0.8
6) 6; the ratio is 5 : 6 = 0.833..
7) 7; the ratio is 6 : 7 = 0.851742...
8) 8; the ratio is 7 : 8 = 0.875
9) 9; the ratio is 8 : 9 = 0.888...
10 ) 10; the ratio is 9 : 10 = 0.9
11) 11; the ratio iss 10 : 11 = 0.90...
.
.
.
3000) 3000; the ratio is 2999 : 3000 = 0.9996...
37896544) 37896543; the ratio is 37896543 : 37896544 = 0.9999999736123695078896904169075
As you can see as the numbers get larger the ratio between a natural number x and its successor x + 1, given by x : (x + 1) approaches, in the limit, 0.999...
But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1
In other words, there will come a point in the sequence of natural numbers where a natural number x and its successor will have the relationship x : (x + 1) = 0.999... but since 0.999... = 1, x : (x + 1) = 1 and that means x = x + 1 which basically means there's a natural number which will not increase in size when you add 1 to it. We can't say that x is infinite because if it is then x : ( x + 1) = infinite : infinite which is undefined and can't equal 0.999... Ergo x must be a finite natural number but since adding 1 doesn't get us a number larger by 1, it follows that there is a largest natural number.
Replies to this post are to be addressed to @TheMadFool
To simplify:
[math]\lim_{x \to \infty}(\frac{x}{x+1})=1[/math]
The below is a proof that 0.999... = 1.
[math]\begin{align}0.\overline{9}\stackrel{def}{=}\lim_{n\to\infty}\sum_{k=1}^{n}\frac{9}{10^k}=\lim_{n\to\infty}(1-\frac{1}{10^n})=1-\lim_{n\to\infty}\frac{1}{10^n}=1-0=1\end{align}[/math]
To prove it wrong you need to show that its definitions are wrong (within the domain of mathematics, not within the domain of British English) or that its inferences are invalid.
I think you're referring to the discussion I was having which was in ?.
"What I'm particularly concerned about is the ratio between consecutive elements in the set N"
n/(n+1) = 1/(1+1/n) -> 1/(1+0) = 1
Rest easy, mate. Time for a toddy.
Would you agree that the fact that a thing has more than one nam is no argument against the two expressions or representations designating the same thing?
I thought you were making that argument earlier but now you don't seem so sure.
In any event, I offer you this. .999... = 1 is a theorem of ZF set theory; for exactly the same reason that the knight moves the way it does in chess. There is no "truth" to the situation; rather there are only the rules of a formal game. If you made different rules you could defined .999... to be 47 and you could make the knight move differently. it would be fine.
The acceptance that .999... = 1, and of the consequences of the ZF axioms in general, is based on utility. When we accept ZF we can build up most of known mathematics and provide rigor to what the physicists do (usually a century or two after they've already done it). That's a pragmatic argument for accepting the axioms.
If you want to say that .999... = 1 offends your sensibilities, you are free to do that. As long as you are willing to grant the proposition that .999... = 1 is a theorem of ZF. That is a matter of objective fact that could be verified by a computer program. That is, there's a finite sequence of verifiable steps from the axioms to the conclusion.
That's really all it means; and even if you think that somehow "deep down" the equality is false; you must still admit that the statement follows logically from the axioms of ZF. So that you'd have to conclude either that ZF is inconsistent (which as far as we know, it may be) or that it's simply the wrong set of axioms for mathematics. In which case you're free to propose different ones.
Your tightening up of the mathematics is exemplary. The result will be to show in even greater relief that this is a thread about @Metaphysician Undercover, not about maths.
You don't seem to understand. "One" does not represent a quantity which can be divided. Any multitude such as two, three, or four, can be divided, because being a multitude means that it is composed of parts and therefore can be divided into those parts. If one could be divided, then you are saying that it is made up of parts and is therefore a multitude, and not one. If it could be divided in two, then you are saying that it is made of two parts, but that would mean that it's two, not one. If it could be divided in three, then that would mean that it consists of three parts, and is really a quantity of three.
If you think that the quantity represented by "one" can be divided in any way that you please, then you deny the meaning of "one" as a single thing, because you are saying that it's really a multitude of as many things as you want it to be, existing as a unity. But that's nonsense, because that's what the other numbers represent, multitudes which have a quantitative value. If you say that one can be divided any way you please, then you are saying that "one" represents a multitude with no particular quantitative value, it is however many things you want it to be. But that's nonsense, because we all know that "one" represents a single thing, not a multitude of however many things you want it to be.
Quoting InPitzotl
It appears you just haven't taken the time to understand what I was saying.
Quoting InPitzotl
Again, it appears you haven't taken the time to understand what I was saying. As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division.
Concise Oxford,1990, p813.
That the mathematical definition of "number" changes like the weather is good evidence of what I've been arguing. We do not have any logically rigorous definition of "number", and mathemagicians just use the term however they please, referring to whatever they want as "a number".
Quoting Banno
The issue, as I said, is the op's question: "As a matter of representing numbers, wouldn't most be fine with...". I'm not fine with it, because as I said 1/9 is not a number,.it is a ratio. The op doesn't ask for proofs or any such nonsense, it asks if you are satisfied with that way of representing numbers.
Quoting Banno
I couldn't care less about the proof. The op asks, "as a matter of representing numbers, would most be fine with...". If 1/9 is not a number, then we ought not be fine with this, as it is presented as a representation of numbers. If 1/9 is not a number then the presentation, as a representation of numbers, is false.
Quoting Michael
That 1/9 is a representation of a number.
Quoting Michael
The question of the op is are you fine with this, as a representation of numbers. It is not, do you believe that .999...=1. I really do not care whether .999...=1 or not, or how many proofs there are concerning this. I'm concerned about the question of the op, is this an acceptable way of representing numbers.
So the issue I've pointed to is whether 1/9 is a representation of a number, or not. I've argued that it is a representation of a ratio and therefore not necessarily a number. Some ratios are impossible to represent as a number. That is where we get the term "incommensurable".
My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage.
Quoting Banno
Flattery will get you nowhere.
Indeed; none of us understand, Meta. No one but you.
Now, what does that imply?
I could prove from first principles that .999... and 1 refer to the same real number. You choose not to engage with the argument. Nowhere to go with that.
But [math]\frac{1}{9}[/math] is commensurable. It's the fraction of two integers.
Also it seems to me that what you call "numbers" mathematicians call "natural numbers" (or maybe "integers"; do you consider negative numbers as numbers?). There's more than just natural numbers in mathematics; there's rational numbers that include the commensurable fractions like [math]\frac{1}{9}[/math], real numbers that include irrational numbers like [math]\sqrt{2}[/math], and more.
I don't see what purpose there is in saying that non-natural numbers aren't numbers, and latching onto the OP saying "as a matter of representing numbers" completely misses the point of this discussion. @jorndoe wants to show that 0.999... = 1. That's what the attached PDF tries to show. Whether or not you want to call 0.999... a number is irrelevant.
Yes, indeed. Meta cannot see this.
Quoting Metaphysician Undercover
This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math.
Okay, let me show you where the multiplication is. Let's revisit this:
Quoting Metaphysician Undercover
(a) 1/9 of nine is [math]\displaystyle\frac{1}{9}\times 9 = 1[/math]
(b) 1/9 of eighteen is [math]\displaystyle\frac{1}{9}\times 18 = 2[/math]
(c) 1/9 of 27 is [math]\displaystyle\frac{1}{9}\times 27 = 3[/math]
(d) 1/9 of thirty six is [math]\displaystyle\frac{1}{9}\times 36 = 4[/math]
Do you see the multiplication now?
Since I think we've finally nailed down the problem, I'll keep to just the key parts.
Quoting Metaphysician Undercover
Two major problems with this MU:
(Of course, you could always deny pizzas are real).
What you fail to understand, MU, is that many things can be divided, even if you count one of them. Also, lots of things have whole-part relations; given a loaf of sliced bread with 24 (equal) slices per loaf, I can give you 3 loaves, or 3 slices... I'm still doing nothing but counting, but I'm giving you different "particular quantities" of bread. The slice quantity is much smaller than the loaf quantity. This is what's known as a unit. If I give you 3 slices, I'm giving you 3/24 loaves. We might also say 3/24 of one loaf = 3 slices. We can also apply units to continuous measurements, such as lengths along those dimensions you alone denied exist.
On an interpersonal note:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
^^-- this makes you look irresponsible and lazy. You're blaming me for not understanding you, and blaming me again for you not understand me. This conveys the message that you think your time is extremely valuable and my time is worthless. That's... not great optics.
As The Princeton Companion to Mathematics says:
Wouldn't we have to do that to even be able to talk about 0.999...? Or can we somehow deal with "infinite" sequences without the axiom of infinity?
I can't answer this question without an acceptable definition of "number". It seems mine has been rejected. But as I just described, one cannot be divided by nine, because it means that one is a multiplicity, when it is defined as a single, or simple.
Quoting Michael
The point is that a ratio is not a number, it is a relation between two numbers. And some ratios cannot be expressed as a number, the relation expressed by pi for example. Which ratios can and cannot be expressed as a number is a matter which might be discussed. I am taking the extreme position to claim that none of the following 1/2, 1/3, 1/4, 1/9, etc., can be expressed as a number, because it contradicts the definition of "one" to say that one is composed of a multiplicity which can be divided.
If the others, in this thread, insist that all ratios can be expressed as a number, then I want to see that definition of "number" which allows for this. Or is "number" just some meaningless word which mathematicians can use however they please, in any random way?
For example, if the quantity expressed by "one" can be divided however a mathematician wants to divide it, then it must be a multiplicity composed of an infinity of parts. But that contradicts its definition of "single", and makes "one" into a meaningless term. Having no restrictions derived from what it means to be "a single object", or some such thing, a person might refer to any multiplicity whatsoever as "one". And this leaves "one" as an absolutely nonsensical term.
Quoting InPitzotl
Of course. If you're just now noticing, I refuse to use that deceptive language, loaded with contradiction in its axioms.
Quoting InPitzotl
That is just deception. To divide nine into nine parts, or to divide eighteen into nine parts is very clearly division. To express this as multiplying nine by another number, "1/9", or 18 by "1/9", is an act of deception. Claiming that division is multiplication is deception.
Quoting InPitzotl
No it isn't it has been divided. Either it is one object, or it is eight objects. To claim both is to claim contradiction.
Quoting InPitzotl
If this is true, then we need to define how to distinguish a whole from a part, so that we are not referring to the part as "one", when it is really 1/8 of the whole, and we are not referring to the whole as "eight" parts when it is really one whole. I would enter a discussion of parts and wholes with you, so long as we have principles whereby we can distinguish one from the other, and not just randomly decide to call this a part, and that a whole, at will throughout the discussion, because that would get nowhere.
Quoting Michael
That's putting the cart before the horse. Before deciding which items go into which set, we need to define the conditions of the set. No one puts a whole bunch of random terms into one set, then names the set "numbers". if that were the case, why wouldn't we put "house" and "car" into that set called "numbers" as well?
This still begs the question about the difference between a limit and an actual infinite sum. Your reasoning shows that you don't run out of natural numbers 'until' infinity.
They start by defining the following set:
N = {0, 1, 2, 3, ...}
Then they define the set "Z" as the set that contains the elements in set "N" and also their additive inverses, i.e. {-1, -2, -3, ...}.
Then they define set "Q" as the set that contains the elements a/b where a and b are elements in set "Z" and b is not 0.
Then they define set "R" as the set whose elements are the limit of a convergent sequence of the elements in set "Q".
Then they define set "C" as a + bi where a and b are elements in set "R" and i is a formal square root of ?1.
(There are more formal ways of defining these sets than the above, but the above is easier to understand for someone who doesn't know much about maths. See here if you want something more exact).
None of this depends on there being a formal definition of the English language word "number" which is what I was talking about and which is where you're getting lost.
If you absolutely must have a formal definition of "number" then lets go with "any member of [math]C[/math]" so we can get back on topic.
Quoting Metaphysician Undercover
^-- One of those two things is a lie. Most charitably, you're incapable of using the language.
Quoting Metaphysician Undercover
^-- This is straight up paranoia. Deception has two parts... the advertised meaning, and the true meaning... the advertised meaning must be what you want to trick the other person to believe... the true meaning must be something different. We don't have that here... we only have one part... the usage.
The problem is that you don't understand the language, therefore, you spin this meaningless narrative that people are trying to deceive you (sprinkled with paranoia). It's meaningless, because there's nothing you can say that you're being deceived to believe... it's a working language, you just don't understand it. This is trivial to show in your debates; you don't even bother to debate what doesn't work. This is perceived on your part as non-compliance: "I refuse to use that deceptive language", but in terms of truth, that's hollow... your only possible genuine complaint is that the language doesn't work... to show it doesn't work you have to know how to speak the language, in order to construct the contradiction.
Otherwise, the only possible complaint you have left, the one you keep whining about, is that it's not the same language as your uninteresting one.
Quoting Metaphysician Undercover
Sure, but that's just division:
(e) [math]\displaystyle 9 \div 9 = 1[/math] <- "divide nine into nine parts"
(f) [math]\displaystyle 18 \div 9 = 2[/math] <- "divide eighteen into 9 parts"
...and this is using a fraction:
(g) [math]\displaystyle \frac{1}{9} \times 9 = 1[/math] <- " 1/9 of 9"
(h) [math]\displaystyle \frac{1}{9} \times 18 = 2[/math] <- "1/9 of 18".
Going back to our loaves:
(i) [math]\displaystyle 3 \div 24 = \frac{3}{24}[/math]
There's no deception here, there's only confusion... on your part. Nobody who uses this language is confused. This is just how the language works. A ninth is the multiplicative inverse of nine. A twenty fourth is the multiplicative inverse of twenty four. Dividing by nine is equivalent to multiplying by a ninth. "A ninth of" is multiplying by a ninth; just as "five ninths of" is multiplying by five ninths. There's no problem here.
Quoting Metaphysician Undercover
We do it by applying a unit. A slice is a part of a pizza. One pizza. Eight slices. It's so easy, everyone but you does it all the time!
Quoting Metaphysician Undercover
A yardstick measures 1 yard. It has 3 feet in it. Each feet has 12 inches. Those 12 inches usually are marked in fractions of an inch; typically at least an eighth of an inch. Now don't get scared... an eighth of an inch is part of an inch which is part of a foot which is part of a yard. Parts are transitive; an inch being part of a foot being part of a yard means an inch is part of a yard. We call the "whole" we're talking about a unit, and we just specify it... that's all there is to it. I say the yardstick is one yard long. That is three feet long, 36 inches long, and 288 eights of an inch long.
Why not? That's how you use language. You have to specify the thing you're talking about, even if it's a part. I drive my car. I drive it into traffic. I turn the steering wheel. There's no problem doing this, outside of you having a problem with it, but that's not our problem. Let me rephrase this so that it sinks in:
If the only problem with the language is that you have a problem with it, then you are the problem.
Nobody has to talk to you before they use an IEEE-754 64-bit float in a program and, even if they do talk to you, you're giving them absolutely no reason to care. Let me rephrase that... nobody cares about your impoverished language. The reason I'm talking to you is that I care about you.
Quoting Metaphysician Undercover
I almost agree... your whining about something that works gets us nowhere. The only part where I disagree is that your whining about something that works has negative effects.
Curious about your 1/9 concerns. A while back you told me you believe in rationals but not sqrt(2). But now you don't seem to believe in rationals. What's up?
Secondly, can you give me a yes or no response to this question? Do you agree, either by personal understanding or by taking my word for it, that regardless of whether .999... = 1 is "true" in any metaphysical sense, it is still the case that it's a formal consequence of the axioms of ZF set theory?
@Metaphysician Undercover's in my head for sure. But I think it's a good clarifying question. Whether he accepts the formalism on its own terms even if not in any ultimate sense.
...and here is the gift the crackpot gives to the world. Occasionally.
Quoting tim wood
I'm a grizzled veteran of .999... threads. I was dismayed to find one had sprouted up here, but I'm powerless to resist. I've sworn off them many times without success.
With this, I have great sympathy. I compared it earlier to the compulsive tonguing of a broken tooth.
This is where I have a disagreement. There are many instance of a/b, which cannot be called an element. As I described already, in many cases a cannot be divided by b, it is impossible. One might express the ratio a/b, but the operation which is required to produce an element from this ratio cannot be carried out, therefore there is no element in these cases. So we have a faulty set here consisting of some necessarily non-existent elements.
Quoting InPitzotl
There is a problem, dividing is clearly not the inverse of multiplying. The evidence of this is the existence of irrational numbers, which are derived from dividing, but not derived from multiplying. For a mathematician to say that dividing is simply the inverse of multiplying is like a physicist who says that time can be modeled as going either way, future to past, or past to future, one is just the inverse of the other. There is ample evidence that this is not true, and those who overlook the evidence, like yourself, start making false claims.
Quoting InPitzotl
This is all wrong. These are measurements, and what you are describing is equivalencies. A
"yard" is equivalent to three feet, and a foot is equivalent to twelve inches. Each term refers to a particular length, and the length is one unit, without parts. If a yard, or a foot consisted of parts, there would have to be something within that unit to separate the individual parts, one from another. Clearly there are no such separations within a yard or a foot, and there are no such parts within these units. What would that separation be made of? And without the separation there are no parts. Do you know what a "part" is?
Quoting InPitzotl
I hope you realize that this is a very selfish expression. And I really hope you don't behave this way in your common interactions with people.
Quoting InPitzotl
Oh sure, the person who's trying to convince me that division is really just inverted multiplication is doing this because they care about me. I think you're like Plato's philosopher king, with the noble lie. You actually believe that your lying to me is for my own good. Or are you so naive to actually believe that there is no more to division than an inversion of multiplication?
Quoting InPitzotl
Whether or not it "works" is not the issue. I have no doubt that it works. What is at issue is the truth. You know, until they're exposed, lies and deception work. Don't you?
Quoting fishfry
Different thread, different argument. What makes you think that I believe in any sort of mathematics? What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms.
Quoting fishfry
Sure, why would I deny this? It's been shown to me in so many different ways. But if you have good reason to believe that the consequence is a falsity, then it's just evidence of the faults of those axioms. Do you agree, that if the the formal consequence of the axioms is to produce a falsity (whether or not you believe the present example is a falsity), then there is likely fault in the axioms?
:up:
It must be brutal that few in the mathematical community seem concerned. But I do agree that the axiom of choice is an unhealthy pathology. :cool:
By definition, division is the inverse of multiplying.
Quoting Metaphysician Undercover
Silly MU. Given any integer a; and any nonzero integers b, c, d:
[math]\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a\times d}{b\times c}[/math]
...and since a, d are integers, and b, c are nonzero integers, the result is also rational.
This means three things:
Quoting Metaphysician Undercover
This being a false analogy, we can ignore your conclusions, except insofar as they reveal your state of mind. But for that, I'll just let your post speak for itself.
Quoting Metaphysician Undercover
Yes, they are measurements.
Quoting Metaphysician Undercover
Yes! Let's math this using the equivalence symbol.
[math]1\ yard \equiv_{?} 3\ feet[/math].
There. Now what is the question mark here? We need the sense of equivalence... I wonder where that comes from? :chin:
Quoting Metaphysician Undercover
Thank you! So let's math that:
[math]1\ yard \equiv_{length} 3\ feet[/math]
Quoting Metaphysician Undercover
Without parts you say? Interesting:
[math]1\ yard \equiv_{length_{w/o\ parts}} 3\ feet[/math]
Great! With you so far. But one more thing... we already know these units indicate length. So we can drop the qualifications here and just say:
[math]1\ yard = 3\ feet[/math]
See anything interesting? We're using the number 3 in a sense that, by your own admission, does not require parts by your definition (you said it yourself).
Quoting Metaphysician Undercover
But by your own words, we have an equivalence relation without your parts. So we have something that works already.
Quoting Metaphysician Undercover
Not true. 1 yard = 3 feet without your parts. There is a different sense of part that is in play here, though. The particular length that is 1 yard is length-equivalent to 3 feet in a specific way... there are two positions (particular points) along a yard-length section that separate a yard-length into 3 contiguous equivalent lengths. Each of these three contiguous length has the particular length of a foot. Conversely, if we take three foot-lengths so arranged such that they are laid out end to end meeting at these two points, then the total distance covered by these three foot-lengths is itself that same particular length we call a yard. So in this sense, a yard-length is composed of three foot-length partitions, each of which we can call a part. Note that you can slice the ruler at this point if you choose and make separable parts, but that does not in any way affect the invariant condition of being a particular length measured by these particular quantities (1, 3) of particular length-units (yard, feet).
Quoting Metaphysician Undercover
Nope. The above suffices to make 1 yard equivalent to 3 feet without needing your parts. Given it works without your separable parts, your parts are superfluous.
Quoting Metaphysician Undercover
You misunderstand MU. You are the problem, and you are suffering because of it. You have chosen to pit your views against math. But you've handcuffed your own personal identity to your views; and, you're here in this thread sharing them. Because of the nature of the battle you yourself picked, it's you versus math. So if there's no problem with the math, you're going to suffer. And that's exactly the situation you're in... there's no problem with the math, and you're suffering. Take another look at the reactions your getting and tell me I'm wrong.
Quoting Metaphysician Undercover
You continue to misunderstand. I don't care if you believe division is inverted multiplication or not; that's not what's hurting you. What's hurting you is the fact that by pitting yourself against the theory that defines division this way using your worthless theory, you're defacing your own image in the eyes of others who know better. There's a severe risk that people will equate your value to the value of your views, because your views are total garbage. But you're not. My goal here is simply to give you some perspective so that you can see what I see... that you're just hurting yourself.
Quoting Metaphysician Undercover
Dysphemisms and appeals to my alleged gullibility isn't an argument.
Quoting Metaphysician Undercover
There's no sense of math being "true" other than that it works. You're basically trying to sell us a belief. Math is a language that does what it says on the tin... this follows; that is consistent, and so on. The truth of math is measured by what it says on the tin, and the fact that it does that. And here you come dressed in salesmen clothes peddling this new theory, telling us how math has led us astray. How pray tell? It does exactly what it says on the tin. Of course that's the issue. What sort of "truth" are you pitching?
Quoting Metaphysician Undercover
Deception working isn't a truth criteria for deception.
So clearly you don't understand mathematics. Let me set the record straight; mathematicians can do calculations with any kind of [math]\frac{a}{b}[/math] where [math]a[/math] and [math]b[/math] are integers and [math]b\neq0[/math]. That [math]a[/math] can be divided by [math]b[/math] just is that they can do these calculations. I'm not exactly sure what it is you even think it means for [math]a[/math] to be divided by [math]b[/math].
Point1: Ok a fair answer but still a deflection. The question is why you earlier believed in the rationals, but now do not believe in 1/9. Since 1/9 is a rational number, being the ratio of two integers, 1/9 is rational.
If I've caught you in a little inconsistency, or your thinking has changed, or if I'm misunderstanding this point, I'd like to understand. Specifically with respect to 1/9 and its rationality.
Point 2: Now your deflection is interesting. You changed the subject to claim that I have no reason to believe you believe in mathematics. But it's perfectly obvious that you do. I don't believe in tennis. It doesn't interest me. I don't hang out on tennis forums and tell players that their game is nonsense and their rules are unsound. I simply don't watch tennis matches and don't click on tennis-related news. The last tennis match I paid attention to was Bobby Riggs versus Billie Jean King. So when I see you passionately arguing your point -- whatever it may be, since your mathematical nihilism is hard to fathom -- I assume you must care a lot about mathematics.
Point 3: Do you regard the rules of chess as needing a "good dose of skepticism?" Why or why not? Perhaps you are putting more ontological certainly into math than math itself claims. I personally don't think that .999... = 1 is "true" in any meaningful sense. In the real world the notation isn't defined at all, since there are no infinite series because as far as we know, the axiom of infinity is false.
So YOU are the one setting up strawman claims on behalf of math, that math itself doesn't claim.
How can you complain about the rules of a formal game? How could one be "skeptical" about the rules of baseball? What does that even mean?
Quoting Metaphysician Undercover
But that's great. Then you and I are in absolute agreement. The proposition here is that ".999... = 1 is a logical consequence of the ZF axioms." You are agreeing with this. From a formalist perspective, it is no different than saying that a particular chess position can be legally reached.
But if you agree with this, then you and I have no disagreement. Because I make no other claims!
I wonder what claim you think it being asserted by .999... = 1. It's a statement in the formal game of modern math. You can no more object to it than you can object to the rules of chess.
So tell me: What extra secret sauce are you imbuing the symbolism with? Why do you think there's some "true" meaning out there in Platonic land? Do you think there's a real way that the knight moves and the rules of chess have got it wrong?
Can't you see that if you agree that the formalism is valid, then we're in agreement. I myself make no claim of the soundness of mathematics; only its validity. You're arguing against a strawman of your own invention.
Quoting Metaphysician Undercover
No. Math isn't true or false any more than chess is true or false. If you criticize math for having rules that are not technically true of the world, you must make the exact same criticism of chess. Do you?
Quoting Metaphysician Undercover
Suppose for sake of argument I say yes. The axioms of math are faulty by virtue of not being true of the world. Will you then grant me that the rules of chess are likewise faulty by virtue of not being true of the world?
Sweet.
I've never seen any such definition of "division". The usual definition involves dividing something and this has nothing to do with multiplying.
You turn a blind eye to the evidence, to insist on a falsity. Take the circumference of a circle, and divide it by the diameter, the result should be pi. But to start with the same diameter, and multiply it by pi, will give you a different number as the circumference, because you'll have to round off pi. This is the same situation in the op. Start with one, divide it by nine, and you get .111.... Start with .111... and multiply it by nine, and you do not get one, you get .999.... In these cases, when you take a number and divide it by another number, then take the quotient and multiply it by the divisor, the product is different from the original number. Therefore division is not a direct inversion of multiplication.
Of course you insist that .999.. is the same as 1, and therefore division is simply an inversion of multiplication. But this is just begging the question. Your false assumption that the two are the same thing, supports your conclusion that division is an inversion of multiplication, and the false assumption that division is an inversion of multiplication supports the claim that the two are the same.
You ignore the evidence of the fundamental difference between multiplication and division.. This evidence is that when you carry out an operation of division there is often a remainder. There is never a remainder in multiplication, nor do you start with a remainder, There is no place for a remainder in multiplication, yet there often is a remainder in division. Therefore division is not simply an inversion of multiplication.
Even if you provide examples where one is a direct inversion of the other, (eight divided by two equals four, and four times two equals eight, for example), this is not sufficient for the inductive conclusion that division is the inverse of multiplication. All it takes is an example or two, such as the ones I provided, to invalidate such a conclusion. Whenever there is a remainder, there is evidence that your conclusion is invalid. So you make the inductive rule (division is the inverse of multiplication), then when exceptions to the rule are shown to you, which ought to make you think twice about the validity of the rule, you simply deny that the exceptions are real exceptions, by claiming that .999... is the same thing as 1.
Obviously, you think that "the remainder" in an operation of division is not a real thing, that its existence can be denied and ignored, and so we can say that division is simply an inversion of multiplication. You turn away from, and ignore the overwhelming evidence that you are wrong.
Quoting InPitzotl
You seem to be conflating units of measurement, foot, yard, etc., with length, which is the determined measurement of something. So your argument here really makes no sense. You argue that three one foot long rulers makes up a length which is a yard, and you conclude therefore that a yard, as a unit of measurement consists of these three parts. But this is clearly false, because this is just one example of something which measures a yard, three one foot measuring sticks, and it in no way indicates that the unit of measurement "a yard" is actually composed of these parts.
Quoting InPitzotl
I'm not worried about that, because the problems in math are glaring. So if it takes "no problems with math" to make me suffer, I think it will be an extremely long time before I start to suffer.
What's with the appeal to others? Banno was in the same boat as you, implying that if others agree it must be correct. It's as if when someone comes up to you and pats you on the back saying "your right", this makes you right. Then you might have a whole group of people in a big circle jerk, patting each other on the back saying "you're right", and "I know I'm right, and so are you", onward and onward, blissfully unaware of the truth, when they're not really right. And if someone from the outside tries to point out your mistakes you shun them, saying you're not part of our circle, you don't understand our language, go away, we don't want to hear what you have to say, it interrupts our self-congratulations.
Quoting InPitzotl
Oh, poor me. Don't you just feel so sorry for a poor soul like myself? I'm standing up here in front of others, doing whatever I can to make a fool of myself. And you want to shelter me, and protect me. What kind of bullshit is this? You're even worse than Banno.
Quoting InPitzotl
What I am arguing is the lack in consistency in math. How many different "number" systems are there, natural, rational, real? How can you believe that there is any consistency within mathematics as to what "number" refers to?
Quoting Michael
I think it's quite clear what division is, it's to divide something into parts. You think it's to do a certain type of calculation. I would go along with this, as a theoretical type of division, so long as there are some rules involved.
I hope you don't think that division is simply an inversion of multiplication. If you do though, then we ought to adhere to the rule that if there is going to be a remainder in any calculation of division, then this calculation cannot be carried out, because it cannot be inverted into multiplication. This would mean that some numbers cannot be divided by others. But if you insist that any number might be divided by any other number, then we need to accept that division is not a simple inversion of multiplication, because we can have remainders.
Quoting fishfry
I don't think that I said I believe in the rationals. I was arguing using principles consistent with the rationals, so you inferred that I believe in the rationals. But arguing using principles which are consistent with one theory doesn't necessarily mean that the person believes in that theory. So I don't see your point here, I think you just misunderstood.
Quoting fishfry
What I argue against is inconsistency in the rules. And, if someone asked me to play chess, and I noticed inconsistencies in the rules, I would point them out.
Quoting fishfry
As I've demonstrated, we can still object to a specific set of mathematical rules, using a different set of mathematical rules to make that objection. This is due to inconsistency in the rules of mathematics. Look at how many different systems of "numbers" there are. You, in this very post, have accused me of being inconsistent for switching from rational numbers to natural numbers. This is not my inconsistency, it's inconsistency within the rules of mathematics. Imagine if chess were like this, and every time you wanted to play a game with someone you had to discuss all these different and inconsistent conventions, deciding which ones to play by.
Quoting fishfry
I don't agree with this analogy at all. We apply mathematics toward understanding the world, and working with physical materials in the world. This is completely different from the game of chess. If the principles of mathematics were not to some degree "true of the world", they would not be useful in the world. There is no such requirement in the game of chess. So it's completely acceptable to criticize the principles of mathematics when they are not "true of the world", because mathematics is used for purposes which require them to be true of the world. But the game of chess is not used in this way. So if I were to criticize the rules of the game of chess, it would be if I thought they were deficient for serving their purpose.
Quoting fishfry
This is a nonsensical analogy. The rules of mathematics are used for a completely different purpose than the rules for chess. And the rules of math, to whatever degree they are not true of the world, lose there effectiveness at serving their purpose. The rules of chess are not used in that way.
However, modern, abstract mathematics may be more like the game of chess and less likely to describe our world. For a number of years it's been fashionable to move away from the kinds of mathematics that we normally associate with physical reality and into "higher" levels that are increasingly abstract and generalize concepts and processes to the extent that ideas and technicalities peculiar to classical math don't even appear on the radar.
But then I wonder, perhaps what seem like total abstractions really do point to some underlying aspects of reality that we have not reached the point of comprehending. One might think this possibly true with regard to QM, although most of the math used there is fairly traditional. Even oddities like virtual particles are really just mathematical terms in the series solution of difficult integrals. Or so I am told.
Maybe there is a mathematical universe, and somewhere, through all the "chess game rules" mathematicians study, a path to understanding it can be found.
[math]1\div2=0.5[/math]
[math]0.5\times2=1[/math]
What's the problem?
Color me surprised.
Quoting Wikipedia
Quoting Metaphysician Undercover
You're only demonstrating your incompetence, over and over. You're just proving you don't speak the language.
Quoting Metaphysician Undercover
Wrong. The exact ratio between your circumference and diameter is pi. c/d = pi, pi*d = c. If your circumference is 1, your diameter is approximately 0.318310. If your diameter is 1, your circumference is approximately 3.14159. Division's role here is a red herring; you have to round off for both operations because pi is irrational.
Quoting Metaphysician Undercover
Wrong. That's just the decimal system. In base 3, divide one by 9 and you get 0.01[sub]3[/sub]. In base 9, you get 0.1[sub]9[/sub]. In dozenal, you get 0.14[sub]12[/sub]. 0.01[sub]3[/sub]*9=1[sub]3[/sub]. 0.1[sub]9[/sub]*9=1[sub]9[/sub]. 0.14[sub]12[/sub]=1[sub]12[/sub]. 1 in each of the bases is the same as 1 in decimal. The reason 1/9 is a repeated decimal has to do with the way placement systems work and the fact that its radix is 10, not some ill-placed conspiracy theory about mathematical deceptions of division.
Quoting Metaphysician Undercover
In these cases all you're doing is tripping over your confusions of the decimal system representation of numbers. But clearly you're convinced these are truths.
Quoting Metaphysician Undercover
I'm not ignoring your evidence; I'm collecting it. But the evidence doesn't point to your conclusions; it points to your being confused.
Quoting Metaphysician Undercover
Integers don't form a field under addition/multiplication; but rationals do.
Quoting Metaphysician Undercover
Remainders aren't fractions. But they do indicate the numerator of the fractional part of a mixed number. You have no real point here, though. No amount of confused gibberish you spew prevents me from sharing two pizzas evenly between three people, nor does it change the method by which I do so. All you're doing is inventing fake contradictions.
Quoting Metaphysician Undercover
Wrong. Conflating requires confusing two unrelated ideas... the units of measurement of lengths are lengths.
Quoting Metaphysician Undercover
Of course, because you're confused.
Quoting Metaphysician Undercover
Wrong. I never mentioned foot long rulers... I mentioned foot long lengths. You could use a 50 foot tape measure to mark off these lengths starting from a point in the center of a 12 foot board. You don't even need to use that clumsy folding metal thing at the end of the tape... the distance from the 2 inch mark to the 14 inch mark is a foot. You can use foot rulers if you like, but all you need to measure a particular length is something that has that particular length, such as two marks on a tape measure.
Quoting Metaphysician Undercover
I argued that it was by definition, so I provided you the definition.
Quoting Metaphysician Undercover
Wrong; see above. This is a generic description. It's not about the ends of foot long rulers; it's about the particular length that is a foot. We don't need an 8-inch long ruler to measure 8 inches, nor do we need eight inch-long rulers. We just something 8 inches long, like marked partitions on a bigger ruler.
Quoting Metaphysician Undercover
Fine. Worry about being permanently trapped by the unfalsifiability of false narratives that you've spun out of straw man while being blissfully unaware of this condition.
Quoting Metaphysician Undercover
What problems? Zero of the things you've pointed out so far have been problems; all of them without fail have been confusions.
Quoting Metaphysician Undercover
It's about a lack of meta-cognitive awareness on your part of your low degree of expertise on the subject being made apparent to people who actually know about it.
Quoting Metaphysician Undercover
It's not just @Banno, though I have to say based on his posts (in every thread I see him in) I generally love the guy. There is a difference though... I'm giving you the benefit of a doubt; he's ruled you out years ago. I factor that in, but choose to give you the benefit of a doubt anyway. Right now, though, you're stuck in your own web. I don't think much is going to come from this conversation, because you have rigged the false game you're playing. But I don't mind fiddling with the puzzle.
Quoting Metaphysician Undercover
Nice narrative... why do you suppose you're spinning it? I've been on this forum for less than a year. I learned the math here over 3 decades ago in high school... before my BS math minor. Banno and I agree because we know the material, not because I'm his puppy. In contrast, by your own admission, you have never heard of the definition of division.
Narratives are not arguments.
Quoting Metaphysician Undercover
Yes.
Quoting Metaphysician Undercover
No. I want you to realize you're in a trap of your own making, and not as you perceive at the crux of a great uncovering. I don't want to protect you, you're a grown man. But you care about truth. So long as you do, you're harmed by your web.
Quoting Metaphysician Undercover
Okay, and I should care why? I don't need anything from you, MU. You're the one who needs this.
0.999... = 1, so you do get 1 by multiplying 0.111... by 9.
[math]0.111...=\frac{1}{9}\\\frac{1}{9}\times9=\frac{9}{9}=1[/math]
Ooooh... that's sweet.
Yeah, I gave up on @Metaphysician Undercover about a twelve-month ago. I'd previously thought that there was something interesting going on, but it's not forthcoming. Even here, there is something about definitions, meanings, that is just wrong. Here it is something about what it is to be an individual that excludes its being divisible. that sits just behind his thinking but is never articulated.
In the end that is not a philosophically interesting position, but an odd piece of personal psychology that prevents Meta from participating in the discussion. Instead of progressing our thinking about mathematics we find ourselves stuck on the Meta Treadmill, pointing out the same errors repeatedly.
A beginner question...
If this proof is fine, then why the proof in the PDF?
I think because it doesn't explain what is meant by 0.111...
Quoting jgill
If there is such a thing, then it is part of "our world". And so the mathematical axioms must be "true to it", in order to be correct. Then the chess game analogy fails.
Quoting Michael
This is where the illusion is created, in incidences where there is no problem, just like my example of eight divided by two. The illusion takes the form of a general rule, that division is the inversion of multiplication. However, the cases of division in which there is a remainder demonstrate that the inductive reasoning which creates the general rule is faulty, if we allow that division can be carried out in these cases.
Quoting Michael
This depends on how one deals with "the remainder" in division. I was following InPitzotl's principles to demonstrate the inconsistency in what was argued. If one is divisible by nine, as InPitzotl claims, then division is not a direct inversion of multiplication because there is a remainder signified with "...".
One way to resolve that inconsistency, which I've been arguing for, would be to establish the true nature of a mathematical element signified by "1" as a unit which cannot be divided, as the common definition of "one" implies. It is not comprised of parts like two and three are, and therefore is not a multiplicity. If it's not a multiplicity it cannot be divided into constituent parts. But this principle would deny your other example, of one divided by two, as well, and in each case where there is a remainder, the proposed division would be denied as impossible . This would allow for the truth of the inductive principle that division is the inversion of multiplication..
.Furthermore, this does not mean that fractions are not valid mathematical representations. It just places them into a category other than numbers, so that they do not get conflated and confused with one another. It is to recognize, maintain, and uphold the real difference in meaning between symbols like this, "1", "2", "3", which represent a number (quantity), and symbols like this, "1/2", "1/3", "1/4", which represent a relation between numbers. There is a real difference between what is internal to an object, it's constituent parts, and what is external to an object, its relations to other objects. and this difference needs to be respected.
This distinction must be maintained because numbers are often conceived as Platonic objects, and when they are given such ontological status it is important to recognize that what exists between numbers is not of the same "material" (implying the same meaning) as the material (meaning) which comprises the object, a number. The Platonic object is an element of meaning, so different types must be separated categorically. So for example, "2" represents a number, but what is signified is that there are two parts, which are united by some principle of unity. But "1/2" represent a division by 2, a dissolution of that principle of unity which makes "2" signify one unity. So if "1" represents a fundamental number, with no such parts as a multiplicity has, there is no such unifying principle, and it cannot be divided. If a person wants to divide "1" into parts, this cannot be done by following the same rules which we would use to divide "2" into parts, because the principle of unity in the object "1" is completely different from the principle of unity in the object "2". What I propose is that the principle of unity in "1" implies that it cannot be divided into parts.
If we proceed to deny this distinction then there is no principle by which a number might be an object, and if it is insisted that numbers are objects, there is absolute lawless chaos as to what distinguishes one object from another because the features which separate one mathematical object from another, as the principles of divisibility, are completely ignored. .
There's a remainder in my example:
[math]1\div2=0.5\\0.5\times2=1[/math]
I don't understand why this is a problem.
Quoting Metaphysician Undercover
[math]1\div9=0.\overline{1}\\0.\overline{1}\times9=1[/math]
Again, what's the problem?
Quoting Metaphysician Undercover
[math]13[/math], [math]\frac{65}{5}[/math], [math]XIII[/math], and [math]11_{12}[/math] are the same number. You don't seem to understand that different symbols can be used to refer to the same thing.
And [math]0.999...[/math], [math]0.\overline{9}[/math], [math]\frac{9}{9}[/math], and [math]1[/math] are the same number. You're getting so lost in what the symbols look like that you're not paying attention to what they mean.
Quoting Metaphysician Undercover
I explained here how mathematical objects are separated. I also explained here that mathematicians have moved away from the vague notion of "number" and use different terms instead. Your continued insistence that we must have some formal definition of the English word "number" that allows it to refer to every kind of thing used by mathematicians when they perform calculations is ridiculous. You seem to be reifying. Read some Wittgenstein. Words are just useful tools. Don't make them into something more significant than that.
That's just the point. Perhaps we don't really understand "our world" that well. Odd looking axioms should not be cavalierly discarded simply because they are "not true" to our limited view of reality. You misuse the word "correct" IMO.
Instead of writing virtual tomes about the drivel on this thread you should apply your critical thinking skills to actual controversial items like the Axiom of Choice.
I'm trying to learn the language, and I don't like inconsistency or contradictions within the language I use. Such things lead to misunderstanding and even deception. So I am very careful in learning language
I'm fine with defining division as the inversion of multiplication, if that's what you want, so long as you accept that any instance in which an operation of division would result in a remainder, this cannot be cannot be an act of division. As I've explained, multiplication has no place for the remainder, so under this definition of division, such cases cannot be called "division". Therefore under this definition, 1/9 cannot be a representation of division. Do you accept this. If not, then how do you represent in multiplication, the remainder which results from 1 divided by 9?
Quoting InPitzotl
It really looks like you're the one confused.
Quoting InPitzotl
Whatever you use, sticks or markings on the ground, my criticism holds. You are not distinguishing between a unit of measure, "a foot", and a measured foot on the ground, or foot ruler. We were talking about the units of measurement, "a yard", "a foot", not a marked off measurement. Consider that the number "2" is a unit of measurement, rather than a collection of two things. then you will see the mistake you are making, like referring to the collection of two things (like the markings on the ground) as the unit of measure called "2" (like the unit of measure called "a foot"). Do you see the difference?
Quoting InPitzotl
OK then do you agree to what I stated above? If division is defined as the inversion of multiplication, then any proposed division in which there would be a remainder cannot be a division, because there is nothing in multiplication to account for the remainder.
Quoting Michael
The remainder is not identified or given a specific numerical value. It is hidden to create the illusion that it has been dealt with.
Quoting Michael
You're right back to where InPitzotl first engaged me, and the discussion I had with fishfry in a previous thread. It is not true that "13" and "65/5" refer to the same thing. They are equivalent. Do you not understand the difference between being equivalent and being one and the same thing? "Equivalent" allows that two distinct things have the same value. "The same thing" does not allow for two distinct things. In the example above, "65/5" is clearly not the same thing as "13". It doesn't even have the same meaning. In order that they refer to the same thing, we need to reduce the meaning to a simple numerical value, and apply a principle which makes a value into an object. This way we can say that the value of "13" is the same as the value of "65/5" and since that value is an object, then they both refer to the same thing. But it's doubtful that there is any truth to the premise that a value is an object, or that there is an acceptable principle which turns a value into an object..
Quoting Michael
Oh, it's not me who is not paying attention to what the symbols mean. That's why I offered a definition of "number". It's people like you, who claim that "65/5" represents a number rather than what it really represents, a relation between two numbers, who are not paying attention to what the symbols mean.
Quoting Michael
Again, you've got this backward. It's the people like you, who claim that a numerical value is an object, and therefore "13" and "65/5" both refer to the same "number", who are reifying.
Quoting jgill
"Correct" is a value judgement, and it needs to be grounded or based in some principles. I base "correct" in truth, but Banno clearly bases "correct" in what is conventional. I think that's Wittgenstein's influence, which gives this notion of "correct" which is unacceptable to me.
Quoting jgill
As I explained to Banno, I don't mind discussing trivial things, like the subject matter of this thread. But do you see how seriously some people take these trivial matters, hurling the insults at me as if I've just attacked the most sacred thing in the universe, instead of simply noticing that I have a difference of opinion? In philosophy we respect a difference of opinion. But for some reason in mathematics a difference of opinion is perceived as a threat, so the defenders must attack and belittle the person with a difference. It's as if the mathematicians know and accept that their principles are doubtful, so they are insecure, and therefore they must attack and keep the skeptic away. Can you imagine how offended they would be if I addressed something of more importance?
I know, I know. You'd be driven from your castle in the dead of night by an angry mob of mathematicians waving their torches and holding their frothing mastiffs on chains. They are an uncivilized and ignorant bunch, so it serves them right you are withholding precious knowledge. :scream:
Unfortunately for you, that's not how language works. English is the language we speak, but it's also a relationship with England, and a type of spin imparted upon a ball. Words can have multiple meanings (homonyms), and math is no different in this respect. The precise meaning of the word often depends on context.
Quoting Metaphysician Undercover
Sounds like you're more interested in controlling the language than you are learning it. Unfortunately, that's not how it works.
In the C programming language there is an operator /, which is used to instruct the underlying machine to perform a division. But loosely speaking there are three distinct types of divisions: integral division, floating point division, and complex division. In a well formed program, the type of division being performed in any application of the / operator is defined by the type system as specified by the standard, but it can nevertheless be one of these three types. Now all of this is describing what we call "the C language", and the C language by official definition is the language specified by the C standard. Given this, it would be quite silly of me to argue that the standard is lying to me because, as I rationalize, I like my languages to be consistent and have no contradictions for fear that I might misunderstand what / is or even be deceived. The rationale here is actually irrelevant to what the standard is specifying, i.e. what the language is. It is incumbent upon me as a user of the language to learn what type of division is being performed based on the context. Any misunderstandings is not a fault of the C language; it's a fault of my not understanding what the language is.
We have a similar situation here in math; the meaning of division depends on the context. In the context of interpreting the meaning of 0.999... in the statement 0.999... = 1, we apply field operations under a normative application of addition, subtraction, multiplication, and division, as applied to the reals (or at a minimum the rationals); a definition of a repeated decimal; and the mathematical interpretations needed to apply the definitions. Fractional notation can easily be added and mixed in at will.
Quoting Metaphysician Undercover
You're unqualified to make that judgment. But I'll show you how this works by example. First, let's use integral division with remainders:
(a1) [math]\displaystyle 24 \div 9 = 2\ rem\ 6[/math]
Some terms... 24 here is the dividend, 9 is the divisor; 2 in this form is the quotient, and of course 6 is the remainder. Now let's do the same operation using mixed numbers.
(a2) [math]\displaystyle 24 \div 9 = 2\ \frac{6}{9} = 2\ \frac{2}{3}[/math]
More terms... the top portion of the fraction in bar form is the numerator... the bottom portion is the denominator. Note that the numerator in the fractional part of the first mixed number is the remainder from a1, not accidentally. And the denominator of this fractional part is the divisor, also not by accident. Now I chose this example precisely because it reduces to make a second point... the 6/9 fraction reduces to the 2/3 fraction by means of an equivalence relation. 6/9 is equivalent to 2/3 in a specific sense... it represents the same portion of a unit. The meaning of that equivalence is that if you split the unit into 9 equal pieces and take 6 of those ninths, you wind up with the same portion of a unit as you would if you split it into 3 equal pieces and take 2 of those thirds. In other words, the fraction represents a particular quantity; viz, a specific portion of a unit.
Quoting Metaphysician Undercover
Wrong. Your original criticism was that I require parts in the way you think about it. This is analogous to demanding that the / operator in C must refer to integral division. The standard does not specify such a restriction; I can indeed do floating point and complex divisions. Likewise, the fact that your pet theories of number having no bearing on how people use numbers suggests not that other people are misusing numbers, but rather, that you don't understand what other people mean by numbers.
Quoting Metaphysician Undercover
That's correct, but the problem is on your side. A foot is simply a specific particular length. The foot ruler is just a tool to measure that length. In fact, by the official definition, a foot is 1/3 of a yard; a yard is 0.9144 meters, and a meter is [math]\displaystyle c\times \frac{1}{299792458}\ sec[/math]. Note that a foot is defined as a particular length, but that particular length is not defined in terms of the length of any ruler.
Quoting Metaphysician Undercover
In measuring lengths in feet the unit is known as a "foot", and the number 2 represents the quantity of those units that are spanned by the length; that is, starting at one position going to another position, you count the quantity of foot-lengths. We do the same thing when we drive; we can use our odometer to measure the driving distance... we do that by counting 1/10 of a mile each time the odometer ticks up by a tenth; if we want the result in miles we convert the tenth mile units to mile units. This is perfectly well defined. Your complaint is about an irrelevancy that you want to picture numbers as meaning. Counting isn't necessarily (and therefore isn't fundamentally) counting objects... you can count the number of times a bell rings, can you not?
Quoting Metaphysician Undercover
No, as explained. You need to apply the correct definition for the correct context. The context here is clearly understood by speakers of the language; see above.
I was considering just amending this claim to something agreeable, but as it is presented, I cannot see an easy edit. The important thing to preserve here is the intended meaning, but the meaning isn't so much in the rules for calculation or the representational system, as it is in the mapping of how the division operator transforms the particular quantities of its operands into the particular quantity of its result (or by extension, how the field under consideration works). For example, if [math](1\div 9) \times 9 = 1[/math], and [math]1\div 9=\frac{1}{9}=0.111...[/math], then [math]\frac{1}{9}\times 9 = 0.111... \times 9 = 1[/math].
Quoting Metaphysician Undercover
It appears you don't understand this, since you're repeating the same error. Equality is an equivalence relation, but it's a specific equivalence relation... not all equivalence relations are equality. Take "modulo 4" for example, which is an equivalence relation defined by having the same remainder when dividing by 4. 7 is equivalent to itself, 3, 11, 15, 19, and so on modulo 4. Clearly all these numbers have different values. But 7 is only equal to itself; that is, it's equal to a particular quantity. That equivalence doesn't indicate the same number is irrelevant, because you're presumably talking about not merely equivalence, but equality. The issue isn't whether equivalence indicates the same number, it is whether equality does. Just as the thing that is the same when two numbers are equivalent modulo 4 is the remainder when divided by 4, the thing that is the same when two numbers are equal is the particular quantity that they refer to. So 0.999...=1 does indeed mean they represent the same number.
Just because I brought this up earlier doesn't mean you resolved it. You didn't. Committing the same error twice doesn't make you correct, it just makes you still wrong.
Quoting Metaphysician Undercover
Ah, more narratives, more dysphemisms. The problem here, MU, is that you're derailing a thread and breaking social norms. The paranoid projection that mathematicians are insecure and just can't handle your superior knowledge is a delusion... you have no superior knowledge here. You're not addressing any of the issues with math. You're just confused. But what annoys people here isn't your confusion... it's your attention hogging, derailing, social norm breaking. There's nothing wrong with a good discussion about the limitations of math... about considering say Platonic philosophies, the absurdity of AOC and/or AD, and so on. But this isn't a (mathematically) interesting discussion. It's simply a language barrier.
So, @Metaphysician Undercover, I picture you sitting down with a piece of paper and a pen, and start writing out 1/9 using the simple mathematics you were taught in early elementary school, subtraction remainders repeat all that... 0.1 ... 0.11 ... 0.111 ... and, presumably, you catch on after a short while. "My god, it's full of 1s." (Is Strauss appropriate here?)
It's a fairly simple procedure (incidentally, one that I've had to implement on a computer to calculate digits of ?, like many before me).
The interesting part is now what we can prove about that procedure without even keep running it: following the procedure just results in unending 1s.
Kind of dull I suppose, repetitive, something that most elementary schoolers catch on with quickly, but, anyway, the proof sure saves a bit of paper, so we'll then just write that as "0.111...".
Conversely, once we take the opposite approach, rewriting this result concisely as a sum and a limit, we can also prove that we end up with 1/9 — consistency within the mathematics. (y)
By the way, if you really want something more concise about the numbers themselves, a constructive approach, then maybe check the doc Michael posted, looks neat to me anyway.
The "unending 1s" indicates that there is a remainder. And, anytime we express the inverted division problem as multiplication, the remainder must be added. Example: seven divided by three equals two time three plus one. In this case, one is the remainder, and must be added into the multiplication expression.
In the case of the op "1/9=.111...", the "..." indicates that there is a remainder which has not been stated. So in the inversion, 9x(0.111...)the remainder is not indicated, and not accounted for. Therefore it is not an accurate representation.
Quoting jorndoe
Do you agree with my premises?
P1. Any time there is a remainder in division, that remainder must be added into the equivalent multiplication inversion, in order for there to be accuracy in the equivalence.
P2. No matter how many 1s you write, and however you express this multitude of 1's, there will always be a remainder when you divide one by nine.
P3. The multiplication expression of the op "9X(0.111...) does not add the remainder.
Tell me, what what you think is wrong with my conclusion that the expression of the op is not accurate.
The further point, which is not a matter of fact, but simply an opinion, is that the expression ".111..." creates the illusion that the remainder in the division problem has been resolved, and so there is no remainder. However, we know that no matter how many 1s we put, even after we put an infinity of 1s (whatever that means), there would still be a remainder. So this illusion, that the remainder has been resolved, is quite clearly a matter of deception.
The thread might continue until someone produces an infinity of 1s, and you guys see that there is still a remainder. But then some smart ass will suggest that if we add another 1 the remainder could be resolved, and we'd start all over again and produce another infinity of 1s. And there'd still be a remainder.
Or, another way of writing all that is "..."
Prove it.
Quoting Metaphysician Undercover
You misunderstand — this is about the procedure, not about writing.
Go back, think about the procedure instead.
In fact, we can go much further, though it requires some abstract thinking, e.g.: Repeating decimal (Wikipedia)
Hm regarding abstract thinking, in analogy: suppose we want to prove p; then by some other means we find that we can prove that p can be proven; well, then we're done with our initial task (unless we're curious).
We know that there isn't. I don't know why you think that you know more about maths than generations of professional mathematicians. With everything you're saying about numbers and division and the like, I honestly want to know what is going on in your head. Do you think that there's some grand conspiracy and they're lying to us? Do you think that you're an enlightened prodigy who is able to outsmart the people who have studied this stuff for years, despite probably having little to no formal training of your own? Seriously, I want to know. The psychology of this is fascinating.
@Metaphysician Undercover <- look MU... free attention, freely given!
I think the most important thing here is, what is MU's criteria for truth? MU made an actual truth claim here to counter a proof. Can MU offer a proof in return, or does MU think he has a better truth criteria? Either way, I want to see the proof or this better criteria.
Thanks for the lesson in terminology InPitzotl. But I don't see how expressing the remainder as a fraction resolves the issue of the remainder. The fraction is just an expression of an unresolved division problem. So in expressing "24÷9=2 rem 6", as 24÷9=2 6/9" or "2 2/3", all you are doing is replacing the remainder with an unsolved division problem. It's not really any different than having a remainder because what you are doing is saying the division can be carried out to this point, but the rest remains not divided.
Quoting InPitzotl
It's been proven. It's called inductive reasoning. Every time someone adds another 1, there is still a remainder. And never ever is there not a remainder. And since the nature of the numbers stays the same, we can conclude that this will always occur. I don't see what makes you think that at some point we'll have enough 1s that there'll suddenly be no remainder.
Quoting jorndoe
I don't see your point. The "procedure" demonstrates very clearly that there is a remainder in this division problem. So, it's quite obvious that 1 cannot be divided by nine. As I explained earlier in the thread, some numbers just cannot be divided by other numbers. It's impossible, and we ought to respect this simple brute fact which is inherent to the nature of numbers. I could go back over this again if you'd like, but you'd probably just deny the evidence like InPitzotl, and postulate like Michael, that any number is divisible by any other number. But why employ a false postulate?
Quoting jorndoe
The op asks whether I think this "other means" is acceptable. My answer is no. The reason is that the so-called "other means" does not actually achieve what it is supposed to achieve. That is because the thing which it is supposed to do is actually impossible, by the very nature of numbers themselves, and mathemagicians like to use smoke and mirrors to create the illusion that they have figured out a way to do what is impossible.
Quoting Michael
I suppose we disagree then, on what "we know", and of course that's quite common here at TPF. I've seen some mathematical proofs, and as I've shown that the axioms are full of inconsistencies and contradictions. A lot of these so-called "proofs" are smoke and mirrors built on false premises and therefore unsound.
Quoting InPitzotl
The criteria for truth is honesty. I provided my argument, and you disagreed with the second premise, that no matter how many 1s you place after the decimal point there will still be a remainder. I think that this premise is true and I am honest in this claim. If you claim that you do not think that this premise is true, I think you are being dishonest. If that is the case, then your claim is false.
I agree that just because you argue from certain premises doesn't mean you agree with them. But you are being disingenuous here. I could easily go back to our older discussions and show you where you accepted the rationals in order to deny [math]\sqrt 2[/math]. I don't take this as a serious remark. Your prior posts don't support your claim that "I was only kidding about the rationals." You are retconning your posts and I'm not buying it.
Quoting MetaphysicianUndercover
But that is fantastic! If you have discovered a specific inconsistency in the ZF axioms, you would be famous. Gödel showed that set theory can never prove its own consistency. To make progress we must either assume the consistency of ZF; or else, equivalently, posit the existence of a model of ZF. This by the way is what some readers may have heard of in passing as "large cardinals." For example there's a thing called an inaccessible cardinal. It can be defined by its properties, but it can't be shown to exist within ZF. If we assume that one exists, it would be a model for the axioms of ZF; showing that ZF is consistent.
So this is the state of the art on what's known about the consistency of ZF.
If you have found an inconsistency, you will be famous. I'd be glad to help you express it mathematically and we can both be famous.
The problem is that so far you have not demonstrated an inconsistency in ZF. You've only made a sequence of increasingly bizarre and nihilistic assertions about mathematics, none of which are remotely true as concerning that discipline.
To show ZF inconsistent, here is what you must do: Produce a proposition [math]P[/math], a well-formed formula of the first order predicate calculus plus the axioms of ZF; such that there is a proof within ZF of both [math]P[/math] and of [math]\neg P[/math].
You made a bold claim. That's what you need to do to back it up. I'll be glad to help with the translation of your idea to math; if you actually have an idea.
Quoting MetaphysicianUndercover
I don't think any claim is being asserted beyond the fact that the equation is derivable line by line from the axioms of set theory and predicate calculus. You're the one who thinks it "means" something. I have no idea what you are even thinking. The equation refers to nothing in the real world and I never claimed that it does. You're punching at a strawman.
Quoting MetaphysicianUndercover
Of course. You could use a different model of the real numbers such as the hyperreals. But .999... = 1 is a theorem in the hyperreals as well. You could try intuitionist math. .999... = 1 is most probably a theorem of intuitionist math but I confess ignorance on this point. I can never make sense of the intuitionists and it's not for lack of trying.
So if you want to work in some alternative framework I'm perfectly open to it. There are in fact a number of interesting variants of chess, too. Like the 3D chess they play on the Enterprise.
Quoting MetaphysicianUndercover
There's no general definition of "number" in mathematics.
We do have exact definitions for natural numbers and integers, rationals, reals, complex numbers, quaternions, octonions, p-adic numbers, transfinite numbers, hypereal numbers, and probalby a lot more I don't even know about. But ironically, and confusing to many amateur philosophers, there is no general definition of number. A number is whatever mathematicians call a number. The history of math is an endles progressions of new things that at first we regard with suspicion, and then become accusotomed to calling numbers.
You know @Meta, you seen to deny any understanding of math as a social activity of humans. But that's exactly what it is. Perhaps there's a Platonic math out there and perhaps not; but either way, mathematics included the history of people who do mathematics, going back to the first cavedweller who put a mark in the ground when he killed a mastodon.
in any event there are dictionary definitions of number, but there is no general mathematical definition of number. Particular kinds of numbers, yes. Number in general, no.
Quoting MetaphysicianUndercover
You're confusing pure math with applied math. And it's true that chess doesn't apply to the world; but I could pick a better analogy. Take sailing. Recreational sailors are playing a game that has no actual consequences outside of the game. But their game arises out of the accumulated knowledge of thousands of years of sailing, most of which was done for trade and exploration. So that's a formal game, if you like, with connections to the real world.
But really, you are saying that there is no math other than applied math. You miss a lot from that perspective. And a lot of abstract pure math becomes very practical hundreds or even thousands of years later. So you can't really make the distinction you are making. Euclid studied the factorization of integers into primes; but it wasn't till the 1980s that someone had the idea of applying prime factorization to the security of digital communications. Today number theory underlies the security of the Internet. If you'd been in charge back then you'd have told Euclid to stop fooling around and build a wheel or something, and humanity wouldn't have learned any number theory and would not today be able to secure the Internet.
You're not only a mathematical nihilist. You're a mathematical Philistine. "One who has no appreciation for the arts." You deny the art of mathematics. You know nothing of mathematics.
Quoting MetaphysicianUndercover
You're arguing that BECAUSE math is sometimes useful, it may ONLY exist if it is useful. What's your evidence for that proposition?
That's like saying that abstract art is ok as long as it's useless; but the moment anyone uses a painting to cover a hole in their wall, only practical art is permitted. You know you are speaking nonsense.
Physicists and others find math useful. That doesn't place any limits on what math can be or what mathematicians can do.
Euclid wasn't trying to solve the problem of Internet security 2200 years ago; but that's where his mathematical thinking led. You simply never know when a piece of math will eventually be indispensable, as they say, in the world.
Nobody claims that math = physics. That hasn't been true since Riemann and others developed non-Euclidean geometry in the 1840s. Surely you must know a little about this.
Quoting MetaphysicianUndercover
You're just confusing pure and applied math. And missing the lessons of history that what is abstract nonsense in one era may well and often does become the fundamental engineering technology of a future time.
You know when Hamilton discovered quaternions, nobody had any use for them at all. Today they're used by video game developers to do rotations in 3-space. Did you know that? Are you pretending to be ignorant of all of this? That when you run the world nobody will do any math that isn't useful today?
Man you are a nihilist true.
Good... you're caught up then.
Wrong. The fraction part of a mixed number specifies an exact portion of a unit. We can only say that 3/9=1/3 insofar as both of these fractions represent that specific quantity of a unit.
Quoting Metaphysician Undercover
Wrong. Assume I start with 24 pizzas. 2 rem 6 simply means that if I give each of 9 people just 2 pizzas, that I would have 6 left over.
By contrast, the fraction specifies an exact quantity. It means a specific thing to give one person 2 2/3 pizzas. If I give each of 9 people 2 2/3 pizzas, then I have none remaining.
Quoting Metaphysician Undercover
You didn't provide reasoning until just now; you just asserted it. Now let's go over your argument:
Quoting Metaphysician Undercover
Correct. But also, every time someone adds another 1, is a time. Algorithmically, that time is a step; we can count the steps. Specifically, each step is a finite step. Might I remind you, though, that your truth claim is explicitly about "even after we put an infinity of 1s (whatever that means)".
"never" applies to all steps in the process. And all steps are finite. So this does not apply "after" we put an infinity of 1s. (I would argue there's no such thing as that after).
We can conclude that for all steps. But we cannot conclude that "after" we put an infinity of 1s, which is the very thing you're making a truth claim about.
I don't think that at some point we'll have enough 1s. I think you're once again speaking about something you have no clue about. There is no last finite counting number; there's no "point" "after" you have an infinite number of 1's. But there are an infinite number of counting numbers. I don't think you proved anything, except what you explicitly admitted to here... that you don't know what this means.
How can you say you proved something when you don't know what it means?
Quoting Metaphysician Undercover
You're by context including infinite strings. The literal string .111... refers to an infinite string starting with .111 and followed by a 1 for every finite ordinal position; that is, if you count the first 1 as 1, the second as 2, and so on, there is no finite n such that the nth position does not have a 1 in it. There is no point in this string that is "the last 1" for the same reason there is no last counting number. Your argument hinges on the hidden assumption that there is such a step... but that's a confusion on your part.
Quoting Metaphysician Undercover
Your honesty and sincerity is not in question; your claims are. Your proof falters because it does not apply to the one thing you're making a claim about. .111... is an infinite string; that is the thing under discussion. Your proof applies only to finite steps, which ipso facto is not infinite. By the way, because I use correct reasoning, I will not claim that your proof being wrong means there's no remainder; it does not mean that at all. What your proof being wrong means, instead, is that your reasoning that there is a remainder is invalid. I explicitly mention that because you make that mistake here:
Quoting Metaphysician Undercover
My honesty and sincerity is not in question; your claims are. Your proof still falters for the reason specified above.
Quoting Metaphysician Undercover
Wrong. Your claims stand or fall on their own merits; it has nothing to do with me. This isn't a relevant argument, it's a psychological response. You cannot conclude anything about the veracity of your claim based on presumed character flaws you guess I have.
Sorry, but I have no idea what you're talking about fishfry. The stuff you claim here makes no sense to me at all. When did I say I was just kidding?
Quoting fishfry
You know, ZF is only one part of mathematics. If axioms of ZF contradict other mathematical axioms, then there is contradiction within mathematics. In philosophy we're very accustomed to this situation, as philosophy is filled with contradictions, and we're trained to spot them. So we might reject one philosophy based on the principles of another, or reject a part of one philosophy, and so on. There is no reason for an all or nothing attitude. Likewise, one might reject ZF, or parts of it, based on other mathematical principles.
Quoting fishfry
So, mathematicians can call whatever they want, "numbers", but not philosophers? Whenever a philosopher uses the word "number", the mathematician has the right to say "your wrong, because you are a philosopher not mathematician", yet the mathematician can make "number" refer to whatever one wants, especially something different from whatever the philosopher wants it to refer to. Isn't "number" a weaselly little word? Whenever the philosopher comes close to nailing down a definition, the mathematician says no that doesn't suit me right now, I still want to be able to use the word in other ways.
I get the picture, the mathematician doesn't want "number" to be defined, in order to proceed in using the word however the mathematician pleases, in acts of deceptive equivocation. This is why philosophers are trained to recognize such inconsistencies, so that we can address such sophistry.
Quoting fishfry
Actually I love the art of mathematics. You even said so yourself in this thread, that I obviously care very much about mathematics. Notice I didn't disagree with that #2. But people are always doing something with their artwork, and mathematicians like to "prove" things. And the nature of that art of mathematics is that when it is applied it is extraordinarily persuasive. So when mathematicians use their art for deception, I especially despise that, because it gives them an extraordinary power to succeed.
Quoting fishfry
Actually, until you demonstrate the validity of your supposed distinction between pure math and applied math, you have no argument here. The fact that someone discovers something which is useless to the person at the time, because they may have been doing something else at that time, does not mean that they were not involved in some application at that time. So, when a principle is discovered, and not put to use for hundreds or thousands of years, this does not mean that the person who discovered it wasn't involved in application at the time.
It would seem so. Your comments are still off topic.
Back to the topic here:
We can prove that all the procedure does here is give us 0, decimal point, followed by endless 1s.
And we can prove that without writing down 0, decimal point, followed by endless 1s — it's an artefact of the procedure, and the proof involves mathematical induction and such.
Doesn't really matter much whatever anyone makes of it, that's how the arithmetic works.
That was the topic brough up, though we can prove more than just that (repetend length is 1).
But, proof or not, this should be intuitively clear. You understand? If yes, then you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like [math]0.\bar{1} = 0.(1) = 0.111\ldots[/math].
You didn't really answer the question though. Do you believe that generations of mathematicians are aware of this, and yet for some reason continue to use them, or are they unaware, and you're just smarter than everyone else?
I disagree with this fundamental point, and reasserting it will not persuade me to agree. The fact that there is a repeating decimal when we attempt to divide one by three demonstrates that the unit cannot be divided in three exact portions. There is a remainder. Therefore it is impossible that 2/3 represents an exact portion of a unit. What you have argued is that you can define "one" or "unit" however you please, as consisting of three parts ( the same as "three"), or consisting of nine parts (the same as "nine"), or whatever number you want, and that's just contradiction plain and simple. In no way can "one" represent whatever number you want, without contradiction.
Quoting InPitzotl
You're intentionally avoiding the point, and I must say, lying, when you say 2/3 of a pizza is an "exact quantity". Sorry, no offence meant, but I feel it's necessary to point this out. You have no qualifications here to stipulate the size of the pizza and whether it might be divided in thirds, so it's impossible that this represents an exact quantity.
Quoting InPitzotl
OK, I agree with you here, so at least we agree on something. There can be no "after" we put an infinity of 1's, because it is impossible to put an infinity of 1's. If it were possible to do that, then someone might do it, and then there would be an "after:" it was done.
Quoting InPitzotl
Actually my truth claim was that no matter how many 1's we put, there is still a remainder. So we can remove the needless qualification of "even after we put an infinity of 1's", since we both agree that this is impossible, and just adhere to the basic premise. No matter how many 1's we put, there is still a remainder.
Quoting InPitzotl
OK, that's fine, I'll accept that as an honest answer. Now, can you give me an honest answer to how you think the remainder is dealt with then, such that we can end up with an "exact quantity".
Quoting InPitzotl
Now, here you go and contradict the only thing we could agree on. We agree that one cannot put an infinity of 1's, and now you are claiming that ".1..." means that an infinity of 1's has been put there. Don't say I do not understand the language, because it's right there in English. Do you not apprehend a contradiction here? Or, are you saying that you're putting an infinity of 1's there, and insisting that there is no "after" this?
Quoting InPitzotl
No, .111... cannot refer to an infinite string, because we've agree that we cannot put an infinite string there. Now if you go and put an infinite string there you've reneged on our agreement, and I'll insist that there is still a remainder even after you've put your infinity of 1's there.
You are now claiming to do what we've agreed is impossible. Which do you accept as the truth, can we put an infinite string there or not? If you say that .111... refers to an infinite string that is somewhere else other than there, then how is it relevant?
Quoting InPitzotl
Is that so? You've refuted my proof by proposing that there cannot be an "after" one puts an infinity of 1's there, and then going and putting an infinity of 1's there. Now we are at the point of after you put the infinity of 1's there, so all you have done is disproven the premise of your refutation.
Quoting jorndoe
You keep referring me back to the same post, so that I've read it numerous times now, and still don't see the point. You claim it ought to be "intuitively clear" but I'm sure my intuition is quite different from yours.
All I can say is that you seem to contradict yourself. First you say "Doesn't really matter much whatever anyone makes of it, that's how the arithmetic works.". Then you say "you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like 0.1¯=0.(1)=0.111…0.1¯=0.(1)=0.111…".[/quote]
Doesn't the second statement directly imply the falsity of the first?
Quoting Michael
I don't think you can cast a net of generality on all mathematicians in that way. Some follow the discipline in such a way that they would apply the principles without being aware of the underlying issues. Fishfry might call this applied math. Some question the underlying principles, as indicated by jgill. Fishfry might call this pure math. If I understand fishfry's proposed divisions.
Furthermore, there are multitudes of complex problems involved with what might be called "pure math". If there is such a thing as "pure math" it would involve analyzing these problems. And those who are interested in addressing the problems direct their attentions toward the issues which interest them. For instance, jgill suggested I direct my attention toward the axiom of choice, but it's not my interest right now.
So there's no issue of anyone being smarter than anyone else, I don't know how you would even judge such a thing. It's a matter of where one's attention is directed. I happen to have an interest in music, and musicians work with a fundamental unit called an octave, along with divisions and multiplications, using frequencies to produce harmonies and dissonance. So the matter of what can and cannot be divided into equal parts is interesting to me. The issue of the acceptable divisions of a unit has never been resolved. And to claim as InPizotl seems to, that a unit can be divided in any way one pleases is totally unrealistic. However, notice that my interest in the problems of division is piqued by my interest in music, such that the pure side of my math interest is still guided by the applied side. And this is why I do not accept fishfry's proposed division.
This doesn't answer my question. Do you think that mathematicians are aware that "the axioms are full of inconsistencies and contradictions. A lot of these so-called 'proofs' are smoke and mirrors built on false premises and therefore unsound"?
If they are then why do they use them and not "fix" them? If they're not then how are you, a mathematical layman, able to notice what the experts can't?
There is of course a simpler explanation. You don't know what you're talking about.
Hmm... No pattern recognition...? Odd.
Quoting Metaphysician Undercover
Nope. Arithmetic works fine regardless of notational conventions.
Intuitions and conventions aside ...
We can prove things about switching 1/9 to decimal form without doing it (? stands on its own).
You understand...?
I wonder, do you ever balance checkbooks, file taxes, etc? :)
(Side note: like @fishfry, I don't know if intuitionist mathematics blocks anywhere, but offhand I kind of doubt it.)
We've been through this MU. We're not debating... you're under the delusion that we're having a debate... that my goal is to persuade you, that I'm trying to do so, that something is riding on your agreement, and that it actually matters that I persuade you. We're not, I'm not, I'm not, it isn't, and it doesn't. But I have to say... it's all kinds of adorable that you think we're debating!
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
The decimal is a red herring; 2/3 is a fraction, not a decimal point number. You're conflating notation with representation.
Quoting Metaphysician Undercover
But I can. I can use a pizza cutter, I can slice a pizza into 3 equivalent parts, and then I have thirds of a slice.
Quoting Metaphysician Undercover
It's no contradiction; we just specify the units. I slice 6 of the pizzas into three slices each. Now I have 18 pizzas and 18 thirds-of-pizzas. I give each of 9 people 2 pizzas and 2 thirds-of-a-pizza. This is something I can actually do in real life. Keeping track of two kinds of things (pizzas, slices-of-pizza) is child's play. If you have difficulties doing that, that's your problem not mine.
Quoting Metaphysician Undercover
Well likewise no offense intended MU on my part but, I can't possibly take any of your arguments seriously, including your ad hominem conspiracy theories about me. So there's no way you offend me by this... all you're managing to accomplish is to expose your own irrationality.
Quoting Metaphysician Undercover
That is literally untrue. I have a BS minor in math; I'm pretty sure that qualification covers mixed numbers, since that's a grammar school topic.
...is exactly as relevant when talking about thirds as it is when talking about wholes. Now that I described what that process is, let's compare notes. There is a real thing I can do to distribute 24 pizzas among 9 people (above). Your objections fail to describe or affect that procedure; worse, they fail miserably to account for the fact that I wind up with no pizzas instead of six.
Quoting Metaphysician Undercover
There's no disagreement about finite decimals, but it's irrelevant to 0.111....
Quoting Metaphysician Undercover
There is no "then" here... belief is not a mandate. You cannot defend a false belief on the basis that nobody offered you an alternative...
There's nothing special about the decimal system with respect to number values. 1/9 is 0.01[sub]3[/sub] exactly, no remainder. 1/9 as a fraction is a value in and of itself. The question under concern is whether 1/9 can be represented exactly by the decimal system. It can, if your decimal system includes (or is extended to include) repeated decimals, and if you use the definition I supplied earlier for repeated decimals. The OP provided a proof of this.
But for now you're choking on the fact that we can meaningfully talk and reason about infinite strings/repeated decimals (ironically, while talking about and reasoning about such things). So let's do this.
Imagine we write a computer program to calculate using standard long division on decimals. As a primer, let's do 1/8 first. Our program then winds up doing the following (if the following confuses you, you know what long division is (?)... do long division yourself and read along and you should see what's going on):
This program is now done; it has emitted 0.125. So we say that 1/8=0.125.
Take the same program and use it to calculate 1/9. Here's what happens
At this point we can pause the program, because we note that the machine is in the same state twice. Because programs are deterministic, if a machine gets to the same state twice, we immediately know it's in an "infinite loop" (that's literally the jargon). But in getting from that state to the next instance of the state, the program will emit another "1". For that reason we can say that the program will emit 0.11 followed by an infinite number of 1's. We know this immediately because we can reason and we can recognize symmetric recursion.
We can represent what the first run does as emitting the string "0.125". We know that's the complete output of the program because we can wait for it to halt. We can write down "0.125" because it's just 5 symbols. The "last digit" here is 5, because that is the thing that the program emitted just prior to halting.
The second run is qualitatively different, but we can still represent what it does. We know that there's no complete output of this program because we know it will never halt; but we know the program keeps generating 1's in perpetuity. We know we cannot write down the full output here, because we know it is infinite, but we know that its output will keep spitting 1's because it did so for a couple of steps and because the nature of the infinite loop is that of symmetric recursion. So we can represent the output as 0.111... meaning it never stops, and will always spit out 1's. "An infinity of 1's" is just a shortcut for saying the same thing. "One repeating" says the same thing as well; 0.(1) refers to the same thing.
Quoting Metaphysician Undercover
Oh, thank you MU. It saves a lot of time when you make a claim but accidentally prove by contradiction that it's false (underlined). If you can describe this string as "an infinite string" and reason about what that implies, then I can refer to the same string as "0.111..." and reason about what that implies. As a bonus points, you've demonstrated that you yourself are just confused about this, which is something I keep saying.
Quoting Metaphysician Undercover
No, you're confused. I'm claiming to do what you (accidentally) proved by demonstration is possible.
Quoting Metaphysician Undercover
That's easy. "We cannot put an infinite string there" is a true statement.
Quoting Metaphysician Undercover
Silly MU. I only claim that "..." refers to an infinite string. Call it 0.(1) if that helps. It all refers to the same idea... that the program generating this string never halts and always repeats as demonstrated by symmetric recursion.
It's trivially relevant, because it's the infinite string referred to by "0.111..." that you are trying to object to.
Quoting Metaphysician Undercover
(a) yes, the program never halts. (b) Not really; we use "0.111..." to refer to the fact that it never halts (and that all of the symbols in it are 1's). I got to this from the program by running just a few steps, and recognizing symmetric recursion was going on. (c) No, we used reason to conclude (a) and (b) and call this an infinite string of 1's. And I'll add (d), that your proof only applies to terminating decimals, because we can only say that a result has a remainder if we're "left" with one "after" we're done, and there's only such a thing as "done" for terminating decimals.
Quoting Metaphysician Undercover
If there's no after putting an infinity of 1's, there's no such thing as the remainder you're left with when you do. To talk about such an entity you have to either reify it, or prove it actually exists. Good luck talking about the remainder at the final execution step just prior to halting, in the context of a program that never halts.
You have this backwards. I don't have to prove your proof doesn't prove something, your proof has to prove it. We have an object 0.111... that describes the output of a program that never halts. You applied fallacious reasoning akin to my previously mentioned troll proof that infinity is finite; "it has a remainder at each step, therefore the infinite string has a remainder" is exactly analogous to "it's finite at each step, therefore infinity is finite". We trivially know this doesn't apply, because there is no such thing as a last step to have a remainder at. There is no counting number that represents the count of the counting numbers. There's no "after" to writing an infinite number of 1's. There's no "end" to a program that never halts.
ETA: Or try this one. 0.111... represents a string with infinite 1's. Let's "prove" that there's a remainder:
"Proof" A: 0.111... is the result of dividing 1/9. When dividing 1/9, we get a remainder of 1 at step 1 (the tenths digit). At any step n, if we start with a remainder of 1, then there is a remainder at step n+1. Apply infinite induction, and we generate 0.111..., and are left with a remainder of 1.
...now let's prove the exact opposite:
"Proof" B: 0.111... is the infinite result of adding 1 digits after "0.". When we follow this procedure, at step 1 we wind up with 0.1=1/10 exactly, no remainder. At any step n, given the value is p/q exactly, adding a digit gives us (10p+1)/10q exactly at step n+1, no remainder. Apply infinite induction, and we generate 0.111..., and are left with no remainder.
...compare to the troll proof:
"Proof" C: 1 is a finite number. If any number n is finite, n+1 is also a finite number. Apply infinite induction, and we get that infinity is finite.
The same problem occurs with proofs A, B, and C. Infinite recursion is fine for proving a property exists at all finite steps (in Proof A, all steps have a remainder of 1; in Proof B, all steps have a remainder of 0, in Proof C, all such numbers are indeed finite as is the following number), but cannot prove anything (at least in this fashion) about the property of the infinite extension (in Proof A, you cannot say 0.111... has a remainder of 1, just as in Proof B, you cannot say 0.111... has a remainder of 0, and in Proof C you cannot say infinity is finite).
Some are, some aren't.
Quoting Michael
Some are actively trying to fix them. There's not universal acceptance of all mathematical axiom because some mathematicians propose alternatives. They are trying to fix the problems.
Quoting Michael
I'm a metaphysician, and some mathematical axioms are derived from metaphysical concepts such as the concepts of unity and continuity, which are features of "being", a subject of metaphysics. So I'm not exactly a layman on these issues.
Quoting jorndoe
This is exactly what I've been arguing for the entire thread. What we can prove is that it can't be done.
Quoting jorndoe
I've already communicated the point quite clearly throughout the thread. Some specific numbers cannot be divided by other specific numbers, that's a fundamental feature of "numbers" which is very evident, and we ought to respect it. However, the convention in mathematics is to use postulates such as every number is divisible by every other number (except zero perhaps), then dream up unsound axioms to support these postulates.
Quoting InPitzotl
What are you doing then? Do you not see that such discourse with a delusional person is pointless?
Quoting InPitzotl
And what do you say when we weigh the slices and find out that they are not exactly equal?
Quoting InPitzotl
I've never seen a pizza sliced in exactly equal pieces, and despite your minor in bs, I don't believe that it can be done. Sorry, but your example is what fails miserably.
Quoting InPitzotl
Sure, but as I said, there's still a remainder which hasn't been dealt with, even if you represent the situation as ".111..." Do you not comprehend that? There's something left which hasn't been divided. The machine keeps spitting out 1's forever, and the division problem is never solved. So representing 1/9 as .1... is the same as saying that this is an unresolvable, or impossible division to do. All ".111..." represents in your example, is that the machine could keep adding 1s forever, and the division problem would still not be completed.
Quoting InPitzotl
You seem to misunderstand. I'm not arguing that it's impossible to represent an infinite string of 1's, that's simple to do. What I'm arguing is that an infinite string of 1's. following a decimal point, following a zero, does not represent a solution to one divided by nine. There is no solution to one divided by nine, it is an impossible division. But instead of facing this very simple, and straight forward fact, which is nothing other than the way that numbers are, you and other mathematicians will argue to wits end, providing all sorts of smoke and mirrors illusions, claiming that you have actually resolved this impossible to resolve division.
Quoting InPitzotl
Exactly! Without an end the problem is not resolved. The division has not been carried out. That's because it is impossible to do. The program never halts because the division is never completed, because it is impossible to do.
Hey InPitzotl, there doesn't seem to be anything new in your post. And, as you say we are not debating this, nor are you trying to persuade me of your point of view, so why continue? Are you learning anything yet? Would you consider the proposition that certain numbers just cannot be divided by each other? It's just something that's impossible to do.
Quoting Banno
You know that the value of "one" is that of a whole, a single unit, do you not? If it is divided in half for example, then the two halves together can not have an equal value to the "one" which is a single, not a double. If there is such a thing as "pure mathematics", then the unit which is represented by "1", being simple, must be distinct from the unit represent by "2", or "3", being multiplicities. The need to divide the fundamental unit "1" is a feature of application. Only in reference to the particulars of the application can the divisibility of that which is represented by "1" be determined. In other words, the divisibility of "1" is dependent on, and determined by the divisibility of the object which it is applied to in application.
Most folk can manipulate "one" in quite complicated ways. They learn to speak of one dozen, for example, understanding that they can treat twelve things as if they were an individual. They can have half a glass of water without having an existential fit about the non-existent other half.
Realising this sort of thing happens as one moves to the concrete operational stage according to Piaget. There's a phenomenon called "irreversibility", in which "two rows containing equal numbers of blocks are placed in front of a child, one row spread farther apart than the other, the child will think that the row spread farther contains more blocks". Pushing the block together seems to the child to decrease the number of blocks. Similarly,
Talking to you has similarities to talking to a pre-operational child.
Multiple things. Playing a game. I'm trying to see how much perspective I can give you about your lack of competence in this area... that you're uncooperative makes it a bit challenging. But I'm being quite honest here; I don't take you seriously.
I'm not trolling though... I'm just not debating. If you like you can treat this as a debate; the form is the same (to some approximation). But I don't want to give you the impression that I actually believe you can be convinced if I give you good reasons, nor that I really need you to "believe in math".
Quoting Metaphysician Undercover
Well seeing as the pizzas themselves wouldn't be exactly equal either, why would we care? It doesn't affect the definition of fractions, and 9 people are getting more pizza than they would if you threw 6 in the trash. I'm pretty sure 6 pizzas in the bin because you have some sort of deep rooted aversion to fractions is more significant mathematically speaking than guy 3 getting a few tenths of an ounce more pizza than guy 4 because we're approximating fractions (not to mention hand waving that whole pizzas weigh the same, for some mysterious reason).
Quoting Metaphysician Undercover
Your criteria for failure amuses me. I have 9 happy people. You have 6 pizzas in the trash.
Quoting Metaphysician Undercover
Don't I?
Quoting Metaphysician Undercover
But you failed to prove there's still a remainder in an infinite string of 1's following a decimal point following a zero.
Quoting Metaphysician Undercover
Without an end, when do you have a remainder? (Did you not see where I pointed out the flaw in using your argument to show there was a remainder? That's "Proof" A, it's still in the post, countered by "Proof" B, and satirized by "Proof" C).
Quoting Metaphysician Undercover
I'm doing multiple things at once; debating just isn't one of them. I'm trying to see, as a challenge, how much perspective of math community you will take in while being paranoid about it. I'm attempting to reverse engineering your jaded views of the math community.
Quoting Metaphysician Undercover
Yes; I'm learning about how you think.
Quoting Metaphysician Undercover
If we're talking about integers, sure. If we're talking about fields, no. It's intriguing to me that you take this sort of integral and/or whole and/or counting number realism to such extreme deepisms that you both transport the properties of such things into other number systems and trick yourself into thinking you've done something profound, but I have no actual interest in the broken theories that lead to this. I am however interested in the psychological aspects of why you're so committed to these deepisms... but not being a psychologist I'm content with just what I can piece together with reverse engineering.
Quoting Metaphysician Undercover
If pure math can have no fractions, what is this?:
[img width="50%"]https://upload.wikimedia.org/wikipedia/commons/1/14/Boundary_mandelbrot_set.png[/img]
Quoting Banno
A child would be taught reversibility by being given different examples of this sort, until they learned to talk about the number of blocks in a suitable way. Meta is like a child who, when shown the blocks pushed together, insists that "Yes, I see that there are the same number of blocks when they are spread out and when they are pushed together. But there are still more blocks when they are separated".
"You seem to misunderstand", the child says. "Look, when they are spread out you can see that there are more of them. Are you learning anything yet?"
1. the arithmetic procedure gives 0 decimalpoint and endless 1s (provable by, say, mathematical induction, reductio, whatever)
2. say, [math] (4 + \frac{1}{2}) \times \frac{1}{9} = 0.5 [/math], and [math] 9 \times \frac{1}{9} = 1.0 [/math]
?
This is applied mathematics. I was speaking about what fishfry called "pure mathematics".
Quoting Banno
In case you haven't noticed, that is my intent. I'm sure you've read Wittgenstein's "Philosophical Investigations". What he demonstrates is that to properly understand the nature of fundamental, basic concepts, upon which knowledge is built, an individual must get one's mind into that same condition which it is in when one learns those concepts naturally. This is the condition which you call "a pre-operational child". The time when a person learns such concepts naturally is the time when the "understanding" of the concepts occurs. Later, we take the concept for granted, and claim to understand it. The role of the skeptic is to analyze the actual "understanding" of the concept, which is performed by pre-operational children. The difference between the skeptic and the pre-operational child, is that when we revisit this condition, we can revisit it as an observer, and thereby learn something about the actual process which is called "understanding".
Quoting InPitzotl
In other words, you're trying to persuade me.
Quoting InPitzotl
This is why the example, as proposed, is not useful. We are talking about what fishfry called pure math, not the application of principles to pizzas. We are dealing with numbers, not with pizzas, and discussing the basis (principles) upon which we divide quantitative values. I've tried to make this clear to you, but you keep going back to these examples. As soon as we come to a mutual agreement about the divisibility of quantitative values (abstract numbers), we can move on to examples of application. What I am trying to impress upon you, is the simple fact that some quantitative values cannot be divided in certain proposed ways. That's a fundamental feature of what a quantitative value is, being based in "the unit".
Quoting InPitzotl
I gave you an inductive proof and you refused it. I accused you of lying in denying the truth of my inductively derived premise. What else might I do?
Quoting InPitzotl
Each time the machine is forced to "loop back" it is because there is a remainder which must still be divided. The machine does not stop looping back because there does not stop being a remainder. In learning long division, we are instructed to round off at some point, carry it to two decimals, three, whatever.
Quoting InPitzotl
Good, we're making some progress toward principles of agreement. If you recognize that there are some restriction which may apply to the division of a unit, due to the nature of the unit, then you ought to understand that the conditions are derived from the real particulars of the application. So for example, one pizza might admit to certain equal divisions, and one octave might admit to other equal divisions. The divisibility of the unit, (the restrictions on how it may be divided), are dependent on the nature of the unit being divided. Why would you think that there is any type of thing, like a field or whatever, which would admit to any possible division imagined, whatsoever?
Quoting jorndoe
This is what I disagree with. Instead, I think that one divided by nine is an impossible procedure, provable by induction.
Quoting Metaphysician Undercover
Can you show me a mathematician who has questioned rational numbers like [math]\frac{1}{9}[/math]?
Wrong. To persuade is to convince someone that something is true. You are, in my estimation, unpersuadeable; you've invested huge chunks of your time developing your weird theories and creating narratives to rehearse... your own personal thought terminating cliches (I've seen them), and you're not going to give that up. To see a perspective is entirely different; that is simply to understand what another's view is. As I've said multiple times, I could care less whether you believe the math or not. The only thing I'm giving a shot at is for you to see how the math works.
Quoting Metaphysician Undercover
Quite the contrary... it's the epitome of utility. Each of 9 people are getting dramatically closer to an equal portion of the 24 pizzas with this method than they are with 6 pizzas in the bin.
Quoting Metaphysician Undercover
Wrong. You are dealing with integers or some subset thereof, arbitrarily calling that numbers, ignoring the concepts laid out before you while making deepist excuses and deluding yourself into thinking that by doing so you've actually made some sort of interesting fundamental point.
Quoting Metaphysician Undercover
There's nothing to make clear to me; this is illusory insight. The examples demonstrate that there is another concept here. Along with those six pizzas with not-quite-equal slices going into the bin you're chunking out perfectly valid mathematical ideal slices of ideal equal weight into the bin, with excuses. The excuses give you the illusion that you're being rational, but they are irrelevant with respect to throwing away the principles of rationals. They are, however, relevant to what mathematicians talk about.
Quoting Metaphysician Undercover
But you have an illusory insight with no valid truth criteria. You're in essence making an idol of integers, arbitrarily calling that number, and pretending you've done something fundamental.
Quoting Metaphysician Undercover
The problem isn't that I refused it. The problem is that it didn't prove what you claimed it proved.
Quoting Metaphysician Undercover
Try this... instead of 1/9, let's do 1/7. Now our description has to change, because we get 0.(142867). So yes, each "time" the machine is forced to "loop back" it's because there's a remainder. But what is the remainder to 0.(142867)? Is it 3, 2, 6, 4, 5, or 1? Note that "each time the machine is forced to 'loop back'" it is because there is exactly one of these left as a remainder. Is there exactly one of those left as a remainder to 0.(142867)? Can you even answer these questions... do they have an answer? I'll await your reply before commenting further.
But if we can't say which remainder this is, we can still talk about the same thing using an alternate view. Suppose we run our long division program and we're told that the result is 0.125. Then what can we say about the ratios it was dividing? I claim we can say it was dividing k/8k for some k. Now likewise suppose we run our long division program and we're told the output is 0.(142857) using the description given by a symmetric recursion and infinite loops. Now what can we say about the ratios it was dividing? I claim we can say it was dividing k/7k.
Quoting Metaphysician Undercover
No, we're not. No mathematician denies that division is not closed in the integers; if you look back, you'll see where I actually posted the same thing in a prior post. You're denying that we can divide at all, and field division by definition can do so. The real discussion then is whether we're doing integral division using decimals or rational division, and since decimals are driven by powers of tens (including powers of tenths), it's immediately apparent it's rational division. But because you worship the idol of the integers, you're incapable of using the appropriate language for the appropriate context.
Quoting Metaphysician Undercover
You've got it backwards. They're derived from the axioms of the system you're using. The axioms define various relationships between undefined terms. The application demands use of an appropriate axiomatic system whereby the mappings of the undefined terms have the relationships described by the axioms.
Quoting Metaphysician Undercover
Because we define it. Incidentally in terms of application we can use this in arbitrarily complex ways. There are some 10[sup]80[/sup] atoms in the universe, but we can practically get far smaller than 10[sup]-80[/sup] by applying arithmetic coding to text. Note also that machines can far exceed what we can do, so the limits of what we can do are not bound by some smallest unit of some extant thing... they're bound by the furthest reaches of utility we can possibly get from machines. We can get much further not limiting our theories in silly inconsistent ways. But even without all of this, just for the math is all of the required justification.
And yet 1 is deductively provable. even went through the troubles of outlining the start of a proof by induction. Thus, your disagreement ain't right.
What we can't do here, is write down 0 decimalpoint and endless 1s on paper. And we don't have to, because we can reason about 1/9 nonetheless as shown, like we can for other numbers.
Is writing down 0 decimalpoint and endless 1s on paper the whole of your troubles/denial here?
(Don't confuse/equivocate base 10 algorism, decimal representation, and numbers; it so happens that 1/2 = 0.5[sub]base 10 (decimal)[/sub] = 0.222...[sub]base 5[/sub], artefacts of procedures.)
[math]\frac{1}{9} = 0.111..._{10} = 0.1_{9}[/math]
That may be true of multiplication, exponentiation, tetration, etc, but the inverse operations break that closure. The numbers you can get by starting with 1 and then doing those operations are all the same, the natural numbers.
But if you subtract (undo addition to) a natural from another, you might get something that isn’t a natural: a negative number. So okay, we call the naturals and their negatives integers.
But if you divide (undo multiplication to) an integer by another, you might get something that’s not an integer: a fraction. Okay, so the integers and all their fractions are the rationals.
But if you take the root or log of (undo exponentiation to) a rational, you might get something that isn’t a rational... etc.
[math]\underbrace{y+...+y}_y=-\frac{1}{2}[/math]
Solve for [math]y[/math].
There're two questions there really.
One is: can you define all operations in terms of addition? This is the question of whether you can generate a bunch of operations (like +, -, times, divide, raising to a power, taking a root) with one operation? In other words; can every member of a select collection of operations be defined in terms of one of the operations within it?
For the natural numbers, addition generates (addition, multiplication, raising to a power)
a times b = a + a + ... + a, b times
a^b = a times a times ... times a, b times
Another is: is a mathematical object closed under a select collection of operations?
Closure of an object under an operation is when you can apply the operation to any appropriate collection of its elements and get a result which is still a member of the object. EG, adding two natural numbers {0,1,2,...} will always get you a natural number, so it's closed under them.
In order to define a bunch of operations on a mathematical object, you usually have to insist upon the closure of that object under the operations; you need to make sure you can apply the operation to everything and get something familiar out.
@Pfhorrest's example illustrates that since the natural numbers (or integers, or rationals, or reals) are not closed under the operation sqrt(-x) ( sqrt(--2) isn't rational, i=(sqrt(-1)) isn't real) but they are closed under addition, so addition alone can't generate the operation sqrt(-x) (for every x anyway).
So I'm shaky on the next bit.
If you move to the complex numbers, which is closed under that operation, the answer is more tricky (@jgill would know much better than me). In the complex numbers, you can define any holomorphic function as an infinite sum. Since the sqrt(x) function isn't holomorphic; there's a discontinuity at 0;, but f(a,b)=a+b where a and b are complex looks like it is... And the composition of holomorphic functions is holomorphic, it seems like you can't end up with a non-holomorphic function by arbitrarily composing a finite collection of holomorphic functions. IE, you can't get sqrt(x) defined everywhere in terms of addition alone, even using complex numbers.
You can’t get -1 by adding natural numbers to each other. You have to do subtraction, and then that takes you out of the naturals to the integers.
You can’t get 1/2 by adding (or subtracting) integers to each other. You have to do division, and then that takes you out of the integers to the rationals.
You can’t get the square root of 2 by adding (or subtracting or dividing) rationals to each other. You have to take a square root, and then that takes you out of the rationals and into the reals.
Etc.
Quoting Michael
Clever! :up:
The "solve for" operation means you've already put the inverse function of f(y)=k into the mix, its existence is required for that proof to work; that inverse is sqrt(y).
From here:
And exponents can be defined by muliplications and multiplications by additions.
Yes. That is true.
Yes or course, but that relation there is subtraction, not addition. You have X and owe Y, so your net worth is Z = X - Y. So long as X > Y you can start with natural numbers and stay within them, but once X < Y you have to, as you say, invent a new kind of number.
Likewise with division, square roots, etc. They require you to invent new kinds of numbers, because the kinds of numbers you already had aren’t suitable to solve all such problems.
Negative numbers are not the same thing as subtraction. Negative numbers are defined as the additive inverse of the positive numbers, and then subtraction is defined as adding a negative number to another number.
I thought we were discussing the accuracy of tim's claim that "all mathematics is addition"? I don't know about all maths but that seems to be the case for arithmetic. You start by defining the natural numbers and addition. You then define the negative integers as the additive inverses of the natural numbers (the number that when added to natural number n yields zero). You then define subtraction as adding a negative number to another number. You then define multiplication as repeated addition, and division as the inverse of multiplication. You then define exponents as multiplication, and roots as the inverse of exponents.
In the very post I was replying to.
Quoting Metaphysician Undercover
Fine. Find a statement P such that there's a mathematical proof of both P and its negation. That's the only way you can demonstrate that mathematics is inconsistent. I'm still waiting.
Meta's still playing with rocks while the rest of us have pointy sticks.
[hide][/hide]
He sure gets into a lot of people's heads.
I know a technical context in which that's not true.
There's a theory weaker than Peano arithmetic called Presburger arithmetic that allows for only addition. It's strictly weaker than PA; and in fact has the remarkable property that it's logically complete in the sense of Gödel. Every statement in Presberger arithmetic can either be proven or disproven.
When you add in multiplication, you get PA and that is logically incomplete.
It's commonly believed that in PA you recursively define multiplication based on addition. But it turns out that at a technical level (which I haven't yet grokked) the particular use of recursion is not strictly within the allowable rules in Presberger arithmetic and so is more folklore than truth. In fact the theory of addition is strictly weaker than the theory of addition and multiplication.
Regarding @Metaphysician Undercover thought process. From Apocalypse Now:
[i]Capt. Benjamin Willard: They told me that you had gone totally insane, and that your methods were unsound.
Colonel Kurtz: Are my methods unsound?
Capt. Benjamin Willard: I don't see any method at all, sir.
[/i]
I have not as yet acquired sufficient technical understanding to answer this question. In fact it's a point on which I'm stuck myself. It's a point of logic involving induction. At one point I read the Wiki page and a couple of articles about Presburger arithmetic and thought I had a vague understanding of what was going on; but if I did, it's certainly not stored in my brain cells right now.
The best place to start is the Wiki page and see if it sheds insight. Also I remember that in my earlier research, I discovered that the Wiki page on the Peano axioms actually hints at the difficulty with defining induction, but again these are memories of a few months ago.
All I really know is that when you make a recursive definition of multiplication in PA, in the manner I'd always assumed you can do, you are actually adding some kind of secret sauce without realizing it. You can see how little of this I understand.
I didn't say any did. I imagine some have, or maybe not. That's not relevant because it doesn't mean that it's wrong for me to.
Quoting InPitzotl
Well, if you think that I haven't already seen how math works, then you're wrong. And as I've already explained, the conclusion I've made from what I've seen is that a healthy dose of skepticism is needed in my approach to mathematics. That is why I've tried to take the discussion beyond rational numbers, to natural numbers, and number theory itself. It appears like you, and most in this forum believe this to be a pointless exercise. That doesn't really concern me. If that's what you think as well, then you're wasting your time here if your true intent is for me to see how math works. I've already seen it. That does not mean that I understand it. If you want me to see why math works, then drop your presuppositions and come to the bottom with me.
If it's just that I am providing entertainment for you and the others, at least it's of a healthier sort than that provided by the president of the USA.
Quoting InPitzotl
Oh, so you do not see any difference of type between the object we call a pizza, and the object we call a number. That's revealing.
Quoting InPitzotl
I can't see how this makes any relevant point. You've just demonstrated another smoke and mirrors method to hide the fact that there is a remainder. If one expression is more vague than the other, then it may or may not be a better way of hiding the fact that there is a remainder. To see what the remainder is at any given time, all we have to do is look to see at what point the machine is at when it loops back. Where's the problem?
Quoting InPitzotl
I don't deny any of this, that's how math works, conventions are followed, and that's what convention has us call "dividing". The question is on what principles do we say that the conventions are right or wrong. Do you agree that for any particular way that an action is carried out (an action being the means to an end), in this case a mathematical operation, it is possible that there might be a better way? So even if following the conventions works, there is quite possibly still a better way. We are inclined to say that the conventional way of doing things is "the right way" simply because it is the conventional way, but then what do we say when a better way is shown? One might follow a trail, between the residence and place of work, to and from, day after day, and following that trial always works to get the person where they are going. The person says it's the right way to go to get to and from my work. But that doesn't mean there's not a shortcut. How does a shortcut make the right way into the wrong way?
Quoting InPitzotl
This is an abysmal straw man.
Quoting InPitzotl
You're making the same mistake as fishfry. I do not worship any numbers. In the other thread I was using principles from the rational numbers to attack the real numbers. and for some reason fishfry got the idea that I strongly believed in the rational numbers, just like you think I strongly believe in the integers.
Quoting InPitzotl
No, the discussion is whether rational division, as the inverse operation of multiplication, is a true form of division.
Quoting InPitzotl
This is the root of the difference between us. You seem to think that mathematics works because people dream up random axioms, then the axioms are applied, and voila, mathematics works. I think that mathematics works because people design the axioms so as to be applicable to the real world. So from my perspective, the real world puts limits on which axioms ought to be accepted. From your perspective, so long as the axioms are coherent and consistent, the mathematics ought to work in the world. Do you see how you are the one who has it backwards?
So I start with the fundamental principle of "pure mathematics", which states that a "unit", as a simple, cannot be divided. However, I qualify this by saying that whenever the "unit" is applied to the real world, in "applied mathematics", the nature of the object, which the unit represents in that application, determines how the unit might be divided, depending on the object's parts etc.. So the divisibility of the unit is dependent on the object it is applied to.
You start with the opposite (and what I claim backwards) position, that the fundamental "unit" is divisible any way one can imagine, an infinity of different ways. First, I will argue that this annihilates pure mathematics and number theory, making "one" signify a multitude. Second, I will argue that it leads you to believe, as you've demonstrated in this thread, that any object is divisible in any way imaginable, i.e. an infinity of different ways. So this backward conception of "unit", which you hold, misleads you in this way, actually deceiving you to the point that you will argue persistently that any object can be divided in an infinity of different ways.
Therefore, the approach which takes as fundamental, that a unit might be divisible in an infinity of different ways, and then might qualify this in application, tailoring divisibility to meet the specifics of the object, is the wrong approach. It is the wrong approach because it has misled you, and others of course, into thinking that mathematicians can produce axioms and the world will exist in the way that the axioms dictate. But when we hold in theory that the "unit" is fundamentally indivisible until its divisibility is proven through practice, we avoid this problem.
Quoting InPitzotl
This demonstrates my point.
Your replies are vague and hard for me to understand. That the procedure proves what the procedure is supposed to prove is not the issue. Of course it will do that or else it would not be an acceptable procedure. The question I thought, was whether there are doubts about the procedure. As I explained above, doubt arises if one believes that there might be a better way. To doubt in this way does not require that the skeptic produce the better way, only that the skeptic demonstrate issues with the accepted way, which might be improved upon.
Yes, that is what is at issue here, the validity of such inversions, when the inversion turns up something which is outside the rulebook of what it is supposed to be an inversion of.
I already answered this for you. Your request is outside the range of what I asserted, so not relevant.
I found this, but need a translation.
Here's the problem.
Think I pointed this out before. And I was not alone.
Don't recall what exactly this was about. Feel our convo is at a plateau at the moment, will be taking a break from our back and forth.
So it's just you against the world of mathematics. If that can't convince you that your views are the problem, not mathematics, then I don't think anything will.
What I mean is something like this:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
...
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
...no; blaming the mathematicians for your not finding the remainder is not healthy skepticism.
Quoting Metaphysician Undercover
You didn't answer the question, and I think the reason you didn't is because the question doesn't make sense. That carries over to your previous claim that 0.111... has a remainder.
Quoting Metaphysician Undercover
...so... 0.(1)=1/9?
Quoting Metaphysician Undercover
Logical deduction based on the axioms.
Quoting Metaphysician Undercover
Sure, why not? Something like: 97*104=97+(100-3)(100+3)=97+10000-9=10088?
Quoting Metaphysician Undercover
That's quite a fair description of Finite Geometry.
Quoting Metaphysician Undercover
No it doesn't:
Quoting Pure mathematics (wikipedia)
Part of the reason I post this definition (and rearrange this) is context for the response below:
Quoting Metaphysician Undercover
Examples of pure mathematics becoming useful (exact opposite of what you just said) here.
Quoting Metaphysician Undercover
I start (frontwards) with the axioms.
Quoting Metaphysician Undercover
I wouldn't go that far... it's just useful.
Quoting Metaphysician Undercover
Interesting... you claim that I've demonstrated that I believe any object can be divided an infinity of ways, and yet, in the same post, you quote me as saying there's about 10[sup]80[/sup] atoms in the universe. Ladies and gentlemen... MU's healthy skepticism!
Quoting Metaphysician Undercover
No clue what you mean by that, so I'll just generically offer that arithmetic encoding works based on the same concepts we're discussing here... converting fractions (representing ranges of relative symbol frequencies) into placement systems (representing the coding).
Quoting Banno
Having an idea which is inconsistent with the conventional demonstrates "a problem". I agree.
Quoting Michael
I told Banno already in this thread, I do not believe in mob rule. I'm an individual, and what makes an individual an individual, is to not be identified as a part of a group. Therefore it's natural that I be different from the others. This is what makes "one" fundamentally different from "half of two". The former identifies the thing being spoken about as an individual, the latter identifies the thing being spoken about as a member of a group. The mistake which many people partaking in this thread make, is that they think that because "one" is equal to "half of two", in mathematical applications, they both mean the same thing. But I believe myself to be "one", an individual, and my identity is not based in being a member of that group. Therefore I need not partake in their mistake.
Quoting InPitzotl
You seem to always misunderstand, or misrepresent what I say. I didn't ever blame the mathematicians for "not finding the remainder", I blamed then for hiding the fact that there is a remainder. And this is clearly evident from what is expressed in the op, when it is asserted that the follow is an accurate representation: 1/9=.111..., and .111...X9=1. And, I suggested that mathematicians ought to respect the fact that it is impossible to divide one by nine equally, instead of using smoke and mirrors tactics to make it appear like this impossible thing is possible. To me, it just makes the mathematicians look bad, more like mathemagicians.
Quoting InPitzotl
You're ignoring a key part of your Wikipedia definition, "These concepts may originate in real world concerns...".
Quoting InPitzotl
If your desire is to dispute what I have stated as a fundamental principle of "pure mathematics", which you have defined through your wiki quote as "the study of mathematical concepts independently of any application outside mathematics", then to simply assert "no it's doesn't" is completely insufficient. But I don't see any point to trying to dispute any stated fundamental principle of pure mathematics, without having a good reason. That such and such a principle is not supported by such and such application as is your demonstrated mode of arguing, is not a good reason. Your reason must be based in logic, as the reason for my stated principle (which you dismissed for no reason) is.
Quoting InPitzotl
Your logic is way off bud. This is what you've argued. An object, represented as "1", can be divided in an infinity of different ways. Here's one of that infinity of different ways that an object might be divided. Now you seem to be claiming that by providing one possible way out of an infinity of possible ways, you have demonstrated that you do not really believe that an object might be divided in an infinity of different ways.
Hey, that looks like me, trying to get these guys to see their mistakes. At least the exercise is good. Now, you need a picture of two, one going each way, the other will be InPitzotl trying to get me to see my mistakes. Isn't philosophy fun?
Mathematicians aren't making mistakes. [math]\frac{1}{9}[/math] is a number and [math]0.999...=1[/math]. If you don't understand this then you don't understand mathematics. I suggest you study more before wildly claim that mathematics is contradictory and derived from false premises.
It's been mentioned before but it's worth mentioning again.
[math]\frac{1}{9}[/math] in base 10 is equal to [math]\frac{1}{10}[/math] in base 9, so [math]0.111...[/math] in base 10 is equal to [math]0.1[/math] in base 9. It's divided equally.
Explain to me what you disagree with.
[math]9_{10}[/math], [math]10_9[/math], and Roman numeral [math]\mathrm{IX}[/math] all refer to the same number. You're getting lost in what the symbols look like.
So, the answer to the question of what the remainder of 0.(142857) is, is that there is in fact a remainder, it's clearly evident, mathematicians ought to respect that it is impossible to divide one by seven, and mathematicians are using smoke and mirrors to hide the fact that there is a remainder?Quoting Metaphysician Undercover
You're ignoring a key part of your Wikipedia definition: "may", not to mention the bolded part.Quoting Metaphysician Undercover
Multiple examples provided in link of utility following math.
Quoting Metaphysician Undercover
Of course. But the object represented as 1 is a mathematical object, not an onion. Nobody is claiming you can chop an onion into infinite pieces. But we can subdivide 1 indefinitely; there's no "math-atom" we run into. We can apply arithmetic coding for example to encode 10G of text, which means we can generate an arbitrary precision number with on the order of ten's of G's of symbols. So it turns out, you're the one confused, not me; you think 1 is a pizza or an onion. It's not. It's a number.
Regardless, I think I have found the core issue here. Your theories of where math originates suggests you think math is about just physical objects and, since it's not, you find counterexamples. But rather than taking this as being proven wrong, you double down, positing that mustachioed mathematicians conspire to lie about the nature of physical objects to themselves and others... something like that?
It's not the case that I don't understand, it's the case that I understand but disagree. You've been at tpf long enough to know that this is common, people understand but disagree. Why would you think that principles of mathematics have special status such that if you understand them you'll necessarily agree with them?
Quoting Michael
I don't see that your making a point. Base 9 is going to have its own numbers which are impossible to divide into each other. So this just emphasizes my point, what can and cannot be divided is dependent on the application.
Quoting InPitzotl
Great, now we're making some progress. You see that your pizza analogy is completely irrelevant, and we are talking about dividing the number represented by "1", not some physical object. Is the number represent by "1" a single unit or a multiplicity of units? Since it is not a multiplicity, as it is defined as a single, then how do you propose that it might be divided. You cannot take a knife or a pizza roller to it. What do you think, that you can imagine that it's really made of parts, a multiplicity, and you can divide it according to those parts? Of course that image would contradict the definition. So I really want to know what principles you are applying to divide 1, because you seem so insistent that you can divide it however you please.
Oh boy, made it to 400 posts!! That's progress! :nerd:
With the Platonic form of a pizza roller??
It is the case that you do not understand.
Quoting Metaphysician Undercover
By just doing so. I gave you an example, which is quite relevant, to help you understand. You ignored it. But it's still there. If you're going to ignore what I say, I'm not going to pretend we're having a conversation.
If you're genuinely interested in this:
Quoting Metaphysician Undercover
...then you need to understand that example.
It doesn't have its own numbers. It has its own numerals.
1111 in binary, F in hexadecimal, and XV in Roman numerals aren't different numbers to 15 in decimal. They're the same number.
[math]1111_2 = F_{16} = \mathrm{XV} = 15_{10}[/math]
Quoting Metaphysician Undercover
If we can talk about dividing a single cake into nine equal slices then we can talk about each slice being one-ninth of a cake, and if we can talk about each slice being one-ninth of a cake then we can talk about [math]\frac{1}{9}[/math].
Here's a cake, it's 90cm x 10cm x 10cm. I cut it into nine equal slices of 10cm x 10cm x 10cm and share it between my friends.
Does it make a difference if I describe the measurements in cm rather than in michaelmetres, where 1 michaelmetre = 90cm, and so the cake is 1michaemetre x [math]\frac{1}{9}[/math]michaelmetre x [math]\frac{1}{9}[/math]michaelmetre? Why?
And does it make a difference if I describe it as 1 cake and [math]\frac{1}{9}[/math] of a cake rather than just 1 and [math]\frac{1}{9}[/math]? Why?
I ignored your example for two reasons. It doesn't answer my question, and it's false. First, my question concerns the principle by which you divide a number, not the act by which you represent this, which is what your example describes. Second, your example is false and invalid because "ten's of G's of symbols" is not the same as infinite.
I have no doubt about your capacity to represent "1" as being divided, we do this with 1/2, 1/3, 1/4, etc., and with .5, .3, .25, etc.. And this is what your example is, an example of a machine making a representation. What I am doubtful of is the "principle", the rule, which says that "1" is a number which can be divided. We can say, and represent whatever we want, but what I want to know about is the rule which makes the representation a valid representation. What rule makes the mathematical object represented by "1" divisible?
Here's another related question. Why is 1 not a prime number? I would say that 1 is excluded from the list of prime numbers by designating that it is something other than a number. If this is the case, then what is the relationship between 1 and all the numbers, 2,3,4, etc.? They are distinctly different types of mathematical objects. And back to my original question, if 1 is something other than a number, let's suppose it's called a "unit", on what basis can the unit be divided? That's the rule I'm asking for.
Quoting Michael
I have no problem talking about 1/9 in that application. In applications, if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems), or we say "I can't do the task I'm being asked to do" (like if you asked me to cut the cake into three million equal pieces).
Where the problem is, is in what fishfry called pure math, which is when we are working solely with abstract concepts. In abstract math the thing being divided into nine parts is the "number" one, or the "unit" one, and this division is said to give a "number" with the value of "0.111...". This is where I see a problem , as I've tried to explain.
And as we've tried to explain, it isn't a problem.
[math]\frac{1}{10_9} = 0.1_9[/math]
Is there a problem with the number [math]0.1_9[/math]?
I can't say that I completely understand your representation so I can't give an honest answer here. Perhaps you could explain better.
It's [math]\frac{1}{10} = 0.1[/math]. I'm just specifying that I'm using base 9 numerals instead of the traditional base 10. Although in this case it isn't strictly necessary as [math]\frac{1}{10} = 0.1[/math] in every base.
And do you understand what bases are? Do you understand that [math]9[/math] in base 10 and [math]10[/math] in base 9 are the very same number? To explain that, let's count the number of apples in this picture:
If I were to count the number of apples in base 10 I would count "1", "2", "3", "4", "5", "6", "7", "8", and "9".
If I were to count the number of apples in base 9 I would count "1", "2", "3", "4", "5", "6", "7", "8", and "10".
[math]9[/math] in base 10 and [math]10[/math] in base 9 are the very same number: the number of apples in this picture. And this is true even if we're not counting apples; it's true when we're doing "pure" maths.
So we can't divide a pizza into 9 slices because the slices don't weigh the same, and we can't divide 1 into 9 because 9 isn't infinity.
Sorry MU, but I'm not interested in playing Calvinball with you.
The problem is in dividing "1". In different representations the problem will appear in different ways, as I explained before. The manner of representation is a matter of application, and to show that the problem takes a different form when we change from this application to that, does not make the problem go away.
There is no problem. If you accept that [math]9[/math] in base 10 and [math]10[/math] in base 9 are the very same number (which they are), then you must accept that [math]\frac{1}{9}[/math] in base 10 and [math]\frac{1}{10}[/math] in base 9 are the very same number (which they are).
I don't see how this discussion of pizzas or apples is relevant. You're just distracting from the topic.
Quoting InPitzotl
This is the subject, 1 as a mathematical object, not pizzas.
The problem is in the supposed equivalence between the fraction and decimal representation. Do you understand, that by moving to base nine, you are actually removing the possibility of dividing one by nine, because nine has been excluded as a number? So all you are doing is obliging me, giving me what I asked for, making one divided by nine impossible. But that's the point of my argument in the first place.
The real problem though, is that one divided by numerous other numbers is also impossible. To demonstrate that you have actually dealt with this problem, show me the decimal representation of 1/7 and 1/8 in base nine. If there is no problem, then we can proceed to the other fractions in base nine as well, just to confirm that there are no such problems in base nine.
If there is still a similar problem in base nine, we might try base eight, and if a problem presents itself we could move to base seven etc.. Or, we could skip all that and just start at base two. Can you show me how to divide 1 in binary?
[math]\frac{1}{10} = 0.1[/math].
This is equivalent to
[math]\frac{1}{2} = 0.5[/math] in base 10.
Quoting Metaphysician Undercover
Nine hasn't been excluded as a number. There are nine apples in the picture above regardless of what base you use to count them. This is exactly what I mean by saying that you don't understand maths.
You're just getting confused by what the numerals look like. Whether I use [math]0.111...[/math] or [math]0.\overline{1}[/math] or [math]\begin{align}\sum_{n=1}^{\infty}\frac{1}{10^n}\end{align}[/math] or [math]0.1_9[/math] or [math]\mathrm{MichaelI}[/math], I'm talking about the same number.
Despite what you seem to be a saying, a number doesn't have to be representable as a terminating base 10 decimal. There are an infinite number of numbers that can't be represented this way. Some can be represented as terminating decimals in other bases. Others can't be represented as a terminating decimal in any base. And they're all still numbers.
Oh that's rich.
Quoting Metaphysician Undercover
^^ 1/9 is that thing.
Quoting Metaphysician Undercover
^^ distraction.
FYI, under the rationals, there's no such thing as 1/infinity.
:D We're no longer talking mathematics. (An acute case of ?-phobia?) Maybe we could call it metamathonomy or something.
Quoting Metaphysician Undercover
As an aside,
Quoting Metaphysician Undercover
... doesn't seem right. You can be both honest and wrong.
So, @Metaphysician Undercover,
Quoting Metaphysician Undercover
Mentioned procedure just writes 1/9 as 0.111... (in the common decimals). You have to understand what you're objecting to first.
I don't think you will succeed in showing an inconsistency in @Metaphysician Undercover's mathematics that he will recognise. Rather, we have a choice between two mathematics.
In one, we can divide 1 into fractions, and hence 0.999...=1, and infinities can have differing cardinalities, and i² = -1, and we can use maths to do velocity, navigation, electronics, engineering and so on.
In the other, 1 cannot be divided. And thats all.
In the end it is the poverty of Meta's mathematics that we leave behind.
This is not true at all. If 1 is divisible, its divisibility is different in base nine from what it is in base ten. That's why I asked you to look at base two as an example, because it becomes very clear there, that if one is divisible, changing the base changes the divisibility of one. Therefore, if fractions are numbers we cannot transpose these numerical values from one base to another in the way that you propose.
Quoting Michael
Again, examples of objects only confuse the issue, because we a talking about the numbers themselves. And "nine" has a different meaning in base nine from what it has in base ten, especially if we allow that one is divisible, so your example is just an example of equivocation.
Quoting Michael
Due to the fact that what can and cannot be represented is dependent on the mode of representation, this claim employs equivocation in the term "numbers". This is the reason why we use different numbering systems, natural, rational, real, for example, and that "number" has a different meaning in each of these systems, just like the base unit "one" has a different meaning in each base system, if "one" is divisible. To claim that "they're all still numbers" is just a matter of equivocation, similar to saying that all uses of "right" refer to the same type of thing a right, unless you can demonstrate a definition, or category of "number", which encompasses all the different numerical systems. Under this definition of "number", we could say that they are all numbers without equivocation. But what I've been trying to demonstrate, is that if we allow that one is divisible, such a definition will prove to be impossible. Your example of using different bases should make this very clear to you, especially if you consider base two.
Quoting jorndoe
I touched briefly on the lack of an acceptable criteria for right and wrong on this thread, in my discussion with Banno and Michael. They seem to think that to act according to the convention is to be right. If this were the case, there would be no sense in discussing the op, because it expresses the convention, and asks if this is right. If we define right and wrong as consistent with the convention, there is nothing to discuss here.
So in order to have anything to discuss on this topic we need to get beyond the idea that the convention is necessarily right. Therefore we must define "right" in relation to something else. I proposed that we define it in relation to what one truly believes. This allows not only that the conventions might be wrong, but also that it would be wrong to use the conventions deceptively. One of the problems with defining right and wrong in relation to conventions is that it makes it extremely difficult to demonstrate that a person using conventions deceptively is actually wrong.
Quoting jorndoe
Right, this is the convention which I object to as a convention which facilitates dishonesty. That dishonesty is demonstrated when people who know that ".999..." does not means the same thing as "1" insist that it does.
It is true. You just don't understand maths.
If [math]9_{10}=10_{9}[/math] then [math]\frac{1}{9_{10}}=\frac{1}{10_9}[/math]
If fractions bother you then we can use exponents instead.
If [math]9_{10}=10_{9}[/math] then [math]9_{10}^{-1}=10_{9}^{-1}[/math]
The symbols don't matter. How many apples are in this picture?
[math]9[/math] if I'm using base 10. [math]10[/math] if I'm using base 9. [math]\mathrm{IX}[/math] if I'm using Roman numerals. [math]\frac{90}{10}[/math] if I'm using base 10 fractions. [math]?[/math] if I'm using Arabic. [math]?[/math] if I'm using Chinese. Different symbols, same number.
It doesn't. 1 is the same in every base.
I'm giving up now. Clearly nothing I can say is going to teach you. Go take a math class. Maybe a professional will have better luck getting through.
Meanwhile, we all understand that half a dozen is six, and what's meant by a third of the area of the lawn, so that works fine (presumably for you as well). But of course, we don't speak of a ninth of dislike for pizza with pineapple, at least not without some further clarification.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
That is, there's something to round off. Seems you've already presupposed what you want to deny. (The division procedure isn't really the problem here.)
Call me crazy, but why isn't a base 9 and a base 10 representation of the same number a base 9 and base 10 representation of the same number?
It is the decimal equivalence which you are claiming that is what bothers me. That is the issue of the thread. You said 1/10 in base nine can be represented as .1 in base nine, and this is equivalent to .111... in base ten. I don't think you can represent a solution to a division problem in base nine, as a decimal, (.1), because decimals are proper to base ten, and that would be to conflate base nine and base ten representations. So your argument here, is nonsensical:
Quoting Michael
And you continue on with this nonsense, as if 1/10 in base nine could be represented as .1, but 1/10 is only .1 in base ten.
Quoting jorndoe
As I've already explained, I have no problem with these divisions in application. We know that the representation of "one" in the case of "one dozen" is a multitude of twelve, and therefore can be divided accordingly. We know that the area of a lawn is going to be represented in a multitude of square meters or some such thing, and therefore can be divided accordingly. And we know that "one octave" consists of a multitude of frequencies which can be divided The problem is when we talk about "one" in the abstract sense, as a "number", or "unit", in which case it is defined as a single, and not as a multitude which can be divided. "One" only submits to being a multitude when it is applied to a thing which can be divided.
Quoting jorndoe
As seems to be the case often, you don't seem to be able to express your point very well, and you leave me wondering what you're talking about.
Quoting InPitzotl
It is only different if the fundamental unit "one" is considered to be divisible. Take a look at the divisibility of "1" in base two, and compare it with the divisibility of "1" in base ten, for a good example of how the divisbility of "1" changes from one base to another. If each unit in a base ten number is divisible in a particular set of ways, and each unit in a base nine number is divisible in a different set of ways, then we cannot say that the representation is of the same number. But If the premise is that the base unit. "one" is not divisible, then there is nothing different about the number being represented in the different base representations.
Additional ad hoc hypothesis used to prevent falsification of the core assumption.
Lousy example. The number's representation is no more the number than you are a white M in a pink rectangle.
You keep telling people to take a look at binary. Okay. 1/9 = 0.(000111)[sub]2[/sub]. And? That's just another name for 1/9. Do you have a real point or a confused one? What do you think 1/9 being 0.(000111)[sub]2[/sub] proves? Five bucks it only proves you're confused.
ETA: If you really want to learn what the fuss is about, try this.
^^-- here's a picture using the number line abstraction.
I'll just comment on this. You're right that decimal representation is exclusive to base 10 because that's what the word "decimal" means. Poor wording on my part. What I meant to say is that base 9 (and 10 and 2) fractions can be re-written using a radix point to separate the integer part from the fractional part. In base 10 we call it the decimal point, in base 2 we call it the binary point, and so on.
But nomenclature not withstanding, my point stands. In both base 10 and base 2, [math]\frac{1}{10}=0.1[/math], and [math]0.1_2=0.5_{10}[/math], and as the linked page shows, [math]1101.101_2=13.625_{10}[/math].
This isn't me making stuff up. These are mathematical facts. As I've said before, it's you against the entire world of mathematics. That you think that you're right and everyone else is wrong amazes me. You're not the next Newton or Einstein.
Sorry, but we haven't resolved the question of whether "1" is the representation of a number or not. If you think it represents a number, then why is this number not a prime number?
Quoting InPitzotl
The point is that 1/9 is not a name for anything. It's a bunch of signs which have meaning in a conceptual scheme. Your notion that a mathematical expression names a thing, is the problem you need to deal with. This idea allows people like fishfry to argue that "2+2" refers to the same object as "4". But in this argument, fishfry neglects the meaning, or role, of the operator represented as "+". Thus we have the false premise that an expression with an operator has the same meaning (expressed as 'refers to the same thing') as an expression without an operator. I see the very same problem when it is assumed that "1/9" names an object, the meaning of "/" is not accounted for. Therefore it is false to say that 1/9 names the same thing as .111..., or any other numerical representation in another base.
Quoting Michael
It is the meaning of what is being represented which we are discussing, and the meaning of what is represented by .1 differs from one base to another. So your argument makes no point. If .1 in base nine has the same meaning as .111... in base ten, then you haven't resolved anything by changing the means of representation. You just show that "1/10X10/1=1" in base nine, represents the same thing as "9/1X1/9=1" in base ten. But that does not capture the issue expressed in the op.
The point I am arguing is "1/9" does not have the same meaning as ".111...", or ".1 in base nine", or whatever base you want to represent it. The reason is that in the expression "1/9", the symbol "/" has a role which is not represented in the other representation. By convention, we say that 1/9=.111..., just like the convention allows that 1/10=.1 in base nine. And, the convention allows that "=" expresses an equivalence of value, the two have the same value according to the convention. But if we desire to make the conclusion that because "1/9" and ".111..." are expressions of equal value, they are therefore referring to the same thing, we need a further premise. This further premise, that two things of equal value are the same thing, is what I dispute.
This is the same argument which I had with fishfry on the other thread. Fishfry insisted that "2+2" refers to the same mathematical object as "4". But this assertion neglects the role of "+", just like the assertion that "1/9" refers to the same mathematical object as ".111..." neglects the role of "/".
What the op demonstrates is that by the conventions of modern mathematics, division is not an exact inversion of multiplication. If we start with one, and divide it by nine, then take the solution and multiply it by nine, we come up with something different from one. Further conventions implore us to accept that division is an exact inversion of multiplication, therefore the two are equivalent, ignore the difference. Thus we are inclined to ignore the difference. We assume 'a difference which doesn't make a difference', and get on with the calculations.
But whether 'a difference which doesn't make a difference' is an acceptable principle in mathematics, which strives for exactitude, is another question. And, ignoring the difference does not make it go away. To argue that there is no difference, like participants in this thread do, as if this argument could make the difference go away, is not the same as ignoring the difference. So if you choose this option, you'll have to discourse with people like me who will look for whatever ways possible to bring attention to the difference, trying to refute the false assumption that you can make a difference go away through argumentation. In reality such argumentation only brings attention to the difference.
Sorry, but we haven't resolved that there's an actual problem here (not to me, or to anyone else here that I've seen).
Quoting Metaphysician Undercover
If turtles are animals, why do they lay eggs? Since when does being called a prime have anything to do with being a number? The very fact that you even asked this question and actually think it's relevant shows that something is majorly wrong with your "problems".
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
1/9 and 1 sure look like they name points on the number line to me. So where's the actual problem again?
Quoting Metaphysician Undercover
We already know what numbers are and what expressions mean. The only problem here is you, and we don't even have to deal with that problem. But you've diverted 11 pages on this thread so far on your ego tripping delusions of having a problem. That's the problem.
Quoting Metaphysician Undercover
Looks like the same point on the number line to me. So where's the actual problem again?
In the eleven pages of your rantings, I have yet to see an actual problem.
Not much to it.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
What exactly are you rounding off to decimal notation...? 1/9 ? ?2 ... You already acknowledge those numbers that you round off, only to go ahead and deny them. Inconsistent.
Numbers in the abstract are quantities of whatever we may want to examine, where the rules of mathematics are invariant (e.g. division) or otherwise set out. Whatever that "One" you mention is, it's apparently not among them, perhaps like distaste for pizza with pineapple. That's something you've added here.
So, I ended up thinking that you're no longer talking mathematics.
Denial is one of many possible responses.
Quoting InPitzotl
What kind of nonsense is this? Birds are animals too. What does laying eggs have to do with this?
Quoting InPitzotl
They are called "prime numbers". And "one" fulfills all the conditions of "being called a prime", except that it is not a number. Therefore the only reason why "one" is not a prime number is that it is not a number.
Quoting InPitzotl
You simply assume that one is a number, and class it with the other numbers. But it's not a number otherwise it would be one of the prime numbers, not divisible by two other numbers. Once you've made your faulty assumption that one is a number, you proceed to call fractions numbers too. Clearly you haven't got a clue what a number is, yet you keep insisting that such figures represent numbers.
Quoting InPitzotl
This is very clearly not true, as I think everyone else on this thread has admitted, except you. There is no clear definition of what a number is, and there are supposed to be different sorts, natural numbers, rational numbers, real numbers. What the other participants in this thread have indicated is that "number" is just a vague term with no real defining features. That's why they rejected the definition I proposed at the beginning.
Quoting InPitzotl
Sorry, but I have no idea what your little diagram is supposed to be showing. It's obviously not providing a definition, or any sort of indication as to what a number is. So how is that diagram supposed to argue your case?
Quoting jorndoe
We've been through this already, application is different from theory. There is no inconsistency in using a theory which one recognizes as less than exact (eg. having to round off), and still arguing that the theory is less than ideal. One can use a theory, and also at the same time, recognize and argue that the theory needs to be improved on. The problem is when someone like me recognizes that the theory needs to be improved upon, but others argue that it is already ideal.
Quoting jorndoe
There is an abundance of evidence which demonstrates that the rules of mathematics are not invariant. First, you can look at the history of mathematics and see how the rules have changed. Then you can look at the rules which exist today and see variance and inconsistency between one branch of mathematics and another. Clearly "the rules of mathematics" are not invariant.
Quoting jorndoe
If mathematics to you, is a subject where the rules are invariant, I never was talking mathematics.
Still attacking those windmills with your insightful lance, eh? I have to admit, you've got gumption! :nerd:
And you're immune to it?
Quoting Metaphysician Undercover
It was meant to be an analogy... primes are numbers, but not all numbers are prime, was the point. But apparently you're even more messed up than this:
Quoting Metaphysician Undercover
Utterly wrong. There is a history to the concept of prime numbers... after some time in the development of number theory, it was quite apparent that it would be more useful to exclude one from the definition of primes in particular to avoid having to keep making exceptions for it, especially in the fundamental theory of arithmetic which is heralded as being an especially important theorem. That has nothing to do with considering one as a number though... that ship has long since sailed:
Quoting one
...but ultimately it's just a loss of religion. There's no actual deep reason to not consider 1 (and 0) a number, except a bunch of meaningless mumbo jumbo.
TL;DR version: That one is not considered prime has nothing to do with the consideration of one being a number. It's just yet another confusion of yours.
Quoting Metaphysician Undercover
You still have no idea what you're talking about... consistent with everything I've been saying for 11 pages, this is a language barrier and you're still confused.
You're referring to the fact that @Michael listed some categories of numbers here; namely, N (the whole numbers), Z (the integers), Q (the rationals), R (the reals), and C (complex numbers). Those are indeed categories, but there are more; beyond C, there are quaternions and octonions. In contrast to R, there are surreals and hyperreals. Take just Z into the complex plan and you get Gaussian Integers. This is not an exhaustive inventory. All of these things have their own kinds of numbers, and we can even make up new kinds of numbers on the fly.
I'm far from unaware of this MU... in fact, we've both gone through this. Here is the post where you said you were "trying to learn the language". And here is the reply I gave you seven days ago. Numbers defined differently is not a problem for math; it's just homonyms... just a feature of languages. To avoid the issues a language speaker just applies context.
Quoting Metaphysician Undercover
Vagueness is not transitive. An animal can be anything. My pet is an animal. But my pet cannot be anything; my pet is a cat. A number in general likewise could be just about anything. But 1/9 is a fraction, and 0.(1) is a repeated decimal. Generally discussions of such things are in R, though Q suffices.
Quoting Metaphysician Undercover
That doesn't surprise me, but I gave you a link to it. So I guess a bit more spoon feeding you is in order:
Quoting number line (wikipedia)
On the same page:
Quoting number line (wikipedia)
Compare to here.Quoting Metaphysician Undercover
The diagram tells you how you're supposed to play the language game with real numbers. 0.(1) is a real number.
If you actually knew what your nonsense babblings were trying to whine about, then you should recognize that in this picture:
...we have a number line (top), whereby we add 2 (purple) to 2 (blue) by applying the addition rules (see links) to get 2+2 (green), and that refers to the same number (point on a number line, black, circled) as 4 (same point, literally). This counters your idea that 2+2 and 4 don't refer to the same number.
Silly MU, there is no case. You have no jurisdiction, the defendant is a non-entity (language), there is no standing, and there is nothing actionable.
You're objecting to the rules of a language game by playing different language games, and pretending you've said something meaningful. That's all there is, except for the fact that there are 11 pages of it.
Your so-called history of prime numbers is backward compared to what Wikipedia has to say:
So, according to Wikipedia, and contrary to your claims, 1 was first considered as other than a number, therefore not a prime number. Then, in more modern times mathematicians wanted to treat 1 as a number, so they had to include it in the prime numbers and this created a problem. Now they've excluded 1 from the prime numbers, by definition.
Quoting InPitzotl
On the one hand you say mathematicians "keep having to make exceptions" if one is a prime number, and one the other hand you say that there is "no actual deep reason" not to consider one a number. It's starting to become crystal clear which one of us is actually the confused one.
Let's see what's the case here. We apply a rule, the rule of primality, to the whole infinity of "numbers", and find that there is one exception to the rule, the exception is "1". The rule applies to all the numbers, allowing mathematicians to create theories based in that rule, therefore we can say that it is a defining feature of "numbers". However, the rule does not apply to 1, as 1 needs to be excluded from these number theories. In your mind, what's the logical thing to do, make an exception to the rule, to allow that 1 is still a number despite being an exception to this defining feature of numbers, or conclude that 1 is something other than a number?
Quoting InPitzotl
I know that "numbers defined differently is not a problem for math". What is a problem is conceited people making the universal, uncategorized statements like "we already know what numbers are", when it's very evident that they haven't the foggiest idea of what a number is.
Quoting InPitzotl
No, an animal cannot be anything, a rock is not an animal, a plant is not an animal. Likewise, a number cannot be anything.
Quoting InPitzotl
It is not logical to refer to a property of a special type of number (real number) to demonstrate what a number is in general. This is like referring to your cat's meow to say what an animal is. In philosophy we call this the difference between an essential property and an accidental property, and being able to make the distinction is fundamental to proceeding with deductive logic. That 1 can be represented on a number line as a feature of real numbers, is an accidental property, specific to one type of number, real, and not an essential property, describing, or defining numbers as a whole.
Quoting InPitzotl
I apologize for not joining your little game, but I see no reason to restrict our discussion of "numbers" to real numbers.
Quoting InPitzotl
That's 11 by your convention, not by mine.
Obviously you haven't read those pages.
Don’t use that word! It’s @Metaphysician Undercover’s and it has a technical meaning with unique connotations.
...
Quoting Metaphysician Undercover
...while we're on the subject, what does the very next paragraph say?
Quoting Metaphysician Undercover
Welcome to the year 2020. So what's the problem?
Quoting Metaphysician Undercover
...so where does that leave you? Do you have the foggiest idea what a number is? Do you make universal, uncategorized statements about numbers?
Quoting Metaphysician Undercover
I think you're lost, MU. This is supposed to be a thread about 0.(9)=1.Quoting Metaphysician Undercover
Well that's really easy MU. Here's the primary motivation, in your words:
Quoting Metaphysician Undercover
The way to avoid inconsistencies and contradictions that lead to misunderstandings and deceptions (aka, amphibolies/equivocations) where languages have homonyms is to restrict the conversation to applicable shades of meaning. When in a pool hall and someone talks about how to sink the 7 without sinking the 8, English should be regarded as a pool-technique, so it simply means to invoke a spin on the ball... countering a discussion invoking the use of English with debates about how some hypothetical guy from England might sink the 7 is a meaningless distraction. In this context, we're supposed to be talking about 0.(9)=1. 0.(9) is a repeated decimal. Repeated decimals are special cases of fractions, suggesting a treatment of at a minimal Q, though decimals just commonly invoke R. So to avoid misunderstandings and deceptions, to meaningfully talk about Q and R, we should be employing the context of one of these two things.
Incidentally MU, even if we don't restrict our discussions to the reals, 2+2 and 4 refer to the same object in the reals, and you claim they don't refer to the same object (again, in case you missed it, "Do you make universal, uncategorized statements about numbers?"). This implies you're flat wrong in at least one context. According to your pretended concerns about the inconsistency or contradictions with language leading to misunderstandings or deceptions... according to your pretense of avoiding smoke and mirrors, just having this single context in which you are wrong is challenge enough to warrant a clarification of your claims anyway... you know... to... avoid misunderstandings and deceptions?
TL;DR, we should restrict our discussion to the reals because that's the context within which 0.(9)=1 and 0.(1)=1/9 are meant to be discussed; i.e., it is this context from which the meaning of such things derives. Ranting and raving about what some guy in 300BCE would have called 1 is a meaningless distraction.
Your previous side-track doesn't really matter much here; it's about the numbers, 1/9 ? ?2 ... By rounding them off, you've already admitted them. Denying them is hence inconsistent; you wouldn't have anything to round off in the first place.
Quoting Metaphysician Undercover
Saw the word "invariant" and took it for a ride? Having five fingers on each of your two hands means having ten fingers on them, not none, not a dozen. 5 + 5 = 10 = 2 × 5 (and 5 < 10 by the way). Notice how that goes for toes and claws as well? Whether yours or mine or the Pope's? You don't mysteriously get a dozen fingers in that case. That's what's meant by invariance here, + - × /, and what you tried to dismiss with a casual handwave. Oh, also, ?2 × ?2 = 2 (and 1 < ?2 < 2), irrespective of your rounding, so there. ;)
Quoting Metaphysician Undercover
As mentioned, whatever your "One" is, this is something you've added here, much like I added distaste for pizza with pineapple. Your "One" apparently does not figure as the number 1 does in arithmetic.
Stick to the topic.
Has anyone ever persuaded a change of an opinion or belief you've held?
Are you solipsistic, by any chance?
Quoting tim wood
I've already made the point numerous times. The op asks: "As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ?" I agree that most would be fine with that, but I am not. If you are interested in the reasons why, you can read the thread. I started with the need for a definition of "number", as necessary in order to determine the acceptability of a matter of representing numbers.
Quoting InPitzotl
Your reference to the history of the prime numbers neglected the fact that for millennia 1 was not considered to be a number. It was only in relatively modern times that mathematicians wanted 1 to be a number, and this created the problem which required an exception to be added into the rule of primality.
Quoting InPitzotl
I provided a definition at the beginning of the thread, this was my idea of what a number is, an arithmetical value representing a particular quantity. It was rejected, and then it was explained to me that "number" is not a defined term in mathematics. So I concluded that no one really has the foggiest idea of what a number is. Then you contradicted this, claiming that we know what numbers are.
Quoting InPitzotl
This is the point I've argued from the beginning of the thread. To know whether the op offers an acceptable representation of numbers, we need a working definition of "number", and restrict the conversation so as to use "number" only in that way, and thereby discuss whether the op offers an acceptable way of representing numbers or not. As I stated earlier in this thread, I don't think that .111... is acceptable as "a number" because it does not represent a particular quantity. But it was claimed that my definition of "number" was unacceptable.
Quoting InPitzotl
Clearly "2+2", and "4" do not refer to the same "object" by any conventional definition of "object". So I think it's time for you to start learning the language.
Quoting InPitzotl
It ought to be clear to you by now, that I do not accept "the reals" as a representation of numbers. Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously.
Quoting jorndoe
That's not true. Due to the nature of representation, using symbols which represent quantities does not necessitate that the person believes in the existence of numbers. I can ask for two coffees for example, using the term "two" to get what I want, without believing that "two" represents some sort of mathematical object called a number. So I can do all sorts of arithmetical operations, using those symbols in the way that I am taught to, including the rounding off of quotients, without believing that there is any such thing as numbers. There's no inconsistency between using those symbols and denying the existence of numbers.
Quoting jorndoe
But your claim was "the rules of mathematics are invariant", not the number of fingers on my hand is invariant. I gave you a clear explanation of how the rules of mathematics are not invariant. Your logic is appallingly bad similar to InPitzotl's. You give me one example of an invariant rule and conclude therefore all the rules of mathematics are invariant. It's as if you are arguing that "5+5" is equal to "10", in all instances, therefore all the rules of mathematics are invariant. Look at the conventions for multiplying negative integers, and imaginary numbers as an example of how mathematical rules are not invariant.
Quoting jorndoe
I'm still waiting for someone to explain to me how the so-called "object", or "number" which is represent by "1" and is by definition not a multitude, and therefore not composed of parts, can be divided into nine parts. Care to explain how the division might take place? I'm not asking for a demonstration in symbols, because it's easy to represent something with symbols, which is actually impossible to do, just like we can talk about doing things which are impossible to do. I'm asking what makes it possible to divide a unit which is not composed of parts?
Quoting dex
My beliefs change like the weather. But to be honest, I wouldn't say that it's others who persuade me to change.
In that case, may I ask why you're arguing your position here? If you yourself can't be persuaded by others, what makes you think others will be persuaded by you?
What are you talking about, "problem" and "required"? The fundamental theorem of arithmetic states, in the modern reading, that all positive integers can be represented as a unique product of primes (barring order). That's perfectly phraseable with prime including 1, it's just clumsy: "All positive integers can be represented as a unique products of primes, barring order, excluding from said product the number 1". Both phrases describe the same fact. One is just clumsier.
Quoting Metaphysician Undercover
Pretty much.
Quoting Metaphysician Undercover
Sort of, but not really. "Number" applies to a lot of things. But that's not a problem; it's actually a benefit. The definition of number should not merely not be nailed down; it should be open. But part of the point of categorizing these numbers is so that we can give particular kinds of numbers names.Quoting Metaphysician Undercover
This is jargon... they refer to the same mathematical object.Quoting Metaphysician Undercover
Anyone who uses the decimal system to count above 9 shouldn't take your pronouncement seriously.
It is worth learning just how far @Metaphysician Undercover is willing to go; and how far his pretence of rational conversation drags us into his madness. He should have admitted his error, one supposes - and yet he hasn't, so there must be something he has misunderstood, something that will trigger Meta's realisation... and so the conversation continues.
It becomes harder for @InPitzotl and @jorndoe to walk away the more they invest.
I disagree. Just because he's illogical doesn't mean his psychology isn't interesting. He's yet to answer the question I posed most likely because it isolates the underlying hypocracy of his debate stance -- apparently truth has little to do with his posting motivation -- so it's only pointless to argue against that which his hypocracy is productive of. His whole intellectual orientation is faulty. But it's for some reason been useful enough for him to maintain it.
Something tells me that it's partly solipsism, partly an expression of aggression against the imposition of an external control over his thinking. So he's in company with the likes of Kanye West when he decided to support Trump after the media made out it was a bad choice.
People like this aren't nut-cases; they're protesters.
Yes, it's about vulnerability, not truth. Meta sees his argument as invulnerable, others see it as contradictory and infertile. More over the dance gets attention.
I think you are being needlessly complicated. He's simply seeking attention, and his approach works well.
I mean you don't typically use mathematics for no reason there generally is a precision that is lost otherwise that is fundamentally detrimental but.. for most yeah why not.
I pay for a 15 minute massage and for some reason I time it and it only turned out to be 14 minutes and 59 1/2 seconds. I wouldn't call that a scam or even anything.
Again mathematics generally has a purpose. It's not like fashion. Say I pour you a glass of water. Nice of me, right? Now if I put- pay attention now- less than a billionth of a gram of polonium in it. You would be dead in minutes. Just saying.
I haven't checked out this thread in a while, sounds like it's about to get spicy. Alright!
Quoting Outlander
You're confusing pure math with applied math. A common category error. Math is not subject to any standard of applicability. On the contrary, the only criterion for the worth of a piece of math is whether it's regarded as interesting and beautiful by mathematicians. But to satisfy the Philistine in you (one with no appreciation of the arts), be aware that the most abstract and useless mathematics of one era often becomes a core engineering discipline of another. Number theory provides a striking example. Regarded as supremely beautiful yet utterly useless for over 2000 years; number theory is now the foundation of Internet security and cryptocurrencies. Mathematicians let their minds wander, and they let others care about purposes.
MU has a metaphysical theory of numbers, he's a believer in them in the full b-word sense (it's part of his identity... almost literally), and modern math is kind of a heresy wrt it. That's my take. I personally envision his theories as being roughly of both the form and value of Eric the half a bee.
I can't help being struck by the amount of mindshare @Metaphysician Undercover holds here.
Cyril Connolly?
That makes sense. The arrogance is still pretty weird, though. All the same, math noobs like myself can learn a bit from the counter-effort. This thread was a heck of a read.
LOL
I partake in this forum to learn, and to help others learn. Learning is a communal process requiring sharing and consent. I apprehend a difference between understanding and being persuaded. One can very often persuade a person to act in a particular way, without the person understanding the need to act that way, or the reason why that action is being called for by the other. This difference is what allows for the existence of deception.
"Learning" is a broad term which is used to refer to both, understanding, and being persuaded to act without understanding. So for instance, as children we are highly susceptible to being persuaded, and our capacity for understanding is quite limited. We are taught principles, like arithmetic, and are persuaded to behave in a particular way, without understanding the reasoning, which is the theory behind those principles. "Learning rules" of arithmetic, and even the "rules" of higher mathematics is not actually a case of understanding principles, as "learning rules" seems to imply. Fundamentally it is a matter of being persuaded to act in particular ways in response to specific situations, and develop particular habits, like training a dog. We are persuaded to act in a particular way without understanding any actual rule, though the rule might be produced as a description of that behavior. Wittgenstein is relevant here. When we are older, and our capacity for understanding is increased, one might delve into theoretical mathematics, what fishfry called pure mathematics, in an attempt to understand these actions.
What is important to apprehend, is that in the general sense, understanding follows from acting, it doesn't precede it, as we learn from experience. So theory follows practice. We find a practice which works, and we employ it, then we develop the theories to account for why it works. In this theoretical process, which is the "understanding" of the practice, it is of the utmost importance to determine the faults of the practice, exceptions, places where the practice produces less than perfect results. This is where proper understanding, and formulation of theory in a way which accounts for these discrepancies can lead to a better practice in the future.
So for example, the ancient practice of astronomy was to map the orbits of the planets as circles. This practice worked very well, and provided very good prediction, as Thales apparently predicted an eclipse. But there were slight imperfections. What was required was theoretical analysis of the slight imperfections, to produce a true understanding of the real orbits of the planets. That new theory produced a whole new set of practices which today extend far beyond the solar system. But the new practices have demonstrated their own imperfections. Therefore we need to revisit all the theory from bottom up to understand and account for these imperfections.
Quoting dex
To answer your question now, I believe it's a faulty goal to partake in this forum with the intent of persuading others. We are here as philosophers with the goal of understanding. We cannot treat each other as children to be persuaded, and even a minimal degree of participation will reveal that persuasion is never forthcoming. My goal in arguing the position I have argued in this thread is to bring to the attention of others, the slight imperfections which I've observed to exist within the practicing of mathematics. We can only move forward, collectively, by acknowledging, and accounting for these imperfections. To me, the imperfections are glaring, but every person perceives and apprehends things in one's own way. So some people cannot even see the imperfections, and others who see them dismiss them as being so minor that they're irrelevant (a difference which doesn't make a difference), so they end up denying that the imperfections are even imperfections. This is what I refer to as contradiction, to say that there is a difference which is not a difference, as the difference between ".999...", and "1".
Quoting InPitzotl
And what about 1? Is it excluded as a positive integer, or natural number? Or have you made the fractions into integers? Where does 1 fit in this theorem?
Quoting InPitzotl
That's simply an assertion. I have yet to see a definition of "mathematical object" which allows for the application of the law of identity. And the law of identity is what identifies an object as an object. To say that they refer to "the same mathematical object", says nothing more than that they are equal. And two distinct objects with the same value may be equal, and clearly not the same object according to the law of identity. So the phrase "they refer to the same mathematical object" is nothing but a deceptive use of jargon.
Quoting InPitzotl
Sure, leaving the definition of "number" open is a "benefit"; to those who want to expand mathematical theory in any imaginable direction, like fishfry promotes, and also for those who argue by equivocation. For those who want to develop clear and consistent mathematical theory with universal applicability, it is detrimental.
Quoting fishfry
This is the fantasy that aesthetics is valued over and above good. It is a fantasy because we can only passively enjoy beauty for an extremely short period of time before our bodily needs get in the way and we are urged to act. The natural human condition is to act, so even the purest forms of theory are influenced by the urge to act.
Quoting InPitzotl
No. I don't seem to have a metaphysical theory of numbers, because I do not understand numbers well enough to create such a theory. What I do understand though, is that there is no metaphysical convention, and therefore no ontological coherency, in modern math. You might say that I believe in metaphysics, and modern math demonstrates a poverty of metaphysics.
Quoting fishfry
I would blame Banno, for declaring that this thread is about me. I'm just doing whatever I can to live up to Banno's expectations of me. See my respect for you Banno?
Quoting Banno
Good question. 1 is the product of zero primes: https://en.wikipedia.org/wiki/Empty_product
Quoting Metaphysician Undercover
Where have you looked? Or am I your personal search engine now?
Quoting Metaphysician Undercover
We have the terms rational and real, so, you're just whining.
Quoting Metaphysician Undercover
I beg your pardon:
Quoting Metaphysician Undercover
I note here that you're not just asking for a definition of number. You're asking for a definition of number that has these properties.
Also:
Quoting Metaphysician Undercover
First class example of deception.
Quoting Gary M Washburn
This is not necessarily a fallacy. It may be the case that the principles required to establish compatibility between the two, such that we can conceive of the two coexisting, as "what is real", have not been discovered.
Quoting tim wood
If you think that you can explain the meaning of "2+2=4" by saying that "2+2", and "4" represent the same mathematical object, and "=" represents "is the same as", then I would say that you sorely misunderstand mathematics, and you ought not be taken seriously.
Do you understand that for an "equation" to be at all useful in honest mathematical practice, the right side must necessarily represent something different from the left side? If not, the equation would be a useless tautology. But that is how some people might present "2+2=4" as an example of a useless tautology in which "2+2" represents the very same thing as "4". But clearly that is not how equations are used by scientists and mathematicians. It is only how sophists who are attempting to persuade someone that "2+2" represents the same object as "4" might use an equation, and this is surely not honest mathematical practice.
It is.
"the procedure proves what the procedure is supposed to", here, here, ...
Inconsistent. Recycle.
Quoting Banno
You're right. Isn't the adventure into @Metaphysician Undercover's Wonderland oddly fascinating though? :) I guess it becomes trite after bit.
Quoting Banno
By the way, @Metaphysician Undercover, despite having been given references, you may of course ask for definitions of definitions of ... but I doubt anyone is going to teach you elementary school material on up. If you don't (or won't) get it, then so be it.
Yes, but it's quite ineffective... we already knew you weren't here to learn. Along the same lines, I never heard that paragraph after the one you quoted about the history of primes (gee, I wonder why, wink wink... o/c I know why and everyone with a browser can find out why in 30 seconds). Also along the same lines, that wiki page on mathematical objects that you could just as easily have looked up as the primes is still there.
That the procedure produces an answer to the question, and the fact that the person uses the procedure to produce an answer, does not prove that numbers are objects. The procedure is designed to resolve a specific type of problem, not to prove that a number is an object.
Repetition: a person does not need to believe that a number is an object to carry out mathematical procedures.
Quoting InPitzotl
You mean, I am here to learn, and not to be deceived, don't you? To fall for a deception which has been proven on others to be an effective deception, is not an instance of learning, even if the others believe it to be an instance of learning.
Alright I'll play. What is the nature of this deception?
Once apprehended, it should be incarcerated and prosecuted to the fullest extent. Understanding abets acting and is equally guilty.
Quoting Metaphysician Undercover
Perhaps. But there are instances where it arises, like non-standard analysis which incorporates Leibniz's infinitesimals - which I claim are metaphysical actualities. And from my perspective, modern transfinite set theory seems somewhat metaphysical (others will probably disagree). The higher one goes into the thin air of mathematical abstraction the more likely one will encounter metaphysics - in my opinion. For example, one new developing area is that of "magnitude" in abstract spaces. Although the groundwork has been laid, this concept seems to me metaphysical. :cool:
The idea that one is a product of zero.
By "poverty of metaphysics", I mean poor metaphysics. And I consider infinitesimals as poor metaphysics, being a compromise between the incompatible principles of continuity and discrete units. So to me, it's like a monism which instead of respecting the reality of the two distinct and incompatible aspects of reality, which dualism recognizes, the metaphysics of infinitesimals blends the two together in an unintelligible vagueness where the two are assumed to be one.
Oh you silly confused soul, seeing liars behind your eyelids. I suppose you also see lies in the fact that 3*3=9, given we're multiplying two threes and getting an odd number? Maybe this whole math thing isn't going to work for you.
And I consider it the best of metaphysics, existing solely in the mind but useful in developing the mathematics describing physical phenomena.
Carson Chow (Scientific Clearing House, 2012):
"While metaphysics as science is a dead-end for me, metaphysics as mathematics is ripe for very interesting insights. Instead of asking directly about “our” reality, we should be asking about hypothetical realities."
But, then, I am not a philosopher and must bow to your competence in this area, as I have to your competence in mathematics. :cool:
(Refer to Metaphysics Defined in this forum)
So you're not talking mathematics, or even logic for that matter, don't understand the formal expressions. (Which was observed earlier I guess.) Still going downhill. In a manner of speaking, proofs explicate tautologies.
Quoting Metaphysician Undercover
Prove? Objects? The numbers are already operands in the procedures. There isn't anything to round off (you claim (that you believe)), but do it anyway. Inconsistent. What exactly are you rounding off if not 1/9 ? ?2 etc? Recycle.
I guess you don't believe in pocket calculators, which do not list kilograms, claws, or square miles, for example (cf mentioned invariance). You should at least understand what you're talking about before objecting and proclaiming (vast) conspiracies. :D References have been posted.
Sarcasm, ...? :)
If the left and right sides were different then they wouldn't be equal. What do you think the equals sign is?
You don't seem to understand the difference between expressions and values. [math]2+2[/math] and [math]4[/math] are different expressions with the same value. [math]9_{10}[/math] and [math]10_9[/math] are different expressions with the same value. [math]0.999...[/math] and [math]1[/math] are different expressions with the same value.
Or for a non-maths example, "water" and "H[sub]2[/sub]O" are different expressions that refer to the same thing.
It’s not that one is a product of zero, it’s that one is the empty product.
Take any set of factors and multiply them together. Say for example {2, 2, 3, 5}. The product of those is 60, right? Now put a 1 in there, to make it {2, 2, 3, 5, 1}. Still 60 right? Put in another 1, and another, and another, and it doesn’t change anything right? Take away a 1, and another, and another, and it doesn’t change anything, right? Including or removing ones makes no difference to the product.
So if you have the product of {2, 2, 3, 5, 1, 1, 1}, and you get rid of the 5, then the product is 12 right? And if you get rid of a 1 it’s still 12. If you get rid of the 3 then the product becomes 4. If you get rid of a 2 then the product becomes 2. We’re down to {2, 1, 1} now if you’re having trouble following along.
Get rid of a 2 and the product becomes 1. Get rid of a 1 and the product doesn’t change — still 1. Get rid of another 1 and the product still doesn’t change. We’re down to the empty set here now, {}. That has the same product as {1} or {1,1} etc because including or removing 1s doesn’t make any difference.
If you put a zero into any of those sets, the product would become zero, yes; but we’re not doing that.
"Multiplicative identity" is more appropriate. But whatever. It appears this thread will go to infinity without ever leaving the starting point. Paradox?
Being persuaded your logic is wrong; persuading others their logic is wrong. Say your understanding of something is off-base: how is it wise to ignore all logic from more educated sources? Did you teach yourself the alphabet, too?
Learning is a kind of changing of minds. Persuasion in debate isn't surreptitious or whatever. But you don't appear open-minded enough to consider that the consensus is right about your being wrong. How do you hope to learn anything here?
Tbh it comes over like you're sticking your finger up while elevating how visible you are (Banno has a point) -- if that weren't the case then your language wouldnt be as blindfold-defensive as it is. Logic helps us become humble to our own bullshit but you aren't using it that way.
Hang on a sec, you're not trying to avoid those old record long threads of yours getting beaten? :)
Yeah, the comments here (one sub-thread in particular) have run their course.
Oh no! And we are so close to half a millennium! Don't stop now. This may be a record for total nonsense. :scream:
There's no easy way to find longest threads that I am aware of; but one would have to go a long way to beat the Trump thread. My record on this forum is only 1.5k replies. On the old forum it was many times that.
I'd still like to have a thread reach over 100 replies from a single post, though. You could easily have done that here, had you not replied to your own OP.
Fixed in the PDF.
Quantification --- Forming Propositions from Predicates — Shunichi Toida et al, Old Dominion University
[math]{1\over3} = 0.333...\\3 \times {1\over3} = 1\\3 \times 0.333... = 0.999...\\[/math]
Let's not distract from supertasks by questioning very simple mathematical facts.
That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is.
On the other hand, it does show that looking at a problem another way might show that the problem is an illusion. But that would be philosophy.
Well it's not a mathematically rigorous proof as it doesn't prove each of the three steps. A mathematically rigorous proof is much more complex, as seen with TonesInDeepFreeze's answer.
But it's a simple proof for those that accept each step individually. If you want a proof of these then that's a topic for another discussion, probably on a forum dedicated to maths.
There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1
Of course, one could (with undesired consequences) reject the first two steps of this proof.
There is also the sum of an infinite geometric progression of term a = 9*10^(-n):
0.999 = 0.9 + 0.09 + 0.009 +...
= 0.9/(1-0.1) = 0.9/0.9 = 1
Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate.
Quoting https://files.eric.ed.gov/fulltext/EJ961516.pdf
Mind you that this article is not written by experts.
I can see that point. I didn't look at the issue in the light of infinite series or take on board that it was a question of the sum of an infinite series. I apologize for the distraction.
Quoting Deleted user
That's very neat.
I do appreciate your help.
But this doesn't seem to work with other similar sequences, such as 0.333... or 0.444... or 0.1212....
What have I got wrong?
10a=3.3333333
9a=3
a=3/9
a=1/3
1/3 = 0.33333
b = 0.12121212
100b=12.12121212
99b=12
b=12/99
12/99=0.121212
But now I think we'll all agree that as you divide a number by smaller and smaller numbers your output gets ever greater.
So:
1/0.1 = 10
1/0.0001 = 10,000, etc.
But since 0.00...001 = 0 and 1 / 0.000...001 = ? then 1/0= ?.
Now it is also true that 4/0=? and 9.7181=?. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin.
:razz: :nerd: :up: :100: :heart: :strong: :strong: :strong:
Oh I see what happened. @Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread.
Ludwig, let me put to you a question.
Suppose that .999... is not 1. If they are different numbers, then there must be a third number strictly between them. What is it?
Put another way, and echoing the point you made in the other thread, suppose I have the sequence
.9, .99, .999, .9999, .99999, ...
What number can possibly get between ALL the terms of that sequence, and the number 1?
Well maybe it's .95. No, that's smaller than .99.
Ok maybe it's .995. No, that's smaller than .999.
Ok then maybe it's .9995. No, that's smaller than .9999.
You see how this works? You can't find any number to stick in between ALL of the elements of the sequence, and 1.
Since you can't find a number between them. the limit of the sequence is 1. Or putting it another way: .999... = 1.
It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.
Quoting fishfry
Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.
However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit.
But an infinite series never reaches its limit. To put it another way, "=" in this context (an infinite series) does not mean what it usually means.
Thank you very much for those.
If I've understood, your argument shows what the sum of the infinite series is.
Right?
I look forward to mankind's return to the Garden of Eden.
It's probably just that everyone who joins needs to be taken through it. Each person has to learn everything for themselves.
It's typically a land mine of ignorance and confusion. Mathematically there is no question whatsoever. .999... = 1 is a theorem of ZF once the appropriate definitions of the real numbers, limits, and infinite series are made. It's like looking at a chess position and saying yes or no, is this a legally reachable position according to the rules. .999... = 1 is a legally reachable position in ZF.
Quoting Ludwig V
There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1.
Quoting Ludwig V
Quoting Ludwig V
We had this conversation. 1 is the limit of the sequence .9, .99, .999, ... "reaching" is just something people say to confuse themselves.
Quoting Ludwig V
It means exactly what it usually means. The limit of .9, .99, .999, ... is 1. Or equals 1.
I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread.
The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2.
Yes. I assume you mean all the terms of the infinite sequence?
Quoting fishfry
And I'm puzzled why you think I'm disagreeing with you.
Quoting fishfry
So it is. But what is the element of the sequence immediately preceding 1?
Yes. "Sequence" and "infinite sequence" are basically synonymous, since finite sequences aren't of interest in this context.
Quoting Ludwig V
It's late, time for bed. I don't think you're disagreeing, but possibly misunderstanding.
Quoting Ludwig V
[/quote]
There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point.
For me, this issue has a wider context.
This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable.
The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.
But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability,
I think this is a mistake, because it neglects context. But it is new angle on the mistake.
I'm basing this on an assumption that both theses are correct - in their context.
Quoting fishfry
I think it follows that "0.999...." does not equal 1.
Sadly, my best time for philosophy is first thing in the morning...
It shows how we get the fraction representation of repeating decimals.
OK. I wondered if it worked a bit more widely than that. I don't think that it would work for sqrt2, since Aristotle could prove that it was "incommensurable" without involving decimals. What about ?? I was taught that it was 22/7 or 3.14....?
Thanks.
I never gave any thought to the relation of truth and probability. Probability is just a number we assign to an event. Before you roll a die there is no truth to its outcome, it hasn't happened yet. You do know with probability 1 that it will turn up 1, 2, 3, 4, 5, or 6. And that if you roll it a million times, about 1/6 of the time it will turn up each number. I don't know how you relate that to truth.
Quoting Ludwig V
I don't understand what you're saying. I have 1 apple in the fridge, and there is 1 president of the United States, but there is no deep philosophy there. 1 is a number that has many applications.
You are still (I think) applying some mysticism to .999... = 1 but there really isn't any. It's a theorem of ZF. If you wanted .999... to be 47 you could make up a system in which that's a theorem.
1 is a probability and 1 is the number of stars in our solar system. I simply do not see the point you're making.
Quoting Ludwig V
Well if I have a hammer I can use it to pound a nail or go out into the parking lot and smash everyone's windows. A hammer is just a tool with many distinct and unrelated uses; and if you think of it that way, the number 1 is also a tool with many uses.
The hardware store owner has no use for hammers, to him it's the buyer who supplies the use. Likewise the pure mathematician has no use or application for the number 1; he just makes sure all the numbers are nice and shiny and logically constructed, for others to use.
Any of this make sense? I don't get what you are trying to say.
Quoting Ludwig V
Hammers and numbers. Tools for most people, objects of interest in and of themselves to hardware store owners and mathematicians, respectively. I am baffled at where you are going with this.
Quoting Ludwig V
In what system of rules? In ZF? You are wrong. In the "point-9-repeating equals 42" system? You're right. What underlying assumptions are you making?
If you assume the axioms of ZF, then .999... = 1 can not be challenged or disputed, any more than you can argue with how the knight moves in chess. But if you make up a chess variant in which the knight goes, say, three steps vertical or horizontal and two step diagonal, then that's how the knight moves in his alternative variant.
Make sense?
Now, if you would like to chat about why .999... = 1 in ZF, I am trained to know this. I had it beaten into me by professors at some of our finest universities. But if you prefer the "point-9-repeating equals 42" system, I'm perfectly happy to work with that as well.
Quoting Ludwig V
I've always been a night person. I generally post in the evenings US left coast time.
Just an aside. You probably know this stuff. But others might not. This is not a rigorous presentation.
When you talk about the probability of something, that needs to be defined as an event. Which is a particular kind of mathematical object. It does not tend to be the kind of mathematical object that a formula in a mathematical argument is. Eg the probability that it will be raining in 2 hours given that it is raining now makes sense. The probability that 2+2=4 doesn't make too much sense.
However. If a statement A is provable from a statement B and concerns a quantity [hide=*](in some amenable sense I won't specify)[/hide], the probability of A given B is 1. As an example, what's the probability of X+1=4 given that X=3? Probability 1.
Another fact like this is that if A and B are mutually contradictory, the probability that A occurs and B occurs is 0. That also works with entailment. Like the probability that X=3 given that X+1=2 is 0, since X+1=2 implies X=1, and there's "no way" [hide=**](in some amenable sense I won't specify)[/hide] for X to be 3 given that assumption.
The same holds for statements [hide=***](in that same amenable senseI haven't specified)[/hide] you can derive from B using classical logic and algebra and set operations. eg if the probability that X=3 is 0.3, what's the probability that (X=3 or X!=3)? 1, since those are exhaustive possibilities. The latter does have a connection to truth, as if you end up asking for the probability of something which must be true, its probability is 1.
For folks like Fishfry, I'm sure you can make the amenable sense I've not specified precise. Logical, algebra and set operations which can be represented as measurable functions on the sample space work like the above. "no way" corresponds to the phrase "excepting sets of measure zero". Which is the same principle that stops you from asking "What's the probability that clouds fly given that x=2?", as there's no way of unifying both of those types of things into a cromulent category of event.
The latter also blocks a more expansive connection to truth. Since the kind of things that humans do while reasoning from premises typically aren't representable as measurable functions. Maths objects themselves also have plenty of construction rules that behave nothing like a probability - like the ability to conjure up an object by defining it and derive a theorem about it, there's just nothing underneath all maths that would take take a probability concept which would usefully reflect its structures I believe.
Actually I never formally studied any probability theory. I've seen measure theory but not the fine points of probability.
Quoting fdrake
I've never seen probabilities assigned to mathematical facts like that. Not sure what it means.
Quoting fdrake
Same remark. Don't follow this at all. If you pick a random real in the unit interval, the probability that it's between 0 and 1/3 is 1/3. That I understand, from measure theory. But I don't follow assigning probabilities to equations at all.
Quoting fdrake
You are giving me too much credit. I have no idea how to assign a probability to an algebraic statement. I've never seen that.
Quoting fdrake
I don't know if reasoning from premises is amenable to probabilities. I may have missed much of what you said in this post.
P(X=1|X+1=2). Where X is a random variable. That'll give you probability 1.
I apologize for this post. I'm just flailing around. Actually, I'm still not sure where the best place to begin is.
This is about categories or conceptual families or language-games and the importance of context and use. I won't try to give a general characterization of this. I think it will help more if I focus on something specific.
Quoting fdrake
and
quote="fdrake;916313"]The probability that 2+2=4 doesn't make too much sense.[/quote]
What fdrake is saying (I think) is that probability is inapplicable without a context of argument and evidence and has much to be said for it.
Quoting fishfry
Neither am I. But if probability=1 and true=1, then fdrake's conclusion follows.
Quoting fishfry
These are different uses of "1", in different contexts (language-games).
(Iadded this later, to try and clarify). Compare a traditional example:- "John came home in disgrace, a flood of tears and a wrecked car." "In" is ambiguous, because "disgrace", "flood of tears", and "wrecked car" are different kinds of thing, are pieces of different language-games and "in" is polymorphous and has different senses, or uses" in each of them. That's the theme of this whole argument.
Applying numbers to objects in the solar system is one kind of language-game. Applying numbers to probabilities is quite another. Actually, there are (at least) two ways of using numbers in the context of probabilities. There are 6 probabilities (I prefer "possibilities" or "outcomes" as less confusing) when throwing a die, each of which can be assigned a probability of 1/6, and if the 6 comes up we can, I suppose, assign a probability of 1 to that outcome.
So I would prefer to say that probability is not applicable to either 2+2=4 or (x=3)&(x+1=4). Why? Because there are no other possibilities. Probability of a specific outcome is only meaningful if there is a range of possible outcomes. 1 is conventionally used as the range of the outcomes. Assigning a probability to one outcome and then to another without outside that context is meaningless. 1 isn't counting or measuring anything - it's just the basket (range) within which we measure the probabilities (in relation to the evidence and if there is no evidence, then equally to all). (fdrake is right to emphasize the role of evidence - especially in the context of Bayesian probability) We use 100 as a basket in other contexts when it suits us. In the case of the die, P(1v2v3v4v5v6)=1 is just reasserting the rules.
In the case of truth, the language-game that provides the context is different. In a sense, when we assign 1 to truth, it is not a number at all. We can equally well use "T" or a tick if it suits us. This reflects the point that "true" is one of a binary pair. Probability isn't. I want to say that probability and truth are different language-games.
But that would be too quick, because they are related. Probability is what we retreat to when we cannot achieve truth, one might say. There are others - "exaggerated", "inaccurate", "vague", "certain", "distorted", "certain". I would be quite happy to say that truth is not binary, but multi-faceted; the language game of truth has more than two pieces - probability is just one of them. Probability itself has more pieces than are usually recognized. In the context of empirical probability, we find ourselves confronted with "likelihood" and "confidence" and, sometimes, "certainty" and, of course, in the context of Bayesian probability, "credence" - "degree of belief" turns up from time to time, as well.
@fishfry There's one other point I would like to make, in the context of our previous discussion about time in mathematics. Given that, probability is a bit of a problem, because it seems to me that it has time, or at least change, built in to it. (I have seen it said that probability is inherently about the future). We build the table around the outcome, in the context of a thought-experiment such as tossing coins or throwing dice or drawing cards lotteries or roulette wheels. (I expect you know that Pascal built the theory around a desire to help his gambing friends) We expect an outcome, when everything changes. Time isn't essential. The outcome could be unknown, for example. Even if it is known, we can pretend that we don't know it. But there is an expectation of change, without which probability makes no sense. So the timeless present does not describe what is going on here.
One could regard probability theory as applied mathematics, but probability isn't a prediction. Probability statements are neither confirmed nor refuted by the actual outcome. (That's not quite black and white, because we do use deviations from probability predictions as evidence that something is wrong. But still...)
I prefer to say, however, that the probability table does not change when the outcome is known. It describes a situation and that description is correct even after the outcome is known - it just doesn't apply any longer. So probability = 1 doesn't really apply.
Ok. That will do. Maybe some of that is helpful.
Full disclosure - I haven't formally studied probability either, any more than I've studied mathematics. But I have discussed both and thought about both a good deal, in various philosophical contexts.
Yes ok, a true proposition has prob 1 and a false one 0. I don't see how intermediate probabilities could apply. Unless, say, we could poll a bunch of mathematicians and ask them to assign a probability to the Riemann hypothesis being true. That would be one example I suppose. But I think that's credence (degree of belief) rather than probability (whatever exactly probability is).
Earlier you said there was something off about using 1 as a probability and that .999... = 1. But that's two uses of the same number 1. So I don't see your point. Of course 1 has many different uses. Why is this nontrivial or interesting?
Quoting Ludwig V
It's another. It's not "quite" another. You seem to be saying that it's not only a different usage; but a super-different usage, if I'm understanding you. And it's not. It's just different.
In fact let me tell you what a probability is. It's just a real number between 0 and 1, inclusive. So it's the same real number one in the context of probability or anything else.
Quoting Ludwig V
Yes ok. You don't have to suppose. Probabilities are additive. That is, if the events are independent (meaning that one is not dependent on the other) then you can add the probabilities. It's one of the axioms of probability. Or one of the consequences of the axioms, depending on how you state the axioms.
Quoting Ludwig V
I see the point you are making but it doesn't seem right. If we roll a die the probability that it's either 1, 2, 3, 4, 5, or 6 is 1. There is no other possibility. In fact that's another one of the probability axioms: That the total probability of the entire event space is 1.
Quoting Ludwig V
1 is a probability. 0 to 1, inclusive, is the range of probabilities.
Quoting Ludwig V
Well, "evidence" is a term in the philosophy of probability, I suppose. But it's not a word in the formal mathematical theory of probability. In any event, I don't think that's right. Evidence can change the credence of an event -- your subjective degree of belief. But it doesn't change the probability.
I'm in way over my head on the philosophy of probability actually.
Quoting Ludwig V
Well I certainly agree with you. I am not the one trying to apply probability theory to true/false propositions. @fdrake is doing that. I'm a bit baffled by the attempted connection.
Quoting Ludwig V
Are you perhaps referring to credance, or the degree of belief? I can't really debate these issues, I know nothing about them. Truth in mathematics is binary. In real life, not so much. Also in intuitionist logic, where we reject the law of the excluded middle. That's another complication.
Quoting Ludwig V
Uh-oh. Was all the preceding not for me? Probably wasn't since it's not about anything I can sensibly talk about. To me a proposition is true or false. That's the definition of a proposition.
Quoting Ludwig V
Philosophical probability, I suppose. Mathematical probability has no time element in it. A probability measure is a function from some event space to the set of real numbers between 0 and 1, inclusive, satisfying some additional rules. That's it. No time involved.
Quoting Ludwig V
The mathematics of probability is abstracted from all that. No time element.
This article gives the mathematical definition of probability.
https://en.wikipedia.org/wiki/Probability_axioms
You don't need to follow the symbology. The point is that time is not mentioned. Probability is a mathematical function that outputs a real number in the range [0, 1] and satisfies some rules.
Now particular applications of probability often involve real life, temporal events, such as tomorrow's weather or the next card dealt from a deck. The underlying theory is abstracted from that.
Quoting Ludwig V
Probability theory is abstract. Applied probability is applied.
Quoting Ludwig V
I don't know why you have that hangup about probability 1. Probability 1 is just the probability of the entire event space. It's the claim that out of all the possible outcomes, one of them will occur. After all, in any situation, something must happen, even if we don't know what. The probability that something, anything at all will happen, is 1. That's one of the rules of probability in the Wiki article.
Quoting Ludwig V
You have thought a lot more deeply about the real-world meaning of probability than I have. The math is just math, as in the article I linked. It's very mathy as you can see.
Quoting Ludwig V
It's the philosophical contexts that I don't know much about.
Yeah no I ain't assigning random variables to generic mathematical expressions.
Yes. Is that a definition or an axiom? Whatever it is, it isn't just another assignment of a probability because it enables the actual assignments to the outcomes to be made. But I don't see that anything is wrong with representing them as percentages, in which case the probability of the entire event space is 100. Meteorologists seem to be very fond of this.
Quoting fishfry
Timeless present? It looks like it. In which case it is what I'm looking for.
Quoting fishfry
Yes. Most of the discussions I get involved in are at the applied level. But I have seen some posts that are completely abstract. So I think I understand what "event space" means. It is a metaphor to describe a formulation that doesn't identify actual outcomes, but only gives, for example, E(1), E(2)... - variables whose domain is events. In particular applications, that domain is limited by, for example, the rules of the game. That's not a complaint - just an observation.
Quoting fishfry
Yes. But the mathematical table you draw up doesn't change when it does happen. Assigning a probability to the outcome that happened isn't a change to the table, but just a misleading (to me, anyway) way of saying "this is the outcome that happened (and these are the outcomes that didn't happen)". The table doesn't apply any more.
Quoting fishfry
Yes. It's a rule, not an assignment of a probability.
Quoting fishfry
Yes. To be honest, the value, throughout our dialogue, is the opportunity for me to see how mathematics reacts to these questions. So the difference is the point. I'm very grateful to you for the opportunity.
To be honest, the use of "probability=1" is so widespread that it seems absurd to speak as if it should be banned. So far as I can see, it doesn't create any problems in mathematics. But in the rough-and-tumble of philosophy, it's a different matter. People asking what the probability is of God existing,
Quoting fishfry
Neither do I. But given that intermediate probabilities don't apply, I would say that probability in this case doesn't apply. Probability theory has no traction. Perhaps that's too strong. So I'll settle for a philosopher's solution. Philosophers have (at least) two ways of describing statements like this - "trivial" or "empty".
But now consider "There is one star in the solar system". Given that there is just one star in the solar system, intermediate probabilities don't apply. So assigning a probability of 1 is trivial or empty.
But, once I have won the lottery, intermediate probabilities don't apply.
Quoting fishfry
Yes, and I once I realized that, I withdrew. Perhaps I wasn't clear enough.
Ok. But you know in this case we can. We can interpret probability as credence, the subjective degree of belief, which is an epistemological claim rather than an ontological one. Pretty much any mathematician in that field would be glad to offer a number. Most believe it's true. I'd guess Riemann has better than a 90% credence among specialists.
With this interpretation, we free ourselves from having to give an account of what probability "is." We just talk about our own subjective degrees of belief. Sort of removes the mysticism from interpretations of probability.
This way we can reason mathematically about our beliefs, using the technical apparatus of abstract probability theory.
I'm just trying to interpret this question. About applying probabilities to predicates, I don't know anything about that. But I do think that people could "vote" on predicates, even in situations where you can never know the truth.
This is one of those times a def is an ax and vice versa. You can say probability is the study of measurable spaces with total measure 1; or you can say that this property is one of the axioms of a probability space. It's the same thing, really.
The point, or my point anyway is that the mathematical theory of probability is entirely abstracted from any meaning or interpretation or philosophy of "probability" that anyone has ever had.
In math, a random variable is just a measurable function on a probability space; which defined as a measure space with total measure 1. It's all very technical and precise, and completely avoids all of the murky metaphysics of randomness. In that sense, my view of probability is not overloaded with philosophical interpretations. Whether that's good or bad I'm not sure. :-)
Quoting Ludwig V
Oh ok.
I'm still antsy about assigning a random variable to the truth of a theorem. How do you sample from mathematical theorems? What would it even mean for a mathematical theorem to be expected to be true 9 times out of 10? How do you put a sigma algebra on mathematics itself...
Credence, or subjective degree of belief. You ask 10,000 specialists in analytic number theory whether they think the Riemann hypothesis is true. You take the percentage of yesses out of the total to be the credence of the group.
OR you ask each mathematician what is their subjective belief that it's true; and you average all those individual credences.
If I'm understanding your objection, the idea is to replace the idea of probability, with that of credence.
With probability, we have no idea what it "means" to say that a theorem might be 75% true. But with credence, we do. Even though the theorem itself must be either true or false; still we can each have a fractional "subjective degree of belief" that it's 75% likely to be true.
In this way we can apply the mathematical techniques of probability theory, sigma algebras and such, but without having to figure out what we even mean by probability. We go from the objective to the subjective. From ontology to epistemology. X may be true or it may be false. and no other outcomes are possible. Yet, I can still have a subjective belief, based on what I know, that it's 75% likely. It's just a guess, but it's objective. We ask everyone what they think.
Better clarify that. Everyone's personal opinion is subjective, that's the beauty of the concept of credence. But the FACT that 75% of them think X and 25% think not-X, that's objective. So we can use the rules of probability without having to do metaphysics.
https://en.wikipedia.org/wiki/Credence_(statistics)
Yes, I get that. In the sense that we've discussed, it is a speech act either way. However, axioms and definitions are not the same kinds of speech act. I expect there's a mathematical explanation of the difference. But they are both setting up the system (function?) - preparatory. So they are both different from the statements you make when you start exploring the system, whether proving theorems or applying it.
Quoting fishfry
This is a different speech act, even though it may be the same sentence. The context is different.
Quoting fishfry
So what does it mean to update the table? Are you correcting it, or changing it, or what? It seems like something that happens in time. You might be constructing a new table, I suppose.
Quoting fishfry
In a way, yes.
Quoting fishfry
We are very close here. However, I take the fact that intermediate probabilities don't apply to mean that in a context where intermediate probabilities don't apply, "probability = 1" is empty.
Quoting fishfry.
It depends whether you are a mathematician or a philosopher.
Quoting fishfry
Hardly irrelevant. I think I understand your point about abstract systems and I am interested in interpreting or applying the abstract formal system; but that begins with the system.
However, I can't help remembering that Pascal was interested in helping his gambling friends, so the application drove the construction of the theory. In the same way, counting and measuring drove the construction of the numbers - not that I would reduce either probability theory or numbers to their origins.
But I do think that interpretations and applications are not an optional add-ons to an abstract system.
Quoting fishfry
Yes, I get that. There are even some beautiful arguments in philosophy. I'm sometimes tempted to think that the beauty is the meaning. I would, sometimes, even go so far as to agree with Keats' "‘Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know." But only if all the philosophers are safely corralled elsewhere.
I'm not a normal philosopher, with a fixed (dogmatic, finalized) doctrine. I'm exploring, with a view, if I'm successful (and I rarely am), I'll be able to understand how these concepts are related and maybe even construct some sort of map or model of them. (I'm heavily influenced by Wittgenstein, I'm afraid, though I'm incapable of imitating him. But that is why I don't do metaphysics.)
Quoting fishfry
I've lost the context of this. I do hate the way that some people talk of chance and probability as if they were causes. Most philosophers (after their first year or two) will jump on that very firmly and, yes, the conventional doctrines about causation have little to recommend them. As for real world applications, they are derived from the mathematics, but heavily adapted. For one thing, they don't atually assign probabilities, but estimate them, and buffer them with likelihoods and confidence intervals. Almost a different concept, linked to the mathematics by the "frequentist" approach.
Probability is the main way that we try to limit uncertainty, find some order in the chaos.
Quoting fishfry
You're welcome. I agree that there is something universal here. It is the faith that there is order to be found in the chaos we confront in our lives. Some people think that is a truth about the world, but I'm not at all sure it is that. The evidence points both ways. However, chaos is worse than anything. We will do anything, think anything, to achieve some way of organizing the world. Probability is not ideal, but it is better than nothing.
Quoting fishfry
If you think about why you select specialists to ask, you will see that your are not escaping from the serious difficulties about achieving knowledge, in particular, the fact that conclusive proof of anything is very hard to achieve (not impossible, I would say, but still difficult). We have to weigh one argument against another, one piece of evidence against another, and there seem to be few guidelines about how to do that. Eliciting the consensus of those who are competent is one way of doing that - although far from certain. Asking 10,000 random people in the street what credence they have in the Riemann hypothesis won't help much, will it?
Quoting fishfry
Oh, I agree that there is a fact there. The question is what it's value is and that takes us back to the evidence.
So - the great virtue of Bayesian probability is that it will give you a probability for a single case, which neither mathematical nor empirical probability can do. I still have a problem, because we normally express a probability in terms of the number of times it can be expected to show up in a sequence of trials. But that limitation, strictly speaking, means that its application to a single case, which we very often want to know, is extremely murky. Expressing it in terms of making bets helps.
Quoting fishfry
But each of those people, if they are rational, will be assigning their credence on the basis of the evidence. But in this case, and many others, the issue is what counts as evidence and how much weight should be placed upon it.
We started off talking about "probability - 1" and in order to understand that, we've explored the construction and meaning of the probability table. I think that was all constructive, but we've got as far as we can with it. Now we are talking about Bayesian probability and what credence is.
I know that I can be a bit relentless. If I'm boring or annoying you, please tell me and I'll shut up.
Sometimes they are pretty much interchangeable and other times not. It depends on if it's an "if and only if" definition or not.
The axioms of group theory are the definition of group theory.
Quoting Ludwig V
You keep trying to frame this discussion in terms of speech acts. I'm not sure what point you are making.
Quoting Ludwig V
What table? Lost me on that.
Quoting Ludwig V
I don't know what you mean that a probability can be empty. A probability is a real number between 0 and 1 inclusive.
Quoting Ludwig V
Ok. So please remind me of what point we are trying to discuss.
Quoting Ludwig V
Applications are always at the historical origin of every abstract theory. Not specific to probability.
Quoting Ludwig V
Yes of course, no issues there.
Quoting Ludwig V
They are not optional add ons. So they are mandatory add ons? Or not add ons at all? Didn't understand that.
Quoting Ludwig V
I'm not saying there's no meaning in math. I'm saying that the math itself doesn't refer to its meaning when we're doing the formalizations. The meaning is not to be found in the math, but rather in the minds of those who do or use the math. Is that better?
Quoting Ludwig V
I know that whereof I cannot speak, thereof I must put a sock in it. That's as far as my knowledge of Wittgy goes. Also, that he thoroughly misunderstood Cantor's diagonal argument. I seem to recall that.
Quoting Ludwig V
Me too, for sure.
Quoting Ludwig V
Right. Well that's the beauty (or the flaw I suppose) of mathematical abstraction. Mathematicians just think a probability distribution is a particular kind of function on a probability space. There is no meaning or metaphysics.
Quoting Ludwig V
Why are you telling me this? I don't know what we are talking about.
Quoting Ludwig V
I'm a new mysterian. I don't think we're going to know. We can't know any more than an ant on a leaf in on a tree in a forest can know about the world as we understand it. But the ant knows warm from cool, what to eat and what eats it. It has a metaphysics!
https://en.wikipedia.org/wiki/New_mysterianism
But I'm not sure why you mentioned this. The point was that the concept of credence lets us apply the mechanics of probability theory, without regard for the metaphysics. Because even though I don't know what's going on, I can have an opinion about it. And we can tally people's opinions to quantify their frequency.
Quoting Ludwig V
No, not at all. Instead we ask a hundred million people in the street to vote on how we should run our society! I believe it was Socrates who distrusted democracy. "In Plato's Republic, Socrates depicts democracy as nearly the worst form of rule: though superior to tyranny, it is inferior to every other political arrangement." So says Wiki. We can certainly see his point.
Quoting Ludwig V
Ah. No. Not the point I'm making. I'm saying we can substitute credence for probability, so that we can apply the techniques of probability without being burdened by metaphysics. I didn't say it was more true, only more workable. A pragmatic shift in view.
Quoting Ludwig V
Yes ok. If a baseball hitter has a batting average of .250, we would say he has a 1/4 chance of getting a hit on his next at bat. But of course this is absurd, the specifics of his next at bat are subject to all kinds of variables, how he's feeling, how the pitcher's feeling, the humidity and temperature of the air, etc.
But I don't follow your point in bringing this up. And betters use credences! The odds are based on the credences of the betters, and NOT on any metaphysics of what is really going to happen. That's a good point. Gambling odds are based on collective credence, along with an attempt to judge "objective" reality. It's a bit of both.
Quoting Ludwig V
Yes. That's why we aggregate everyone's subjective opinion and evaluation. These are situations whee nobody can know all the evidence. Like a murder mystery with only circumstantial evidence. We can't know for sure but we can use our best judgment and have a credence.
Quoting Ludwig V
I don't know what you mean by probability table.
Quoting Ludwig V
I've said nothing about Bayesian probability. I like credence because we can always have one, even when we can't know enough to assign a metaphysical probability.
Quoting Ludwig V
Well I'm concurrently dabbling in the political threads in the Lounge, so this all seems like light recreation by comparison.
But your idea about the nonexistence or vacuity of probability 1, that I don't follow.
Quoting fdrake
Quoting fishfry
We are very close here. However, I take the fact that intermediate probabilities don't apply to mean that in a context where intermediate probabilities don't apply, "probability = 1" is empty.
— Ludwig V
I was building on his point and your reply. We have somewhat different opinions. I'm not sure that anything important hangs on it, so perhaps we should leave it at that.
MATHEMATICS
Quoting fishfry
Quoting fishfry
I'm interested in the relationship between the purely mathematical abstractions in the context of what I'll call the everyday world. I'm not trying to undermine the concept of mathematics in any way.
Quoting fishfry
Yes. Not perfect, but better. I understand meaning to be the use of a symbol, in the context of related symbols. So I would say that pure mathematics does have a meaning, defined by the interacting concepts in play. When the interpretations and applications come into play, we have a new context. Since the context of the use of the concept has changed, the meaning of the original concepts may or may not have changed, but may well be seen differently. Does that help?
Quoting fishfry
That's a very good question. What I said was not quite right. I refer you to what I said about meaning and use above.
Quoting fishfry
Well, I've explained what I mean by meaning. I hope that meets the case. But I'm not at all clear what you mean by metaphysics. I would hope that nothing that I say is metaphysical, but the word is so badly defined that I might have erred unwittingly.
POSTERIOR PROBABILITY
Quoting fishfry
I read the Wikipedia article. The context seems to be Bayesian probability, which is a different kettle of fish. It's not, if I understand you right, about the basic mathematical function, but about the inputs to the function, so we're talking about an application, right?
Quoting fishfry
[quote=Wikipedia]The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating.[/quote]
OK. It's a small point, but wouldn't be clearer to say and more consistent with the timelessness of mathematical functions, to say that when new information becomes available, a new probability is established, which is substituted for the old one? I think that's compatible with what Wikipedia says.
Bayesian probability is a scenario, or posits a scenario. There's nothing wrong with that. Traditional probability does the same thing with its reliance on gambling scenarios. You're right that it is not a question of truth or falsity, but of enabling us to apply an existing concept in a new way - and one that is particularly interesting in view of the fact that we do ask about the probability of single cases.
I don't see the metaphysics in standard versions of probability. Can you explain?
BAYES
Quoting fishfry
This way of articulating chance or probability depends on a "frequentist" concept of probability. One can then understand what the probability means as a phenomenon over a number of cases. But that makes it difficult to see how it applies to a single case. I guess a way of making it concrete is to see it is a question of the odds on a bet. That'll work for insurance and precautions in general, and in planning to take account of possible eventualities. But that only has application in the context of balancing risk and reward - decision theory. Maybe that's all there is.
MISCELLANEOUS
Quoting fishfry
Yes. Public/political life - the "state of the world" - has all the ghastly fascination of watching a shipwreck. I expect you know that there's been a change of government in the UK. Suddenly I found myself unreasonably optimistic. Well, until I heard about the events in Pennsylvania.
Quoting fishfry
Yes. If you expect the democratic vote to determine policy, you are going to come unstuck. Whether it was Socrates or Plato who rejected democracy is underdetermined and likely always will be. Small correction. The view in the Republic is that democracy will always turn into tyranny, because demagogues will take over and establish themselves. Say no more. The thing is, Plato blocks a proper discussion of the issues by positing someone who gets the answers right. But sometimes there is neither right nor wrong and sometimes actual people get things wrong. So his appeal to the philosopher-kings avoids the real issues. Popper says that the vital thing about democracy is that you can get rid of the ruler when they screw up.
Well, perhaps one can quote the old saying that those who do not understand history are doomed to repeat it.
Quoting fishfry
Yes, that bit of the Tractatus is much misunderstood. There are suspicions that he was flat wrong, but that would be heresy. He is, perhaps, a rather specialist taste. Yes, his interpretation of Cantor and Godel is vigorously contested. I have the impression, however, that almost everything about those two is contested. I'm not taking sides yet.
Quoting fishfry
H'm. Metaphysics again. Ants know what they need to know. There's a concept of the "lived world" that's quite useful in cases like this. Sure, whether you call it a metaphysics or a lived world, we have one too.
But there's a difference. We contemplate Euclid's geometry and start wondering whether the parallel postulate is really necessary. Next thing you know, whole new worlds have opened up. Or Mercator realizes that conventional maps are all wrong and works out how to project a spherical surface into two dimensions. So something new happens. We can do this in a generation or two, whereas evolution can take a very long time indeed.
We'll never know everything because we'll always find new things to know.
There are too many people around who think that science has the answer to everything or can discover the answer to anything. That view is overblown and we do need a more tempered attitude to it.
I haven't explained what I mean by a probability table. I meant something like this. (Forgive my primitive graphics)
Probability
{E(1) v E(2)} 1
Possible outcome E(1) 0.5
Possible outcome E(2) 0.5
not{E(1) v E(2)} 0
When the outcome is known, all that is required is a foot-note - "The outcome was
Anyway, spiral:
Check.
You keep saying that. You have not yet articulated it in a way that makes me believe you are saying anything sensible.
Quoting Ludwig V
Well I'm cycled out on this I think. At the end of most of the convos I'm in. I could let this go soon.
Quoting Ludwig V
That's a tall order. You mean differential geometry, the super-abstract geometry of Riemann, applied to general relativity? Or the math of quantum field theory?
Or do you mean something far more prosaic?
The math of the everyday world is to be found in the grocery check-out lane and the baseball scores.
Quoting Ludwig V
No, I think you obfuscated the point.
I said there is no meaning in math. That when we manipulate symbols according to rules, there is no meaning that's part of the formal game.
But of course "in the back of our minds," we do know what it all means. We have some every day experience in mind, even though that has no bearing on the symbology we write down.
Quoting Ludwig V
Ok.
Quoting Ludwig V
You're the pro, so when I say metaphysics it just means, "What's really true about ultimate reality." Or something like that.
Quoting Ludwig V
Well there's abstract and applied probability. I knew a grad student who got a Ph.D. in abstract probability and got a job as an actuary at an insurance company. The insurance companies know more about probability than anyone, it's their business. And bookies. Sports book operators know the theory and the practice.
Quoting Ludwig V
I'm talking about credence, not Bayesian probability.
The metaphysics is that when we say, "The probability of rain is 25%," we're making a statement about the REAL WORLD. When I say that "My credence it will rain is 25%," I am making a factual, verifiable statement about my subjective state of mind. I don't need to know anything about the real world, though I do base my credence on the available evidence. Clouds in the sky, for example. But in credence, I'm not making a claim about the world. I'm making a claim about my own subjective degree of belief.
Quoting Ludwig V
Ok. Not disagreeing.
Quoting Ludwig V
I read Spiked Online (https://www.spiked-online.com/) as my main source of British politics. They're slightly right of center. I gather Starmer is a typical collectivist leftist, but that the so-called "conservatives" mucked up their own charter so badly they deserved to go. Maybe he's a better guy than I've heard.
As more news continues to come out, the Pennsylvania deal looks like an op. An operation. It is not as we are being told, and we will likely never be told. If it was some weirdo 20 year old kid with access to his father's gun, so be it. But the numerous incomprehensible malfeasances of the Secret Service raise many questions; and the Biden administration is actually stonewalling and slow-walking the case, raising even more suspicions. I'm not saying one thing or another, just that transparency and accountability are in short supply from the government this week.
Quoting Ludwig V
Is Charlie someone's idea of a philosopher king? Poor guy, his entire role in life from the time he's a child is wait for his mum to die, then she turns out to have great genes and lives till 96. And a year later the poor guy gets a serious cancer. Feel bad for him. I always like Liz, she was a very great lady.
Winston Churchill said that the greatest argument against democracy was a five minute conversation with the average voter. I believe that!
Quoting Ludwig V
My sense is that he just didn't get it. That he was wrong, not just having a side. But I could be wrong too.
Quoting Ludwig V
Yes agree. Science versus scientism. Science is using experiment and rationality to understand the world. Scientism is the belief that science is infallible, or that "trust the science" was ever anything other than an authoritarian political slogan. Covid lockdowns were scientism, not science. Science as a means of social control, not as a path to enlightenment.
Quoting Ludwig V
Ok, list of events and their associated probabilities.
I don't think I have any more to say about probability = 1. So let's agree to disagree. I think I understand at least where and why we disagree. I'm sorry I can't make myself clear to you.
Quoting fishfry
If I were qualified to tackle those areas, I would take them on. But I know better than to talk about them without a reasonably thorough understanding of them - which I don't have. I have to settle for the prosaic. Which matters too, I think.
Quoting fishfry
I can see why you think that. But I'm fascinated by the fact that we can posit some relatively simple rules and draw such startling and unexpected conclusions from them. How is that possible? For you, that's your home, but for me it is foreign - and confusing - territory.
Quoting fishfry
To me, that's paradoxical. But, from another perspective, very helpful.
Quoting fishfry
That's good enough for this discussion.
Quoting fishfry
OK. It's just that a link to the real world (whatever that is) is what makes the difference between something interesting and useful and a fantasy.
Quoting fishfry
For me, the formal representations in decision theory do have the prospect of articulating our decisions more precisely and enabling us to make more coherent and better balanced decisions.
Quoting fishfry
"Slightly right of centre" is about right. "typical collectivist leftist" sounds like slapping a conventional label on something without thinking about it very much. So it's very likely that he is better than you've heard. Most of the British media is right wing, so most of what was written was, essentially, political. (Perhaps the most significant thing about our election is that the normally right wing press abandoned the Conservative party. That's not happened since Blair got elected in 1997.) You have to realize that our right wing political people have no hesitation about government action when it suits them; but they often disguise it so they don't have to take responsibility for the outcome. Starmer's programme is very moderate and addresses areas where almost everybody agrees that existing, supposedly free market, structures have completely failed to deliver.
Quoting fishfry
I'm not surprised. It's clear that there was a major screw-up on the security front. So the Government was bound to take some flak. So it went in to self-protection mode. All Governments do that. It doesn't usually work very well. It seems likely to reinforce Trump's lead in the election stakes. Biden must surely wish it had not happened.
Quoting fishfry
I also feel sorry for Charlie. He's never been comfortable in his role. No, he's nobody's idea of a philosopher-king. He's there to be the unity that ties us all together, despite our disagreements and whatever happens in politics. Simply by existing. A philosopher-king would be completely unsuited to the role. It needs someone who doesn't think. He does, though not very well. That's one big reason why he's not suited to the role. But he will do his best, and I'm sure it will serve. In the US, that role was served by the Constitution. That seems to have become a political and legal football too, which really does not help.
Quoting fishfry
He did, and he's right. But the full quote is:- [quote=House of Commons, 11 November 1947]Many forms of Government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of Government except for all those other forms that have been tried from time to time.[/quote]
He also said: -
[quote=House of Commons, 8 December 1944]My idea of it (sc. democracy) is that the plain, humble, common man, just the ordinary man who keeps a wife and family, who goes off to fight for his country when it is in trouble, goes to the poll at the appropriate time, and puts his cross on the ballot paper showing the candidate he wishes to be elected to Parliament—that he is the foundation of democracy.
And it is also essential to this foundation that this man or woman should do this without fear, and without any form of intimidation or victimization. He marks his ballot paper in strict secrecy, and then elected representatives together decide what government, or even in times of stress, what form of government they wish to have in their country. If that is democracy, I salute it. I espouse it. I would work for it.[/quote]
Great man. But his record before WW2 was, let's say, mixed.
Quoting fishfry
Covid wasn't dangerous enough. When people realized that it wasn't the plague or Ebola or HIV, they felt, not unreasonably that the risks and benefits were not sufficient. They were misapplied as a result of a political miscalculation. IMO.
The problem got serious in the two world wars 100 years ago. It was very successful in developing new weapons - arguably, it was a major factor in winning them. And, then, of course, "science" got taken up by institutions that were not capable of grasping what it was all about and misused in the service of other interests.
Quoting fishfry
I wish I had thought of that. But I do think the layout is significant. But I think that's over.
We can agree to disagree, but I don't understand why you think probability 1 is "empty."
Quoting Ludwig V
Well, the prosaic applications of math to everyday life are not really what mathematicians do.
Quoting Ludwig V
Law of unintended consequences is a rule of general life too, right?
Quoting Ludwig V
I don't see why. The importance to some people of the world chess championship is not inherent in the rules of chess. Symbolic systems have no meaning in them. It's the people who supply meaning.
Quoting Ludwig V
Credence is not fantasy.
Quoting Ludwig V
That was my three-word summary of everything I know about Starmer. I agree I haven't thought about him much.
Quoting Ludwig V
Well if he's not free-market he's a collectivist! Generally speaking.
I did hear that he wants "closer cooperation with Brussels," meaning that he'll be yet another British PM stabbing Brexit in the back. I think it might have had a chance to produce good results if the politicians had respected the will of the people.
Quoting Ludwig V
The security incompetence is of a degree that invites suspicions of complicity. Just as in the JFK assassination, where the Secret Service was likewise grossly incompetent. Biden has other problems this week. Rumor has it he's dropping out of the race this weekend. But that might just be spin from his enemies (in his own party) leaking to the press to weaken him.
Quoting Ludwig V
LOL. My impression too.
Quoting Ludwig V
Don't think I've heard that before, that the Constitution is the US analog of a hereditary monarch. Maybe there are some parallels.
Quoting Ludwig V
I believe he said that "History shall be kind to me, for I shall write it."
I believe he said that in the context of WWII. Most of Chamberlain's bad reputation is due to Churchill. Chamberlain was actually a pretty good guy, and his appeasement of Hitler was both rational and very popular at the time. So I've read from some alternative views of history.
Interesting that after the war, the British people showed Churchill their appreciation by voting him and his party out of office at the first opportunity.
Quoting Ludwig V
Miscalculation or malevolence, take your pick.
Quoting Ludwig V
Yes.
Quoting Ludwig V
Ok.
This is a bit embarrassing. I was using a bit of philosophical jargon, which seems to be out of date. You must have been wondering how empty sets were relevant. The expression derived from the logical positivists who classified tautologies as empty or trivial because, although they are not false, they do not assert anything. For them, proper, non-empty, statements were those that could be verified or falsified. The idea is used in Peter Unger's book Empty Ideas See Review of Unger "Empty Ideas (I don't recommend the book. For all that the review talks about philosophy being fun, which I approve of, this book is hard going for rather small rewards.) I don't agree with this application of the argument, but the idea can be useful.
Some examples may help.
1) An obvious case is "This sentence is true".
2) If I assert that snow is white, it is empty for me to assert in addition that I believe that snow is white.
3) Tarski's redundancy theory of truth (which, in case you don't know, is popular among philosophers) says that "snow is white" is true iff snow is white.
4) The probability of p = 1 iff p is true iff p
Quoting fishfry
Yes. But I thought that unintended consequences were events in the empirical world.
Quoting fishfry
We're using "meaning" in slightly different ways. The paradigm case of a symbolic system for me is language, and that has meaning - if it didn't, it wouldn't be a symbolic system. A symbol is created by setting up rules for the use of an arbitrary character or object. So the rules of chess set up rules for the use of the various elements of the game. I'm inclined to say that establishes the meaning of the symbolic characters within the game, and I would agree that that meaning is "in the minds of" the players and spectators.
I also agree with you that the significance of the game (e.g. its interpretation as a war game, suggested by the names of some of the pieces, or the value attached to titles like "grandmaster") is not established by the rules of the game. So there are layers of meaning (or significance), depending on context.
Quoting fishfry
Yes, I'm agreeing with you. But I want to distinguish between the two by saying that credence should be based on evidence or at least plausibility and that fantasy has neither of those. That's all. How else would one separate them?
Quoting fishfry
Yes, I remember the JFK story. I was once, briefly, an auditor (annual accounts for companies and other institutions). They drummed into me that when something was wrong, cock-up was more likely than conspiracy. But that doesn't prevent suspicions.
Quoting fishfry
The days of dogmatic nationalization of the means of production are long gone. Nowadays, at least in the UK, it's a pragmatic issue and we have a number of half-way houses and regulators for specific areas.
But isn't the free market a collective social institution? One of the basic functions of the state is to supervise and enforce contracts, and the companies and other collectives that operate in the market are themselves collective institutions - and they aren't accountable to voters.
Quoting fishfry
There was a lot of back-stabbing in the aftermath of the referendum. It was not pretty. But I don't think any of the Prime Ministers intended that. Brexiteers told everyone that the EU could be adjusted to suit what they wanted. The EU were reluctant to do so - and why should they? It's not as if public opinion in
the EU thought Brexit was a good idea. Brexiteers labelled any compromise as "stabbing Brexit in the back"; it seems they didn't grasp what negotiation is all about. The only people who were stabbed in the back were the Northern Irish who were thrown under a bus by Boris Johnson.
I voted remain, but had serious doubts about the ultimate EU project ("ever closer union"). Europhiles didn't pay enough attention to the longer-term history of the UK (since, say, 1700).
Quoting fishfry
Forgive me, but I can't think of anyone, malevolent or not, who actually benefited from the lockdowns apart from the vulnerable groups - older people, people with health issues. I plump for miscalculation, in spite of the UN warnings, so by British politicians.
Quoting fishfry
Yes. The conservatives thought they could go back to the way things were before the war. The voters wanted a fresh start. They got it - even the conservatives had to accept the new ways. It took them 50 years to unpick it and they're still not done.
Quoting fishfry
Well, people were kind to him for quite a long time. But that's changing now.
That makes perfect sense. I understand you now! Yes I was thinking of the empty set. Perhaps what you're describing I might call, "invalid," or an error condition in programming. In programming, an operation can succeed; or it can fail; or it can blow up entirely and throw an error condition.
A tautology that doesn't assert anything is kind of a dead end in the reality tree, if I may wax poetic. If we're processing it, it's an error. It doesn't have any meaning.
Is that about right?
But if that -- then I still don't get it! Probability 1 says that something is certain to happen. If I add 1 plus 1, I am certain to get 2. If we have a slow computer, we put in 1 plus 1 today, and we are certain, with probability 1, that the computer will output 2 tomorrow. What's wrong with that?
Note that I introduced time into math, by imagining we're doing math on a slow computer!
Quoting Ludwig V
Since you don't recommend the book I will not dispatch a clone to read it!
But surely, putting 1 + 1 into a computer and expecting to get back 2, is not an empty idea! I don't see that.
Quoting Ludwig V
I took a MOOC in mathematical philosophy, and the prof showed us that "Snow is white" is true if snow is white. That was several years ago, and to this day I don't really get it.
But anyway, mathematicians are trained to get used to empty objects. There's the empty set, and the empty topological space, and so forth. You get used to accepting vacuous arguments. So I don't see empty ideas as a problem. An empty idea is still and idea. The empty set is a set.
Quoting Ludwig V
That was about something interesting but I forgot and didn't feel like tracing back :-)
Quoting Ludwig V
When the knight is captured it doesn't feel good or bad. The player may feel good or bad. I'm back to the Chinese room. Searle says the room doesn't know what any of the Chinese sentences mean. So if you agree meaning is in the mind, that's what I believe also.
Quoting Ludwig V
Yes, it would be the same game if you called the knight the frisbee.
Quoting Ludwig V
I say that it is NICE if my credence is based on some evidence. Maybe I put some work into forming my opinion.
But maybe I just didn't have the time to get a Ph.D. in quantum field theory in order to have any credence at all that there are quarks. I believe there are quarks, 100%. I believe in that particular science. But I would be hard pressed to lay out the mathematical theory. I don't know the evidence. I only know that if Sean Carroll tells me there are quarks, I believe him. Actually Veritasium has an awesome video on quarks, that's where I learned that mass comes from the binding energy that keeps the quarks from flying apart.
I believe what I just wrote. I have zero evidence for any of it. Binding energy is analogized by a rubber band, that's what I know about it.
I hope I'm making my point. We are all obliged to place high credences on many things that we can't possibly have the slightest idea about. The electric grid will be up tomorrow. How the hell do I know? Did I personally inspect every faulty transformer that's about to blow, and take down half the county with it?
Here is my thesis. For every proposition P, I have a credence credence(P), whether I know the first thing about the topic or not. I think there's a 10 percent chance the Royals whacked Diana. I've seen enough hit man movies to know that when you die in a car crash, it might or might not have been an accident!
Point being that I have a credence, which I found by simply thinking about it for a moment, about a situation in which I can't possibly know the first thing, and actually I haven't looked into it much. So I know nothing. But I have an opinion!
Isn't having opinions about things that we know nothing about, one of the most human things we do?
Quoting Ludwig V
The US government was up to its eyeballs in chicanery that would have shocked the naive America of the 50s and 60s. Assassinating foreign leaders. Interfering in foreign elections. Running sick mind control experiments. Business partners with the Mafia in plots to kill Castro. Controlling what the news media reported. I agree that just because they covered up the assassination, doesn't mean they did it. Doesn't mean they didn't, but doesn't mean they did. But under the law they are accessories after the fact, and just as legally accountable as the actual perpetrators.
Likewise this week. The Biden admin, Mayorkas and that clown Cheatle, are embarrassed at their gross incompetence on display. By the way I am not one that says Cheatle is a clown because she's female. I say she's a clown because she's an idiot. Whether she actually believed that nonsense about the sloped roof preventing an agent from being up there, she was stupid enough to say so in front of a camera. That's a firing offense for any bureaucrat. Because it totally destroys the public's trust.
Quoting Ludwig V
It's a continuum, to be sure. Individual versus the collective.
Quoting Ludwig V
That last bit I didn't know anything about, the Northern Irish.
I went on a business trip to Cork once, it was so lovely.
Quoting Ludwig V
I don't like the idea of giving up national sovereignty to such an undemocratic institution as the EP. "Brussels" has become a pejorative and not just the name of a city.
Quoting Ludwig V
I would say at the least, that many of the authoritarian types in our society took advantage of the situation, in a manner not supported by the science. And anyone who pointed that out, was cancelled, had their career ruined, their jobs or professional licenses taken away.
I do not regard that as miscalculation. I regard that as evil, cynical calculation.
Quoting Ludwig V
Well the recent batch of conservatives have been useless. May was terrible. Was Johnson next? Then Sunak? They're what we call RINOs, Republicans in Name Only. Squishy liberals with no convictions calling themselves conservatives. Well here's to Starmer, he's got his work cut out.
Quoting Ludwig V
Is that right? Is there Churchill revisionism about?
:smile:
Quoting fishfry
Yes.
Quoting fishfry
Empty sets, etc., are defined in a context, which assigns a use to them (though perhaps not a meaning!). So that's different. The criticism is directed against ideas or uses that are not in a context that gives a use to them.
Quoting fishfry
Quite so. I'll overlook the intrusion of time. My point is different.
The use of "probability=1" is defined in the context of the table (function), that is, in context where a range of possible outcomes is given, one of which will turn out to be the outcome. Outside that context, it's use is not defined. Or rather, its use is defined as "= true". That is quite different from "probability (A v B vC..) = 1" meaning "the total of the probabilities of A v B v C... is 1", that is, its use in defining the range of the probabilities of the outcomes. So it serves no purpose, apart from confusing me.
Quoting fishfry
Meaning is a slippery word. One might want to object that the meaning of the word "table" is an object in the world. But we make the words and we use them.
Quoting fishfry
I would put it stronger, but it is true that credence is not necessarily based on conclusive evidence, and may be not be based on evidence at all.
Quoting fishfry
You're right. Most of what we know, we know at second hand. If we had to prove everything ourselves from scratch, we would be very limited. Standing on the shoulders of giants and even midgets is essential. Philosophers like to brush that aside and only pursue the gold standard. There shouldn't be any problem about assigning a credence to what we are told by others. I would count it as evidence. Why not?
Quoting fishfry
Quite so. We react instantaneously and without conscious thought to most of what's going on around us. We would never keep up if we had to sit down and reason everything out.
But, if I've got any sense, I will give more credence to credences assigned by someone who knows what they're talking about over credences assigned by someone who doesn't. That's reasonable, surely?
Quoting fishfry
Well, the opposition in the UK were certainly not silenced. Their voices were heard throughout. The problem is that without an estimate of what would have happened without lockdowns, we have no way of assessing their success. It's has always been regularly used with Ebola outbreaks, so it must have its uses. But those incidents have been relatively contained. I think the scope and duration of the COVID lockdowns was the problem.
Quoting fishfry
It's not that simple. Every time you sign a treaty, you give up some sovereignty. It's a question of balance - quid pro quo.
Quoting fishfry
It's long and peculiar story. There'll be lots of stuff on the internet if you want to look it up. The problem was that it needed free access to both UK and Republic markets. While both were in the EU, it wasn't a problem. But when the UK left, it was not possible for them to continue free trade with both and yet could not give up either. It was obviously insoluble from the beginning, but nobody bothered until the reality hit.
They seem to be reasonably satisfied with the most recent arrangements, but they are a bit of a lash-up.
Quoting fishfry
I'll bet. It's a very beautiful place. The whole island is - outside Belfast.
Quoting fishfry
Well, there's always been a counter-narrative. The left wing have never liked him. There was the Sidney Street siege, Gallipoli, the famine in Assam in 1943, and pet research projects that wasted a lot of money and it took a lot of persuading to get him to accept the invasion of France. No financial scandal that I know of, which makes a nice change. I think most people accept he made a critical difference in WW2.
We should go back to agreeing to disagree, since your understanding of probability 1 is contrary to, well, everyone else's. Is this a standard philosophical view? IMO you are choosing to confuse yourself about something very simple.
Quoting Ludwig V
There's no meaning in symbols. That was the thesis. Mine and Searle's, at any ratet.
Quoting Ludwig V
Then you agree with my point. Credence could be influenced by evidence but need not be.
Quoting Ludwig V
They might have little or no evidence themselves. Credence is just what people believe, evidence or not. If you redefine evidence as "what my friends believe," that way lies mob rule.
Quoting Ludwig V
Right. Hence credence. I think you are agreeing with me. Credence is a nice concept because we can apply the rules of probability to it, but we needn't know anything about the world to have subjective beliefs.
Quoting Ludwig V
Sure.
Quoting Ludwig V
Ok well I should terminate my own thread hijack about this subject. But a quick lookup showed that Ebola lockdowns were only in two regions. The covid lockdowns were virtually global and were not a good idea.
Quoting Ludwig V
Some Europeans are getting restless, are they not?
Quoting Ludwig V
Ok, I should look that up.
Quoting Ludwig V
Yes I'd love to go back.
Quoting Ludwig V
He all but fired the torpedo himself at the Lusitania to get the US into WWI. The Admiralty records are sealed to this day.
Fair enough. We're obviously not going to reach agreement. For what it's worth, my diagnosis is that we disagree about the boundaries of the relevant context. For you, the assignment of 1 to the probability of an outcome which has actually occurred, is sufficiently defined by the context of probability theory. For me, it isn't.
I'm perfectly happy with our agreement (?) that it can be described as empty. That bothers me, but not you.
Quoting fishfry
I can understand, roughly, why you (plural) believe that and there's a sense in which I agree. I just don't think it is the whole story.
Quoting fishfry
Yes, I do. But I also think that credence should be influenced by evidence.
Quoting fishfry
Careful.
Quoting fishfry
I hope you weren't just appealing to a vote. But if you mean that I should take more seriously the opinion of others who can be expected to know what they are talking about, then your question is valid. My view is not at all standard. That doesn't bother me. What does bother me is that the orthodox view is comprehensible and so not irrational. I'll have to reconsider.
Quoting fishfry
Yes, I can see that. If you'll forgive me, I think that mathematicians and especially logicians tend to be to keen to get to the formalization and too quick to move from setting up the formalization to exploring it. I get stuck on the question what the value is of beliefs that have no connection with the world. To believe something is to believe that it is true.
Quoting fishfry
I agree with that. I went looking for the UN policy statement about this, but couldn't find it. But I did find a string of warnings about the dangers. Whatever went on in the US, disagreement was not suppressed everywhere.
Quoting fishfry
Certainly. Their problems are different, but nonetheless based on their history. Like the Brexiteers, they want to have their cake and eat it. The difference is that their ambivalence is the question of Russia. The problem exists, but less acutely, for the whole of the mainland. Geography is inescapable, even in these times.
Quoting fishfry
Quite so. I forgot about the Lusitania.
Well ... letting the matter drop would be for the best. But ... can you explain to me how today being Sunday, the probability that yesterday was Saturday is anything other than 1?
I mean, suppose that you had to place a bet on the proposition. What do you think the odds should be?
That is, if today is Sunday, what is your credence that yesterday was Saturday?
Quoting Ludwig V
Well, nothing is the WHOLE story. Life is complicated.
Quoting Ludwig V
You have already stipulated to the opposite. You have, if I have understood you correctly, agreed that it's NICE if credence is influence by evidence; but sometimes there's not enough evidence or I'm not qualified to evaluate the evidence. Then I must NECESSARILY form a subjective degree of belief without benefit of evidence. It's your use of "should" that I object to. I could live with "preferable," but not "should."
Quoting Ludwig V
If you were a betting man, what would you bet that today being Sunday, that yesterday was Saturday? What is your credence for that proposition? How should a bookmaker set the odds?
Quoting Ludwig V
Don't you like fiction? Do you have the same complaints about the novel Moby Dick ("He tasks me. He heaps me.") and the game of chess; one a work of fiction, and the other a meaningless formal game with entirely made-up rules?
Quoting Ludwig V
In the US it was ugly. People lost their medical licenses for expressing scientific skepticism. Teachers, workers of all kinds lost their jobs. People lost friends. The mass formation, as some called it, was terrifying. For the first time I truly understood Nazi Germany. Excuse the argumentum ad Hitlerum. I saw how a society goes mad. I was immune by personality. I kept my head down, and since I'm not much involved in public society, I didn't have to risk anything. I just waited it out. But I'm wary of my fellow Americans now in a way I previously wasn't.
Quoting Ludwig V
Immigration is an issue on both sides of the pond. Liberty versus top-down control. The wokesters versus the people who never voted for the woke policies. The double standard of justice, a kid jailed fo making bicycle marks on a Pride crosswalk, while Antifa defaces statues. Don't get me started. LOL.
Quoting Ludwig V
I saw a really good tv movie that cut between the action on board ship, on the German U-boat, and in the halls of the British admiralty. I read up a little afterward. A distinguished jurist was appointed to lead the inquiry. Afterward he called it, "a dirty business," and retired. They screwed the captain to cover up Admiralty complicity.
I'm not saying that if to-day is Sunday, the probability that yesterday was Saturday is anything other than 1. Of course not. I'm saying that because today is Sunday, probability doesn't apply to the proposition that yesterday was Saturday.
(Yes, I'm writing this on Monday, but it is simpler to pretend that it is Sunday for the sake of simplifying the discussion.)
Quoting fishfry
That bet is money in the bank, so long as the odds show a profit. But who would take the other end of it? I suppose you might find a taker who would give you your money back. But that would be an empty ritual. There's a good reason why bookies close their books when the race is over.
More politely, where there is no risk, there is no bet.
Quoting fishfry
Ah, this is a different can of worms. "Credence" is degree of belief, isn't it? And belief concedes the possibility of falsehood. So that makes sense.
But when there is no possibility of falsehood, we do not speak of belief; we speak of knowledge. So if you assign a credence to "Yesterday was Saturday", you are allowing the possibility that it wasn't. But if you assign a credence of 1, you are excluding that possibility.
In my book, credence doesn't apply.
I agree that it feels tidier to express the outcome by saying P(outcome) = 1.
Quoting fishfry
I think "preferable" is better than "should". I'm happy with that.
Quoting fishfry
H'm. That's a new take on mathematics. I can understand the idea that axioms and definitions can be posited in a spirit of exploration. The point in that case is to work out the implications of certain ideas. But the axioms and definitions, even if they are, in some sense, provisional, need to be clear and consistent, don't they? Anyway, you're not telling me that the axioms and definitions of probability theory are in some sense provisional, are you?
Quoting fishfry
It looks as if I have got you started. There are real and serious issues at stake in these disagreements. (You didn't mention climate change.) The biggest problem is that the parties have given up listening to each other. Meanwhile, Putin and Xi Jinping with Kim Jong Un and Ali Khameini are calculating that the West is so divided that they can re-make the world in their own image. There's a serious need for some waking up on all sides. Perhaps one day, the threat will be so great that we'll be forced to recognize that the things that we share are more important than the things we disagree about. I hope we don't wake up too late.
I don't see why not. If a bookmaker had to set odds on the proposition, he'd assign it as 1. Anyway let's agree to disagree.
Quoting Ludwig V
Nobody. That's why it's got probability zero!
Quoting Ludwig V
Hmmm. "No action." As a bit of a gambler back in the day, I understand that!! Probabilities 0 and 1 are no action. Not a valid bet! So I can sort of relate to your point.
Quoting Ludwig V
I better quit while I'm behind here.
Quoting Ludwig V
Right. To account for situations where we can't possibly have any evidence, or know what's going on, but we have a subjective opinion and degree of belief anyway.
Quoting Ludwig V
Fictionalism. It's all fake. But interesting and useful, so why not do it anyway and enjoy it.
https://plato.stanford.edu/entries/fictionalism-mathematics/
Quoting Ludwig V
Clear, with some study. And consistent, well we often can't even prove our axioms are consistent. Nobody knows for sure if the axioms of set theory are consistent.
And paraconsistent logic is a thing these days. Logic in which we can allow a certain well-controlled amount of contraction.
https://en.wikipedia.org/wiki/Paraconsistent_logic
Quoting Ludwig V
Provisional. Explain what you mean by that word. They're seemingly sensible, but they immediately lead to anomalies like the famous non-measurable set and the Banach-Tarski paradox.
The probability axioms are highly useful and natural, but they bite.
https://en.wikipedia.org/wiki/Vitali_set
Quoting Ludwig V
I think the eco-loons are self-centered virtue signalers . Every time you make energy production harder you starve a few hundred thousand third-worlders to death. The Green agenda is starting to crack in Europe. We all like clean water and air, but destroying our economy in the name of "the planet" is suicidal and cruel. The billionaires flying their private jets to climate conferences give the game away. The Obamas own beach front property in two states. They must not be too worried about the seal level rising.
I'm for nukes. Environmentally clean and abundant energy to run our world. Some of the eco-loons would have us living in grass huts. Of course THEY wouldn't live in grass huts. The rest of us would.
I see in England that they threw a few highwayblockers in prison for 4-5 years. Did you see the story? A good start, I say. And the next time some trust fund vandal glues their hands to a museum floor, just leave them there.
https://abcnews.go.com/International/wireStory/british-climate-protesters-plotted-highway-shutdown-record-harsh-112069763
Aren't you glad you asked :-)
Quoting Ludwig V
Is the Biden coup getting much play where you are? He published a letter saying he's dropping out of the race. But there's no video or photos of him signing the letter, and he hasn't been seen for five days. I have no doubt the global competition is taking note. The leader of the free world is the victim of a coup by his own political party.
I'll settle for that. It's a very marginal point, anyway.
Quoting fishfry
Both this argument and the one about evidence are actually aimed at the same point. Once you have introduced probability as an interpretation/application of the formal function, it is very difficult to ignore reality - metaphysics.
Quoting fishfry
Yes, all true.
Quoting fishfry
I have read about this, but didn't realize that's what you meant. It's an interesting take on the idea that we construct mathematics - and some other things as well. However, if it is fiction, it is not the same kind of fiction as literary fiction. However, fake means pretending to be something you are not. Neither is doing that.
Quoting fishfry
I'm sure that many of them - especially the loonies - are virtue signallers. It doesn't follow that they all are. There is a real issue here.
Quoting fishfry
There's a line of thought in eco circles that accepts that the world will not be able to make the changes quickly enough to make much difference. I think that's right. The thing is, the disruption and costs of serious climate change will be greater than the costs of changing now. If we could change now, and do it right, the disruption could be kept to a minimum. There'll be lots of work in the new industries.
Quoting fishfry
Fusion could do it, and it seems to be getting closer. Fission leaves waste. There used to be a lot of concern about what to do with it. I think the plan now is to bury it and leave it alone - for 100,000 years. You can't say those guys are not ambitious.
Quoting fishfry
Don't worry. The last Government passed a new law, restricting free speech to ensure that all protest can easily be ignored. I doubt that the new Government will prioritize repealing it. The people who've been imprisoned will become martyrs - and the whole thing will escalate.
Quoting fishfry
Oh, yes, it's all over the media. From here, it seems that the chaos will continue and spread. I don't think it will end with the election, either.
Ok glad we got to the bottom of that!
Quoting Ludwig V
Ok.
Quoting Ludwig V
Fictionalism is a useful point of view. Avoids having to defend what math "means."
Quoting Ludwig V
Oh you baited me about the eco-loons. I'm all for clean air and water. I'm also for modern civilization. The point is to strike a sensible balance, not to throw tomato soup on paintings.
Quoting Ludwig V
We'll all toil in the windmill factories? I think I better quit while I'm behind here. Eco hysteria is a luxury belief. Green policy hurt third world is one random link I found.
Quoting Ludwig V
The ITER project had another setback. Nine more years delay, another five billion dollars over budget. Fusion would be nice if they can make it work. The fission waste is a problem, but you can't run the world on windmills.
Quoting Ludwig V
I support free speech. Blocking roads is not speech.
Quoting Ludwig V
I hope things don't get too much worse. At the moment nobody knows if we have a president.
Yes, Meaning is as ill-defined as metaphysics. It's usually easier not to mention it.
Quoting fishfry
It is ironical that his most Presidential act has been not to stand for his second term. He'll be a lame duck until January, but that's normal. I expect the system will survive.
There's a paradox about democracy which "of the people, for the people, by the people" misses. It is crucial that the people who lose the election accept the result. That means that the process has to be very carefully organized so that there's as little excuse for contesting it as possible. So the events in January 2021 are a concern. I've also heard that some Trump supporters have said that they will refuse to accept the result if they lose; (I assume they will accept it if they win!). That's absurd. It puts Trump in the same territory as Putin and Xi Jinping. Bluntly, hypocrites.
Quoting fishfry
The people who were jailed were convicted under existing laws. The new law is an opportunistic grab by those who want to ensure that free speech is allowed, so long as it cannot be heard. It's a difficult balance to strike. My complaint about those protests is that they were too effective because they produced more opposition without taking their campaign forward. Protest needs to attract attention - especially media attention, of course - without creating more opposition for the cause.
Quoting fishfry
Probably not. There's also solar panels, hydro-electric, tidal, wave, and volcanic. Still, it's pretty clear that lots of batteries will be needed. China has quietly cornered the market in the rare earths that are needed for them. Now, that's a sensible way to approach the issues. The rest of us will have to pay their prices or find alternatives.
Quoting fishfry
There are two distinct problems. One is enabling as many people as possible to find decent jobs. That's a problem anyway. The other is enabling people whose jobs are phasing out to find alternative employment. That's more difficult. There have been many cases in the past (like phasing out coal) which have not been well managed. But it doesn't seem impossible. At least we could try harder.
It quite likely that third world countries will suffer more. There's a lot of talk about providing additional help to them. That seems like a no-brainer, since unless they join in it will be hard to restore stability. But, curiously, it seems to be very difficult to make progress. Why? Who could possibly be opposing that?
Does your link compare the damage to third world countries with the damage that will be caused by climate change? Or perhaps with the damage caused by existing free trade treaties?
Quoting fishfry
Clean air means less carbon dioxide and methane. Clean water means less plastic. Amongst other things.
Sensible balance is good. But big corporations always end up defending their shareholders' interests and fail in the end. They just waste time and money.
At least they threw tomato soup, which is easier to clean than pain. Paint has been used in the past for similar escapades. It can be cleaned off, but it is much more difficult to do so.
Sorry I mentioned it.
Quoting Ludwig V
Oh please. He left as gracefully as Caesar did.
Quoting Ludwig V
The Dem hysteria that started on election night of 2016 has been extremely damaging to the country.
Quoting Ludwig V
Well toss a can of soup on a painting then. You lost me here.
Quoting Ludwig V
And the third world can suck eggs so that upscale liberal virtue signalers can feel good about themselves.
Quoting Ludwig V
Let's agree to disagree. Sorry I brought it up. No wait, you brought it up and I let you bait me for a while.
Quoting Ludwig Vher
Ok. Well I'm all talked out here. I think we've long forgotten the topic.
Did you hear about that windmill that fell apart, closing a beach during the height of tourist season? Fiberglass shards everywhere.
https://www.cbsnews.com/boston/news/broken-wind-turbine-blade-atlantic-ocean-nantucket-massachusetts/
ps -- This just came in. Couple of soup throwers were convicted, they're going to jail. So never mind on the soup. Looks like England has had enough of the eco-loons.
https://www.thetimes.com/article/67d7c8f4-fbed-4ea1-94f3-252caa171723?shareToken=0b96051817d3dc87ee7ba226d3a18e34
I agree. But I would like a better mutual understanding before we move on. I don't know for sure about you, but my comments were intended to provoke a reply, but only in the interests of a discussion. I thought you were doing the same. I didn't realize that you thought I was baiting you, which is a different kettle of fish. So I apologize.
Quoting fishfry
I wasn't complaining that you did. In fact, in our disagreement, the vagueness of meaning enables a diagnosis of what we disagree about, so it was actually useful. (I'm not sure whether the same applies to the concept of metaphysics.)
Quoting fishfry
Quite so. I'm afraid I was guilty of irony, which is always dangerous. His inability to recognize when the game is up is not particularly unusual. I can think of other examples.
Quoting fishfry
I agree with you that the hysteria around everything is very damaging. But I think both sides are to blame. Each side thinks that it can win by escalating the emotional temperature; the media feeds on that and joins in. The question who started it is a good one - unless the answer is to be used as a weapon of further escalation.
But neither side is really to blame. There can be little doubt that a large part of the problem is systemic - the whole set-up encourages escalation and the desire to win, rather than compromise. Again, it's not unique to the US. It's not difficult to think of other examples.
Quoting fishfry
Quoting fishfry
Quoting Ludwig V
I see from the reports that the soup did actually damage the paint of the frame, so I was wrong about that.
Responding to those protests with outrage and attempts to suppress is exactly what they want - to attract attention and controversy. Difficult as it may be, the only thing that would persuade them to stop is ignoring them. But it is also important to reward them when they do the right thing, there should be a reasonable response to civilized and legal protests.
Failure to recognize when one is being baited is very common and failure to deal with it rationally - by not rising to the bait - frequently underlies escalation.
Quoting fishfry
I'm not sure what you expect me to say. It's definitely a bad thing. Needs to be checked out and any problems resolved - and any parties who haven't been doing their job properly held to account.
But does it show that wind farms should be abolished? I don't think so. The fact that so many people dislike them is much more relevant and it's right to be cautious about setting them up. Off-shore farms seem to be more acceptable, so it's better to be content with them. (There's the question of bird strikes as well, though I've heard that they may have found a solution to that.) I think it's unlikely that that on-shore farms can be a major contributor to the project of finding renewable sources of energy. For on-shore generation, solar farms may be more appropriate.
Not necessary, I willingly took the bait.
Quoting Ludwig V
Over on the political forums in the Lounge, someone actually suggested to me with a straight face that Biden willingly stepped down. I confess I don't understand that degree uncritically parroting propaganda.
Quoting Ludwig V
The Dems refused to accept the result of the 2016 election and have been causing mischief since then, with Russiagate, two fake impeachments, lawfare, and then weakening Trump's Secret Service protection to the degree that he almost got killed. I doubt they're done yet.
Quoting Ludwig V
I hope these court cases will deter some of the vandalism.
Quoting Ludwig V
You can't ignore people blocking major highways. Glad those leaders are going to prison.
Quoting Ludwig V
LOL I'm baiting you!
Quoting Ludwig V
Point being, EV's are a disaster. Green energy is a disaster. If the eco measures actually worked, I'd support them. They don't. They're a scam, and their negative impact falls mostly on the poor of the world, so that the upscale can feel better about themselves.
Quoting Ludwig V
Many of those projects should be abolished, for good and sound reasons.
Quoting Ludwig V
Bird stew?
Quoting Ludwig V
You're halfway to my point of view. And now that Germany, for one, is starting to see the economic downsides of their green energy programs, the tide is turning.
I'm not going to argue the rights and wrongs of all of that. I don't know enough. But I don't believe that Trump's hands are clean, either. Even if Trump himself didn't intend to encourage them, which is very hard indeed to believe, his supporters invaded the Capitol on Jan 6 2021.
Quoting fishfry
I don't think the courts will deter anyone. The protesters are getting what they wanted. Publicity, fuss, arguments.
Quoting fishfry
Maybe so. But not because a single blade on a single tower snapped off.
Quoting fishfry
Well, I'm not keen on any of it. Not least because I'm not anywhere near wealthy enough to avoid the negative economic impacts - and you are right, it will not be the wealthy who bear the brunt of them. On the contrary, they are quite likely to make money out of it. But I don't see any evidence that the whole thing is a scam. True, we're not having much effect yet. But we are nowhere near the level where we might actually slow climate change down. All I see is oil companies defending their profits and nuclear companies returning to profitability by polluting the planet for the next 100,000 years.
Quoting fishfry
That made me laugh. A lot of those birds taste and smell very strongly of fish. Not surprising. They mostly eat fish and that makes them very unappetizing. They reckon that painting one of the blades black, instead of white, makes them flicker, which is enough to deter them.
Quoting fishfry
China has invested a great deal of money and years of effort in cornering the market for rare metals. They must be very confident about where we are going in the long run.
Spare me. In 2020 leftist BLM/Antifa mobs killed 20 people and caused $2B with a 'B' in documented insurance claims. All you're doing is throwing out leftist talking points. Videos show that the Capitol police let the protesters in, they were all unarmed, and most of them calmly wandered around, often escorted by the Capital police. Enough with the leftist propaganda. This is not productive.
Quoting Ludwig V
It's a metaphor for the whole enterprise.
You J6'd me? Are you kidding?
Quoting Ludwig V
Had enough. J6 was the end. Take it to Lounge where the TDS sufferers hang out. Do you honestly believe the J6 propaganda?
Quoting Ludwig V
Interesting if true. No more politics please. J6 is like argumentum at Hitlerum. Terminal point of any conversation.
Quoting Ludwig V
They're also bring coal plants online like nobody's business.
J6. Jeez man that's all you got?
This is not a political thread and actually I've had quite enough of the TDS over on the Lounge. No more please.
Quoting fishfry
Quoting fishfry
I can see you are serious. But I have no idea what you are talking about.
Right you are. I drank my own Kool-Aid. J6 is an emotional topic for me. Thank you for giving me a chance to gather some of my thoughts. You don't need to agree, but at least this is what's on my mind about J6.
J6 was a Reichstag fire for our times. A psy-op, a mass propaganda event. It was no "insurrection." The protesters weren't even armed. There was no intention to "take over the government" nor could they if they'd wanted to. An emotionally troubled guy in a fur vest and a horned helmet sitting in Nancy Pelosi's chair is not an insurrection. It's just a politicized word to make half the country hate the other half. Poor deluded bastard got three and a half years.
There was a lack of security, caused when Pelosi didn't support Trump in calling out the National Guard. Things got out of hand. People who were violent should be prosecuted and given the same slaps on the wrists the Floyd rioters got. George Floyd, by the way, died of a fentanyl overdose. The medical examiner said that if he'd seen Floyd's body dying peacefully, he'd have no trouble calling it a fentanyl overdose. He had a fatal dose in him.
That doesn't make Derek Chauvin officer of the year. He didn't kill Floyd, but he's in prison for smirking.
The Feds have thrown the book at little old ladies who walked peacefully through the Capitol, invited in by the Capitol police. We've seen the videos. People who didn't even go inside got tracked down and prosecuted.
Meanwhile the Floyd rioters caused two billion dollars in insurance-confirmed damage, so the real number's higher. 20 died. Kamala supported a bail fund for rioters who got out and committed far worse crimes. In New York City a pair of lawyers tossed Molotov cocktails into cop cars and got slaps on the wrist.
As we speak, there are hundreds of J6 protesters still in jail. There are stories that the temperatures in the cells are in the 40s. That's Fahrenheit, that's 4.44 Celsius. People denied access to their medications. The Feds, on behalf of the Democratic party, are running a political Gulag. It's utterly shameful that Democrats and liberals cheer this on.
The Feds had informants and provocateurs in the crowd.
The J6 committee was a complete fraud. They didn't allow the Republicans to choose their own members. Tens of thousands of hours of video remain locked away, never seen. The committee put together a Hollywood production of selected excerpts from the videos. They lied, cheated, and perverted the US criminal justice system. That will have long term repercussions that are not good. Once the rule of law gets perverted to political purposes, a nation does not recover.
So for those reasons and many others, I strongly oppose the Democratic spin on J6. I want the people languishing in jail right now to get the same lenient treatment as the Floyd rioters. I want all the video released to the public. I want members of the committee prosecuted for destroying records. I want the undercover provocateurs exposed. I want the whole sordid, evil propaganda op exposed and the people responsible held accountable.
So, whether you agree with my points or not, this is why I reacted as I did. J6 is a crime perpetrated on the US by the Democratic party. They need to be held accountable. The truth needs to come out.
OK. I didn't grasp the significance of J6 until later. I'm sorry I upset you. It wasn't in any way intended as baiting, or even provocation.
Thanks for explaining. It would be absurd for me to argue with you. I don't know anything like enough. It is indeed to be hoped that (more of) the truth, or, maybe a better balanced account, will emerge one day. I accept that Trump did not intend to overturn the whole constitution, so calling it a coup, in the normal sense, is an exaggeration. But it does seem inescapable that he was not prepared to accept the election result until he had tried everything possible to overturn it.
But, if I may, my perspective is that all politicians will play dirty when push comes to shove and the opportunity arises. There's no point in moralizing about it, that's how the world is. So there's no reason to think that Trump (or his supporters) are an exception. That's not an unreasonable view, is it?
It's good. I needed to rationally state my position, not just get upset. It's frustrating because J6 is a massive article of faith on the left. And I used to be on the left. That's what drives me nuts. I just don't know what's gotten into my former fellow leftists. They went insane when Trump got elected. I don't love Trump, I see his many flaws, but he's the only alternative to what's been happening to the left. Perverting the criminal justice system for political gain. If this stands, we are no longer the same country. We meaning the US of course. I suppose our cousins across the pond can only watch in bemusement and horror as the US comes apart at the seams.
Quoting Ludwig V
But "he was not prepared to accept the election result until he had tried everything possible to overturn it" is just what Al Gore did against Bush in 2000. Hillary paid for the Steele dossier and created the Russiagate nightmare that wrecked Trump's presidency. The Intel agencies said Hunter Biden's laptop was Russian disinformation, even after they knew it was real. Stacey Abrams still thinks she's governor of Georgia even though she lost by 50,000 votes.
When Dems try to overthrow or deny elections, it's ok with them. When Orange Hitler does it, they weaken his Secret Service protection. Ok that's a strong charge. I think the case can be made.
Quoting Ludwig V
Perfectly reasonable. In fact my position on electoral cheating is that the GOP needs to learn to do it better. If they got out-cheated in 2020, they shouldn't whine. They should cheat better themselves. As the saying goes: Politics Ain't Beanbag. GOPs better wake up. The Dems may be evil, but the GOPs are hopeless at best, and often complicit.
ps -- I can't find what category this thread is in. It says .999... = 1 but I can't find this on the main page or in the Lounge.
We have been pushing the boundaries for a long time. I'm finding the thread via the list of "mentions". I think they are trying to persuade us to move to private discussion or stop. I'll send you my response to this post in that way. If you really want to stop, just tell me. But I think we've just opened up another layer of discussion.
Ok I see. Well I'll respond to the PM when I get a chance. We're having an interesting discussion, but I prefer for such discussions to be in the public space. I am on a bit of a mission, which is to slowly and painfully try to get some checkbox liberals and TDS sufferers to, if not see things my way, to at least agree that I have a rational position.
Of course it's a lost cause. In the political threads if you express a thought contrary to their doctrine, they just call you names. It's quite frustrating.
On the other hand, our convo is helping me to at least articulate some of my thoughts. Especially about J6. J6 is an article of faith for the True Believers. That's why I reacted as I did.
On the private thread you referred to the rise of Hitler. But I did say that I see J6 as a Reichstag fire for our time. It's the Democrats making up an insurrection to get their rabid followers to hate the likes of me, their former ally who has dared to think an independent thought.
So I am really primarily motivated to write on this forum for the benefit of my liberal tormentors, the ones who call me names and say I get my ideas from Sean Hannity. It's too stupid to bear. But these people must wake up, for the good of the nation. I might as well do my part, since I was a checkbox liberal myself till 2016. I'd been wavering for a long time ... it's an interesting story, how I came to be a fallen liberal.
So anyway ... the answer is that we should probably wrap it up here ... and I don't know what we should do over there. You're open-minded, you're not the person who needs to hear what I have to say.
I'll sit with this for now.
I have no objection to the public space. But it seems that this is no longer really a public space.
Quoting fishfry
Yes, it's helping me in the same way. It's rare to find people who are willing to emerge from their bunkers and actually discuss things. If it has to be in private, so be it.
So have the mods actually hidden this thread? Without saying anything?
Still, I'd rather move our chat back to here. I think of the private messages as more short-term communications.
Quoting Ludwig V
Yes I think the politics threads are a loss anyway.
OK. I don't feel entirely comfortable about private threads. It's just that I've picked up references and deduced that some people take their discussions to private threads to avoid intrusive or annoying comments, which you can get on public threads.
I found this thread in "All discussions". But it says the last post was 24 days ago. It's way down on page 2. But when I look at your last post in the thread, it says you posted it 11 hours ago. Whatever the reason, pushing it down the list means that fewer people are likely to visit it.
Test. Political discussion so often turns into Punch and Judy. There are several reasons for that. But it is often not helpful but actually harmful.
Real discussion is not possible unless one is willing to endanger oneself, by allowing one's own position (and self-esteem) to be on the table. That applies to all philosophy and possibly even more widely.
Yes, I take your point. Really, I do. I don't know how to open up a discussion about this without seeming to trigger the righteous anger, not only of victims, but of many decent citizens as well.
There are issues that need to be recognized, and I hope that you will be able to see them. I do not mean to deny righteous anger, which is expressed in the desire for revenge and to exclude the offender from one's society.
First, there is the familiar problem of the cycle of revenge - the blood feud, continuing a cycle of violence which can even be inherited for generations. It is the result of what is often forgotten, that what we think of as punishment may not be "accepted" by the criminal, who then gets angry and seeks revenge in turn.
Second, there is the issue of proportionate revenge. The traditional "eye for an eye and a tooth for a tooth" is an early attempt to limit the revenge response, which can easily go way beyond what is reasonable.
Third, handling the revenge response is not the only issue when a crime is committed. There is the question of prevention and the question of what happens after revenge is exacted. This is where the issues arise. But let's pause there and see how far we agree thus far.
Or perhaps I should start another thread?
Evidently nobody can see this anyway. It's weird that the moderators did that without notifying anyone.
Quoting Ludwig V
Too bad. I can't buy an intelligent conversation in the political threads. It's all partisan nonsense.
I try doing it over there but to no avail. May have to let it all go.
Nobody is reading this.
I'm sure you know about the riots in England over the subject of immigration, relative to that awful killing of three little girls.
I happened to see a picture the other day of Keir Starmer taking the knee during the George Floyd protests. So when liberals are burning down the country, he supports them. And when people get angry that thee children were slaughtered, he comes out four square against the protesters. Never mind the stabbers.
To be fair, the stabber in this case is born in Wales to immigrant parents. But that only makes it worse in some people's eyes. Why let in the foreigners in the first place? Without endorsing that sentiment you can see why people are upset; and threatening to jail the protesters seems to miss the point.
As far as I can see, Starmer is living down to my worst fears about him.
Quoting Ludwig V
If you don't deny the righteous anger, Keir Starmer might not be pleased! You are NOT ALLOWED to be angry at the fatal stabbing of three little girls at a Taylor Swift dance class. If you are angry at the stabber, you are a right winger. This is the official policy of your government as far as I can tell.
Quoting Ludwig V
So why allow people into the country who may harbor ancient ethnic or religious grudges? I'm not arguing that but it's an argument put forth by the protesters. And frankly it's not a bad question.
Quoting Ludwig V
Yes well these things do tend to escalate.
Quoting Ludwig V
Three little girls are stabbed and Starmer wants to put the protesters in jail.
I suppose this should go in the Lounge, at least someone would see it. Doesn't matter. Seems to do me the same amount of good to write it whether anyone reads it or not.
I started on this caper two years ago. I've found that there is some fun and instruction to be found, provided one understands how the game is played and doesn't take it too seriously. But every so often, one finds a more constructive engagement. It doesn't necessarily last forever. So it is important to recognize when one can go no further.
Quoting fishfry
OK. There are many environments in which I don't make that point. However, I don't think that one can simply let anger rip. My main reason is pragmatic. It so easily feeds on itself and becomes destructive. It is important to be sure that one has the right target. But the worst effect is that it can so easily provoke a response in kind and a spiral of violence.
Quoting fishfry
That's the reason that Governments and similar authorities get so exercised about it. They need to stay in control, and not just because they are taking sides. (Though there is an element of that, of course.)
Quoting fishfry
Good question. It is true that it not wise to ask it in many environments. It does have some traction, though it is more complicated than it seems. (This is a different issue, though it is tangled up in the Southport business,)
How would you feel if the UK banned immigration from the USA because there are white supremacists there? Over-reaction, I think. One has to try and weed them out. Same for Muslim fundamentalists. In the UK, there is a lot going on to try to do that - most of it secret, so it is hard to know.
Here's a couple of politically incorrect thoughts of my own.
1. I would not be happy to live in Cairo or Dubai. I've been there, though not lived there. So I would not be happy if those social and religious norms were imported to the UK. Same for many other countries. There is a fear that a substantial minority arriving in the UK will introduce ideas and practices that I don't like. I expect people coming to live here to assimilate.
But there are also ideas and practices in the UK that I don't like. So it's a question of balance - accommodating new ideas and practices and assimilating to what already exists here. It's not a black-and-white question. (Actually new ideas and practices are often harmless or even beneficial. Moreover, societies do better if they are willing to change and adapt.)
2. Then there's the question of economic impact. There is an opinion, here, at least, that immigration drives wages down by increasing the supply of labour. No-one wants to end up with third world conditions in their own country (though they're quite happy to take the benefit of cheap imports). Economists insist that's not the case. I don't know the truth of it, the claim that it makes no difference seems implausible to me. There's also an argument that the UK benefits because immigrants also contribute to the economy. Which is true, so far as it goes.
I don't know the answer, but I'm inclined to think that, again, it's a question of balance. It may seem feeble, so I should emphasize that I'm very happy to rigorously exclude people who are going to deliberately spread disinformation and provoke violence, for the same reason that when UK citizens do those things, they should be repressed.
I did no understand what you are referring to. I just meant the political threads over at the Lounge.
Quoting Ludwig V
I have been following this, it's really blowing up. Starmer is cracking down hard and calling them right wingers, but they're mostly working class folk whose live are being impacted by immigration promoted by the government, despite the will of the people. And the British cops stand down in the face of Muslim protests. There's a photo of Starmer kneeling for George Floyd. The hypocrisy.
They're calling him Tw-Tier Kier. And a lot of people are upset. Three little girls got stabbed and the stabber happened to be a first-generation Brit, and it's sparked a lot of pent-up anger. The government has been unresponsive on the issue of immigration.
I'm hearing talk of a "civil war" in Britain, but I can't tell if this is overblown or not.
Quoting Ludwig V
It's the taking sides that's blatant here. Starmer took a knee for the American BLM/Antifa riots, and his police stand down in the face of violent Muslim extremists; then call out the dogs, running courts 24/7 to arrest and convict and imprison anyone who expresses a word of dissent.
I see Starmer blowing this tremendously. The partisan application of justice is a step down a very slippery slope for a nation. We're seeing a lot of it in the US as well.
Quoting Ludwig V
I'm all for ethnic diversity. You know what changes? It's like the first big wave of American immigration from Europe in the early 1900's. They all assimilated. I think what went wrong was people not wanting to assimilate. It's not clear if you can run a nation like that. We're all finding out.
Quoting Ludwig V
The Muslims have a bad track record. The religion and state are intertwined. They are fundamentally incompatible with western thought. Many integrate very successfully. I'm for human movement. Governments should set and enforce their own laws, not have open borders like the US and western Europe.
We don't even know if the stabber is Muslim. That case is just a flash point for a lot of other issues that have been going on a long time.
Quoting Ludwig V
Some say that the open-mindedness and acceptance of the West is exactly why they will be conquered by the East. I'm not wise enough to know. But it's a possibility. You see the liberals in cities voting in soft-on-crime prosecutors, then being overwhelmed by the crime they voted for. Islam does no seek to coexist. It seeks to conquer. I believe this is just how it is. Am I wrong?
Quoting Ludwig V
Cheap labor is always good for business. Cheap labor that can't complain about exploitation, lack of safety, and being cheated, because they are illegal, is even better! There are many powerful interests perfectly happy with the corrupt and immoral system we have now.
But at some point, when you have imported the Third World into your formerly First World country ... how do you think that's going to work out for you?
You have to set some limits, you have to have some laws that you are willing to enforce, you have to try to reduce the corruption and brutality and evil in the system.
I'm for serious immigration reform in the US, whatever that may look like.
Quoting Ludwig V
Who exactly are the people who "deliberately spread disinformation and provoke violence?" How do you know who they are? Does Keir Starmer tell you? He kneeled for BLM/Antifa. But didn't BLM/Antifa also deliberately spread disinformation and provoke violence? Twenty people died. There were two billion dollars in property damage. George Floyd was a violent career criminal who died of a fentanyl overdose. That doesn't make the cop officer of the year. But if all you know is the mainstream account, it's all a lie. The cop was following department protocol. His knee was on Floyd's upper back. Floyd did not die from strangulation, he died of an overdose. The police department threw Chauvin, the cop, under the bus and let him take the fall. Then the liberals unleashed their shock troops on the country.
And Two-tier Keir kneeled.
OK. I took it that you were referring to all the threads. Sorry.
Quoting fishfry
Are you sure it is not a fake?
Taking the knee is not the same thing as burning down the country. You can't infer from the fact that he takes the knee against racism (or even against what happened to George Floyd), that he doesn't oppose burning down the country.
Quoting fishfry
I haven't heard/seen any reports of any violent Muslim extremists.
Quoting fishfry
Yes. I broadly agree with that.
Quoting fishfry
The issue is that you can't enforce immigration laws unless most ordinary citizens will help you. Most ordinary people in the UK (and, so far as I can see, the US) will not (or perhaps cannot) help enforce the rules. It does mean something much more like a police state than we are happy to live with. But you can't have it both ways.
I remember, back when UK was in the EU that the middle class (not just the rich) were delighted with the cheap Polish plumbers and builders that they could employ. UK plumbers and builders were somewhat less enthusiastic. Now, plumbing and building are much more expensive and difficult to get done. Again, after COVID there was a serious shortage of HGV drivers which resulted in rapid increases in transportation costs (and delays in supply chains). Manufacturers and customers alike were very unhappy. HGV drivers were too busy making lots of money by driving to tell anyone how happy they were. Nobody thinks about how things affect other people.
You can't expect to tell the world how well you are doing economically and expect people who have no prospects where they are not to come and join in the feast. The root cause of immigration, legal and illegal, is the unequal distribution of wealth across the world. The only way to stop it is to make sure that international trade benefits everyone.
Quoting fishfry
I think it became very clear during the last few days what the people think, don't you?
Quoting fishfry
It is indeed grossly over-blown.
I hear a lot about the possibility of civil war in the US. What do you think?
Quoting fishfry
Yes, I know that what happened to George Floyd was contested and I don't really know what the truth of the matter was. How do you know that the mainstream account is all a lie? Everyone lies, not just the Government.
I agree that things are different in the Middle East. But religion and state are also intertwined in the West. The relationship works differently, that's all.
Quoting fishfry
I think you are paying to much attention to the fundamentalists - who are a problem, but not an existential threat, I think. The biggest threat is not from Islam, but from Putin and Xi Jinping. Putin is (officially) Christian and Xi Jinping (officially) communist. Both are actually old-fashioned imperialists, just like the West was in the 19th and early 20th century.
I don't know for sure who will win. But I think the West has a very good chance.
You joking?
https://www.telegraph.co.uk/columnists/2024/01/23/keir-starmer-take-knee-culture-wars-blm-rnli-national-trust/
Plus you know things have gotten worse the past few days. Starmer's throwing protesters in prison for long terms. He threw some old guy in prison for a Facebook post. He's threatened to arrest and extradite Americans for exercising our free speech rights. He can't do that, we have the First Amendment here. He's gone mad as far as I can tell. His double-standard with respect to violent Muslim rioting is obvious.
Quoting Ludwig V
Like Walz burning down Minneapolis?
Anyway the point of the knee is the two-tier justice to rioters. And, the British establishment's ignoring the opposition of Englishmen to rampant uncontrolled immigration the past decade or two.
Quoting Ludwig V
Oh please. Floyd was a violent career criminal who died of a fentanyl overdose. He took the knee to violent leftist rioters and throws the book at violent rightist rioters. Two-tier.
Quoting Ludwig V
You joking again? I didn't feel like digging out specific news stories, but there are plenty. People aren't complaining about two tier justice in a vaccuum.
https://www.theguardian.com/uk-news/article/2024/aug/11/uk-two-tier-treats-far-right-attacks-less-harshly-islamist-violence-rusi
Quoting Ludwig V
Trump implemented a Stay in Mexico policy that was effective in cutting down the problem. On his first day in office Biden overturned that and about 40 other Trump immigration policies. Then the last few months of this election year, Biden has tightened the border and cut down on his immigration problem. The government can turn illegal immigration on and off like a faucet. The current hordes coming in, in the US and in England, are a matter of government policy.
Quoting Ludwig V
A government can enforce its borders. Biden and both parties in England have chosen not to. It's not a police state to prevent people from entering your country illegally. You don't need "average people" to patrol the border. You simply need to have the border guards do their jobs instead of telling them not to.
Quoting Ludwig V
Cheap labor is always popular. But who gets hurt? The people legally here, the natives, who are perhaps in the trades themselves and who can't compete with the cheap labor.
In any event no nation, can import massive hordes of third-world immigrants with no understanding of and no respect for that nation's culture and laws.
Quoting Ludwig V
Yeah yeah root causes. I'm all for addressing root causes. Meanwhile control the border. Because if you don't, you won't be able to suppress enough free speech to stop the tidal wave of resentment that's coming. Didn't the Tories just get swept out because they FAILED to deliver on their promise of controlling immigration?
Quoting Ludwig V
I wouldn't know ... I actually don't know what you're referring to. People who speak out against immigration are being thrown in Starmer's prisons. So clearly we are not hearing people's true feelings.
Quoting Ludwig V
I think that's overblown too!!
Quoting Ludwig V
Floyd was a violent career criminal who died of a fentanyl overdose. His police force threw him under the bus. That's supported by the facts.
We shall see. My understanding is that over the long term, Islamists seek to take over the west. Maybe that's just right wing propaganda.
Quoting Ludwig V
I'll grant you than in the 1970, US and European radical leftists set off a lot of bombs and killed a lot of people, but the rest of us survived.
So, how many Islamic terror bombings are ok with you? They just tried to pull off a terrorist attack at a Taylor Swift concert. Myself, I am not a big fan of Muslim extremism, and on average these days, there's way too much of it.
Quoting Ludwig V
The Chinese are not US and British domestic terrorists. And I'm sure you know how Xi handles his Muslims. He puts them in concentration camps in western China. I don't support him in that. I support the plight of the Uyghurs. These are all complicated issues. I don't have to be ignoring Xi just because I'm opposed to Islamic terrorism.
Quoting Ludwig V
People have been predicting the fall of the west for about a century now I think. Spengler, the Decline of the West.
Sorry, I think you are a bit confused. He can arrest and deport (i.e. send back home) US citizens who misbehave. The UK also has free speech, but bans incitement to riot. That seems perfectly reasonable to me. They are lucky that he doesn't apply UK law and throw them in jail.
I haven't seen anything about violent Muslim rioting recently - not in the UK, anyway. Obviously, if no Muslims are rioting, he can't throw them in jail.
Quoting fishfry
Well, you know best about what's going in the USA. In the UK, the Government has been trying to prevent immigration across the Channel for decades. You would think it was easy enough. But they've failed.
Quoting fishfry
People are who prepared to die to get here are very difficult to stop.
Quoting fishfry
Who employs the cheap labour? When those people are not prepared to employ them, the incentive will disappear. That's what I meant about lack of public support. People are happy to make a fuss, but not willing to pay a bit more for labour. You can't have it both ways.
Quoting fishfry
You're begging the question. The courts think that those people are rioting, and that's not free speech, it's violence. As for people's true feelings, you seem to trust the Telegraph.
Daily Telegraph Southport Counter-demonstrations
Quoting fishfry
Yes, but that was just one aspect of their failure to deliver any public services at all. Health, Education, Justice, Defence, not to mention the housing crisis - the list is endless. Obsessed by in-fighting and tax reduction, failed to do their job.
Quoting fishfry
I'm very glad to hear it.
Quoting fishfry
I agree it is supported by some of the facts. But surely the police are not supposed to throw people under buses - arrest and fair trial?
Islam is a missionary religion. It seeks to become the universal religion. The idea of the theocratic Caliphate is an aim that some fundamentalists are committed to. That's true. It's just that I don't think they will succeed. Sadly, they can do a lot of damage while they are trying.
Christianity has the same ambitions. They are not terrorists, of course. Nonetheless, while I respect their right to campaign for their views, I object strongly to their desire to impose their views on me and suppress mine.
Quoting fishfry
Oh, come on. I think that Islamic fundamentalism is not an existentialist threat to the West. That doesn't mean that terror bombings are ok with me
Quoting fishfry
I agree with you that they are complicated. The desire to suppress IS and similar groups is perfectly reasonable. But the means employed against Uighurs are grossly disproportionate.
Quoting fishfry
You're missing the problem. People who are willing to die to get in to UK or US are very hard to stop. Public opinion won't support extreme measures (which would probably not work anyway)
Quoting fishfry
Strictly speaking, they are not terrorists. But both of them operate in secret in the UK and elsewhere.
Quoting fishfry
Fair enough.
He explicitly threatened non-Brits in their home countries. I am not confused about this, it has been extremely widely reported.
"London’s Metropolitan Police chief warned that officials will not only be cracking down on British citizens for commentary on the riots in the UK, but on American citizens as well.
“We will throw the full force of the law at people. And whether you’re in this country committing crimes on the streets or committing crimes from further afield online, we will come after you,” Metropolitan Police Commissioner Sir Mark Rowley told Sky News."
https://nypost.com/2024/08/10/media/uk-police-commissioner-threatens-to-extradite-jail-us-citizens-over-online-posts-well-come-after-you/
Quoting Ludwig V
Some 60 year old was just jailed for a Facebook post. I can't argue with you about British politics, you being there and all, but we seem to have very different information. You can be jailed for just reposting info about riots, not inciting them.
Quoting Ludwig V
Well I've said my piece on this.
Quoting Ludwig V
By incompetence or design? Either way, the people seem unhappy about it.
Quoting Ludwig V
Who is prepared to die? Impoverished peasants streaming across the US southern border?
Quoting Ludwig V
Agree on that. Lots of people benefit from broken immigration systems on both sides of the pond.
Quoting Ludwig V
The British courts don't have the US First Amendment, which provides legal protection for the most appalling expressions of ideas. I read that Prince Harry has called the First Amendment "bonkers." The US has very strong protections for speech not found in most other democratic nations.
Quoting Ludwig V
And now you've got Starmer. Good luck! I should talk, right? We're about to have Queen Kamala.
Quoting Ludwig V
During the summer of Floyd that's exactly what they did. There was another case of the three Georgia guys thrown in prison for decades for the accidental death of a known burglar.
Well if Islam seeks to become a universal religion, what happens to your nation when there are enough of them to make a political difference? It's no hypothetical.
Quoting Ludwig V
Who is suppressing your views?
Quoting Ludwig V
We shall see.
Quoting Ludwig V
I agree.
Quoting Ludwig V
Nobody's "willing to die," they're just walking across an open border in the US. When Trump had his Remain in Mexico policy, the problem was greatly reduced. These aren't armed hordes "willing to die," what are you talking about? These are illiterate peasants walking across an open border that can be closed if the leadership wills it.
Quoting Ludwig V
Ok. China has its own problems though. I hear they're in demographic collapse.
You're right. I was confused. But it is quite simple. If you break British law in Britain and go home, Britain can sue in US courts for extradition, take you to back Britain and try you. If you break US law in the US and go home, US can sue in British courts for extradition, take you to back to the US and try you. Seems fair enough to me. Most countries in the West have the same arrangement - by treaty, i.e. international law.
Quoting fishfry
Info or Incitement?
There's an interesting question about people who are US citizens in the US posting something to Britain that is within US law but banned in Britain. There's a suggestion that they can be extradited, but I find it very hard to believe.
There's a new law in Britain that if you re-post an illegal post by someone else, you are also guilty of incitement. I agree that's pushing it a bit, but if someone is inciting violence and you join in the incitement, I think there's a case for it - if you can prove it. After all, if you help someone committing a theft, you are also breaking the law. No?
There's a big push in the UK and Europe to get the internet under control. You may not be aware of how much the big internet companies are resented over here. They have a very poor reputation. One has to give them credit for taking the issues seriously, but they don't take effective action. They plead free speech, but no-one believes that. It's about the bottom line and that's not acceptable.
Quoting fishfry
I'm not sure who you trust on this. But Reuters have a pretty good reputation.
Reuters on deaths on US-Mexico border
Certainly, people die in the Channel regularly. BBC on migrant deaths in the Channel
I don't know how many, if any, are illiterate. Why does it matter?
Quoting fishfry
Yes, that's true. The UK does have protection for free speech. Just not as much as in the US. People resent they way the the US internet companies impose your law on us.
However, I really don't care at all what Prince Harry's views are; he has no special knowledge or authority that I'm aware of. I can't understand why people in the US get so excited about our royal family. They are an embarrassment in a supposedly democratic country.
Quoting fishfry
Starmer is at least less of a joke than the other lot. Rishi Sunak was better his immediate predecessors, but was undermined by his own party. I have the impression that Trump is still likely to win.
Quoting fishfry
Hopefully, by that time, there will be more home-grown imams and fewer radicals imported from back home. There are already a good many of them (home-grown imams) - they just don't get the news coverage. Plus, generations born and brought up here are, on the whole, often atheists or moderates. I think they will settle down. If the other immigrant communities are anything to go by, there'll be a lot of inter-marriage with the general population, anyway.
Quoting fishfry
Sorry, I wasn't clear. No-one is suppressing my views. Fortunately, I'm pretty much mainstream. I've tried to clarify what I was trying to say and failed, so I'll have to let it go.
Quoting fishfry
So are many other Western countries, including Britain, not to mention Japan and Korea. There's a lot of argument about the reasons. Most plausible explanation is that that a modern capitalist economy makes it too hard to bring up children. Either you live in poverty with children or you work to make the money for a decent life without children. Not to mention the gloomy outlook for the West. That also is one of the reasons why Britain actually needs immigrants and allows many in, legally.
The USA is not doing well but is not in collapse - yet.
US Census Bureau 2023
US Census Bureau 2021
A British official threatened to extradite Americans whose free speech offended him. There is no conceivable way you can spin this. It's disgraceful.
Quoting Ludwig V
In Britain a guy was arrested for "anti-establishment rhetoric." If that doesn't bother you, I won't further argue the point.
https://www.allsides.com/news/2024-08-14-1315/general-news-bbc-court-hears-man-arrested-anti-establishment-rhetoric
Quoting Ludwig V
It's hard to believe they could actually do it; but a British official did threaten it. The British government has gone full fascist. I'm sorry you can't see it. Maybe you're too close.
Quoting Ludwig V
Anti-establishment rhetoric. As an American accustomed to the robust protections of the First Amendment, I'm appalled. You don't seem very keen on free speech as I understand the term.
Quoting Ludwig V
Free speech is under attack everywhere. That's why it's so vitally important to defend it, and to push back on these awful statements and policies of the Starmer regime. I'm sure Europeans have been conditioned to hate and fear free speech, free expression, and free thinking. That's to their own ultimate detriment. Lot of people in the States want the government control the Internet too.
Quoting Ludwig V
You're making an obscure and convoluted point. I'm fully aware of the dangers to illegal immigrants. But most just walk across (in the US) and are welcomed by an administration that refuses to enforce its own laws.
Quoting Ludwig V
I have a theory about why the Americans love the British Royals. We get to enjoy all the pomp, the circumstance, and the salacious scandals. And we don't have to pay for it!
Quoting Ludwig V
Kam's got the media on her side and a newly energized Democratic party. Trump is old, seems confused and out of sorts lately, and IMO may be suffering a touch of age-related dementia himself. The election could go either way.
Quoting Ludwig V
The second-generation native born Muslims seem to manage to get themselves radicalized anyway.
Quoting Ludwig V
You are a glass half full guy! I am not so sanguine.
By the way, 100,000 Hamas-loving maniacs are going to riot at the Democratic convention in Chicago this week. Should be something for the world to see.
Sorry, I wasn't clear. No-one is suppressing my views. Fortunately, I'm pretty much mainstream. I've tried to clarify what I was trying to say and failed, so I'll have to let it go.[/quote]
Of course. You have the establishment view. I often take the anti-establishment view. In your country I'd be subject to arrest.
Quoting Ludwig V
"There's a lot of ruin in a country."
Are you saying that US law should apply in the UK? How is that not imperialism?
Quoting fishfry
Actuallly, it does
[quote=News report]the posts were alleged to contain anti-Muslim and anti-establishment rhetoric.[/quote]
They don't give details (no doubt for fear of being accused of spreading the words more widely), so I can't sort out what's going on. Anti-Muslim is a problem. Anti-establishment is not. It's interesting that the headlines all mention "anti-establishment" and don't mention "anti-muslim". That does puzzle me.
Quoting fishfry
I don't think they could do it either.
Quoting fishfry
Perhaps I am. My parents fought WW2. So I think I have a real understanding of what full fascism is. Believe me, this isn't it.
Quoting fishfry
Perhaps we just have different ideas about free speech. You have yours. I have mine. Why is that a problem? I don't think anyone thinks there should be no restrictions at all. Even the US has libel laws, doesn't it?
Quoting fishfry
Sadly, from my point of view, US citizens have been conditioned to hate and fear sensible controls to minimize the harm that some people will inflict on them by exploiting their freedoms - not only in free speech, but also in the matter of gun control. There may be detriments to control, but there are detriments to unlimited freedom. It's a choice. Nothing is pure benefit.
Quoting fishfry
And have they been conditioned as well? Or just making a different choice from you?
Quoting fishfry
And have they been conditioned as well? Or just making a different choice from you?
Quoting fishfry
Perhaps. So long as you are aware. The problem is that many people aren't as concerned about immigration as you are. So, to enforce immigration restrictions, you would need a police state. Indeed, I rather think that you would not be happy about that.
By the way, why are you so keen on freedom of speech and so much against freedom of movement?
Quoting fishfry
I suppose that works. But they are actually very boring people.
Quoting fishfry
It's true. Kam has managed to revive the Democrats, and now it's more of an actual race. I did wonder, in all the fuss about Biden, whether the issue might come back to bite Trump.
Quoting fishfry
There's not that many of them. There will be fewer in the third generation.
Quoting fishfry
Ever since that business started off, I've been astonished how Israel has mismanaged the propaganda war. They started off with the moral high ground and have surrendered it almost completely.
Quoting fishfry
Sometimes I agree with the mainstream (that's less pejorative than "establishment"), but not always. No, you would not be subject to arrest in this country on the basis of anything you have said to me.
The emerging globalist government is cracking down on free speech. You and I are not on the same side of this issue. Perhaps we can agree to disagree. I'll go with the First amendment to the US Constitution. I'm burnt out on this topic, my apologies.
Quoting Ludwig V
Your government is way over the line these days. But like I say, I have my hands full fighting off the censors in the US. Hoping for the best for our British cousins. I hear Starmer is letting hardened criminals out to make room for the posters of mean tweets.
Quoting Ludwig V
Well authoritarianism doesn't always look like jackboots.
Quoting Ludwig V
Americans have extremely wide latitude for free expression. For the moment, anyway.
Quoting Ludwig V
Well your side is going to soon crush my side. I have no doubt that bad days are ahead. You might call them good days. No unapproved thoughts.
Oh yeah there was that woman arrested for silently praying. That case got dismissed. But still ... arrested for what is in your thoughts?
https://www.bbc.com/news/articles/c4gze361j7xo
I think we should drop this. You know the kind of scurrilous literature I read. Since we talked last I've got 20 articles about the repression of speech in England. I won't bore you with them.
Quoting Ludwig V
Actively trying to destroy free speech. I say that's bad.
Quoting Ludwig V
Bad choices.
Jonathan Turley just wrote a book about all this.
https://www.amazon.com/Indispensable-Right-Free-Speech-Rage/dp/1668047047
Quoting Ludwig V
Not so. Trump's Remain in Mexico policy was keeping a lid on the problem. You don't need a police state to simply defend your own border and enforce the laws already on the books.
I believe it was Milton Friedman who said you can't have both open borders and a welfare state. That's the mistake the US government is making.
Quoting Ludwig V
I can live with open borders as long as nobody gets government services. But that's not workable, because people get sick and need health care. Kids need education. It's a thorny problem.
But your question is analogous to asking, "Since you're against bank robbery, why are you against bank withdrawals." I'm fine with legal immigration.
Quoting Ludwig V
I don't spend much time following the Royals, but they're in the news and hard to miss. Meghan and Harry and all that. England's gift to the US.
Quoting Ludwig V
So far Kam still hasn't announced any actual policy stances, nor sat for an interview or press conference. She might get away with it. Trump looks tired and out of it these days.
Quoting Ludwig V
One can only hope.
Quoting Ludwig V
Agree. Even their friends are upset with them now. It's a tough situation. And very volatile if Iran and Israel go to war.
Quoting Ludwig V
I hope your buddy Starmer is as open-minded :-)
Pardon the scurrilous right wing site link, it's factual info that I'm sure is replicated elsewhere.
https://www.breitbart.com/tech/2024/08/24/justice-europe-style-telegram-ceo-pavel-durov-arrested-due-to-lack-of-moderation-on-platform/
ps -- I'll stipulate that he's charged with all kinds of awful things. The French must think they have a case. Interesting to keep an eye on this one. Another article:
https://nypost.com/2024/08/24/world-news/telegram-founder-pavel-durov-arrested-at-paris-airport-report/
"Law enforcement believe that Telegram’s lack of moderation and the tools it offers, such as cryptocurrencies, make it complicit in global drug trafficking, pedophilia and fraud. "
Yeah the prosecutor's press conferences always make people seem awful. We'll see what they can actually bring to trial. I don't know the guy, not defending anything he may or may not have actually done. If he's enabling illegal activities, that's different than if he's only enabling free speech. We'll have to wait and see.
Evidently he screwed up by landing in France.
OK.
Quoting fishfry
True. But fascism does.
Quoting fishfry
I don't know about that case. I agree it looks bad. But on the principle, the difference between murder and manslaughter is your intention i.e. what is in your thoughts.
Quoting fishfry
Fair enough. I don't expect us to agree about much. I'm quite happy to understand what you think and find out what we agree about. After that, agreement to disagree is fine, and certainly much better than exchanging abuse.
Quoting fishfry
You seem to resent any restrictions on free speech. The classic question here is whether you have no objection to someone shouting "fire" in a crowded theatre or stadium when they know darn well that there is no fire. (Thus causing mass panic and distress, injury and death) Nobody doesn't limit free speech. The only question is what limitations are appropriate.
Quoting fishfry
I gather that the numbers were down and have gone up since. I don't know why.
Quoting fishfry
It is indeed.
Quoting fishfry
That's what I call the honey-pot effect. That's a thorny problem too.
Quoting fishfry
What if you disagree with the existing laws about immigration? People who have a problem with immigration want restrictive laws as well. Most people expect some level of control. The really thorny argument is how much control should there be. (At one point, the law in the UK did not allow any immigration at all. It didn't work very well.)
I may be exaggerating about the police state, but how would you feel about employers having to get government approval before offering anyone a job? Or hospitals having to check your status with the government before treating or even examining you? Or hotels, landlords and restaurants contacting the police before letting you have a room? Schools asking permission before they take on your child? Have a look at what China is doing on the surveillance front.
Quoting fishfry
Whose line is it over? Yours? But you are not living here and you are not a citizen. The job of the UK government in the UK is to keep in line those who are way over the UK lines (by law). That's what they are doing.
Quoting fishfry
There's a paradox. In the UK, there is practically no coverage at all of what they are doing at the moment. They are invisible.
Quoting fishfry
She does seem to have got the Democrates back in contention. She seems to have worked out that joy and confidence are more attractive than fear. It's a brilliant move against Trump.
I'm also wondering if his age is catching up with him, and whether it will create difficulties for him when it comes to voting. That would be ironic. There's a rather old-fashioned phrase in English English "hoist with his own petard" it means roughly "blown up by his own grenade". Very satisfying.
Quoting fishfry
Well it will help if, in the mean time, we do not treat as terrorists people who are not terrorists. Islamic terrorists are a tiny minority of Islamic people. The vast majority of them disapprove of them. Other Islamic people have suffered from them as well, you know.
Quoting fishfry
I'm sure he will, and if he doesn't, there are plenty of his supporters and officials who will sit on his head.
Quoting fishfry
On the contrary, I'm seriously worried that the whole world is moving to the right. The dictators (Russia, China, Iran, North Korea and all the small fry) think things are going their way. They recently had a global conference to swop tactics and sympathy - somewhere in S. America, I think. The UK, I believe, was represented by Nigel Farage! Talk about the emerging global government. It's quite likely to be a right-wing government.
The question is whether Telegram is facilitating free speech (good) or facilitating criminal activities (bad). I think that if he couldn't help the bad people taking advantage of Telegram. But he could at least try to prevent them or at least help police and prosecutors nail them.
Quoting fishfry
Not really, though politics played a big part. Prosecutions in Athens were only brought by private citizens; there was no such thing as Government legal action. It was a very different world. The real problem that many of his followers were right wing. But there's no evidence that he agreed with them and some evidence that he believed in the Athenian constitution, which the right wing opposed.
Quoting fishfry
There's something we agree on.
I find it hard to believe that he didn't realize he was liable to arrest if he went to France. What were his people doing? It looks as if he and they just assumed that because he was OK in the USA, he must be OK in France. That's the kind of attitude that seriously annoys the rest of the world.
Surge the border
This might be a clue. I am enjoying the discussion by the two of you. Better by far than what is found on the visible pages.
Well, I'm not going to spend a lot of time on this. I've looked at some of the clips, none of which I trust because they are clips and context is always important.
It looks much like what goes on here. More and more people come. The policies make precious little difference. Politicians spend their time bickering and trying to invent quick fixes. No-one really cares about the problem.
I'm glad you are enjoying our conversation. We are trying to demonstrate a better way.
Oh no, that's the point. Mussolini defined fascism as the merger of state and corporate power. That's exactly what happened when the US government pressured the social media companies to censor and suppress speech. That's exactly what fascism is. No jackboots. Just the state and corporate power crushing the freedom of the individual. It's rampant these days. Very dangerous. Because it comes dressed as benevolence. "We just want to keep you safe from misinformation." Soft fascism if you like.
Quoting Ludwig V
We talking about the Telegram guy? Brand new story, he just got arrested lately. One account said "... he’s now jailed, and facing 20 years for the heinous crime of “allowing people to speak privately to one another in a manner the EU cannot readily surveil.”
I find that concerning. We'll see how this plays out.
Quoting Ludwig V
But we sure can yak!
Quoting Ludwig V
I don't like online abuse. Or like that great Rolling Stones line ... "I went down to the demonstration, to get my fair share of abuse." Love that line.
Quoting Ludwig V
Resent? Not sure what you mean. I support the First amendment. One of the best things about the US. I believe freedom of expression is one of the most basic and vital of all human rights. It's under attack all over the so-called liberal west. I find that troubling. I see no resentment there. I see the defining political issue of our time. The freedom of the individual to say what's on their mind.
The classic question here is whether you have no objection to someone shouting "fire" in a crowded theatre or stadium when they know darn well that there is no fire. (Thus causing mass panic and distress, injury and death) Nobody doesn't limit free speech. The only question is what limitations are appropriate.[/quote]
I hope you know, and as a professional philosopher you should know, that this is a bad example, was never a principle of law, and isn't about what you think it is. Even Wikipedia has a decent writeup.
https://en.wikipedia.org/wiki/Shouting_fire_in_a_crowded_theater
Another good writeup:
https://www.theatlantic.com/ideas/archive/2022/01/shouting-fire-crowded-theater-speech-regulation/621151/
You are wrong to use that example. It's totally weak and incorrect argument. It does not mean what people think it means. It's not illegal. It was never illegal. The legal ruling in which it appeared has long been overturned.
In the US, direct incitement to violence or unlawful action is illegal. Just about anything else, no matter how vile, is legal. Of course that is under attack these days.
Quoting Ludwig V
Thorny enough that Brits are rioting. Americans haven't gotten to that point yet. America's a big place, you can drop in ten million foreigners and the disturbances will still be local.
Quoting Ludwig V
The Biden-Harris administration had an open border. They tightened it up this election year when it became a political problem.
No nation can have an open border when the people coming in are by and large illiterate peasants with few work skills and massive social needs.
Quoting Ludwig V
Counterintuitively, I'm a libertarian on that. I believe in the free exchange of labor. But Mexico is not exporting brain surgeons. They're exporting illiterate peasants. You have much the same problem in your country, along with a certain degree of anti-western religious feeling. Not by all, but by some. You did hear about that stabbing in Germany. At the "diversity" festival no less. God is a joker.
Quoting Ludwig V
No, I disagree with all of that. I don't claim to have the answers. I'm pro-freedom. If you abolished the welfare state I'd be for open borders. Some happy medium. Fewer social services in order to discourage people from showing up who can't support themselves. With that proviso, I'd let everyone in who can make a contribution. That's actually my belief, not that I'm certain it would work.
Quoting Ludwig V
Not yours? People thrown in prison for tweets the government doesn't like?
Quoting Ludwig V
Not that benign from what I hear. We'll have to see how it plays out. You know the censors never stop with the people YOU don't like. When they came for the trade unionists I said nothing, etc.
Quoting Ludwig V
Is that right? It's all Meghan and Harry all the time over here. Probably because the New York Post is all over it. Another scurrilous right wing gossip rag I read every day.
Quoting Ludwig V
For sure she's a big upgrade over senescent Biden. She can stay up past 4pm and whip up a crowd. We'll see how long she can get by without ever having a press conference or an interview. She can read a teleprompter very well, but she's often a disaster when speaking extemporaneously.
Quoting Ludwig V
Trump is old and seems tired and out of it. No question. He's 78, nobody should be running for president at that age. And whether you think the lawfare and impeachments and Russiagate have been justified or not, he's been under enormous stress for eight years. Most humans would have long since been broken.
But yes he is the old and feeble one now.
Quoting Ludwig V
I agree. Then again there's that German stabber. Islamic terrorists have taken credit. People don't ike that kind of thing and it only takes 1% to ruin it for the rest. Not very fair to the 99% of hard working, loyal, peaceful Muslims in Europe.
Quoting Ludwig V
I see no checks on his power at the moment.
Quoting Ludwig V
I like Nigel. He's fighting the emerging globalist government, as is Trump. The globalists talk like leftists and rule like ... well, fascists. Without jackboots.
Comes down to what responsibility platforms have. Being litigated all over the world at the moment.
Quoting Ludwig V
I don't really know much about it. I heard he got a bad deal. Still, corrupting the youth. That's the kind of charges agains Telegram and other social media companies. "Disinformation." Who decides what that is?
Quoting Ludwig V
Ok!
Quoting Ludwig V
Nobody can figure out why he landed his private plane in France. Perhaps he expected to get arrested and wants the legal fight. Personally I spent a night in jail once and did not like it. I wouldn't go to jail to prove a point.
Quoting fishfry
Point taken. If Government and Corporations are collaborating, normal people don't stand a chance.
Quoting fishfry
There was a landmark case in the US about this. The difference is that platforms (internet, phone, slowmail and, I think, couriers) are not responsible for the content of what they carry, only for delivering it. But Government can intercept and read them. Newspapers and publishers in general (broadcasters as well) do have responsibility for the material they publish; I think the difference is that they have editorial control over it, i.e. pick and choose what they publish. The point about platforms is that they don't pick and choose. The internet providers won the case, and have been dodging the small print about Government access ever since.
Quoting fishfry
Nigel is indeed very likeable when you first meet him. When you get to know him better - not that I know him, but I have followed him and had him pushed in my face for quite a while - you may well decide that he is a sleaze-bag. I doubt if he seriously cares about anyone but himself.
Quoting fishfry
Checks on the power of the Prime Minister in the UK are mostly behind the scenes.
Quoting fishfry
I'm very mindful of that.
Quoting fishfry
That would be worrying. But people setting up a meeting with the intention of rioting - those I worry less about.
Quoting fishfry
Happy medium is exactly right - but also the problem. You do know, don't you, that illiterate people can also make a contribution? Not sure that reducing welfare for everyone in order to discourage immigrants would play very well in politics.
Quoting fishfry
IS have claimed responsibility for events that they had no hand in. On the grounds that anyone who does something they approve of is a supporter. I'm not sure where that issue has got to now.
Quoting fishfry
Yes, indeed. It's not a popular theology, but the ancient Greeks believed it and the Vikings had a special god, Loki, for mischief. They reckoned that one of the primary functions of human beings is to provide amusement for the gods. Not a bad idea. Conventional heaven seems rather boring.
Quoting fishfry
I'm not surprised. But once you have conceded that, it's just a question of what and where. Not that it's an easy question.
Quoting fishfry
Well, I was never talking about the law as such. I didn't know about the Supreme Court. My intention was to use a cliche as a quick way of making a point.
This was more what I was gesturing at, but more as a moral criticism that a matter of legal action.
I do have a problem about restricting that. Freedom of speech includes the right to give offence.
Quoting fishfry
And I agree with that. It's not contradictory. The reconciliation is that it seems only natural that if someone insults and abuses me, I would want to deck them, but that would be to lose the argument, so instead I would try to make them shut up. In a democracy, if that's the will of the people, I won't object.
Mark Zuckerberg was in the news today, sending a letter to Congress admitting that he was pressured by the government to help cover up the Hunter Biden laptop, which probably swung the 2020 election to Biden. He said he regretted being part of that cover up. Too little too late but better than nothing.
The Internet, which we all naively thought would be a tool of our liberation, instead turns out to be the instrument of our enslavement. China's social credit system on steroids, coming to a gulag near you.
Oh well you'd probably just say I "resent" that boot stomping on my face, forever. I should just get with the program and love Big Brother. I have no other choice anyway.
Quoting Ludwig V
Right. Section 230.
https://en.wikipedia.org/wiki/Section_230
Quoting Ludwig V
Right. But it's tricky. Nobody, not even freedom-loving and rule-resenting me, thinks online platforms should be allowed to carry criminal material.
You know the reason I'm a little triggered by you saying I resent rules is because it's true. I've always been this way, always a rebel against authority.
Quoting Ludwig V
I admit to being taken in by his superficial charm. Plus the UK is getting pretty stabby lately and the public is not happy when the only people going to prison are the ones calling attention to it.
I see no checks on his power at the moment.
Quoting Ludwig V
I'm not talking about people actually inciting riots. There are old ladies being tossed in jail for much much less.
And again -- in the US, the ruling class cheered on the Floyd riots and threw the J6'ers in solitary. So it's two-tier policing again.
Quoting Ludwig V
Milton Friedman said you can't have open borders and a welfare state. That's the point I'm making.
In fact in the abstract, I'm an open-borders type. I say let everyone go where they like, but don't give anyone handouts. Then the productive people would gravitated to to the most free-market jurisdictions.
But of course that's not practical, because when people show up you can't just let them starve in the streets. So my solution is purely theoretical and idealistic. In real life, I'm just glad I'm not a big-city mayor, I'd have no idea what to do.
Quoting Ludwig V
Yes true. But the stabber was an Islamic refugee. And the German people are unhappy, hence their own anti-immigrant movement.
You know I like immigrants. If the government would impose some order on the system, it wouldn't be creating a right wing backlash. I don't like racist hooligans. But we have to try to grapple with the government policies that they are reacting too.
Quoting Ludwig V
Yes definitely. God has a sense of humor.
Quoting Ludwig V
Pretty vile speech is affirmed over and over again by the US Supreme Court. It's a principle not often supported any more even in the US.
Quoting Ludwig V
Yes sorry hope I didn't overreact. I did happen to read about the fire in a crowded theater example, and it turns out it was never against the law, and it was only kind of a sidebar issue to some legal case that's long since been overturned anyway. So everyone uses the example incorrectly.
What you can't do is incite everyone to murder the theater manager. That's a direct incitement to violence.
Quoting Ludwig V
Yes I'm sure they'd throw the book at someone for doing that. So maybe it is illegal after all. I have no idea.
Quoting Ludwig V
Yes right. Just don't let Two-teir Keir hear you say that :-)
And believe me, with Harris and the Dems a pretty good chance to get elected, free speech will be over in the US soon enough.
Quoting Ludwig V
I'm thinking of online mostly. I'm on Quora a lot arguing about the JFK assassination, and people just get vile about the most trivial differences of opinion. And sometimes I do the same thing. I'm trying to be nicer and more civil online. Been at it for about 24 hours now :-)
Some of the public are quite likely not happy. Others are more bothered by the rioting and are perfectly happy. Starmer has read the mood perfectly.
Quoting fishfry
Yes. I won't use it again. And I'm all ready to slap down anyone who tries to.
Quoting fishfry
I don't think he cares much what I think, and anyway, I don't think he's listening. But you never know. Everything leaks in the end. But I do choose carefully about who I raise it with.
Quoting fishfry
I can see your point. The problem is that whether you cheer on the rioters depends on whether you agree with them. You and I don't have to be impartial, so that's ok. Law enforcement does. But it's nigh on impossible, but I think most of them do try.
I do think it is hilarious to hear Trump bleating on about how all the prosecutions against him are political. I don't know whether or how much they are influenced by political considerations. The thing is, he wants to make all prosecutions political, by appointing people who agree with him politically to, for example, the Supreme Court and throughout the legal system. What matters is whether he is guilty or not - the fair trial. He does the same thing about elections. If he likes the result, he accepts it. If he doesn't he decides that the ballot was rigged. His losing the election is not evidence that the ballot was rigged. He's not the only one, but he's the most prominent one.
Quoting fishfry
I agree with all of that. The liberals focus too much on the individuals and the hard-liners too much on the numbers. There's a real need to balance and consensus.
Quoting fishfry
Where would we be without rebels against authority? But choose your issues.
Quoting fishfry
You do like the contentious topics. Yes, some people are very trigger-happy. I find "Let's agree to disagree" followed by ignoring them works quite well.
I've seen a bit of Quora (and Reddit). They look a bit too much like snake-pits for me.
Quoting fishfry
Don't we all? But sometimes there is a deeper issue - the arrogance of the opinion or its wilful blindness, for example, rather than its content.
Quoting fishfry
The first day is the hardest. The hard thing is to disagree nicely - especially with sensitive people. But if you can, you might actually persuade the other side to move a bit.
"If you object to stabbing six year old girls to death, you just might be a right winger."
Quoting Ludwig V
Not today. Today, he's putting people in jail who express ideas you don't express. So you let me know when an authoritarian regime has ever known when to stop. As he was consolidating power, Stalin killed his most fervent supporters. Hitler did the same.
Quoting Ludwig V
What makes you think that? All digital communicates get stored. Nobody looks at them till your friend's friend's friend's friend whom the government doesn't like, steps out of line. Then they roll up the whole chain. Like I say. Find me an authoritarian regime that ever knew when to stop.
Quoting Ludwig V
Doesn't matter. Some friend of a friend might say something the government doesn't like. Your argument here is, "Who cares if someone else goes to jail for saying something the government doesn't like. They won't do that to me." History has not been kind to that argument.
Quoting Ludwig V
I don't want to keep discussing this. Floyd versus J6 is just as blatant an example as you can find. Two billion dollars in property damage and twenty dead; versus a few old ladies wandering aimlessly around the Capitol building. Many MANY completely nonviolent J6 protesters have been in solitary confinement for three years. This is an outrage; and a bigger outrage is that it's not generally recognized as such.
Quoting Ludwig V
I absolutely and without reservation share his bleats. Even liberal legal scholars have been outraged by the New York 34-felony case. It's a legal travesty, the kind of thing you see in banana republics.
Quoting Ludwig V
100%. None of those cases would ever have been brought if Trump weren't Trump.
Quoting Ludwig V
Bullshit. You're just spouting leftist propaganda. It's not worth my time to have these arguments.
Quoting Ludwig V
Stop. Please. Just stop.
Quoting Ludwig V
Man I've been hearing this leftist claptrap since 2016. Enough already. I don't begrudge you your beliefs. I do choose not to engage with them.
Quoting Ludwig V
Ok whatever.
Quoting Ludwig V
I am. Today, these ain't them.
Quoting Ludwig V
I like the math and computer sections of Reddit. Quora is a pale shadow of its former self.
Quoting Ludwig V
I would say you have much willful blindness about the Democrats' corruption of the justice system to go after Trump. But then I'd be arguing this tedious subject again.
Quoting Ludwig V
I have never persuaded anyone of anything in decades online :-)
I saw this today.
https://www.telegraph.co.uk/news/2024/08/26/violent-offenders-increasingly-let-off-with-apology/?ICID=continue_without_subscribing_reg_first
It's about how the Brits let stabbers go if they apologize to their victims. Meanwhile, old ladies who say the wrong thing online go right to prison.
Maybe it's all lies. How would I know, right?
Quoting fishfry
Quoting fishfry
Yes. I do worry about that argument. But since Stalin was on the left and Hitler on the right, it seems like there's no safety anywhere. Any more than there is against the possibility of all-out nuclear war (or indeed against the reality of climate change) These things are hard to predict.
Quoting fishfry
Yes. I expressed myself badly. Perhaps I was in a bad temper. My point was that most people are sore losers and it's very hard to tell when a protest like that is valid.
Quoting fishfry
I'm afraid the Telegraph has been tracking my viewing of its articles. There's a limit on free views of them and I've hit it. But I do know that there was a case like that and there was a lot of reporting of it. I don't pretend to know the rights and wrongs.
Quoting fishfry
Quoting fishfry
Quoting fishfry
I rather think you have a bad day. I'm sorry about that.
There's safety in free speech and a limited, Constitutional republic. Me and Thomas Jefferson against the world.
Quoting Ludwig V
Sigh. I probably shouldn't reduce your esteem for me any more than I already have, but I'm not much of a climate fanatic, either. The question is whether we should wreck our economy and throw billions into poverty to effect a hypothetical fraction of a percent change in the average global temperature, which is ridiculously hard to measure anyway.
The air and water are a lot cleaner than in the 1970s, so I'm all for the environment. I love the environment. Just not the radical environmentalists.
Besides, the Obamas own beach front property in two states (Massachusetts and Hawaii), so clearly they're not too concerned with the rise of the oceans. Besides, warmer temps are GOOD for life and colder temps are BAD for life. So a lot of what passes for environmentalism these days is ass backward.
The world is stumbling into nuclear war. US foreign policy is a bloody disaster.
Quoting Ludwig V
The lawfare against Trump is wholly illegitimate and many liberal legal minds have so opined.
I am a disillusioned liberal. Still a registered Democrat. I'm just horrified by what's become of my former fellow liberals and Democrats. Some of them see it and most of them don't.
Quoting Ludwig V
I don't either. We'll all find out how this plays out in the next few years. No question that the liberal governments of the West have decided to throw open their borders to hordes of people who don't share their traditional values.
You know Christopher Lasch's book. The Revolt of the Elites? The idea (I haven't read the book -- I no longer read books, only Wiki pages and articles in scurrilous right wing rags) is that rather than the people revolting against the elites, these days the elites are revolting against the people. Hard to argue with that thesis, we see it all around us.
https://en.wikipedia.org/wiki/The_Revolt_of_the_Elites
Quoting Ludwig V
LOL. For a while I was trying to engage over in the political threads in the Lounge, but it's just a bunch of mindless checkbox liberals throwing insults. So I gave up. I am a little burnt out on the standard liberal talking points against Trump. Heck I don't even like him much, he's old and tired and bitter. But he's all we've got against the continuation of what's been going on.
Apologies for getting triggered :-)
There is, indeed. It may not be perfect, but some arrangement like that is all there is.
Quoting fishfry
Don't be ridiculous.
Quoting fishfry
It was always obvious that dealing with climate change would be a mess, and that it might well be ineffective. We can probably organize some response after the event. There will be some mitigation, but nothing less that world-wide panic will trigger serious attempts at mitigation and that won't happen until serious climate change has kicked in. As usual, the poorer countries will suffer most, and much of their population will leave, looking for somewhere safer to live. There'll be a lot of trouble.
Quoting fishfry
Fair enough. We can achieve things. It's just that it takes a disproportionate amount of shouting and shoving to make things happen. It helps when people can see the effects themselves. (see above)
Quoting fishfry
Yes. Temperate. So too hot and too cold are both problems and climate change will cause more of both. But the temperate north and south of the world will be less badly affected than the equator and tropics - apart from the effects of sea level rise and the increase in extreme weather events.
Quoting fishfry
No. I looked at the wikipedia article. It seems quite plausible. But I'm very difficult to convert. I'm going to be reading "Techofeudalism" soon, in a futile attempt to keep up to date.
I don't know about you, of course, but I was liberal when liberals were a minority and thought to be insane. Then things starting going our way. Splendid - until I realized that younger generations would want to push everything further. I've gone some way with them, but not all the way. Much of what they are pushing for now seems to be dubious, at best. They don't remember what it was like to be what it is to be an oppressed minority, so they feel no need to compromise and make room for different views. But hey! no-one listens to doubts and compromises any more.
Quoting fishfry
Now that Biden has gone, the context has changed. He looks different in a different context. I think you'll find that the right wing will get some of what it wants - not all. That's what's happened to liberalism. Life has to go on and forces compromises. Remember, liberals are as fearful as conservatives.
You are agreeing that free speech is a virtue then. Yet you don't seem too bothered by the globalist war on free speech.
Quoting Ludwig V
It's a great heresy to be against the environmentalists these days. But of course IMO one can be against the environmentalists yet for the environment. That would be me.
Quoting Ludwig V
The poor countries suffer from radical environmentalism. When you raise the cost of energy, the limousine liberals aren't affected. The poor are. And the third world suffers the most.
Quoting Ludwig V
The effects are virtue signaling among the first world elite; and terrible suffering in the third world, out of sight. This is my point. I oppose the environmentalists.
Quoting Ludwig V
I don't know how we got here but environmentalism isn't one of my favorite conversational topics. I know what I think and I don't bother to talk about it much.
Quoting Ludwig V
I shall read the Wiki page :-)
Quoting Ludwig V
Right. But most longtime liberals haven't noticed. They've gone from gay rights (good) to transing the kids (bad) without missing a beat.
Quoting Ludwig V
Liberals are stupid and mean these days.
Did you see the Kamala "interview?" If the Democrats get away with this the country is doomed. Not just policy-wise. But that Americans would have validated the four year Biden swindle, propping up a senile candidate who campaigned from his basement; and then swapping in the historically unpopular Harris, hiding her from the press while her fans swooned. It's very bad if they get away with this. And honestly, not too much better if Trump wins. He's past his prime for sure.
That's not quite fair. I do agree that free speech is a Good Thing. So I am bothered by Putin and Xi Jinping. But I don't think that criminals should be allowed free access to their victims
Quoting fishfry
The song but not the singer. I don't disapprove of some enivironmentalists, but I do get bored with them.
Quoting fishfry
The truly depressing thing is that the poor are screwed by climate change and by the attempts to reduce it.
Quoting fishfry
I'll shut up about it then (after this reply!)
Quoting fishfry
Well, not to go on about it, I can accept that there is some work around trans people to be done. But the recent publicity has been provoked by some thoroughly objectionable trans people (and some "trans" people). My partner has some acquaintance in those circles and tells me that many trans people just want a quiet life and are horrified by them.
Quoting fishfry
The really basic question is why there is no decent candidate on either side. All the people who might have make a good shot at an impossible job seem to have taken a back seat.
Driving the other day a car revs up behind me while I am going a little over the speed limit, then barrels around me into the oncoming lane on a curve. I wonder if the answer to your question has something to do with a general lack of patience. Everything has to be done as quickly as possible it seems. Patience is no longer a virtue. Just a thought.
Maybe. Impatience is a big driver of the way that debates go. The media (or their readers) do not have the patience for going slowly and paying attention to detail. Everything has to be a slogan - three words - preferably monosyllables and no more than two syllables.
My theory is that the people who might make a fist of the job are reluctant to take it one. One of the things that has changed in the last 3 or 4 decades is that the media scrutiny is much more effective and much, much noisier.
Recent developments in the West are very concerning. Robert Reich, Clinton's Secretary of Labor, just called for "reining in" Elon Musk.
https://www.theguardian.com/commentisfree/article/2024/aug/30/elon-musk-wealth-power
Famous law professor Erwin Chemerinsky just published a book calling for dumping the U.S. Constitution.
https://www.theguardian.com/books/article/2024/sep/01/erwin-chemerinsky-no-democracy-lasts-forever
There are many other examples. You talk about Putin and Xi but you don't seem concerned about the creeping -- actually now galloping -- authoritarianism and censorship in the west. I'm very concerned; you much less so. So I don't think my point was unfair. For a Brit to ignore these issues lately I find very strange. They're putting people in jail in your country for very anodyne online comments.
Quoting Ludwig V
With you there.
Quoting Ludwig V
My very point. Environmentalism is elite virtue signaling.
Quoting Ludwig V
Well yes, of course. It's always the extremists who make the news.
But doctors are doing double mastectomies on perfectly healthy 12 year old girls. That's something tht needs to be pushed back on.
Quoting Ludwig V
Longstanding problem. Bush-Kerry. Trump-Clinton. Trump-Biden. Trump-Harris. etc.
I'm kind of running out of steam on this site. Might need to wrap this up soon.
Why does that concern you? Everybody who has power has an opposition. The opposition always thinks that those with power should be "reined in" or crushed. (Actually, if you think about it, that's really a very mild comment compared with what some people say). Most people with power are either "reined in" by the opposition or their own failures. I've no idea whether Musk will be reined or crash and burn. At the moment, it's impossible to tell which it is to be. The sooner the better, as far as I'm concerned. There'll only be another like him afterwards.
Quoting fishfry
It depends what you think is anodyne. Compared to the way that some people carry on (without being thrown in jail), it probably is anodyne. But most people's comments are just hot air - unpleasant, but not harmful. Look at the consequences.
There was a famous speech in the 60's by a Conservative politician named Enoch Powell, in which he drew everyone's attention to the flood of immigration into Britain, painted a terrible picture of the abolition of the "British way of life" and announced that there would be "rivers of blood" in the end. Was he reporting? Or was he inciting? I don't know what his motivation was, but I know what happened as a result. It wasn't rivers of blood, but it did involve bloodshed and it was very ugly.
You may have seen the reports of the report released about the fire in Grenfell Tower. Everybody is very shocked and horrified. In a way, so am I. But I have known it was coming ever since the then Government relaxed the building regulations. It was only ever a matter of when and where. It was obvious. It was also always obvious that when it happened most people involved would say it was not their fault, even though it is obvious that they all contributed. No clean hands.
There has never been a golden age when there was no censorship, no authoritarian squelching of opposition. It was ever so, it will always be so.
I'm a somewhat old-fashioned middle-of-the-road liberal and I felt more comfortable 20 or 30 years ago. I grew up in the post-WW2 consensus/settlement. It was never what it seemed to be and it fell apart anyway. (If you want a date, it was the election of Margaret Thatcher in 1979 that did it.) Once that has happened to you, you never, ever buy in to anything else with the same innocent, deluded conviction.
Quoting fishfry
If you do decide not to continue, that's fair enough. I wouldn't want to (couldn't) detain you if you have better things to do. So long as you aren't leaving for the same reason that you left the Lounge. Better to let me know when you make your decision, so's I know what's going on. If and when I make the same decision, I will let you know. OK?
I'm beyond explaining this. Let's agree to disagree.
Quoting Ludwig V
Eminently sensible and moderate.
Quoting Ludwig V
You're trolling me now. I'm kind of done here. I can always tell when I'm at the point when I have nothing else to say without repeating myself.
Quoting Ludwig V
Then why is Starmer throwing pensioners in jail for remarks that are unpleasant but not harmful? But like I say, I'm repeating myself.
Quoting Ludwig V
Yes I remember Enoch Powell. Don't recall the incident you're referencing.
Quoting Ludwig V
Now we're into building regulations? Not following. The US infrastructure is likewise decrepit. Gotta fund the wars, you know.
Quoting Ludwig V
Ok. I can't respond with anything I haven't said before. You are justifying evil by saying there's always been evil. Fine.
Quoting Ludwig V
I'm a big fan of Maggie as you might imagine. Though I wasn't at the time. Both she and Reagan look much better in retrospect. I used to be a liberal too. Something happened over the years.
Quoting Ludwig V
Well ... they're a lost cause over there.
In this case, I just see that I haven't said anything new in quite some time.
Quoting Ludwig V
Well ... I guess I'm done. But I've never had a long private convo like this. You could post something on the public area, at least that way we'd get some fresh meat once in a while.
I literally can't think of anything to say that I haven't already. Kamala came out today railing against Elon Musk's freedom of speech. What should I do, say I object and plan to vote against her? Doesn't matter anyway, I vote in California which will go for her by millions of votes. My vote literally doesn't count.
So I guess I'm done. More for lack of anything new to say than any other reason. Appreciate the chat. But I hope we can engage in public where others can jump in. I think it's weird that the moderators buried this chat.
ps -- I haven't anything better to do!! LOL. Am I leaving too soon for your taste? I don't mean to be short. I just haven't got anything else to say. Maybe my concerns about the creeping authoritarianism of the globalists is misplaced. I can sum it up in a cartoon I saw the other day.
Not intentionally. If I've upset you, I apologize.
Quoting fishfry
Yes, that would be good. But maybe other people prefer something noisier - more exciting.
Quoting fishfry
That's what happened to me.
Quoting fishfry
True. I still vote, but my expectations are low. It's more of a ritual than anything real. And yet...
Quoting fishfry
It's just that I'm so angry about the total mess and the expectation it won't be solved.
Quoting fishfry
Here's my most depressing thought. Tyranny and freedom are not opposites. What's tyranny to you is freedom to someone else. What's freedom to you is tyranny to someone else. Oversimplified, I know - there's always compromise. Which is not a solution, just a way of making do.
Quoting fishfry
NOT justifying, I'm trying to work out how to live with omnipresent evil, without indulging in cop-out evasions - blaming Government or Capital or Original Sin. I think I'm closest to Voltaire's "Candide"? Or Kurt Vonnegut's "so it goes" - or perhaps Hamlet's "The rest is silence". Yet obstinately and stupidly, life goes on. It's better than the alternative, I suppose.
Quoting fishfry
You've been saying that for a while now. I'm in the same boat. So now we're talking about the fact that neither of us has anything else to say. Absurd, and yet, here we are.
No worries, I no longer remember.
I did happen to run across something yesterday. The British government put out a big report on the Grenfell disaster.
The Spectator put out a summary blaming the incident on "complacency."
https://www.spectator.co.uk/article/the-vital-lesson-that-must-be-learnt-from-the-grenfell-inquiry/
Spiked-Online noted that the reason the tower burned was that it was wrapped in flammable cladding that had been installed for environmental reasons. In other words the building itself would not have burned but for the cladding that had been wrapped around it as insulation. And now the government is busy removing the flammable cladding from other buildings.
So the loss of life was attributable to liberal do-gooding. Needless to say the official report did not make this point. Thought I'd pass this on.
https://www.spiked-online.com/2024/09/05/why-was-grenfell-covered-in-cladding-climate-targets/
Yes. There's been a lot about it in the media in advance.
Quoting fishfry
H'm. The author says that's his view, that's true. But if only it was just complacency. There was a lot worse than that. Gaming the already lax building regulations - next door to fraud. Ignoring tenants complaints. And on and on. But thanks for the link.
Quoting fishfry
If you look a bit closer, it was partly for environmental reasons and partly for economic reasons. Insulation saves money. When they talk about sustainability in these contexts, they often don't distinguish between something that pays back in the long term and something that is needed for climate control. Insulation ticks both boxes, so it can be hard to discern which they mean. But I would bet it was not climate control what was uppermost in their minds.
Quoting fishfry
If only it was. Progress is glacially slow because everybody is arguing about who should pay. The Government thinks that the industry should pay; the industry thinks the Government should pay. Meanwhile, the companies that designed and manufactured the cladding and sold it on the basis that it wasn't flammable are in deep trouble, but paying to put right what they've done would almost certainly bankrupt them - i.e. they can't pay. Some landlords of long-lease flats (their tenants are responsible for maintenance) are trying to make their lease-holders pay.
The police will take until the end of next year to decide whether there will be any criminal prosecutions and nobody will accept liability until that's settled. Then there may be civil suits for damages, which will take more years. Don't hold your breath. (Yes, some building have been done, but very few compared to the number affected.) Government (both parties - the seeds of this were sown in the 1980's under Thatcher and subsequent governments never put it right), the Local Council, the building industry generally, the companies that manufactured, sold, and installed the stuff, and even the fire brigade are all blamed in the report.
The fire itself was spectacularly awful. 72 people died, which is surprisingly low - thanks to the fire service. But the aftermath - if you don't laugh, you'll have to cry. No-one seems to have a shred of decency - always excepting the tenants.
Quoting fishfry
Oh, please! If there had been any do-gooding at all involved, it wouldn't have happened. It was greed and laziness. Complacency, if you like, in that Government trusted the builders to do the right thing.
Thanks for the opportunity for a good rant. I hope I haven't bored you.
It's a general theme of mine that environmental do-gooding generally results in disaster and misery.
Quoting Ludwig V
Your response was interesting. Clearly you're getting more and better info about his tragedy over where you are. I have to depend on my alt-right sources.
Well, change usually brings disaster and misery to the most vulnerable people, and the rich are mostly not the most vulnerable, so you're not wrong. Some environmental do-gooders claim to be trying not to inflict any additional disaster and misery on the poor and vulnerable and claim also to be succeeding to at least some extent. But of course many people approach the whole business on the basis that it's a profit opportunity and act on their priorities. (Did you notice all the reports a while ago about how China has more or less cornered the market for rare metals, and looks like dominating the market for electric cars - which it makes with power from coal?) That's my the-glass-has-a-drop-of-whisky-left message for today.
Quoting fishfry
Well, it did happen here. The full report is over 1,500 pages long. Only fanatics and people paid to read it will plough through that. But I haven't heard a single complaint that it is prejudiced, thought the government is trying to defend itself the best way it can - it makes things easier for the politicians that every government since Thatcher is blamed. The commission's own summary is probably more than you want, but it is at Grenfell tower report executive summary and recommendations
The Telegraph is Conservative aligned. The (London) Times and Financial Times are not too bad, but are Conservative-leaning. The Guardian is liberal. The Independent is not reckoned to be aligned to a political party, but that doesn't mean it doesn't have a view. All of them are reasonably reliable. The BBC tries hard to be impartial so everyone thinks it is opposed to them, which is a good sign, I think. I tend to use that. My reaction was based on their reports. Here's their outline:- [quote=BBC News at 17:06 BST 4th Sept] * The inquiry's chairman says that all deaths in the fire were avoidable
* The inquiry blames "decades of failures" from governments, firms and the fire service for the disaster that unfolded in west London
* Grenfell residents were badly let down by those responsible for fire safety and there was a "failure on the part of the council"
* Manufacturers of cladding products – which were "by far the largest contributor" to the fire – are found to have engaged in "systematic dishonesty"
* The report also says that "incompetent" companies involved in the 2011 refurbishment of the tower – Studio E and Harley Facades – bear "significant" responsibility for the disaster
* The report said there was a "chronic lack of leadership" and an "attitude of complacency" at the London Fire Brigade
* The victims of the Grenfell Tower disaster were killed by toxic gases, not the fire itself.[/quote]
There's also a lot of comment on the slow progress of remediation - seven years after the actual fire. 4,630 residential buildings are involved. 29% have completed remediation. 20% have started remediation. 50% have not started remediation. Tens of thousands of tenants. That puts all the stuff about it not ever happening again into perspective, don't you think?
There doesn't seem to be anything about the races for the Senate and the House. But isn't it just as important as the Presidency? I have the impression that unless the President and Congress are the same party, the President is pretty much hog-tied. What's happening there? Is it as tight as the Presidency?
I'll intrude if you don't mind. The answer is yes, indeed. Here in southern Colorado our representative to the House, Two Gun Lauren Boebert, has moved north into another district. We do have a decent Republican candidate, but I have seen and heard nothing about him. The Democrat running is more in the old fashioned mold and will receive a lot of conservative votes I suspect. Nevertheless it will be one less Repub in the House - and the ratio is tight.
Issues involving new allocations of money begin in the House, so control there is critical.
and I, apart from a mathematical background, align on political matters it seems to me. I am still a registered Democrat, but it has been awhile since I have thought of myself as one.
Pretty tight in both houses. In general, the US economy does better when the opposition party controls Congress. The government can get into less mischief that way.
I'll pass on the details of the tower tragedy. Just thought it was interesting that a root cause was environmental do-gooderism, implemented badly or not.
Quoting jgill
Same here. It's the Dems who changed, not me.
Quoting fishfry
I can't really talk about the Dems, but I have the impression that the Dems, back in the day, were an alliance of (mainly social) liberals and political left wingers; there was also a lot of support in the South, which goes back to the civil war. If that's true, there's a very similar phenomenon in the UK. The Labour party has always been a rather uneasy alliance between those two points of view. It's not unreasonable, because both were in opposition to existing orthodoxy, just on rather different grounds and with rather different aims.
The last ten years in the UK were largely based around this problem, which got hugely focused on Brexit, though it was never just about that. The promises have been revealed as fantasies, and now we have to make the best of a bad job.
The problem arose because of the success of the social liberal movement, which became a new orthodoxy, in many ways, but also transformed. Many social liberals became successful and powerful and so acquired a new point of view. "Socialism" became more of a threat to them and they espoused free market ideologies, which had been their route to success and became their security. This left the left wing isolated and nearly powerless.
Do you (two) think that I'm talking rubbish, or does this fit with what has happened to you?
For me, it does fit, with one further development since the good old days. The liberation movements since the successes between, say, 1950 and 2000 have moved on as the generations have changed, and some of the demands and expectations seem to me more problematic than the original demands. I'm very hesitant about this because I am just an old fogey who has fallen behind the times. Nonetheless, I'm not comfortable. But I'm even more uncomfortable with what the free market ideology has become.
Quoting Ludwig V
Far from it. I grew up in a segregated South and the Democratic party supported that. Political winds finally shifted during the 1960s.
From Wiki:
I was in a math class at the University of Alabama in 1963 when Governor Wallace was asked to step aside and allow two Afro-American students to enroll. He complied and those of us on the sidelines cheered. An old Confederate cannon went off at the time, but I can find no reference to that.
My first vote for President was the 1960 election, and I caste my ballot for JFK. He had been a genuine war hero, and when he extended my tour in the USAF for a year because the Berlin Wall was going up I forgave him. Turned out it worked out well for me.
Yes. I knew that. I'm sorry I wasn't clear.
What I didn't know is that most blacks (I'm assuming you mean most blacks who weren't slaves) belonged to the Republican party before the war. I'm confused now.
Perhaps I'm just making a mistake trying to apply the political divisions that apply now to politics then.
I should read up on this more carefully before trying to talk about it.
Quoting jgill
I remember reading about that. Some of us thought there would be another civil war. I don't remember the reports saying that people cheered when Wallace gave in. That very good to know. It was also my first year at University. Do you think the cannon was a protest or a celebration? Presumably, it didn't have a ball, but was loaded blank?
Quoting jgill
I was almost completely apolitical. That didn't change until 1968. Remembering those terrible yet exciting times makes me a bit less worried now.
The Democrats used to be the party of the working class. They've become the party of the wealthy liberal elites and the poor who benefit from government services.
In other news from merry old England, I hear Labour has it in for the House of Lords.
Don’t ‘reform’ the Lords – abolish it
https://www.spiked-online.com/2024/09/12/dont-reform-the-lords-abolish-it/
There were several Civil War cannons on the lawn of the old armory where the ROTC had made its home since before that conflict. I'm certain one fired a blank at that time. It's funny but I cannot find any reference to it in the media. But I think it would have been a celebration. Most people at the University disapproved of the Klan, and there had been some speculation the KKK might get ugly, but they backed off and were more or less silent.
I had an older cousin who lived in the country and was a member of the Klan. He had worked at some menial job and kept dogs for coon (racoon) hunting. His kitchen sink had a manual pump and chickens ran loose in his dirt yard. When I went out to visit him in 1963 with my fist wife I was astounded in the transformation. He now lived in a nice brick home and when we were met at the door he was neatly dressed and introduced us to one of his best friends, a black man who worked with him at the BFGoodrich plant nearby. The last time I had seen him was in the late 1940s when I was a child. The plant had opened in 1946 and he was still living the rustic life and a member of the Klan then.
Thank you. That may be short, but it gets to what I was trying to say. And then I was trying to say that Labour has exactly the same problem. The working class, represented within the party by the unions, used to be represented by Democrats/Labour. But, since around 1980 (Thatcher/Reagan), that has gradually declined (basically, I think, as the power of the unions declined). The assumption was always that the working class would align with the poor and socially liberal ways, but that was simply false. Many of the working class do not think of themselves as poor and are certainly not socially liberal, and they basically have nowhere to go. Mind you, another dimension of the problem is that most people are not only reluctant to think of themselves as poor, but also reluctant to think of themselves as working class.
Quoting fishfry
Good Lord! You'll be wanting to abolish the Monarchy next! That's not how we do things here! We don't abolish things! The two Houses started in 1341! How could they be abolished? Tradition, you know!
I like your "spiked online" article. It explains things quite clearly.
More seriously, though, Your constitution is designed to make sure that no-one has total power by splitting the power into three parts - legislators, executive, law. We don't have that. But we do have the equivalent. Any Government, no matter how large its majority, is restrained by thinking that there's no point in doing something that can be undone by a later Parliament.
The reason that abolishing the Lords is not on the agenda is quite simple. If Labour abolished the Lords, the Conservatives would recreate it when they get back in. I'm sure that the Labour plans have been discussed through back-channels with the Conservatives, and they have signalled that they would not reverse these reforms when they get back in.
Mind you, there is another problem. If you abolish the Lords, what do you do next? There's no consensus about that - never has been and likely never will be. If there was a consensus, it would be done. Same for the monarchy.
There is one thing that your founding fathers missed. The theatre of the state. Dressing up in funny hats and strutting around like peacocks in gaudy costumes. That's what all this is about. It's gratifying for those in power and entertains the masses - and provides assurance of stability when everything else is falling apart. Like a conjuring trick, it works so long as you don't look too closely.
Starmer's plans are unsatisfactory, but they are very likely practical, so they will have to do. Better than nothing.
The reform I would like to see is the abolition of the life peers (political appointees) or at least their ejection from Parliament. Imagine if the President could appoint people to your Senate! We have a phrase "kicked upstairs", which means being "promoted" somewhere you can't get in the way. That's what the political peerages are all about. The weird practice actually helps to get things done. So there's absolutely no chance of that.
I had no idea. I don't recall the Klan being even mentioned in the coverage here.
Quoting jgill
I bet you were. I don't suppose you ever had a chance to talk with him about what happened. Likely, he just wants to forget it.
From Britannica:
In a nutshell.
So the mid-20th century party is the one you joined?
Quoting fishfry
Do you agree with fishfry about what the party has become?
If so, that's also the story of the Labour party - and it's also the story of the last ten years or so in the UK.
I probably registered as Democrat when I voted for JFK. I didn't give it any thought at the time. LBJ began his political life in the Senate staunchly against civil rights legislation, but reversed his position as the tides began to turn.
Quoting Ludwig V
More or less. Frankly, I don't know what it has become since the "Squad" gained influence.
That's politics for you. But I always understood that he was very effective (more effective than JFK?) in getting Civil Rights legislation through.
Quoting jgill
I think the problem of our times is that the left wing isn't clear what it's about. So many goals were achieved and the fall of the USSR was taken as "disproving" socialism. The right, these days, at least knows what it's about - and is much more ruthless in fighting for it. The left can't form a united front or articulate a coherent ideology.