"Is there a way to describe various finite quantities without going right to numbers?" - answer, yes there is: we can talk about 'lots', 'few', 'big', 'small' etc. It's the same with infinities. We can talk about 'endless', 'unbounded', for example. Is there anything that is endless or unbounded? I think so. E.g. a race round a standard athletic track that ends only when there is no more track ahead of us. But is that what you are thinking about?
Agent SmithDecember 19, 2021 at 10:24#6328150 likes
Ad nauseam seems, is rather, related or is the same as ad infinitum.
Next time you're intoxicated by alcohol and feel the irresistable urge to hurl, you're, intriguingly it seems, experiencing the qualitative side to infinity.
AgentTangarineDecember 19, 2021 at 14:18#6328420 likes
Infinite means in-finito, not finished, never to be finished. What in life never finishes? The finish line can be pulled away from you indefinitely. Indefinitely=infinitely? On can tell a never ending story, play infinite games. The universe goes on forever, as life in it. It never ends. Infinite!
Is there a way to describe various infinites without going right to number lines? Anything in real life to reference in terms of the infinite?
Yes, we constantly refer to infinity in various ways without referencing math. Eternal, forever, and immortal are just some of the ways we express infinity.
I'm not sure if this is what you're looking for. Maybe you could explain more?
Spinoza (arguably) has a concept of a qualitative infinity right at the beginning of his Ethics, propositions 1 through 8 introduce it. Infinity as the total lack of limitations or constraints.
HeracloitusDecember 19, 2021 at 19:33#6329180 likes
Infinite means in-finito, not finished, never to be finished. What in life never finishes? The finish line can be pulled away from you indefinitely. Indefinitely=infinitely? On can tell a never ending story, play infinite games. The universe goes on forever, as life in it. It never ends. Infinite!
The interpretation of infinite I use as a mathematician (complex analysis) coincides with one employed in the physical world: unbounded. Were I to be a set theorist this probably would not be the case.
TiredThinkerDecember 20, 2021 at 06:18#6330660 likes
Is Infinity of running around a track indefinitely differ from running straight and having road added in front of you indefinitely?
Any examples of Infinity other than the suggestion that the universe may go on forever since we can't prove otherwise?
HeracloitusDecember 20, 2021 at 12:46#6331050 likes
Any examples of Infinity other than the suggestion that the universe may go on forever since we can't prove otherwise?
You were already given some examples in this thread. How about the Successor function as another example?
AgentTangarineDecember 20, 2021 at 18:50#6331880 likes
Is it a coincidence that the symbols for infinity and zero are similar? You can run around on both of them forever. But so you can on 8. On 4, 6, and 9, you can at least take a break on the the side.The infinite can never be reached. Every time you think you've reached it almost, it has resided to... infinity. So infinity is the unreachable. Can zero be reached? You can pass it on the way to negative, so I had exactly zero dollar, like I have right now. No dollar to be found in my pockets. Can you actually have zero dollars if you don't have them? Can the nothing be realized? You can take out all apples out of the back, but what if we dìvide them up? We can't do that indefinitely. But what about the vacuum? You can break that up infinite times. It consists of infinite points with an infinity of points in between them. Do you take points away when you divide it? Will dividing it up indefinitely leave you with nothing? Or with an infinite collection of points? If we assign real numbers to the points, how can two points touch? By bringing them infinite close to one another, so the decimal numbers coincide? The infinite line of real numbers needs an infinity of the infinity of the natural numbers to describe it. The continuum line can be said to have cardinal number [math]\aleph _1 [/math], while the collection of natural numbers has cardinal number [math]\aleph _0[/math]. The two-dimensional continuum has cardinal number[math]\aleph _3 [/math], and the 3D continuum corresponds to[math]\aleph _7 [/math].
The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements, which in the case of discrete points is one (so the ordinal in the cardinal number becomes 0). For the continous line, the number needed is two (ordinal subscript 1), for the 2D continuum it's four (corresponding to subscript 3), and eight will do for 3D continuous space (ordinal number 7, so [math]\aleph _7 [/math]).
Now we can play the same game with with the aleph numbers. For discrete aleph numbers, [math]\aleph _n [/math], the natural alephs, we can assign super aleph number [math]\aleph _{S_0} [/math], when the number of natural alephs needed is one. When 2 natural alephs are needed, for a continuous line of alephs, we get [math]\aleph _{S_1} [/math], for a 2D continuous plane [math]\aleph _{S_3} [/math], and for an n-dimensional continuous volume, when [math]2^n [/math] natural alephs are needed we arrive at [math]\aleph _{S_{n^2-1}}[/math].
Now we can play the game again for hypersuper alephs, [math]\aleph_{{HS}_n}[/math]. For an infinity of inf... well, ad inf.
So, the infinite can't be reached like zero can't be reached (if you don't include the negation of the positive real). Both can't be reached, and still some try to reach it while others, mostly unwillingly are pushed towards nothing. The desire to reach for the nothing is not so different from the desire to reach for the infinite, though the implementations of this in material life have quite different implications. Life and death, even.
Infinity is just as useless as nothing. In between is where the action is. Are the things in life that never can be reached infinite? Yes. My wife is one of them. And I have to admit, attempts made by pettifoggers and skirlers on their doodlesack never cease to bumfuzzle me.
Everyday examples of infinity. Maybe falling asleep and waking up. It seems that time tic-toc-ed infinitely fast in between. It seems nothing at all exists in between, another example that nothing and infinite have a close, if not intimate connection.
TonesInDeepFreezeDecember 26, 2021 at 03:09#6350260 likes
The continuum line can be said to have cardinal number aleph_1,
'the continuum' is probably most exactly defined as , but let's simplify here to just say it's R. It is the continuum hypothesis that its cardinality is aleph_1. It is not given or settled mathematics.
The two-dimensional continuum has cardinal number aleph_2, and the 3D continuum corresponds to alelph_7.
If by "the two-dimensional continuum" you mean RxR, then it is incorrect that its cardinality is different from the cardinality of R. If by "the 3D continuum" you mean RxRxR, then it is incorrect that its cardinality is different from the cardinality of R. For any natural number n>0, R^n has cardinality equal to the cardinality of R.
The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements
I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements" but the aleph notation is defined by transfinite recursion on the ordinals. For an ordinal k+1, aleph_k+1 is the least cardinal greater than aleph_k. For a limit ordinal L, aleph_L is the union of {aleph_k | k < L}.
AgentTangarineDecember 26, 2021 at 09:30#6350770 likes
I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements
If you need an infinity of infinites then aleph is 1. If you need an infinity of them, then 2, etc. The cardinal number of the continuum is defined on one dimension only. You really think the cardinality of the 2 or 3 dimensional space and the 1 d are the same? They're not. Aleph line is 1, aleph plane is 2 and aleph space is 3.
SpaceDwellerDecember 26, 2021 at 14:25#6351780 likes
It is a theorem that card(R) = card(R^n) for any natural number n>0. This is known by anyone who has read a basic textbook in set theory. Just read the proof for yourself.
Moreover, you keep claiming that card(R) = aleph_1, thus precluding that card(R) might be greater than aleph_1 and thus precluding that there might be cardinalities between card(N) and card(R) . That is only the continuum hypothesis, not settled mathematics.
AgentTangarineDecember 26, 2021 at 15:16#6351930 likes
Then you have to reread your proofs. Are you seriously implying that the cardinality of the 2-d continuous plane is the same as that of the continuous line? The cardinality of R is the power set of N. The cardinality of RxR is 2. There are obviously more elements in RxR than in R. That's why the cardinality of RxR is 2, of R it's one, and of N it's zero.
There could be cardinalities between 1 and 2. The fractals, with fractal dimension.
The points on the side of a square have c=1. The square has c=2, while a fractal curve in it has c between 1 and 2.
TonesInDeepFreezeDecember 26, 2021 at 18:34#6353020 likes
There are no cardinalities between 1 and 2. And there are no cardinalities between aleph_1 and aleph_2. However, without the continuum hypothesis, there could be cardinalities between card(N) and card(R), as, without the continuum hypothesis, card (R) could be aleph_x for some x>1, as I pointed out to you over and over in the other thread.
You don't know what you're talking about,.
Bret BernhoftDecember 26, 2021 at 20:12#6353640 likes
I have not mentioned continuousness. I have merely pointed out the utterly well known fact that it is a theorem that card(R) = card(RxR),.
Yeah, you have said that infinite times already. It's just not true. There are inf^2 points between 0 and 1. Aleph1. There are inf^4 of them in 2d. Aleph2. In 3d there are inf^8. Aleph3. I'm off. It's boring.
Well, it's not. Between aleph1 and aleph2 lies aleph1.5. A fractal line occupying half the square. There are inf^3 points for this figure.
TonesInDeepFreezeDecember 26, 2021 at 20:28#6353730 likes
"inf^2" is not a recognizable notion. Probably what you mean is 2^N.
And you ignorantly, wantonly persist about aleph_1
The claim that aleph_1 = 2^N is the continuum hypothesis.
The claim that aleph_1 = 2^N is the continuum hypothesis.
Then you have a different notion of aleph one. The ordinal in Aleph one is just related to how many times the infinity is present. For the naturals inf^1, so aleph 0. For the line inf^2, so aleph1.. For the 2d plane inf^4, so aleph2. For a volume inf^8, so aleph3. For a 1d fractal, say inf^3. So aleph1.6, approximately.
HeracloitusDecember 26, 2021 at 20:40#6353840 likes
Anyway, you said you were off, Eliza. Too bad you didn't mean it.
AgentTangarineDecember 26, 2021 at 20:57#6353890 likes
As if you are a mathematician... Keep it up Cantor! Why I write inf^1 is to highlighten the concept of cardinality. FZ lived 120 years ago. The axiom of choice is based on finite sets. Inf^3 might sound nonstandard in your ears but it's just infxinfxinf. What's so difficult about that? You have to raise 2 to the power 1.6 or something to obtain 3. So the aleph is aleph1.6, approximately. The fractal line occupying part of the square has aleph1,6 multiplied by a factor to account for the degree of covering.
AgentTangarineDecember 26, 2021 at 21:49#6353970 likes
It's gotta be understood though that the cover of a square by a fractal curve is not the same as just cutting out that part of the continuous square. That would imply that the covered piece has the same cardinality as the whole square. The fractal line doesn't cover the whole part. Like a fractal collection of points doesn't cover a continuous part of an interval (it can be piecewise continuous though.).
AgentTangarineDecember 26, 2021 at 22:09#6354010 likes
Anyway, you said you were off, Eliza. Too bad you didn't mean it.
Why? It's not as boring as I thought. There are even alephs0.5 and alephs0.99 or alephs0.01. If there are alephs(sqrt2) remains to be seen. A closed interval on the real line, like [0-1] can be fit infinite times on the real line, so in fact the cardinality of the real line is 1.4. That of the 2d plane is about 2.6. That of the 3d volume is about 3.2. That of a fractal line, plane, or volume, lies between these. Cantor didn't realize this yet.
Coming to think about it, of course aleph(sqrt2) exists. And the aleph for [0-1] is in fact aleph1. Cantor overlooked one infinity! Which only shows his genius! Who can overlook infinity...?
That would imply that the covered piece has the same cardinality as the whole square.
This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0
1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)
Extending these ideas shows the cardinality of R^3 is the same as that of R.
TonesInDeepFreezeDecember 26, 2021 at 23:39#6354470 likes
The axiom of choice is not needed for finite sets. Every finite set has a choice function, irrespective of the axiom of choice. In the context of our exchanges, one of the important points about the axiom of choice is that it implies that every infinite set has a cardinality and that every infinite cardinal is an aleph.
That would imply that the covered piece has the same cardinality as the whole square. — AgentTangarine
This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0
1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)
Extending these ideas shows the cardinality of R^3 is the same as that of R.
TonesInDeepFreezeDecember 26, 2021 at 23:46#6354540 likes
By the way, I did look at that Quora page you suggested in the thread that has since been deleted. The first post there is a proof that card(R) = card(RxR), which is exactly what you deny!
TonesInDeepFreezeDecember 26, 2021 at 23:47#6354560 likes
Delusional and disconnected from reality. Check for fever.
That's what is said about geniuses in general. Like being a crackpot. Untill now I haven't seen one bit of math, only parrot references to the net. The cardinality of RxR being the same as R. I gave you a link to a so-called proof of a bijection between R and RxR. A wrong one. I asked you why it's wrong. No reply. Cantor didn't take the infinity of the real line into account in determining how many powers of an infinity are needed. You only replied that you can't raise infinity to a power. You just need inf^3 times to enumerate all points on a line. Saying that the aleph of a 2d infinite plane is the same as that of an infinite line is the same as saying R is the same as N.
TonesInDeepFreezeDecember 27, 2021 at 00:02#6354640 likes
I gave you a link to a so-called proof of a bijection between R and RxR. A wrong one. I asked you why it's wrong.
I don't need to defend someone else's proof that I hadn't referenced. I already know a more general proof, as can be found in a textbook on the subject.
Take away the word 'project' (a projection function is a certain kind of function and it is not needed to mention regarding whether there is a bijection from N onto R). So use 'map' instead'.
Also, trivially we can map N into R one to one. So in this context instead of 'to' we must say 'onto'.
Then your claim becomes:
We cannot map N onto R one-to-one.
And that directly contradicts your claim now that we can map N onto R one-to-one.
You are very very confused.
AgentTangarineDecember 27, 2021 at 00:17#6354690 likes
And that directly contradicts your claim now that we can map N onto R one-to-one.
We can! An infinite times infinite times actually. One time onto 0.1-0.9999999. This can be done an infinite times for [0-1]. And an infinite times for the whole real line. Hence aleph1.4, and not aleph1.
TonesInDeepFreezeDecember 27, 2021 at 00:21#6354700 likes
I already explained to you that proving that there is a bijection from N onto R requires stating your axioms, definitions, and rules of inference and using only those axioms, definitions, and rules of inference to show that there is a function whose domain is N, whose range is R, and is 1-1. All three clauses: domain, range, 1-1. Such a proof, if it were in ZFC, would contradict the theorem that there does not exist a bijection from N onto R, thus proving that set theory is inconsistent, and would make you among the very most famous people in the entire history of mathematics.
On other matters such as alephs, you're proven wrong by simply referring to the definitions and by fhe fact that the assertion that card(R) = aleph_1 is famously independent of ZFC.
AgentTangarineDecember 27, 2021 at 00:47#6354880 likes
1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)
This is the correspondence between infinite and infxinfxinf. And if r goes to 0.04566..., or 0.005667... or 0.000556654... (you get my point, I guess...) even infinite times are included. So the map becomes one between inf^2 and inf^4. Aleph1 and aleph2.
TonesInDeepFreezeDecember 27, 2021 at 00:50#6354890 likes
Show the correct answer then. I already gave you a link to a supposed bijection between R and RxR. How many times can N be mapped on the real line? Just offer a tasty recipe.
AgentTangarineDecember 27, 2021 at 00:54#6354940 likes
TonesInDeepFreezeDecember 27, 2021 at 00:56#6354970 likes
I'm embarrassed. "Heat Wave" is by Irving Berlin not Cole Porter.
TonesInDeepFreezeDecember 27, 2021 at 01:07#6355030 likes
CORRECTION to the post below. This answers proving that card(R) = card(RxR), which is not what was asked of me. Instead, what was asked of me is "how many times N can be mapped on the real line?" If that is taken in the sense of what is the cardinality of {f | f is a function from N onto R}, then of course the answer is 0.
I'll leave my remarks about card(R) = card(RxR) anyway:
I have been referencing the more general theorem that for an infinite set S and natural number n>0, we have card(S) = card(S^n). S = R is a special case of that. I'm not sure, but it seems perhaps the particular proofs mentioned in threads here lately for R don't use the axiom of choice (?). I have not studied those proofs to verify them for myself though I get the gist of them and they seem okay to me to that extent.
The proof I have studied of "for an infinite set S and natural number n>0, we have card(S) = card(S^n)" is in Enderton's 'Elements Of Set Theory'. It is pretty involved, two pages, requiring a number of previous lemmas, a proof that the axiom of choice implies Zorn's lemma, closure under unions of chains, and more (and even an illustration to aid intuition). I would not spend my time and labor composing it all for you in the confines of a post, and it would do you no good anyway since you are utterly unfamiliar with even the basics of set theory that are prerequisite let alone the mathematics of Zorn's lemma, chains, et. al. And I admit that I am rusty myself on some of the details now, though I have previously studied it in every detail to verify for myself that it is perfectly correct.
The best I can do for you is to recommend that you get a textbook and study it from page 1. Enderton's 'Elements Of Set Theory' in particular is widely used, highly regarded, beautifully written, and pedagogically exemplary. Though, I would actually first recommend at least gaining a basic understanding of symbolic logic.
AgentTangarineDecember 27, 2021 at 01:12#6355060 likes
I just asked you the question how many times you can map N on R.
AgentTangarineDecember 27, 2021 at 01:15#6355070 likes
Now that I've answered your question, howzabout you answering a question I've been asking you: Why won't look up 'continuum hypothesis' on the Internet?
TonesInDeepFreezeDecember 27, 2021 at 01:23#6355090 likes
AgentTangarineDecember 27, 2021 at 01:24#6355100 likes
It's not so complicated as they make it. There are basically two types of infinities. Infinite countable infinities, like N and infinite times infinite types. Like the interval [0-1]. Uncountable. All alephs are multiplicities of them.
AgentTangarineDecember 27, 2021 at 01:34#6355130 likes
And that's exactly where the failure lies. The relation between R and RxR is the same as the relation between R and N. R can be viewed as corresponding so a single cardinal. How many lines you need to construct RxR? The same number as the number of cardinals to construct R.
TonesInDeepFreezeDecember 27, 2021 at 01:36#6355140 likes
Every cardinal is either countable or uncountable. Every countable cardinal is either finite or denumerable. There are denumerably many countable cardinals. There are denumerably many finite cardinals. There is only one denumerable cardinal.
That is easy. The hard part I described is PROVING that for an infinite set S and natural number n>0, we have card(S) = card(S^n).
After proving that, cardinal arithmetic is indeed beautifully simple:
If K and L are cardinals, and the larger of them is infinite and the smaller of them is not zero, then
K + L = K * L = max(K L)
TonesInDeepFreezeDecember 27, 2021 at 01:38#6355150 likes
The relation between R and RxR is the same as the relation between R and N.
That is purely arbitrary unfounded assertion.
On the other hand, in set theory, from rigorously stated axioms, definitions, and rules of inference we prove things.
And I answered a question for you, but you still have not answered my question for you. So the next question for you is: Why do you think one would feel need to answer your questions when you don't answer questions yourself? Wait, I have the answer: Because you are a crank.
TonesInDeepFreezeDecember 27, 2021 at 01:48#6355190 likes
You contradict yourself. You claim there is a bijection from N onto R, but above you admit that the interval [0 1] is uncountable. And we don't write '[0-1]' to denote and interval.
You are totally mixed up.
And it is beyond me why you think that the stuff you make up in your own head - without rigorous principles and without any understanding of the mathematics you allude to and the terminology you abuse - is somehow true and correct while the mathematics that has been rigorously formulated, studied, scrutinized, critiqued, and checked and thousands of times over rechecked by professionals and students all over the word is all wrong. In other words, what makes a crank tick?
AgentTangarineDecember 27, 2021 at 01:57#6355200 likes
You contradict yourself. You claim there is a bijection from N onto R, but above you admit that the interval [0 1] is uncountable.
You have to read what I write. There are infinite bijections between between N and [0-1]. Like there are an infinite times infinite between N and R. And there are an infinite times infinite between R and RxR.
TonesInDeepFreezeDecember 27, 2021 at 02:06#6355230 likes
If you consider the infinite line the discrete elements on the plane, like the natural numbers on R, than the relation between the lines and RxR, the infinite plane, is conformal to the relation between N and R.
TonesInDeepFreezeDecember 27, 2021 at 02:07#6355250 likes
Liza is getting tired and reduced to token replies.
Now you take refugee behind empty verbiage. Liza is tired indeed. My wife lies in bed for a couple of hours already... I told her to come too. Three hours ago...
TonesInDeepFreezeDecember 27, 2021 at 02:23#6355420 likes
Eliza is now all tangled up in confusion. My answer was to the question about mappings from N onto [0 1] not about why your claim that I have done nothing but give references is false.
I'm not surprised that your attention span wouldn't provide recalling my question: Why won't you look up 'continuum hypothesis' on the Internet?
AgentTangarineDecember 27, 2021 at 02:25#6355430 likes
Not at all. It's telling that you could figure out for yourself, but you won't, that your claim is false for the simple reason that I have not just given references but have posted formulations for you (as I already mentioned I have done that to your PREVIOUS claim that I only give references) and explanations too.
TonesInDeepFreezeDecember 27, 2021 at 02:26#6355450 likes
I have done that years ago already. I don't agree though. I'm not a parrot like you.
So years ago you looked it up, but still don't understand it now.
You don't agree with "it"? The continuum hypothesis? You have been claiming the continuum hypothesis for a zillion posts yet you say you don't agree with it! You are one really confused crank.
You are parroting your own world now as this is the second time you've said I am parroting, And even though I amply refuted that a while ago.
AgentTangarineDecember 27, 2021 at 02:31#6355490 likes
I can map N infinite times infinite times on R.
AgentTangarineDecember 27, 2021 at 02:32#6355500 likes
So you are in flat out contradiction with yourself.
— TonesInDeepFreeze
So?
AgentTangarineDecember 27, 2021 at 02:45#6355700 likes
I think we have discovered a real infinity in life, which is what this post is about. You and me commenting on each other. I may be in flat contradiction with what I thought previously, that's right. But not with my present self. But for sure with you....
AgentTangarineDecember 27, 2021 at 02:47#6355710 likes
So which is it now? You claim that card(R) = aleph_1, thus asserting the continuum hypothesis? Or you deny that card(R) = aleph_1, thus denying the continuum hypothesis
No. I assert card(R)=aleph1.4. Call me crazy, like you wish. Card(RxR)=aleph2.6, more or less. 2^(1.4)=3, more or less, 2^(2.6)=6, more or less.
So which is it now? You claim that card(R) = aleph_1, thus asserting the continuum hypothesis? Or you deny that card(R) = aleph_1, thus denying the continuum hypothesis?
Hardly matters though, since it is sheer crazy talk to say that there is a mapping of a singleton onto R.
TonesInDeepFreezeDecember 27, 2021 at 02:50#6355730 likes
This is the correspondence between infinite and infxinfxinf. And if r goes to 0.04566..., or 0.005667... or 0.000556654... (you get my point, I guess...) even infinite times are included. So the map becomes one between inf^2 and inf^4. Aleph1 and aleph2
That's not quite right since it has free variables on the right that aren't on the left.
f is onto Y if and only if (f is a function & range(f)= Y)
Note that that precludes Y from being a proper subset of range(f), which is a situation not usually mentioned. I suppose one could reformulate the definition to allow Y being a proper subset of the range(f) for a broader definition.
Also, some mathematicians consider a function to be not just the graph but also that the domain and the range are specified, and sometimes the domain and a co-domain (there may be different co-domains, since a co-domain can be any superset of the range). That could be made rigorous by saying a function is a triple or alternatively . However, in set theory, the function is the graph, period, though of course that graph determines the domain and range.
AgentTangarineDecember 27, 2021 at 03:08#6355790 likes
There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject.
AgentTangarineDecember 27, 2021 at 03:46#6355910 likes
f is onto Y if and only if (f is a function & range(f)= Y)
Indeed. So N can't be mapped 1-1 onto. You can at most map N to infinite points, infinite times. Say 1 into 2, 2 into 4, 3 into 6, etc. But into still. Then you still need to map into all inbetween intervals. On each interval an infinity of infinities of naturals is needed. So you need inf^3 times to map N into R. Pffffff.... I'm done! Thanks for the resistence! :smile:
AgentTangarineDecember 27, 2021 at 03:48#6355920 likes
There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject.
There are a lot of good books indeed. Thanks for the references. I prefer the math use in physics though. And the alephity of the continuum has implications for particles moving in it. That's why I think the aleph of the volume is different from the line and plane. I have booked a hotel for us... Just kidding! Gnight!
TonesInDeepFreezeDecember 27, 2021 at 03:51#6355930 likes
And you desperately need one if you are not to remain mired in your terrible confusions.
Well... I don't take it too seriously... You are probably right. Still, I can't see how R and RxR can have the same cardinality. There are just inf^3 times as many points in RxR as there are in R.
TonesInDeepFreezeDecember 27, 2021 at 04:07#6355990 likes
The problem is not so much that you don't take it seriously, but that you take it seriously enough to stubbornly persist in claims that are false or just ersatz gibberish from your own mind uninformed about anything other than itself.
Well, the point made is that a pair of numbers (x,y) say (0.678567..., 0,98678...) is contained in a single number 0.65456456.... The infinite number behind the 0 should contain both the infinites behind the 0 of x and y. This is not so.
You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.
Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.
TonesInDeepFreezeDecember 27, 2021 at 04:51#6356100 likes
No way. One can offer alternative systems; I enjoy reading about them if they are rigorous. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion.
Saying in a case like this "Oh, it's all opinion anyway" is intellectual dereliction.
TonesInDeepFreezeDecember 27, 2021 at 04:58#6356110 likes
No way. One can offer alternative system; I enjoy reading about them. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion.
I'm currently reading a book on mathematical philosophy. I'm no good at math although I'm fascinated by the subject. I noticed that math, its various branches, start life more as vague intuitions rather than crystal-clear concepts/ideas. Rigor comes much, much later if I'm not mistaken.
Too, the definitions in math give me the impression that true understanding is being sacrificed for logical formalism.
TonesInDeepFreezeDecember 27, 2021 at 05:09#6356140 likes
To know, we would have to have access to the mental states of mathematicians. We would have to know how long was the time between their first pre-formal musings and then putting them down in concrete formulations. There is no reason to believe that for many mathematicians that time might be very brief.
Anyway, if cranks said, "Here are my pre-formal musings, maybe something could come of them", then that would be one thing, but instead cranks insist that their view and only their view is correct; that ordinary mathematics (and even the alternative systems that the crank is ignorant of) are wrong. It is the crank, not the mathematician, who is dogmatic and exclusionary.
TonesInDeepFreezeDecember 27, 2021 at 05:11#6356160 likes
Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many. (same for (.0329576914..., .0925318623...).
Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many.
Give me an example of a number you think is left out. I bet it's not. Avoid the .999... =1.0 thing. Have you figured out what the algorithm is?
Yes, a pair of numbers. Not a number as you wrote.
You keep resorting to saying that I must consider the rest of what you posted. But each time it turns out that the rest of what you posted doesn't actually qualify into correctness the initially incorrect statements you make.
But funny, actually is a number. It's a complex number.
TonesInDeepFreezeDecember 27, 2021 at 05:29#6356210 likes
a number (x,y) say (0.678567, 0,98678) is contained in a single number 0.65456456.
Whatever you mean by an ordered pair being "contained" in a number, what we have is each number mapped to an ordered pair. The claim of the prover is that the whole mapping is 1-1 and onto RxR. All it takes then is to see that no ordered pair is mapped to by two different numbers, and that each ordered pair is mapped to by a number.
TonesInDeepFreezeDecember 27, 2021 at 05:31#6356220 likes
It is the crank, not the mathematician who is dogmatic and exclusionary.
It's you who is the crank. You are exclusionary and dogmatic. And you have no sense of humor. Sense of rigor, maybe. Jgill knew to convince me (well, almost...) in one comment. But he's a real mathematician.
You don't need the diagonal proof to realize that. Every real number can be mapped from N^3. Every real number can be reached from N^3. N^3 can be mapped onto R.
TonesInDeepFreezeDecember 27, 2021 at 06:12#6356280 likes
Just now, and in the other thread that was deleted yesterday
On the contrary. I even told you I contradict myself in previous posts. I never told you to consider other posts. Anyhow... I'm truly tired and my beloved has awoken. Damned! 7 hours about infinities.. I'm off to bed. Gonna contemplate about mr. Gill's procedure. It was fun! :smile:
TonesInDeepFreezeDecember 27, 2021 at 06:19#6356300 likes
There is no map from N^3 onto R. And even if there were, it would prove the countability of R not the uncountability. You are again completely backwards and confused.
AgentTangarineDecember 27, 2021 at 06:20#6356310 likes
You have not shown any dogmatism by my. Nor any exclusion other than of ignorant confusion and misinformation.
Yeah, you are right about that! Sorry that I called you dogmatic and exclusionary! You certainly got me interested in this aleph topic! As a physicist I find it difficult to believe that the number of points on a line is the same as on a plane or in a volume. The number of directions are different though. Or not even that?
TonesInDeepFreezeDecember 27, 2021 at 06:23#6356320 likes
By "the rest of what you posted" I meant the rest of what you posted in that post, just as I was responding exactly to your complaint that I hadn't quoted more of your post.
TonesInDeepFreezeDecember 27, 2021 at 06:24#6356330 likes
You can map N to all reals between 0.1 and 0.999999...
You can do this N times (for all smaller decimals). You can do this for all N size one intervals. Where am I wrong?
AgentTangarineDecember 27, 2021 at 06:28#6356350 likes
The definition you are seeing is the formal aspect. It's a kind of final touch to an idea that began as an interesting notion.
Yep, the formalism is necessary to ground what started off as an intuition, in logic; a necessity no doubt; after all, in math not having a crisp definition trying to find a missing person without either having a good description or photo of that person.
My point is that making a definition precise (usually) means losing some/all of the feelings that go with the intuition. The transition from just a vague notion to a clear-cut, well-defined idea is, for me, a heart-to-brain relocation of an idea and that's what bothers me. Math's link to the heart pops up occasionally though - I've heard of mathematicians being moved by the elegance and beauty of some formulae for example. I have a book titled The Heart of Mathematics which is on my reading list.
TonesInDeepFreezeDecember 27, 2021 at 06:32#6356370 likes
In thinking that the fact that in your own mind you imagine that it must be so implies a mathematical proof. And in thinking that your disconnected and mathematically unsyntactical dribblings are too mathematical proof.
TonesInDeepFreezeDecember 27, 2021 at 06:33#6356380 likes
I think it's the continuum that confuses me, and its break-up into (onto?) its parts. I think the break-up of a square into lines is the same as a line into points. But maybe the break-up of a square into points is the same as a line into points.
AgentTangarineDecember 27, 2021 at 06:38#6356400 likes
That's almost a good example. But it's better described as the centuries-ago formulations being more than vague intuitions yet not adequately formalized. Then, centuries later, it was discovered with mathematical logic and model theory how to vindicate the notion rigorously.
Of course, one can look back centuries, even to the ancients, to see that their mathematics has seen been formalized.
What I thought you had in mind though are cases where mathematicians had vague notions and then they or their contemporaries formalized those notions themselves.
Agent SmithDecember 27, 2021 at 06:42#6356430 likes
Reply to TonesInDeepFreeze It seems, as always, mine is a case of one-sided love. Math, for some reason, refuses to let me unravel its secrets. Cosmic censorship or I'm just plain stupid! Likely the latter. Good day. See you around!
TonesInDeepFreezeDecember 27, 2021 at 06:43#6356440 likes
making a definition precise (usually) means losing some/all of the feelings that go with the intuition.
(1) How do you know that unless you've interviewed mathematicians about it?
(2) I highly doubt that mathematicians very much regret whatever such loss of feelings there might be, as I would think mathematicians are primarily eager to communicate their notions clearly and objectively to other mathematicians and to prove their results.
TonesInDeepFreezeDecember 27, 2021 at 06:45#6356450 likes
Start by adopting a specific definition of 'the continuum'. The term is often used flexibly, but I would settle on the continuum understood to be the pair of the set of reals with the standard ordering on the reals:
c =
Or, if you prefer to think of it as merely the x-axis:
{ | x e R}
Or when we refer to "the cardinality of the continuum" we are thinking of the continuum as merely the set of real numbers.
AgentTangarineDecember 27, 2021 at 06:57#6356500 likes
I told you. You don't have proof of it. You only think you do.
But where am I wrong in my proof? Cannot N be mapped onto 0.1-1? You ñeed N numbers for that: 1-99999999.... What number do I leave out here? Or do I leave numbers out between 0.1-0.9999999....?
TonesInDeepFreezeDecember 27, 2021 at 07:04#6356540 likes
Do you mean to suggest that there is a 1-1 function from N onto 0?
By the way, you claimed that I have no sense of humor. Well, you haven't said anything funny. Neither have I very much, since I don't find this context with you to motivate me to make jokes. That is not a lack of sense of humor. You don't know me.
AgentTangarineDecember 27, 2021 at 07:12#6356550 likes
Given any function f from N to [0 1], the diagonal proof constructs a member of [0 1] that is not in the range of f.
I feel pretty safe in thinking that you don't know the diagonal proof.
Moreover, even if the diagonal proof were found to be incorrect (it won't be) then that would not constitute a proof of its negation.
I've told you a couple of times already: To prove your claim, you must prove that there is a function whose domain is N and whose range is R and that it is 1-1. And I won't even ask that your proof be constructive by showing a particular such function, only that one exists, even though Cantor's proof is constructive: showing, for any given function from N to R, a particular real number not in the range of that function.
AgentTangarineDecember 27, 2021 at 07:18#6356580 likes
Between 0.1 and 0.99999.... you use all numbers of N
What does "use the numbers mean"?
Do you mean they are all in the range of a function from N to some set? WHAT function? You have not adduced any function. For any given k in N, what is f(k)?
So maybe you just mean that all the natural numbers are in the domain? Well, duh, yeah. So what? The domain is not at issue. It is the range that is at issue. You need to prove that there is a function whose domain is N and whose range is R (i.e. every real number is in the range).
So I ask:
Are you clear that your task is to prove that there is a function whose domain is N and such that every real number is in the range of the function?
AgentTangarineDecember 27, 2021 at 07:44#6356670 likes
0.1 connects with 1
0.2 connects with 2
0.3 connects with 3
.
.
.
0.14 with 14
0.15 with 15
.
.
0.53 with 53
0.54 with 54
.
.
0.768 with 768
0.769 with 769
.
.
0.99998 with 99998
.
.
ad 0.9999999999......
Do I have to list all N numbers?
AgentTangarineDecember 27, 2021 at 09:09#6356890 likes
This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology.
AgentTangarineDecember 27, 2021 at 10:04#6356950 likes
You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.
Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.
5h
Yes. I get that. Still... something is nagging. If I map all naturals on [0.1-0.999999] (there you go...) what do I leave out? [0-0.1] seems to contain more numbers than [0.1-0.99999...]:
Which is absurd. Still... On each of the intervals (including [0.1-0.99999...]) you can map the set N directly. Can you break up a continuous interval, like [0.1-1], up in real points? Like 0.1, 0.2, 0.3,...,0.91, 0.92,...,0.110, 0.111,...,0.1222, 0.1223,..., 0.2111, 0.2112,..., 0.24444, 0.24445, ...0,2023432, 0.2023433, ..., 0.655555, 0.655556,.......,0.999999999999....
Every cardinal is contained once. Or is one left out? The diagonal? But how can that be? Say that number is 1000000023432500876.... Isn't that contained in N? But not in R?
If so, can you represent x by 0.5878900... and y by 0.197867....(to name two arbitraries) to map on r?
Or is continuum just continuum, no matter the dimension? No points attached?
Can you knock it off please? The thread title is "Infinites Outside of Math". You've turned it into a factual discussion about math. That Agent has... spicy takes of dubious correctness... on how infinite sets work is besides the point. Please feel free to take it to personal messages, though.
Agent - would it take that much effort to try to fact check what you've written if you're genuinely interested in it?
0-0.1] seems to contain less numbers than [0.1-0.99999...]:
EG: you'd find out that those repeating 9s after the decimal point become the number above in the limit. So if you write 0.999... that already equals 1, or if you write 0.0999... that already equals 0.1.
Can you break up a continuous interval, like [0.1-1], up in real points? Like 0.1, 0.2, 0.3,...,0.91, 0.92,...,0.110, 0.111,...,0.1222, 0.1223,..., 0.2111, 0.2112,..., 0.24444, 0.24445, ...0,2023432, 0.2023433, ..., 0.655555, 0.655556,.......,0.999999999999....
There you've provided at most a countable set of countable sequences, which together turn out to at most countable.
If you're interesting in in these things, I'm sure someone involved would be happy to provide you with study materials.
AgentTangarineDecember 27, 2021 at 11:39#6357260 likes
confidently throwing around specialist terminology.
That's exactly what I not do. I question it. That's all. I have been given no satisfactory answer though. Jgill came close. But the bijection he prescribes is discontinuous and suffers from the same problem as the number of points on the interval [0.1, 0.999999....). You just can't make points touch, or break the continuum up in points. How many points lay between 0.1 and 0.999999...? Do all these numbers constitute the continuous interval [0.1-0.99999...)? (What is continuous? Undivided.) You can assign natural number to each of these numbers. Is a number left out, by the diagonal argument? If so, isn't that a new natural number, contradicting that you left one out?
AgentTangarineDecember 27, 2021 at 12:03#6357360 likes
What if I asked the question, as a new thread, if the continuum can be broken in parts? It's maybe a math question. Maybe not. It will not alter the essence though of this thread, which is life-related.
I think that would be fine. There's been plenty of discussions before about the nature of the continuum. Just try and keep it away from the mathematical equivalent of pseudoscience.
AgentTangarineDecember 27, 2021 at 14:54#6357700 likes
I think that would be fine. There's been plenty of discussions before about the nature of the continuum. Just try and keep it away from the mathematical equivalent of pseudoscience
The "point" is that constructing a continuum out of points seems like a pseudoscience to me. The fact that the line, plane or volume have the same cardinality is because of the attempt to reduce them to points. I'll give it a try later.
This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology.
I'm well aware of the situation. :cool: Having communicated with him I believe he has a graduate degree in physics. However, his concerns are with the "physical" or interpretive aspects of the science rather than mathematical descriptions. Even advanced math in QT seems not to have reached axiomatic set theory to a noticeable extent. Kenosha Kid is another genuine physicist. To the best of my knowledge neither of these gentlemen work in the profession at present.
TonesInDeepFreezeDecember 28, 2021 at 05:13#6359960 likes
Regarding a remark in a previous post, yes I am happy to suggest study materials that would provide understanding of these points:
(1) A bijection between a proper subset, or even the entire set, of terminating decimal expansions (which reduces to the set of finite sequences of natural numbers) and the set of natural numbers is not a bijection between [0 1] (which is represented by the set of denumerable (not terminating) expansions) and the set of natural numbers.
(2) Among the members of [0 1] not represented by terminating expansions are both some of the rational numbers in [0 1] and all of the irrational numbers in [0 1]. Those numbers are not represented in the domain of the bijection.
(3) To answer "What are examples of members of [0 1] that are left out of the domain of the bijection?", we could mention for example: .333.. (i.e. 1/3) and .14159265358979323846264338327950288419716939937510... (i.e. the decimal portion of pi).
Also, the domain of the bijection is only a proper subset of the set of terminating expansions, as for example, not even .01 is a member of the domain. However with a more careful construction, of course, we can show a bijection between the entire set of terminating decimal expansions and the set of natural numbers.
(4) To answer "What natural number is not in the range?", we say that no natural number is not in the range. But so what? Uncountably many members of [0 1] are not in the domain, so it is not a bijection between [0 1] and the set of natural numbers. Of course, any mathematician knows that there is a bijection between the set of terminating expansions and the set of natural numbers. Indeed it is a basic tool of formal languages and computing that the set of finite sequences on a countable set is countable. And, indeed, Cantor proved that there is a bijection between the set of rational numbers and the set of natural numbers, thus, a fortiori, there is a bijection from the set of terminating decimal expansions into the set of natural numbers. But Cantor also proved that there is no bijection between the set of denumerable (non-terminating) expansions and the set of natural numbers. Versions of the proof may be found in any good textbook on set theory.
Suggested study materials (studied in this order).
Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar. I consider this to be the best textbook for learning symbolic logic, which is extremely helpful (for me, crucial) for understanding set theory.
Introduction To Logic - Suppes. This has the best explanation (it is superb) of formal definitions that I have found.
Elements Of Set Theory - Enderton. My favorite textbook on set theory. It is beautifully written.
Axiomatic Set Theory - Suppes. Good as a backup to Enderton.
Philosophy Of Set Theory - Tiles. An overview of the intuitions and philosophy behind various views on set theory and mathematics.
Agent SmithDecember 28, 2021 at 05:36#6360050 likes
The OP's main intention is to see if there's such a thing as nonmathematica infinity. Infinity, as far as I can tell, is an exclusively mathematical object (magnitude/number). So, to ask whether there's nonmathematical infinity is like asking if there's a red that isn't a color?
However, a question has been asked and our task is to answer it as best as we can. Where do we begin?
[quote=Brazil nightclub fire survivor (2013)]I was only there for seconds. It felt like an eternity.[/quote]
A "few seconds" isn't infinity but this hapless burn victim experienced eternity (infinity). This can't be a mathematical infinity for only a "few seconds" had gone by. Ergo, this must be nonmathematical infinity. We're not out of the woods yet for eternity can only be described mathematically. I'm only offering a possible starting point for an inquiry.
TonesInDeepFreezeDecember 28, 2021 at 05:50#6360090 likes
It cause for misunderstanding to say that in mathematics infinity is an object. Granted, there are objects sometimes called 'infinity', such as points on the extended real line. But the more general set theoretic notion of infinity is not of the noun 'infinity' but of the adjective 'is infinite'. Overlooking that distinction often leads to serious misconceptions about how set theory and mathematics treat the subject.
'is infinite', as a set theoretic notion, is a 1-place predicate defined:
S is finite iff there is a bijection between S and a natural number
S is infinite iff S is not finite
S is Dedekind-infinite iff there is a bijection between S and a proper subset of S
Salient about infinite sets is that every Dedekind-infinite set is infinite (this is the other side of the coin of the "pigeonhole principle"), and, with an appropriate choice axiom, every infinite set is Dedekind-infinite.
And mathematical infinity is not just magnitude or number. Yes, there are infinite ordinals and infinite cardinals, which are called "numbers" (there is not, as far as I know, a mathematical definition of 'is a number') but there are infinite sets that are not ordinals or cardinals.
Then there's that hypothetical spaceship we are in, approaching a Black Hole at a constant speed. We sail in and if not annihilated continue our explorations. But a friend is watching our progress from a great distance and sees us going slower and slower until it appears we have stopped before penetrating the anomaly. To him, we will sit there for eternity, gradually inching towards the periphery - like a mathematical limiting process, never quite making it.
(I'm not a physicist, but I recall reading of this. Could be wrong)
This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0
1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)
Extending these ideas shows the cardinality of R^3 is the same as that of R.
Does this kind of reasoning really work? Since at the same time,
r=.xyzabc... <-> p=(.xyzabc..., 0, 0)
and
r=.xyzabc... <-> p=(.xyzabc..., 0, 0.0...01)
and so on?
And moreover,
r1=.xyzabc... <-> r2=(.0....1xyzabc...)
r1=.xyzabc... <-> r2=(.0....2xyzabc...)
...
My point is, every point on the line can be mapped to an edge of the cube. What about all the points in the rest of the cube?
The interleaving algorithm looks good to me. But then, you can map those interleaved points onto a single edge as well. This can go on and on in a cycle. Why is there not therefore a paradox?
Deleted UserDecember 28, 2021 at 21:46#6362210 likes
This user has been deleted and all their posts removed.
Probably because points have no dimension. The real numbers are complicated and fascinating.
Look, suppose I pick at random a point in the interior of the cube: (.251956... , .629435..., .194735...) Then that point corresponds to a unique point on the x-axis edge: .261529194947...
You just alternate digits. Here, you do it by going the other way: find the point inside the cube corresponding to the point on the x-axis: .381402258639...
The process is continuous in the epsilon-delta sense.
English is a real language. The range of sentences expressible in English is infinite. Is that an infinity in the world?
Of course, these sentences can not be enumerated, at least in the world.
I guess whether there are enumerated infinities in the world depends on the nature of the world. Is space-time closed or open? Is there really a unit length, the plank length? Or do lengths truly map to real numbers?
Outside of mathematics, I believe that some people do experience infinity in mediation, other spiritual practices, and with psychedelics. Also, one can experience the law of non-contradiction transcended, so that all dualities are one.
Agent SmithDecember 29, 2021 at 05:10#6362880 likes
Here's another, albeit mathematical, way of experiencing [math]\infty[/math]: Live! If one lives for even just 5 minites, [math]\infty[/math] moments from t = 0 to t = 5 mins would have to flow by. The same can be done with distance, mutatis mutandis.
Given a given line segment has exactly the same number of points as any other line segment, no matter their difference in lengths, and given lifespans can be mapped onto lines, we could say that
1. Everyone one of us - infant, child, adult, man, woman, young, old - has the same lifespan ([math]\infty[/math], measured in terms of moments/instants). We're kinda immortal mortals. The Greeks, it seems, refused to touch [math]\infty[/math] with a barge pole for a very good reason: paradox galore!
2.
Brazil nightclub fire survivor (2013):I was only there for seconds. It felt like an eternity.
This blaze survivor was experiencing instants/moments ([math]\infty[/math]) when the fire was raging around him and not the timespan ("seconds").
Agent SmithFebruary 07, 2022 at 15:11#6523950 likes
I propose a distinction betweem absolute and relative infinities.
As a kindergarten child, I couldn't count beyond 10; the numbers greater than 10 are beyond my ken. This, is relative infinity and since it's in fact not an absolute infinity (more on this in a while), it can be treated as a nonmathematical/qualitative infinity.
Now, as an adult, I could go on counting (the natural numbers), but of course I'll never complete the task; this, is absolute infinity and it's mathematical/quantitative.
Qualitative infinity has to do with limits (our own, our tools'); Quantitative infinity is, at the end of the day, a concept to which limits don't apply.
Alkis PiskasFebruary 07, 2022 at 18:07#6524240 likes
Reply to TiredThinker
When I see your alias, I always wonder how or why you have chosen it ... Thinking should not be tiresome! :smile: Not if directed correctly and it is controlled, and one doesn't "torture" one's mind by chosing to talk about such subjects as infiniteness! :grin:
Is there a way to describe various infinites without going right to number lines? Anything in real life to reference in terms of the infinite?
I'm not sure what you exactly mean by "going right to number lines", but if you imply that "infinity" and "infiniteness" are normally connected to numbers (Math), I think that this is too much restricting. There are a lot of things we call "infinite", even if actually they are not, but only very huge in size, from simple to complex: a line, a circle, love, time, space, the Universe (although its infiniteness is still debatable), ... That is, anything the limits of which cannot be determined.
“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”
? Albert Einstein
Deleted UserFebruary 07, 2022 at 19:24#6524340 likes
Is there a way to describe various infinites without going right to number lines? Anything in real life to reference in terms of the infinite?
An abstraction with utility, but not an actuality. All things are finite, including the universe. There has never been any reason to suggest otherwise apart from our inablity to formulate thoughts on the subject.
You mean because we can travel around a circle forever? I mean infinite more like if absolute coordinates existed we could go straight in one direction without reaching a limit.
Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside.
Potential infinity, as the intuitionists keep stressing and as programmers demonstrate practically, is the only concept that is needed, both inside and outside of mathematics, that refers to finite entities of a priori indefinite size.
The myth of absolute infinity is what give the illusion of mathematics as being an a priori true activity that transcends Earthly contingencies.
TonesInDeepFreezeFebruary 10, 2022 at 02:45#6532370 likes
* There are rigorous definitions of 'limit', in various contexts, in mathematics.
* To my knowledge, there is no general mathematical definition of 'is a number'. However 'ordinal number', 'cardinal number' and the predicate 'is infinite' have rigorous mathematical definitions, and there are proofs that there are ordinal numbers and cardinal numbers that are infinite.
* The lemniscate [here I'll use 'inf'] does not ordinarily denote a particular object. Probably its two most salient uses are for (1) points on the extended number line and (2) in expressions such as "the SUM[n = 1 to inf] 1/(2^n). With (1), inf and -inf can be any arbitrary objects (they don't even have to be infinite) that are not real numbers, serving as points for the purpose of a system. With (2), 'inf' is eliminable as it is merely convenient verbiage that can be reduced to notation in which it does not occur.
So your examples of arithmetic involving inf are not meaningful. However, there are rigorously defined operations of ordinal addition, subtraction, multiplication, and division.
* I agree that you in particular are better off not speaking on 0!, which also has a rigorous mathematical treatment.
TonesInDeepFreezeFebruary 10, 2022 at 03:00#6532380 likes
* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.
* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1.
* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
.
* One may take mathematics as a priori true without commitment to infinite sets.
TiredThinkerFebruary 10, 2022 at 03:35#6532420 likes
Do straight lines exist? And even if you traveled the earth forever you will see the same places more than once.
Take another sip, my friend, then off to bed. :yawn:
Agent SmithFebruary 10, 2022 at 04:44#6532550 likes
Reply to TonesInDeepFreeze I was simply pointing out that basic arithmetic operations are undefined for [math]\infty[/math] which, to me, implies it (infinity) isn't a number like 2, 3, 18986, 0.98457..., 1/8, etc.
Thanks.
TonesInDeepFreezeFebruary 10, 2022 at 05:17#6532640 likes
'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.
In which case, you surely agree that absolute infinity isn't a semantically meaningful assignment to a mathematical entity, for any semantic interpretation of the symbol of infinity as referring to extensional infinity, is question begging.
* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1.
The semantic notion of absolute infinity (whatever that is supposed to mean) isn't identifiable with the unbounded quantifiers used in classical mathematics, logic and set theory, due to the existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas, which usually cannot be fully formulated or solved in those notations due to the inconvenient truths of software implementation and physical reality. Most software engineers don't regard themselves to be mathematicians or logicians, due to historical reasons concerning how mathematics and logic were initially conceived and developed.
* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics, i.e. mathematics with lazy evaluation, that rejects both formalism and platonism, in which unbounded universal quantification is understood to refer to potential infinity, which leads to his formulation of non-classical continuity axioms. In a similar vein, Edward Nelson's Internal Set Theory adds tenses to Set Theory, by distinguishing the elements of a set that have so far been constructed that have definite properties, from those that will potentially be constructed in the future, that have indefinite properties.
A groom, hand on heart, vows sincerely to the bride " I will always remain faithful". Later that afternoon, he runs off with the bridesmaid. Did he really contradict his earlier vows, or does a contradiction exist only in the minds of those who misconceive the nature of infinity ?
Perhaps one might say that to view the groom as contradicting his earlier vows amounts to a definition of 'negated absolute infinity' - but this interpretation is unnecessarily problematic in asserting the negation of a statement that isn't a verifiable proposition with verifiable meaning.
Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas . . .
I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me.
Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside
In the complex plane "infinity" is called "the point at infinity" and correlates directly with the north pole of the Riemann sphere - a specific point. But I've never used this concept.
I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me.
sure, its an overstatement born of frustration with somewhat outdated formal traditions that still remain dominant in the education system.
TiredThinkerFebruary 11, 2022 at 01:57#6534580 likes
I am asking about and infinite landscape. Only new information all the time.
:chin: Such as unknown unknowns which necessarily encompass "knowns":
[quote=Karl Popper]The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know; our knowledge of our ignorance. For this indeed, is the main source of our ignorance - the fact that our knowledge can be only finite, while our ignorance must necessarily be infinite.[/quote]
Agent SmithFebruary 11, 2022 at 13:51#6535280 likes
Are uncountable infinities mathematical?
TonesInDeepFreezeFebruary 13, 2022 at 15:37#6542280 likes
* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
— TonesInDeepFreeze
SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics
That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".
* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.
You simply skipped the point that mathematical definitions are never circular or tautologies.
The semantic notion of absolute infinity (whatever that is supposed to mean)
Whatever what is supposed to mean? "The semantic notion of absolute infinity" or "absolute infinity"? Anyway, in either case, who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?
Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity
Formal classical mathematics is exact in the sense that there is an algorithm to check whether a given finite sequence of formulas is or is not a proof. Any alternative to classical mathematics that also has that attribute would need to be also evaluated for simplicity of the formulation of the system.
TonesInDeepFreezeFebruary 13, 2022 at 16:07#6542370 likes
Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets?
You mean without absolutely infinite sets, presumably. The overall approach would be to stress that the mathematical notion of a derivative approximates the real-life practice of directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function. This is opposite to the conventional way of thinking, which construes the practice of measuring a slope as a means of approximating an ideal, abstract and causally inert mathematical derivative.
With this in mind, the classical definition of df/dx with respect to the (?, ?)-definition of a limit, can be practically interpreted by interpreting ? to be a potential infinitesimal, and ? as representing a random position on the x axis given the value of ? , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded.
What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity.
. . . directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function
dx may be arbitrary, but dy depends on dx and is not arbitrary. That's where the observed function comes into play. Maybe you are talking about something else. You seem confused about these things. Or maybe I misinterpret.
the classical definition of df/dx with respect to the (?, ?)-definition of a limit, can be practically interpreted by interpreting ? to be a potential infinitesimal, and ? as representing a random position on the x axis given the value of ? , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded.
Non-standard analysis is what you are referencing here in a somewhat befuddled fashion. What is a potential infinitesimal?
What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity.
Fractals are generated by simply iterating certain complex functions. They are not "infinite spaces" but images on computer screens. The iteration process is finite, say n=1000. There is no abstract infinity other than implied replications of patterns like turtles all the way down. You never really get there.
It is the reciprocal of a potentially infinite number, e.g. a random value taken from the codomain of the rational valued function 1/x.
'non-standard analysis' is the correct umbrella term, but it is already befuddled by the various alternatives that fall under it, some of which receive rightful criticism for obscuring matters even further, e.g the hyperreals .
TonesInDeepFreezeFebruary 14, 2022 at 02:30#6544690 likes
You mean without absolutely infinite sets, presumably.
Why in the world would I couch anything in terms of "absolutely infinite"? The notion of "absolute infinity" does not occur in classical mathematics since Cantor was superseded by Z set theory, as I had already alluded to:
Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?
And you skipped that question. Would you please answer it?
You keep talking about "absolute infinity" but in absence of any reference to it in mathematics after Cantor, and in absence even of your own definition of it.
[a potential infinitesimal] is the reciprocal of a potentially infinite number
A philosophical view, such that we are already familiar with, on the notion of 'potential infinity' is of course welcomed. And I have little doubt that it has been formalized somewhere. But you again skipped my point that it is only by seeing a particular formulization that we can compare its ease of use with classical mathematics. From the (admittedly not deep) reading I have done, generally alternatives to classical mathematics are much more complicated to formulate.
Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one).
Then, aside from the mathematical and philosophical subject, we now are talking about heuristics in the form of pedagogy. Nothing you have mentioned so far appears to be any more pedagogically promising than the classical definition of 'the derivative'. Indeed, there is no reason to think that a high school or college student would not have a much harder time grasping non-standard analysis combined some way with "potential infinity" than grasping the usual notion of a derivative. Nor have you pointed out any instance in which scientific or engineering calculations would be better enabled by your seemingly personal graft of non-standard analysis with a notion of 'potential infinity'.
TonesInDeepFreezeFebruary 14, 2022 at 02:35#6544700 likes
That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".
you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that.
You skipped that point twice now. Maybe you'll address it later, but as it stands, you fail to substantiate your claim as to what Brouwer or any intuitionist or any constructivist or any mathematician or philosopher ever said.
Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one).
Overall, good observation.
To my knowledge, it isn't possible to point to a complete formalisation of potentially infinite logic, because it doesn't yet exist. All we have are fragments or incomplete axiomatizations of the concept, that have been invented by different logicians over the years with respect to different systems for purposes other than the current discussion. This is somewhat similar to the proliferation of different programming languages. There isn't any inter-subjective agreement as to how to formulate open-world reasoning.
- The I, S and T axioms that Edward Nelson introduced are useful for formulating what it means to reason with as-of-yet unconstructed elements of an unfinished set, in spite of the fact he proposed the axioms in the context of ZFC as an alternative to model-theoretic non-standard analysis in ZFC. ZFC is of course inadmissible for the purposes of this discussion, and is the reason why his formalisation doesn't tend to be associated with formalizing potential infinity. But the I,S and T axioms, divorced from the problematic axioms of ZFC appear to be a relevant fragment of some formalization of potentially infinite logic.
-Brouwer's notion of choice sequences, i.e. unfinished sequences, serve as the template for potentially infinite sequences but his formulation doesn't to my understanding provide what I,S and T does.
Choice sequences allow the expression of unfinished sequences, e.g
{1,2,3,...}, where the dots "..." are understood to mean "to be continued"
Brouwer introduces continuity axioms that define what it means to prove a universal proposition over a domain that consists of such unfinished sequences. However, his concepts, at least to my understanding, doesn't permit direct talk about numbers that we have presently declared, but cannot currently quantify, e.g. "The height of the tallest human being who will ever live"
- Linear Logic, as opposed to intuitionistic logic, is the logic i would associate with Intuitionism and potential infinity, because it is a resource conscious logic. Again, as you might say, it is "not evidently associated with p.i", especially in view of it's exponential fragment. Squinting at axioms to see their practical significance is still an unfortunate necessity.
TonesInDeepFreezeFebruary 14, 2022 at 20:49#6549090 likes
the I,S and T axioms, divorced from the problematic axioms of ZFC
Problematic for some people, not for others. And I am pretty sure that if you take ZFC out from under IST you're left with a theory that accomplishes quite little.
It looks very interesting. But you couldn't seriously propose it as a way for college freshmen to learn calculus.
Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer. And I still don't know why in the world you go on about "absolute infinity", which is a notion pretty much entirely unused since Cantor was superseded by axiomatic set theory.
There are no passages in the article that support your misrepresentations. If you think there are passages that support you, then you can cite them specifically.
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
You fabricated those.
And what was your point arguing about "absolute infinity", which is a notion that has virtually been out of play since axiomatic set theory? For that matter, as I've already asked, what is your definition of "absolute infinity"?
For that matter, as I've already asked, what is your definition of "absolute infinity"?
Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity, relative to which the existence of said entity cannot be independently verified, empirically evaluated, or constructed with respect to a finite amount of data.
Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. Quine's famous example "All Bachelors are unmarried men" can be held as being true by definition in the mind of a particular speaker, but in doing so it can no longer be regarded as being representative of how a community of speakers might use the words "bachelor" and "unmarried man", given the limitless potential of them using the words non-equivalently.
Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension, whereupon it is inconsistently alleged that a finite description of a function can somehow represent a limitless amount of information that is also exact.
The alternative interpretation corresponding to "potential infinity" is to consider the definition of such entities as being vague and verifiable, as opposed to being semantically precise but unverifiable.
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals.
Actually my phrasing is a slightly weaker statement, considering that potential infinity is usually used in the context of monotonic sequences, as in infinitesimals or infinitely large numbers. The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. This constitutes a random stopping event, in the sense that the time of the pause is not defined a priori at the time when dx is declared to be infinitely small or x to be infinitely large.
Defining potential infinity in terms of a "non-terminating" process is problematic however, given the fact that 'non-termination' isn't a verifiable proposition if interpreted literally, which is a concept belonging to absolute infinity. What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. Then consider the fact that any of the finite extensions generated by pausing a process are countable and isomorphic to the integers. These are my considerations when thinking of "potential infinity" in terms of a priori unbounded finite numbers instead of in terms of a "non-terminating" process. However, my definition might cause confusion due the fact it is more general and includes random variables with unbounded values.
Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension . . .
Apart from Cantor's speculations I never came across this idea. It can't be applied to sets so it sits out there in an unattainable splendor, ignored by most in the profession.
TonesInDeepFreezeFebruary 18, 2022 at 22:21#6565310 likes
It would be nice if you addressed my points directly aside from (or at least in addition to) whatever other meandering about various subjects you have in mind.
Instead of defining 'absolute infinity' you give various notes on your ideas about it.
'absolute infinity' is a noun. For mathematics, the definition of a noun requires the definiendum (a noun phrase for defining a noun) on the left and a definiens (a noun phrase for defining a noun) on the right.
So a definition of 'absolute infinity' would be of this form:
absolute infinity = [insert a mathematical noun phrase here]
in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension
There you still don't provide a definition, but you take 'absolute infinity' to refer to a certain kind of "interpretation". Here are the problems for you:
(1) You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it.
Accepting that there is a common notion of an intensional definition as opposed to an extensional definition, we still see that 'an extensional definition' just boils down to the definiens being a finite list. But a set is infinite iff it has no finite listing. So there is nothing gained from saying 'absolutely infinite' rather than merely saying 'infinite'. The definitions:
x is finite iff x is 1-1 with a natural number*
x is infinite iff x is not finite
Then your sense of 'absolutely infinite' is just 'infinite'.
What total function do you offer as an example? 'total' is better thought of as a 2-place relation. A function is total on a set iff the domain of the function is that set. So what exactly do you have in mind regarding totality? And, for example, as to the the infinitude of the set of natural numbers, what total function do you think is improperly employed?
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
— TonesInDeepFreeze
It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals.
No it is not "practically equivalent". You ADDED "finite", "a priori indefinite", "finite entities" and especially "eventually finitely bounded". Those are not mentioned, neither literally or practically, in the article's explanation of the notion of potential infinity (except 'finite' is mentioned regarding the finitude of each natural number). Generally, the article does not say that potentially infinite sequences are finite sequences. Even for an intuitionist, a finite sequence is not a potentially infinite sequence, notwithstanding that only a finite portion of a potentially infinite sequence is constructed at any given point. To really stress the point, especially, the article says nothing like "eventually finitely bounded".
The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time.
Please cite a source where that is mentioned as part of the "common definition".
when dx is declared to be infinitely small or x to be infinitely large.
You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off from them.
/
* That is circular with my definition of 'is a natural number' in another thread, but that can be overcome by using a different definition of 'is a natural number' such as 'an ordinal that is well ordered both by membership and the inverse of membership').
TonesInDeepFreezeFebruary 18, 2022 at 22:40#6565440 likes
Where do you find "eventually finitely bounded" in Brower, or even any secondary source, on potential infinity? Please cite a specific passage.
Again, I'm not asking about your own notions; I'm asking where you got your idea as to what Brower said about it.
And you might hold off snidely telling me to "reread the article(s)" when you still don't cite specific passages in an article and the article you mentioned does not at all support your claim.
What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process.
I think you may have misinterpreted the Wikipedia article:
potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps
There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition.
You have a very sophisticated writing style that I find a bit hard to process, but others may not. And when a person writes well there is a temptation to put aside statements that might give one pause. I'm guilty, but fortunately TonesInDeepFreeze is sharper in this regard.
There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition.
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.
The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future.
Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage.
It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject, and I have already presented my arguments in enough detail as to why it is better to think of potential infinity in that way.
. Quoting TonesInDeepFreeze
You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off fro them.
.
Intuitionism is partially aligned with constructively acceptable versions of non-standard analysis. If you want an more authoritative but easy-read sketch, Read Martin Lof's "The Mathematics of Infinity" to see the influence Choice sequences have had on non standard extensions of type-theory (which still cannot fully characterise potential infinity due to relying exclusively on inductive, i.e. well-founded types.
You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it.
Classical mathematics and Set theory conflate the notions of absolute with potential infinity, hence only the term "infinity" is required there. Not so in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk.
Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory:
1) The Law of Excluded Middle is not only invalid, but false with respect to intuitionism.
2) The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity.
For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..."
3) Choice Axioms obscure the distinction between intension and extension, whereupon no honest mathematician knows what is being asserted beyond fiat syntax when confronted with an unbounded quantifier.
3) The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. Not so according to potential infinity, since it cannot be determined that two functions are the same given a potentially infinite amount of data .
We also have Markov's Principle: according to absolute infinity, an infinite binary process S must contain a 1 if it is contradictory that S is constantly zero, and hence MP is accepted. Not so according to potential infinity, due to the fact that 1 might never be realised. This principle is especially relevant with respect to Proof theory, since any proof by refutation must eventually terminate at some point, before knowing for certain whether an unrefuted statement is refutable. So unless we are a platonist who accepts absolute infinity, Markov's principle isn't admissible.
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.
The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future.
We seem to be looking at a common notion from different perspectives. I assume you are a CS person (?). First of all a student in a math class is more likely to ask, What is an infinite sequence?, whereupon the definition as a function of the positive integers is provided. Then comes the distinction between a sequence in which each term is defined by a formula and one that self-generates by recursion. Examples would make these ideas clear:
Then, of course, there are special and/or more complicated cases, like a sequence which is non-terminating but assumes the same value after a certain point, or a sequence that is convergent in a metric space, etc.
In all my years I don't recall using the expression "potential infinity".
TonesInDeepFreezeFebruary 19, 2022 at 21:23#6568210 likes
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?
Look in virtually any introductory textbook or set of lecture notes on computability. And we don't need a notion of 'potential infinity' to explain the notion of non-termination. The classical treatment of computability is replete with the notion of non-halting. For example, it is a well known simple fact that a program to list the natural numbers does not halt.
Moreover, it is the responsibility of proponents of the notion of 'potential infinity' to provide the needed definitions to support the notion; not the responsibility of people who don't rely on the notion. The fact that you are mixed up about this subject shouldn't entail that you try to patch that up by supplying incorrect and incoherent claims and attribute them to Brouwer and the intuitionists. You continue to say that Brouwer and intuitionists understand potentially infinite sequences to be "eventually finitely bounded". Yet, after multiple requests, you fail to provide a source where Brouwer or anyone said that.
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown.
Classically, it can be explained as not one finite sequence but as a sequence of finite sequences. Or, non-classically, to avoid having an infinite sequence of finite sequences, as anyone can read in virtually any article about 'potential infinity', even at the most basic level: For any finite sequence, there is a finite sequence of greater length, but there is not an upper bound to the lengths of such finite sequences. That is the OPPOSITE of saying that there is a finite bound on the lengths.
"non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.
One can posit that no physical process continues without termination. But, as I've asked you again and again, please cite where Brouwer said that a potentially infinite sequence is "eventually finitely bounded". More generally, if you have no Brouwer source to point to, then you should not conflate your claim that an ideal process is not realized physically so a non-terminating process is "eventually finitely bounded" with Brouwer who, as far as we know, never advocated that a potentially infinite sequence is "eventually finitely bounded", especially as the notion of potential infinity is the OPPOSITE.
Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage.
— TonesInDeepFreeze
It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject
You use terminology in such a sloppy yet grandiose way:
(1) Logical equivalence is a special notion. You haven't shown any "logical equivalence".
(2) By mere fiat you declare a logical equivalence. Still, you do not cite anything Brouwer said that even suggests (let alone is "logically equivalent") to "eventually finitely bounded", especially as it is the OPPOSITE of the notion of potential infinity.
Classical mathematics and Set theory conflate the notions of absolute with potential infinity,
ZFC could not possibly conflate the notions since set theory doesn't even have a notion of 'potential infinity' nor does set theory mention 'absolute infinity'.
Again, 'absolute infinity' is a notion of Cantor that is not used in ZFC. You persist to use 'absolute infinity' in your own personal sense (for you, 'absolute infinity' is the notion that an intensional definition can specify a set that cannot be finitely listed), which is very different from Cantor's use of the term.
And even with your sense, though ZFC does define sets that are not finite, that is not "conflating" with some other notion ('potential infinity') that ZFC does not even address.
in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk.
A rigorous definition would be good for those who use the concept. But ZFC does not use it. And ZFC does provide a rigorous axiomatization for classical computer theory. If you want a rigorous non-classical computer theory, then it's your job to make it rigorous; your lack of doing that is not a fault of ZFC.
And I guess there are some people in computer science who regard ZFC as junk, but that is not at all any kind of consensus or, as far as I know, even a large contingent. You don't legitimately get to speak on behalf of "computer science".
Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory:
Wow. You are so wrong, and so obviously so. A combination perhaps of ignorance and intellectual dishonesty.
The very purpose of Z is to not include axioms that would allow an absolute infinity. Why don't you get a book on introductory set theory and inform yourself on this subject?
The Law of Excluded Middle is not only invalid, but false with respect to intuitionism.
Yes, LEM is not a logical validity in intuitionism. That doesn't contribute to claiming that Cantor's sense of 'absolute infinity' is the same as your own personal sense of it. (Also, there is a technical question about what 'false' in a semantic sense means for intuitionism. In finite domains, LEM is TRUE in intuitionism. Of course, we would have to look at specific intuitionistic set theories to see whether LEM is false in any given model.)
The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity.
I have no idea what you have in mind with the notion of the axiom of regularity "preventing unfinishable sets". ZFC has no predicate "unfinishable sets", so the axiom of regularity couldn't allow nor "prevent" anything about. It is incorrect to say that ZFC makes determinations on notions that are not even expressed in ZFC.
And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.
For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..."
I don't know what that is supposed to mean. But, to be clear "..." is not in the language of set theory, not even as extended by definition, but rather it is informal notation that can be eliminated with actually rigorous notation.
The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function.
No, according to what YOU mean by 'absolute infinity'; it's not what Cantor meant.
And the axiom of extensionality doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.
Even more basically, even though ZFC does capture a great deal of Cantorian set theory (but not including Cantor's 'absolute infinity'), that does not entail that what Cantor meant by 'absolute infinity' is what you mean by it.
TonesInDeepFreezeFebruary 19, 2022 at 21:37#6568250 likes
Condensed:
1. sime's claim of what Brouwer meant is the OPPOSITE of the basic notion of potential infinity, and sime has shown no source that supports his claim.
2. sime is plainly incorrect that Cantor's notion of absolute infinity is the same as sime's, and sime's argument about ZFC in this context is incorrect since ZFC dramatically DIFFERS with Cantor on the matter.
Agent SmithFebruary 21, 2022 at 10:39#6572690 likes
From the mole in chemistry and nuclear physics to the relative nature of time the universe has yet to yield a single finite limit for anything outside of human measurements.
We can measure things and we can make linear statements about those things with mathematics but beyond that we have to use linear algebra. There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics.
Traditionally the first axiom of mathematics is commutative principle. Increasingly, commutative principle is seen as the first derived axiom with the first being the axiom that given any observable set we can assign whole numbers to that set as a form of measurement.
Take it how you will, the simple truth is that it's the psychology of individuals and their personality cults that have had the most influence on how everyday people understand the universe, not the ideas of finite and infinite. Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything.
Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything.
It's interesting you have come upon a well-kept secret within the mathematics community. I, myself, have witnessed such rants at the top echelons of the profession. It's an embarrassing spectacle and rather unexpected within a discipline so rigorous and demanding calm reasoning. Sad. :cry:
Reply to jgill From a psychology (especially evolutionary and developmental) perspective it does make a fair amount of sense.
People gifted in mathematics tend to be (very much a generalization) very judgemental, love symmetry, almost obsessively orderly, emotionally distant, become easily obsessed with problems, stoic in their self image, etc...
Of course there are often exceptions to this as a person can be a brilliant mathematician but not see themselves as naturally gifted. Einstein comes to mind as he forced himself to do math in order to explain his theories but preferred to visualize stuff in his head and then labor to write it down as equations.
Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be.
One of my favorite examples of the difference between linear calculus and set theory elegance is to compare Euler Angles with Quaternions. In linear algebra the quaternion equation iterates over vectors and translates (rotates) their position in 3 coordinate planes. It tells you the new location of each member of a vector. With Euler angles, however, the description of the matrix is what gets rotated, not the matrix.
Reply to SkyLeach Not sure where set theory comes into play here. Quaternions are more algebraic and geometric. But I will admit the part of calculus I least enjoyed teaching is spacial coordinates and rotations and translations. And the closest I've come to exploring Euler angles and quaternions is in SU(2):Dynamics of LFTs of SU(2)
Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be
I can write very concise and precise definitions and descriptions, then use them in sheer babble.
TonesInDeepFreezeFebruary 25, 2022 at 05:55#6591400 likes
There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics.
I'm not familiar with that. Where can I read about it?
Hand them ["famous and influential mathematicians"] a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything.
What famous mathematicians in particular do you have in mind? Where have you witnessed these "meltdowns"?
The use of sets is ubiquitous in mathematics. I didn't know that famous mathematicians were having "meltdowns" about sets.
TonesInDeepFreezeFebruary 25, 2022 at 05:59#6591430 likes
Again, there is no object that is infinity (other than such things as points of infinity on extended numbrer lines). There are sets that have the property of being infinite.
That cardinal arithmetic is idempotent for infinite sets (especially for the set of natural numbers) is not really not problematic if you bother to read the proof.
TonesInDeepFreezeFebruary 25, 2022 at 06:00#6591440 likes
People gifted in mathematics tend to be (very much a generalization) very judgemental, love symmetry, almost obsessively orderly, emotionally distant, become easily obsessed with problems, stoic in their self image, etc...
[note]Developmental, evolutionary and neuronal modeling inferred explanation for how and why the brain needs math for modern communication (and its limits)[/note]
I see that as: mathematics is missing constructs for allegorical analogy which are an essential base component for practical daily use languages.
Language is a serialization of 4D reality as it is "understood" by the speaker that uses shared context and perspective to allow the listener to reconstruct the serialization in the prefrontal cortex as a cognitive narrative (series of causal set functions applied to models).
That process relies very heavily on allegorical analogy to compare properties and functions of sets to completely different sets and infer meaning from the exercise.
Returning to mathematics; the purpose of mathematics is to validate the properties and functions applied to the sets. It makes no assertions, descriptions or assumptions about the nature of the sets. It's intended to strictly regulate validation of function and derived outtcome only.
The entire idea is that if you have applied mathematics to the properties and functions (as many as makes sense at least) and if two individual's precision of definition of a set in question is sufficiently detailed they can be relatively certain of reaching the same cognitive result and thus certain of objective agreement.
One of my favorite examples of the difference between linear calculus and set theory elegance is to compare Euler Angles with Quaternions. In linear algebra the quaternion equation iterates over vectors and translates (rotates) their position in 3 coordinate planes. It tells you the new location of each member of a vector. With Euler angles, however, the description of the matrix is what gets rotated, not the matrix.
- Me
I quoted myself because you seemed to have missed my detailed comparison when you asked where Quaternions came into things.
Jung help us I just skimmed back and now I know why this thread is 10 pages long. We have an ISFJ judging everyone for daring to think instead of accept dogma.
Returning to mathematics; the purpose of mathematics is to validate the properties and functions applied to the sets. It makes no assertions, descriptions or assumptions about the nature of the sets. It's intended to strictly regulate validation of function and derived outtcome only
You refer repeatedly to "sets". Care to expand on that ?
Reply to jgill A set is just a group of things. In the brain every concept is a superset of neural cells and their relationships both to one another and to other associated sets in other regions of the brain. Relationships are also a set of measurable dynamics such as sodium ions, potassium ions, neurotransmitters, inhibitors, hormonal markers, synaptic pathways and their reinforcement with same.
So in this context a set is the boundary of associations encapsulating a concept as distinct from others. It is best defined given current technology by an fMRI showing active regions experiencing state change during specific cognition. Typically it includes the Parietal, Occipital, Temporal and Prefrontal for real-world objects and their allegorical associations. Distinctions between regions are associated with how concepts are introduced (visual, written and spoken instigation as well as direct vs. representational association with cognitive burn-in (reinforced mapping)).
Math is an abstraction of determinism such that you can apply the same series of functions to apples, oranges and the loves of your life because even though two of the former are physical sets of atoms that can be seen and touched while the latter is an allegorical association of evaluated relationship qualifications between a concept of self and a concept of other; all three are treated by the human brain identically. That's because they are cognitively identical processes.
People gifted in mathematics tend to be (very much a generalization) very judgemental, love symmetry, almost obsessively orderly, emotionally distant, become easily obsessed with problems, stoic in their self image, etc...
TonesInDeepFreezeFebruary 26, 2022 at 15:57#6596530 likes
Reply to TonesInDeepFreeze It's the only one you're getting. You're a limbrain ISFJ with an ego the size of a mountain that has repeatedly demonstrated he's here to stroke his ego not discuss philosophy.
And no, I'm not going to ignore you but I'm also not going to argue (as in fight) with your ego instead of your rational mind. It's a waste of my time.
EDIT: I mixed up two replies in one. Glad I caught it on the editorial re-read. That's what I get for typing in the middle of a family in full Saturday boogy.
Agent SmithFebruary 26, 2022 at 21:54#6597960 likes
Again, there is no object that is infinity (other than such things as points of infinity on extended numbrer lines). There are sets that have the property of being infinite.
That cardinal arithmetic is idempotent for infinite sets (especially for the set of natural numbers) is not really not problematic if you bother to read the proof.
I quoted myself because you seemed to have missed my detailed comparison when you asked where Quaternions came into things.
Were (are) you an aeronautical engineer or Naval person? I know little of rotation theory, but the gimbal lock problem can occur in one of these schemes but not the other. The short paper I linked is as close to this as I get. I almost always work in the complex plane.
I was joking about the behavior of fellow mathematicians. In all my days I never saw a rant. And your description of a math person's personality is valid sometimes, but more often they are social animals - the practice of mathematics is a very social activity. I recall being at an autumn meeting at the Luminy campus of the University of Marseilles in 1989 at which there was communal dining and quite a jovial atmosphere. And a summer meeting at the University of Trondheim in 1997 where a member of the royal family attended a convivial banquet overlooking a ski jump where their Olympic team put on a performance. Other international meetings displayed similar atmospheres.
Sorry about the ramble above, but many assume math talent means the sort of personality you have described. Which, incidentally, omitted the fact that many in the profession have musical talent.
Agent SmithFebruary 26, 2022 at 21:59#6597990 likes
Reply to jgill ooh you're right I forgot about that one and yes, it's a problem with rotating space instead of rotating the object in space. You can actually have the problem with either implementation depending on how you're treating your overall scene because quaternions still operate on the three axis and, depending on what you're doing in the scene, become mathematically ambiguous due to loss of precision (i.e. the square root problem and other things).
I am an engineer but software, not mechanical/chemical/structural/etc...
I was joking about the behavior of fellow mathematicians. In all my days I never saw a rant. And your description of a math person's personality is valid sometimes, but more often they are social animals - the practice of mathematics is a very social activity. I recall being at an autumn meeting at the Luminy campus of the University of Marseilles in 1989 at which there was communal dining and quite a jovial atmosphere. And a summer meeting at the University of Trondheim in 1997 where a member of the royal family attended a convivial banquet overlooking a ski jump where their Olympic team put on a performance. Other international meetings displayed similar atmospheres.
...gifted in mathematics tend to be (very much a generalization) very judge...
I am very aware of the problems of bias in generalization and that's why I pointed out it was a generalization. Any time you combine multiple data points in a demographic you wind up with a much flatter distribution curve. "... tend to be ..." drops to 12% instead of the 50:50 median split of a single data point. Most people conceptualize bias generalizations at 50+ during conversation because... well I'll avoid going into why right now...
As soon as you add any other variables you can completely invalidate the metric. I was using my knowledge of Jungian and the MMPI which are the most widely used and thus easiest to draw inferences from. They have their limits, however, so I try to make it clear that I'm generalizing and try never to use them to evaluate any individuals (or prejudice my behavior). The operative word being try, I'm human too.
EDIT: one further thing to add. When it comes to a discipline, generalizations become much more appropriate because we are very much social animals and when we make choices about our own bias which is essential to cognitive function we must choose between social consensus and personal time investment for validation (hopefully those, not the bad ones like bigotry, faith or inanity). Academics is, at its core, an appeal to authority.
Reply to TiredThinker
I have read some of what the others said and my good question is, what kind of infinite are you looking for? As long as you live this mortal life, you will not possess any knowledge of the True Infinite, aka God, at least in the sense that you will have no real evidence of this mysterious Infinity. Any belief in the Infinite must be accepted with faith. And to an extent, even lesser infinities are unknown, until you actually encounter them. Before that, we can only speculate on the nature or existence of infinities.
Reply to Agent Smith ?? If I understand you, the only thing the infinity symbol in mathematics means is that the limit isn't known (thus it can't be summed).
Agent SmithFebruary 27, 2022 at 03:38#6598930 likes
?? If I understand you, the only thing the infinity symbol in mathematics means is that the limit isn't known (thus it can't be summed)
Well, all I meant to ask was whether something uncountable (an example of an uncountable infinity is the set of real numbers R) can be considered mathematical? After all, math is, bottom line, about countability (0, 1, 2, 3,...).
Reply to Agent Smith Not really. That's just the first traditional axiom (self-evident truism in the philosophy of mathematics).
Then again, as I said, there is a very strong and growing debate about that since the argument for sets being the the first self-evident truth has a lot to offer and, of course, can't (or hasn't been) invalidated.
The essential argument is that numbers are simply a form of measurement and mathematics is a precise language for making falsifiable linear statements about those measurements. Numbers are, after all, assigned to representations of sets. Apples/fruit/kilometers/distance/seconds/time... you get the idea.
Agent SmithFebruary 27, 2022 at 07:56#6600090 likes
Reply to SkyLeach I'm somehow not convinced by your words. Uncountability is, in a sense, beyond (conventional) mathematics seen as a counting activity.
True that the variety of numbers has expanded over history. Yet we seem distinctly more comfortable with the category of natural numbers than with any other I can think of.
Continuity is infinite and uncountable. Though you can have one continuity, two continuities, three continuities, more continuities, many continuities, enough continuities, ànd no continuities. How many points reside in a continuity? Is a point a zero dimensional continuity? Does continuity require one dimension at least?
instead of just assuming you should learn a bit more about the humanities and what kinds of research are regularly done
Assuming what? I infer that you don't have any actual studies to cite, since you continue to reply without mentioning them.
And I know a little about the notion of studies in psychology and sociology. Asking you to cite an actual study that justifies your claims is not a fault from my own understanding.
TonesInDeepFreezeFebruary 28, 2022 at 00:53#6605080 likes
You don't know that I am an ISFJ. And even if I were, being any one of the types does not contribute to disqualifying a person from having asked pertinent questions.
with an ego the size of a mountain that has repeatedly demonstrated he's here to stroke his ego
I would guess that most people who post do so with motivated to enhance their sense of self-worth by exercising their prerogative to express their ideas, and many people often with criticisms of the ideas or claims of others. You have no evidence that my motivation is any more for gratification of ego than average. Especially, I come nowhere close to the kind of egotism found in a forum such as this that is displayed by people with terrible grandiosity when they bloviate their personally devised philosophies, with irrational arguments denying any weight to, and insulting other views by people who know something about the subject, and doing that by way of egregious distortions of even the rudiments of the subject.
And even if my motivation were entirely egotistic, that would not disqualify my points themselves.
I am interested in mathematics and philosophy of mathematics. I find that it is worthwhile not to distort or misrepresent the mathematics itself when philosophizing about it. Clearing up misinformation and misunderstanding that is posted about mathematics is a worthy first step toward discussion of philosophy about it.
Mathematics is what a consensus of mathematicians says it is.
I won't argue against the notion that what is the study of mathematics is based on what professionals in departments called 'mathematics' do. But as to whether a purported proof is correct or not (unless it is extraordinarily complicated) is not a matter of consensus. Whatever consensus there might be, if one shows an incorrect inference in a purported proof, then the proof is disqualified from being deemed an actual proof.
TonesInDeepFreezeFebruary 28, 2022 at 01:22#6605270 likes
I don't know of a common definition of 'mathematical object'.
I said that (except in special contexts) in ordinary mathematics, there is no object named by the word 'infinity'. I mean that there is no constant symbol added to the language of ZFC where the constant symbol is rendered in English as 'infinity'.
Instead, there is a predicate symbol that is rendered in English as 'is infinite'. And there are particular constant symbols that are defined as particular sets and we have theorems that those sets are infinite. For example, let 'w' [read as omega] stand for the set of natural numbers. It is a theorem that w is infinite, and it is a theorem that w is countable. Or, let 'R' stand for the set of real numbers. It is a theorem that R is infinite, and it is a theorem that R is uncountable
But as to whether a purported proof is correct or not (unless it is extraordinarily complicated) is not a matter of consensus
It's still a matter of consensus to determine whether the proof is valid. Yes, in some abstract realm a proof is valid or not according to logical principles, but humans have to agree before it becomes an accepted piece of mathematics. Occasionally a proof is so long and so complicated the verification process is difficult.
Whatever consensus there might be, if one shows an incorrect inference in a purported proof, then the proof is disqualified from being deemed an actual proof.
Guess there's not a consensus, then.
TonesInDeepFreezeFebruary 28, 2022 at 01:32#6605330 likes
I meant to ask was whether something uncountable (an example of an uncountable infinity is the set of real numbers R) can be considered mathematical?
The set of real numbers is defined in set theory, which is a mathematical theory. Set theory makes reference only to pure sets. Not sets of apples or nations or thoughts, but only sets provided by the purely abstract axioms. Indeed, the empty set itself doesn't have to be taken as given but is derivable from purely abstract axioms.
After all, math is, bottom line, about countability (0, 1, 2, 3,...).
The natural numbers are foundational. That doesn't entail that mathematics must be limited to the natural numbers. And set theory takes not the natural numbers as primitive, but merely the relation 'is a member of', i.e, membership. And with axioms about membership, the natural numbers are constructed, and from the natural numbers, then the integers, then the rationals, the reals are constructed,.
TonesInDeepFreezeFebruary 28, 2022 at 01:39#6605340 likes
It's still a matter of consensus to determine whether the proof is valid.
Only when the participants haven't themselves checked the purported proof. They may take the word of the referees that the purported proof is correct. My point is that in principle, it is objective to check whether a purported proof is correct, and if an incorrect inference is clearly shown, then no consensus can alter that the purported proof is not correct. (Again, I'm setting aside situations that are so terribly complicated that there is real debate.)
Whatever consensus there might be, if one shows an incorrect inference in a purported proof, then the proof is disqualified from being deemed an actual proof.
— TonesInDeepFreeze
Guess there's not a consensus, then.
No, there is a consensus that the purported proof is not correct.
Agent SmithFebruary 28, 2022 at 03:12#6605730 likes
Reply to jgill Well sure. That was why I pointed out the demographic spectrum and how things are changing, but slowly. There is too much direct control asserted over too much of each generation's career by the previous generation, causing the normal evolution of thought and culture to be retarded in academics.
In addition, the economic and sociopolitical spectrum tends to limit entry into academics as part of it's statistical weighting towards wealth and privilege.
Finally, the social authority encourages ego which, like the other factors mentioned, retards change and adoption of new ideas.
Reply to TonesInDeepFreeze You didn't ask and even if you did I wouldn't for the reasons I already stated.
To remind you, in case you forgot, it's a waste of my time. Don't care what you infer, you've already demonstrated that isn't going to be affected by anything I say or do.
TonesInDeepFreezeFebruary 28, 2022 at 15:11#6609230 likes
Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry!
?SkyLeach
I'm somehow not convinced by your words. Uncountability is, in a sense, beyond (conventional) mathematics seen as a counting activity.
True that the variety of numbers has expanded over history. Yet we seem distinctly more comfortable with the category of natural numbers than with any other I can think of.
Perhaps this will help.
Starting Premises:
Mathematicians earn a PhD just like all other non-medical disciplines
Previous is because their doctorate is in a very specific branch of philosophy
Previous is called the philosophy of Mathematics
Previous is derived from axioms (self-evident truths)
The argument to prepend (as opposed to append) ZFC to the philosophy of mathematics is a rational argument based in a self-evident truth, just like all other axioms of mathematics.
My words are just me talking about that axiom as I understand it.
Look around you (allegorically speaking) and point at anything that is a 1 or a 2 ... or any other number.
Numbers aren't things, they're representations.
Q: What do they represent?
A: Cognitively distinct concepts.
Q: What is a cognitively distinct concept?
A: Anything the rational mind considers a discrete set.
Q: Is there any limit on what the mind considers a distinct concept?
A: Yes, the mind is only able to process allegorical comparisons as set theory and has increasing trouble with concepts sufficiently distant from experience to create one-off pathways (dangling pathways or singular references or [in the case of damage] Disconnection Syndrom
Q: Can you put that in terms I can understand?
A: Maybe. Concepts like massive measurements are very hard for the mind to put into comparative concept and since that's how the mind actually works a great many mistakes are made when doing it until the mind has compensated.
Q: Can you give an example?
A: Yup. Timescales in galactic terms. Distances in interplanetary terms. Infinities. Distance between two points along a curved path like a planet's surface instead of a straight line. Asymmetric periodicity.
Right now, traditionally and with the same consensus as JGill mentioned, numbers are referred to as measurements.
Counting (commutative principle) is just an axiom. Numbers aren't counting except in the sense of combining any two measurements.
When we think of counting apples it can be hard to think it's a measurement. In effect, however, you're measuring volume (just not being very precise). If you're counting apples to fill a pie then the correct answer can vary because the size of the apples definitely will vary based on what kind of apple you are counting. Granny smith vs. red delicious (for example) since granny smith are small green apples and red delicious are quite large red apples.
Ok given that all numbers are just relative measurements between two points in a conceptual context of the mind we can begin to argue for set theory being the first axiom:
All real numbers are representational measurements of similar sets (i.e. apple = set of apple fruit cells) with a contextually regular periodicity (i.e. each set is defined by the apple skin around it or a complex series of functions that eventually lead back there in a well-understood way).
Any set can be defined as a vector (a starting point but no limit) - comes from ZFC
Any set can be defined in terms of its periodicity function (an algorithm, typically a mathematical functioion, that defines the measurement of members of the set) - from ZFC
Any set defined in terms of its periodicity function is effectively infinite unless bounded by logical rules imposed by the algebraic functions of the definition. (also from ZFC)
I know that's a lot to absorb but it's my best attempt at simplifying the entire argument.
Continuous manifolds cannot be represented by real numbers. A continuous manifold is not made up of points. There are tangent spaces defined on them, and even tangent bundles. From where numbers can be made relating to the curvature of the manifold. You can roll a 2d flat plane over o sphere to return to the initial position. The equivalent would be rolling a ball on an equilateral triangle. You will see that an arrow on the plane will have changed its orientation wrt a drawn equator on the ball, signifying curvature. You can't do this with a cylinder. A cylinder is not curved.What happens in the case of a cone?
There is too much direct control asserted over too much of each generation's career by the previous generation, causing the normal evolution of thought and culture to be retarded in academics
Not my experience at all. After my PhD and getting a tenured position I belonged to a small international group of academic mathematicians, all of whom eagerly sought new ideas, novel ways of looking at things, unusual results, etc.
What you might be referring to is when a grad student has to choose a project (dissertation) and is not capable of making that choice, their advisor will guide them onto a path he thinks they are able to follow and hopefully do original research - the fundamental requirement for the degree. And, sad to say, sometimes the advisor will in effect do much or most of the original research and give credit to the student.
However, it is true that some departments are "governed" by a cliche that exerts pressure to push ahead toward certain research goals, having little patience with deviations. I've seen this also. This can occur when the cliche forms around a prominent, celebrated academic. Especially when there are grants to be captured.
Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry!
:lol:
Jokes aside, my example is still in the rational domain. Perhaps I could give an example from construction/engineering: a circular [math](\pi)[/math] dome like the one that tops the Hagia Sophia (Turkey) for an irrational number in the calculation.
Reply to SkyLeach Well, you haven't addressed the issue that something uncountable (the set of reals) could be mathematical i.e., to be precise, numerical in nature. The set of reals, as per mathematicians, can't be counted. How can something that can't be counted be mathematical? Can consciousness be counted?
Do you mean differentiable manifolds? A cylinder created by moving a circle through space is not curved? A sphere in 3-D is not composed of points?
When is something curved? If the value of the tangent differs from place to place? A circle seems curved. So does a cylinder. So does a torus. You can move a circle through space so it becomes a torus. Maybe the way you move points, lines, surfaces, etc. through the higher dimensional space determines if they are curved (apart from Ricci or Riemann tensors). If I move a point wildly through space the ensuing line gets curvature. A circle hasn't though. A sphere does. But the value of the derivative on it is the same everywhere (how do you define a derivative on a manifold?). A torus has Gaussian curvature but it can be defined such that it has zero curvature. There's more to it...
Can you built a line with points? You can move it through space. Or stack lines to form a surface? Planes to build a volume? What if we take a point from a line? Can you still move on it continuously? Math addresses these questions, and confirms, but still. A line made of points? How you glue them together? How are strings kept on 3d while the 3d soars in 4d?
What's that? So a set of random integers is defined that way? Do you speak of a set or a sequence?
It's strange that you read those statements as "all sets are" instead of "any set can be". Maybe my word choice was poor? I tend to think in terms of software, not geometry, so there really doesn't need to be a pattern in the numbers in a set just a relationship between them that associates them, even if they're pseudo-random numbers.
Typically a vector is rationally linked with some spatial coordinate system but in software you just can't limit them by guessing the relationship between them from their values.
This isn't about personal experience or a single academic discipline. When I think of the problems in academia I just don't think about math as one of those. It's not that it isn't one, it's just that the only time the problems ever came up was over the rants against infinity and set theory (which are actually a bit funny to read at times). Mathematics is quite possibly the most empirical of all the sciences.
When I talk about many of the problems in academia I tend to be thinking of cosmology, astronomy, paleontology, the humanities (psych, anthro, socio, etc...) The more empirical and rigid a discipline is the less they seem to get into academic problems.
I double majored but since I graduated I don't think I've read more than a half dozen research papers in mathematics. It's just not one I need to follow closely because new algorithms or solutions sets are rare.
EDIT: damnit I just realized that my perspective will probably get me argued with again...
Set theory isn't just sets of points, it can also be a scene described as a space (hilbert, sobolev, etc...) with objects described functionally instead of sets of points. I deal far more with scenes described rather than sets of points except when rendering a solution set.
How can something that can't be counted be mathematical? Can consciousness be counted?
Are you arguing this way?:
The set of real numbers can't be counted.
Consciousness can't be counted.
Consciousness is not mathematical.
Therefore, the set of real numbers is not mathematical.
Set theory isn't just sets of points, it can also be a scene described as a space (hilbert, sobolev, etc...) with objects described functionally instead of sets of points. I deal far more with scenes described rather than sets of points except when rendering a solution set.
Your perspective of "set theory" is not the normal math perspective. If it works for you, fine.
When I talk about many of the problems in academia I tend to be thinking of cosmology, astronomy, paleontology, the humanities (psych, anthro, socio, etc...) The more empirical and rigid a discipline is the less they seem to get into academic problems
Your perspective of "set theory" is not the normal math perspective. If it works for you, fine
The point was that I have both perspectives. I double majored in mathematics. This being a philsophy forum I default to the more general view (and certainly in this context discussing the mathematics-only perspective would make little sense).
I'm also curious why you just skipped over the whole part about a vecor being defined by element association since that was the mathematics perspective and the practical application perspective.
Reply to EugeneWReply to TonesInDeepFreeze As far as I can tell he's just using the singular form of a word and saying it can't be counted which is kinda like saying "can the number 1 be counted".
The answer is, of course, yes that's the definition of commutative principle we just use a different word with a root modifier for plurality instead of the singular.
That's why I said it didn't make any sense. You can have multiple surfaces, consciousnesses, etc...
I'm also curious why you just skipped over the whole part about a vecor being defined by element association since that was the mathematics perspective and the practical application perspective.
These are vectors as I know them. I guess I don't understand what exactly you are talking about. That could be me. But if you point out your description in the article that would be good.
In mathematics and physics, a vector is an element of a vector space.
Yes, it's literally the first sentence.
Trouble is, then we have to get into what a vector space is...
In mathematics, physics, and engineering, a vector space (also called a linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but some vector spaces have scalar multiplication by complex numbers or, generally, by a scalar from any mathematic field.
And from there we have to get into specific allegorical definitions (while trying to avoid jargon) in order to limit the definition of 'element association'.
Really though, at this point the specifics are specious since the bolded part of the definition of vector space pretty much covers it.
My word "association" is indistinguishable from the definition of mathematic field:
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.
Maybe a real-life example would clear it up better though
In radiology the system that stores images is called PACS. Most of the time, those systems deal with DICOM images. Those images can be regular computer graphics, but not for things like PET. PET is stored as a matrix defined in the sagittal, axial and transaxial planes. The value of each element in the matrix is called a Housfield scalar. It's a relative absorbtion rate (defined by the precision of the PET hardware) between 0-255 (or bigger).
You can't do anything with the matrix in that form, however. You must either convert it to a 2D vector space or a 3D vector space. You can do this directly or by scaling it in a process called fusion with MRI or CT data which is generally stored as a more traditional JPEG2k or MPEG.
Thus, the resulting vector space is either a 3 element vector of (x,y,h) or 4 element vector of (x,y,z,h) with x,y and z mapped to the coordinate space and h mapped to the housfield absorbtion.
It's a whole other dimension, but it's still valid.
Edit the ray stops at 20, but that was to keep the post short.
Edit2: it should be pointed out that since you didn't define the association, the possibilities are infinite. The only way to make it useful is to define how they're associated.
Reply to jgill No, I don't. I have no clue what you're talking about. AFAIK one doesn't use braces for subsets but then since I'm not a mathematician by trade I don't really know anything about how mathematicians do things.
The set of real numbers can't be counted.
Consciousness can't be counted.
Consciousness is not mathematical.
Therefore, the set of real numbers is not mathematical.
More or less. I'm sucker for (good) analogies.
TonesInDeepFreezeMarch 04, 2022 at 18:11#6629020 likes
This distinction between, on the one hand, a set in and of itself without specifying an order and, on the other hand, an ordered tuple is crucially basic to mathematics, especially linear algebra.
Curly braces are used for sets in general. Either by set abstraction such as
{x | x is a natural number less than 4}
or for specifying the members individually such as
{0 1 2 3 }
where the order in which the members are listed, or redundancies are irrelevant, so:
{0 1 2 3} = {3 1 0 2} = {3 3 1 0 2} etc.
Angle brackets [alternatively, parenthesis] are used for order tuples such as
<0 1 2 3 > [alternatively (0 1 2 3)]
and order and redundancy do matter, so none of the below are equal to one another:
<0 1 2 3>
<3 1 0 2>
<3 3 1 0 2>
etc.
/
This is not just a matter of notation, but is a crucial concept, especially in linear algebra. I don't know why someone would be posting such bold claims as yours about mathematics and linear algebra while not even knowing that there is a distinction between merely a set and an ordered tuple.
Saying that linear algebra is the foundation of mathematics while not knowing the basic notion of an ordered tuple is like saying benzene rings are the foundation of chemistry while not knowing what an atom is.
This is not just a matter of notation, but is a crucial concept, especially in linear algebra
Of course it is, but I am not a mathematician and you're literally talking about writing style. I pointed that out already. You're talking about notation which is literally how you write it down by definition so that there isn't a bunch of words needed to explain things.
AFAIK one doesn't use braces for subsets but then since I'm not a mathematician by trade I don't really know anything about how mathematicians do things
See, in the real world people can't specialize in everything. There are limits to how much jargon and special notation and career details a generalist can cover because there just isn't any possible way to read and remember every possible detail of every discipline, even if you spent 100% of your time doing nothing but learning and no time at all in practical application.
And you're just plain wrong about it not being "just a matter of notation". That's bloody well exactly what it is and it's specialized for your particular field because it makes sense in a field where you write down equations all the time. In fields where writing equations is a complete waste of time except for the rare instance when it's going into the documentation it's mostly worthless trivia.
As for knowing it there would be absolutely no difference at all in me saying you don't know linear algebra because you didn't know that what I pasted into my post was set notation for a set of tuples (immutable ordered sets).
If I had thought you'd try such a specious and obvious Ad Hominem attack merely because you don't like my argumet I would have taken the time to run the output through SymPy. I'd have done that because it's a tool for turning functional code notation into symbolic mathematics notation.
The tool's purpose is to ease the grunt work required to turn functional logic into symbolic mathematics for publication because ain't nobody got time for that shit but a mathematician.
I don't know why someone would be posting such bold claims as yours about mathematics and linear algebra while not even knowing that there is a distinction between merely a set and an ordered tuple.
Saying that linear algebra is the foundation of mathematics while not knowing the basic notion of an ordered tuple is like saying benzene rings are the foundation of chemistry while not knowing what an atom is.
Blah blah blah "HAHA! I found you out! Have at the pretender!"
Seriously what the hell is your problem? I don't care if you like the argument for set theory based axioms or not. It's a core concept of the philosophy of mathematics and I think it makes a lot of sense but I'm not on a crusade to make you accept it and you had no call whatever to try to trick me so you could try to shame me.
It doesn't really matter that you failed completely (and made yourself look like a jackass) so much as the fact that I'm a real person. I have real feelings. I like people and genuinely enjoy talking about philosophy and so far all I've gotten since I came here is a bunch of angry people trying to take out their personal issues on random strangers.
Seriously, go see a professional and deal with your issues, not my fault and I don't deserve your shit.
EDIT: I also didn't now that Python borrowed the term "Tuple" from mathematics. Cool. I learned a new etymology. So yeah, I guess I could have answered your question easily if I had known we shared that.
TonesInDeepFreezeMarch 04, 2022 at 21:46#6630030 likes
No, that's the point, I'm not. I'm talking about the very concept, as it was mentioned to you by even another poster. Moreover, even if it were merely (which it is not) notation, then still it is hard to imagine anyone has studied linear algebra without having seen this utterly ubiquitous notation.
Anyway, now that I've given you an explicit explanation, you may understand the difference between a set taken in and of itself and an ordered tuple, and you now have the common notation you may use going forward to contribute to communicating clearly about mathematics with people who have actually studied it and know about linear algebra..
Reply to Agent Smith It's not a joke, he's saying that your semantic mess is ridiculous.
As much as I can't stand that guy, he's right. Your entire argument is semantic and rooted in an incomplete understanding of the philosophy of mathematics and semantic meaning of the words you are using.
It's not a joke, he's saying that your semantic mess is ridiculous.
Pray tell, how is what I wrote/said ridiculous? :smile:
I want to run something by you since you seem to know your way around the philosophy of math.
I recall a video im which Michio Kaku says (paraphrasing) "In blackholes, the (relevant) equations when used result in this: [math]\infty = \infty[/math] and then physicists can't make head or tail of it. Math breaks down"
If [math]\infty[/math] does that to math, is [math]\infty[/math] mathematical?
Once again, my colleague Stephen Hawking has upset the apple cart. The event horizon surrounding a black hole was once though to be an imaginary sphere. But recent theories indicate that it may actually be physical, maybe even a sphere of fire. But I don't trust any of these calculations until we have a full-blown string theory calculation, since Einstein's theory by itself is incomplete.
Combining quantum entanglement with wormholes yields mind boggling results about black holes. But I don't trust them until we have a theory of everything which can combine quantum effects with general relativity. i.e. we need to have a full blown string theory resolve this sticky question.
If I completely avoid theoretical physics questions and stick to mathematics do you notice something here?
I'm specifically talking about MK's approach to theories not his own...
Mathematics is a declarative language. You can make statements. You can make incomplete statements if you provide variables or equations with variables. You cannot, however, speculate or expound on theory. Mathematics is also linear. It makes clearly defined logical statements in a linear series. You can make a linear set of those linear statements. You can stack up as many linear statements describing as many dimensions or universes as you like too.
MK is a theoretical physicist specializing in cosmology and grand unified theory.
The problem MK (and all other theoretical physicists) are running into with their theories is the difficulty of constructing an analogy of sufficiently granular specificity (enough details/variables) to describe the function of the universe linearly and in an unbroken form from quantum theory into relativity (Einstein space-time).
Add to that MK's problem accepting any theory not his own and he says some truly whacked out things on a regular basis. I couldn't find your quote, but just from the ones I did find we already know he's going to utterly disagree with QMRE (quantum-modified Rocharch Equation) and infinite universe because any times infinite anything comes up (or the math showing that super-blackholes just can't form - based on Radjaharma's sp? research) he just says "bullshit" and ignores it.
He doesn't debate. He doesn't offer explanations. He doesn't refute. He just says it's not worth his valuable time. It's 100% ego-driven logic.
Nope. It doesn't do that. That's a linear assumption, NOT an axiom. It's not a proof. It isn't part of mathematics at all. It's just a belief held by some people with big egos and funding to defend.
If ? does that to math, is ? mathematical?
— Agent Smith
Nope. It doesn't do that. That's a linear assumption, NOT an axiom. It's not a proof. It isn't part of mathematics at all. It's just a belief held by some people with big egos and funding to defend.
Can we do math with [math]\infty[/math]? Try some basic operations (+ × ÷ -) on it and check what happens.
Reply to Agent Smith Yes of course. It's just like any other variable symbol.
(I tried bbcode math and mathml and texzilla but this forum doesn't support it :-( )
No work
\sqrt(\infinity*\infinity)=\infinity*1
UTF-8 but no solve 'cause not MathML
?+1= ?+1-1= ?*1
The Fourier Transform (signals math) - edit: this looks horrid. (appended what it should look like)
f(x)=?n=???cne2?i(n/T)x=?n=???f^(?n)e2?i?nx??
edit: and here is what it's doing:
You see what I mean now? You can't sum up infinity so you can't solve the answer to ?*2 but you can work with it like any other variable and to do many modern forms of equation you have to.
Comments (353)
Next time you're intoxicated by alcohol and feel the irresistable urge to hurl, you're, intriguingly it seems, experiencing the qualitative side to infinity.
Eeeeew! Disgusting! :vomit: = [math]\infty[/math]
Yes, we constantly refer to infinity in various ways without referencing math. Eternal, forever, and immortal are just some of the ways we express infinity.
I'm not sure if this is what you're looking for. Maybe you could explain more?
print("I am an infinite loop")}
Little kids love this. No reason philosophers won't too.
Any examples of Infinity other than the suggestion that the universe may go on forever since we can't prove otherwise?
You were already given some examples in this thread. How about the Successor function as another example?
The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements, which in the case of discrete points is one (so the ordinal in the cardinal number becomes 0). For the continous line, the number needed is two (ordinal subscript 1), for the 2D continuum it's four (corresponding to subscript 3), and eight will do for 3D continuous space (ordinal number 7, so [math]\aleph _7 [/math]).
Now we can play the same game with with the aleph numbers. For discrete aleph numbers, [math]\aleph _n [/math], the natural alephs, we can assign super aleph number [math]\aleph _{S_0} [/math], when the number of natural alephs needed is one. When 2 natural alephs are needed, for a continuous line of alephs, we get [math]\aleph _{S_1} [/math], for a 2D continuous plane [math]\aleph _{S_3} [/math], and for an n-dimensional continuous volume, when [math]2^n [/math] natural alephs are needed we arrive at [math]\aleph _{S_{n^2-1}}[/math].
Now we can play the game again for hypersuper alephs, [math]\aleph_{{HS}_n}[/math]. For an infinity of inf... well, ad inf.
So, the infinite can't be reached like zero can't be reached (if you don't include the negation of the positive real). Both can't be reached, and still some try to reach it while others, mostly unwillingly are pushed towards nothing. The desire to reach for the nothing is not so different from the desire to reach for the infinite, though the implementations of this in material life have quite different implications. Life and death, even.
Infinity is just as useless as nothing. In between is where the action is. Are the things in life that never can be reached infinite? Yes. My wife is one of them. And I have to admit, attempts made by pettifoggers and skirlers on their doodlesack never cease to bumfuzzle me.
Everyday examples of infinity. Maybe falling asleep and waking up. It seems that time tic-toc-ed infinitely fast in between. It seems nothing at all exists in between, another example that nothing and infinite have a close, if not intimate connection.
'the continuum' is probably most exactly defined as
Quoting AgentTangarine
If by "the two-dimensional continuum" you mean RxR, then it is incorrect that its cardinality is different from the cardinality of R. If by "the 3D continuum" you mean RxRxR, then it is incorrect that its cardinality is different from the cardinality of R. For any natural number n>0, R^n has cardinality equal to the cardinality of R.
Quoting AgentTangarine
I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements" but the aleph notation is defined by transfinite recursion on the ordinals. For an ordinal k+1, aleph_k+1 is the least cardinal greater than aleph_k. For a limit ordinal L, aleph_L is the union of {aleph_k | k < L}.
If you need an infinity of infinites then aleph is 1. If you need an infinity of them, then 2, etc. The cardinal number of the continuum is defined on one dimension only. You really think the cardinality of the 2 or 3 dimensional space and the 1 d are the same? They're not. Aleph line is 1, aleph plane is 2 and aleph space is 3.
lol man, that's not infinite because if computer loses power, your loop will end as well :razz:
It is a theorem that card(R) = card(R^n) for any natural number n>0. This is known by anyone who has read a basic textbook in set theory. Just read the proof for yourself.
Moreover, you keep claiming that card(R) = aleph_1, thus precluding that card(R) might be greater than aleph_1 and thus precluding that there might be cardinalities between card(N) and card(R) . That is only the continuum hypothesis, not settled mathematics.
There could be cardinalities between 1 and 2. The fractals, with fractal dimension.
The points on the side of a square have c=1. The square has c=2, while a fractal curve in it has c between 1 and 2.
I have not mentioned continuousness. I have merely pointed out the utterly well known fact that it is a theorem that card(R) = card(RxR),.
Quoting AgentTangarine
That is risibly wrong.
There are no cardinalities between 1 and 2. And there are no cardinalities between aleph_1 and aleph_2. However, without the continuum hypothesis, there could be cardinalities between card(N) and card(R), as, without the continuum hypothesis, card (R) could be aleph_x for some x>1, as I pointed out to you over and over in the other thread.
You don't know what you're talking about,.
Came here to say the same thing, but with JavaScript instead.
Yeah, you have said that infinite times already. It's just not true. There are inf^2 points between 0 and 1. Aleph1. There are inf^4 of them in 2d. Aleph2. In 3d there are inf^8. Aleph3. I'm off. It's boring.
Well, it's not. Between aleph1 and aleph2 lies aleph1.5. A fractal line occupying half the square. There are inf^3 points for this figure.
And you ignorantly, wantonly persist about aleph_1
The claim that aleph_1 = 2^N is the continuum hypothesis.
Quoting AgentTangarine
It's not so much boring as it is unfortunate that you persist to post misinformation while you won't even bother to look it up on the Internet.
Which has no apparent meaning.
I guess what you mean is SxS where S is infinite.
But then there are more than card(NxN) real numbers between 0 and 1.
card({x | x is a real number between 0 and 1:) = 2^N, which is greater than card(NxN),
You are abysmally confused.
Then you have a different notion of aleph one. The ordinal in Aleph one is just related to how many times the infinity is present. For the naturals inf^1, so aleph 0. For the line inf^2, so aleph1.. For the 2d plane inf^4, so aleph2. For a volume inf^8, so aleph3. For a 1d fractal, say inf^3. So aleph1.6, approximately.
Indeed I have a different notion from yours! My notion is the usual one in mathematics.
You, on the other hand, are unfamiliar with the ordinary mathematical definitions.
Quoting AgentTangarine
Aleph_1 is the least cardinal greater than Aleph_0. That is the ordinary mathematical definition.
Quoting AgentTangarine
Please stop writing 'inf' that way. It doesn't have any apparent meaning.
Quoting AgentTangarine
No, bad Eliza, bad.
Anyway, you said you were off, Eliza. Too bad you didn't mean it.
Why? It's not as boring as I thought. There are even alephs0.5 and alephs0.99 or alephs0.01. If there are alephs(sqrt2) remains to be seen. A closed interval on the real line, like [0-1] can be fit infinite times on the real line, so in fact the cardinality of the real line is 1.4. That of the 2d plane is about 2.6. That of the 3d volume is about 3.2. That of a fractal line, plane, or volume, lies between these. Cantor didn't realize this yet.
Coming to think about it, of course aleph(sqrt2) exists. And the aleph for [0-1] is in fact aleph1. Cantor overlooked one infinity! Which only shows his genius! Who can overlook infinity...?
This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0
1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)
Extending these ideas shows the cardinality of R^3 is the same as that of R.
In another thread in which you were posting, I already wrote that I am not an expert.
Quoting AgentTangarine
It doesn't highlight any concept. It only highlights that you don't know anything about this subject.
Quoting AgentTangarine
Who is FZ? And why does it matter that he lived 120 years ago?
Quoting AgentTangarine
The axiom of choice is not needed for finite sets. Every finite set has a choice function, irrespective of the axiom of choice. In the context of our exchanges, one of the important points about the axiom of choice is that it implies that every infinite set has a cardinality and that every infinite cardinal is an aleph.
Quoting AgentTangarine
Delusional and disconnected from reality. Check for fever.
Fraenkel and Zermelo. Old-fashioned. They, like Cantor, overlook one infinity.
Quoting jgill
By the way, I did look at that Quora page you suggested in the thread that has since been deleted. The first post there is a proof that card(R) = card(RxR), which is exactly what you deny!
Fraenkel and Zermelo. Aren't they that old vaudeville comedy team that played the Borscht Belt years ago?
But I thought maybe you meant Frank Zappa. He's kinda old school too by this time.
You guys need to get a hotel room.
Step 1. Open browser.
Step 2. In the search field, type:
continuum hypothesis
Step 3. Click on the first link that appears to be an encyclopedia article.
Step 4. Read the part of article that states the continuum hypothesis.
Step 5. Click on bookmark for The Philosophy Forum.
Step 6. Click on thread 'Infinities outside of math?'
Step 7. In the posting box, type:
Now I see, TonesInDeepFreeze. You are right. Thank you.
Step 8. Click on 'Post Comment'.
Step 9. Stop.
That's what is said about geniuses in general. Like being a crackpot. Untill now I haven't seen one bit of math, only parrot references to the net. The cardinality of RxR being the same as R. I gave you a link to a so-called proof of a bijection between R and RxR. A wrong one. I asked you why it's wrong. No reply. Cantor didn't take the infinity of the real line into account in determining how many powers of an infinity are needed. You only replied that you can't raise infinity to a power. You just need inf^3 times to enumerate all points on a line. Saying that the aleph of a 2d infinite plane is the same as that of an infinite line is the same as saying R is the same as N.
So what? In general geniuses drink water. I drink water. That doesn't make me a genius.
In an any case, you show no evidence of genius. Very much to the contrary.
Quoting AgentTangarine
I gave you the primary formulas that you need to start with. You ignore them then complain that I haven't given you any math.
Quoting AgentTangarine
There is no fault in my recommending that you look up the continuum hypothesis.
Quoting AgentTangarine
I don't need to defend someone else's proof that I hadn't referenced. I already know a more general proof, as can be found in a textbook on the subject.
Quoting AgentTangarine
No, I said a lot more.
Disgusting.
Quoting AgentTangarine
Take away the word 'project' (a projection function is a certain kind of function and it is not needed to mention regarding whether there is a bijection from N onto R). So use 'map' instead'.
Also, trivially we can map N into R one to one. So in this context instead of 'to' we must say 'onto'.
Then your claim becomes:
We cannot map N onto R one-to-one.
And that directly contradicts your claim now that we can map N onto R one-to-one.
You are very very confused.
We can! An infinite times infinite times actually. One time onto 0.1-0.9999999. This can be done an infinite times for [0-1]. And an infinite times for the whole real line. Hence aleph1.4, and not aleph1.
"She certainly can-can." - Cole Porter
And anything goes!
I'll give you a point for that one.
Which functions? Of repeated fractions?
I have no intention to be one. But that's what they say. Like being crackpots.
I didn't say I gave you functions. And I have never said anything about repeated fractions. You seem to have confused me with another poster.
You haven't proven any math. But you have proven yourself to be a crank.
Then prove me wrong!
I already explained to you that proving that there is a bijection from N onto R requires stating your axioms, definitions, and rules of inference and using only those axioms, definitions, and rules of inference to show that there is a function whose domain is N, whose range is R, and is 1-1. All three clauses: domain, range, 1-1. Such a proof, if it were in ZFC, would contradict the theorem that there does not exist a bijection from N onto R, thus proving that set theory is inconsistent, and would make you among the very most famous people in the entire history of mathematics.
On other matters such as alephs, you're proven wrong by simply referring to the definitions and by fhe fact that the assertion that card(R) = aleph_1 is famously independent of ZFC.
This is the correspondence between infinite and infxinfxinf. And if r goes to 0.04566..., or 0.005667... or 0.000556654... (you get my point, I guess...) even infinite times are included. So the map becomes one between inf^2 and inf^4. Aleph1 and aleph2.
Incorrigible.
Show the correct answer then. I already gave you a link to a supposed bijection between R and RxR. How many times can N be mapped on the real line? Just offer a tasty recipe.
Okay, N^2, as you wish. Same as inf^2.
I'll leave my remarks about card(R) = card(RxR) anyway:
I have been referencing the more general theorem that for an infinite set S and natural number n>0, we have card(S) = card(S^n). S = R is a special case of that. I'm not sure, but it seems perhaps the particular proofs mentioned in threads here lately for R don't use the axiom of choice (?). I have not studied those proofs to verify them for myself though I get the gist of them and they seem okay to me to that extent.
The proof I have studied of "for an infinite set S and natural number n>0, we have card(S) = card(S^n)" is in Enderton's 'Elements Of Set Theory'. It is pretty involved, two pages, requiring a number of previous lemmas, a proof that the axiom of choice implies Zorn's lemma, closure under unions of chains, and more (and even an illustration to aid intuition). I would not spend my time and labor composing it all for you in the confines of a post, and it would do you no good anyway since you are utterly unfamiliar with even the basics of set theory that are prerequisite let alone the mathematics of Zorn's lemma, chains, et. al. And I admit that I am rusty myself on some of the details now, though I have previously studied it in every detail to verify for myself that it is perfectly correct.
The best I can do for you is to recommend that you get a textbook and study it from page 1. Enderton's 'Elements Of Set Theory' in particular is widely used, highly regarded, beautifully written, and pedagogically exemplary. Though, I would actually first recommend at least gaining a basic understanding of symbolic logic.
That's for countable sets. For R this doesn't hold.
Do you mean N onto R?
The answer is 0.
My many previous remarks amply imply that.
Now that I've answered your question, howzabout you answering a question I've been asking you: Why won't look up 'continuum hypothesis' on the Internet?
Wrong. For any infinite set S.
And that's exactly where the failure lies. The relation between R and RxR is the same as the relation between R and N. R can be viewed as corresponding so a single cardinal. How many lines you need to construct RxR? The same number as the number of cardinals to construct R.
Every cardinal is either countable or uncountable. Every countable cardinal is either finite or denumerable. There are denumerably many countable cardinals. There are denumerably many finite cardinals. There is only one denumerable cardinal.
That is easy. The hard part I described is PROVING that for an infinite set S and natural number n>0, we have card(S) = card(S^n).
After proving that, cardinal arithmetic is indeed beautifully simple:
If K and L are cardinals, and the larger of them is infinite and the smaller of them is not zero, then
K + L = K * L = max(K L)
That is purely arbitrary unfounded assertion.
On the other hand, in set theory, from rigorously stated axioms, definitions, and rules of inference we prove things.
And I answered a question for you, but you still have not answered my question for you. So the next question for you is: Why do you think one would feel need to answer your questions when you don't answer questions yourself? Wait, I have the answer: Because you are a crank.
You contradict yourself. You claim there is a bijection from N onto R, but above you admit that the interval [0 1] is uncountable. And we don't write '[0-1]' to denote and interval.
You are totally mixed up.
And it is beyond me why you think that the stuff you make up in your own head - without rigorous principles and without any understanding of the mathematics you allude to and the terminology you abuse - is somehow true and correct while the mathematics that has been rigorously formulated, studied, scrutinized, critiqued, and checked and thousands of times over rechecked by professionals and students all over the word is all wrong. In other words, what makes a crank tick?
You have to read what I write. There are infinite bijections between between N and [0-1]. Like there are an infinite times infinite between N and R. And there are an infinite times infinite between R and RxR.
I do. You don't conversely.
Quoting AgentTangarine
Read not just what I write, but what is written anywhere you would look: There is no bijection between N and [0 1].
What you wrote is contradictory:
You write that there is a bijection between N and [0 1] and you write that [0 1] is uncountable,
You don't know what you're doing.
If you consider the infinite line the discrete elements on the plane, like the natural numbers on R, than the relation between the lines and RxR, the infinite plane, is conformal to the relation between N and R.
I answered one of your questions. Your turn to answer one of mine.
Yeah, that's the zillionth time you mentioned this. I haven't seen one thing to conclude you do. You only make references.
No one ever told you that mathematics is not throwing around words like 'conformal' while not knowing what they mean.
Not the zillionth time, but it is the second time you have made that false claim.
You haven't answered one!
Everyone knows what conformal means.
Then what is false?
Now you have flat out lied.
Quoting TonesInDeepFreeze
Again now, your turn to answer my question.
Ah, I didn't see that one! 0 times? Are you kidding?
The claim you made, as I quoted it, silly.
But why false?
That you don't understand the answer is not my problem. I did answer it. Now your turn to answer my question,.
Eliza is getting tired and reduced to token replies.
That's no answer why it's false. You seriously believe you can map N zero times on R? What's your question?
Now you take refugee behind empty verbiage. Liza is tired indeed. My wife lies in bed for a couple of hours already... I told her to come too. Three hours ago...
Eliza is now all tangled up in confusion. My answer was to the question about mappings from N onto [0 1] not about why your claim that I have done nothing but give references is false.
I'm not surprised that your attention span wouldn't provide recalling my question: Why won't you look up 'continuum hypothesis' on the Internet?
I have done that years ago already. I don't agree though. I'm not a parrot like you.
Not at all. It's telling that you could figure out for yourself, but you won't, that your claim is false for the simple reason that I have not just given references but have posted formulations for you (as I already mentioned I have done that to your PREVIOUS claim that I only give references) and explanations too.
I really would rather not know about the bedtimes of you and your wife.
Then ignore it. The only riddle you gave as an actual answer is that you can map N zero times on R.
So years ago you looked it up, but still don't understand it now.
You don't agree with "it"? The continuum hypothesis? You have been claiming the continuum hypothesis for a zillion posts yet you say you don't agree with it! You are one really confused crank.
You are parroting your own world now as this is the second time you've said I am parroting, And even though I amply refuted that a while ago.
I don't mean that litteraly.
So what? This is not a game with a scoreboard for how many questions I've answered.
Anyway, I've addressed TONS of your claims.
Then what do you mean? Why won't you now look up 'continuum hypothesis'?
Who says it is? I just said you have answered with a riddle.
Great. And I can make the sun disappear by wiggling my ears.
The number 0 is not a riddle.
I looked it up when I wrote about the alephs, and it was even the reason for my post. Seems years ago.
But if you say it's the number of times you can map it on R it is. I can even map a single element of N to R.
Yet you say you disagree with the continuum hypothesis now, while you have been claiming it in dozens and dozens of posts.
So?
Of course one can map a singleton set INTO R.
But there is no map of N ONTO R.
I can map a singleton on R too. No problem.
So you are in flat out contradiction with yourself.
So?
Either you are insane or you don't know the meaning of 'map onto'.
No. I assert card(R)=aleph1.4. Call me crazy, like you wish. Card(RxR)=aleph2.6, more or less. 2^(1.4)=3, more or less, 2^(2.6)=6, more or less.
Quoting TonesInDeepFreeze
I can send every cardinal on top of R. On top where I send it into.
So which is it now? You claim that card(R) = aleph_1, thus asserting the continuum hypothesis? Or you deny that card(R) = aleph_1, thus denying the continuum hypothesis?
Hardly matters though, since it is sheer crazy talk to say that there is a mapping of a singleton onto R.
As I said, you don't know the meaning of 'onto' in mathematics. Or you're insane.
"The mapping of 'f' is said to be onto if every element of Y is the f-image of at least one element of X."
Y=R, N=X...
Are you off your meds again?
Quoting TonesInDeepFreeze
Pardon me, Ma'am. Here in the bunkhouse on the prairie I tend to forget there are ladies present. :worry:
I now even think R has cardinality aleph1.4!
You guys need to get a hotel room. — jgill
Haha! In Hilbert's hotel!
Haha! In Hilbert's hotel! Wrong post...
That's not quite right since it has free variables on the right that aren't on the left.
f is onto Y if and only if (f is a function & range(f)= Y)
Note that that precludes Y from being a proper subset of range(f), which is a situation not usually mentioned. I suppose one could reformulate the definition to allow Y being a proper subset of the range(f) for a broader definition.
Also, some mathematicians consider a function to be not just the graph but also that the domain and the range are specified, and sometimes the domain and a co-domain (there may be different co-domains, since a co-domain can be any superset of the range). That could be made rigorous by saying a function is a triple
I got it from the net...
Still, I can map N onto R. Inf^3 times even...
Huh?
So? Not everything on the Internet is the sharpest formulation.
There you go.
I can map N onto R infinite times infinite times.
Infinite times onto
[0.1-0.99999...]
[0.01-0.09999...]
[0.001-0.009999...]
.
.
.
So infinite times onto [0-1]. The map even defines [0-1].
Times infinity for all length 1 intervals.
So in total there is an inf^3 involved. Hence aleph1.4. Merry Christmas!
No. You are right. Onto only for N^3! But into inf^3 times. I get sleepy.
Which means aleph1.4 is the one for R, and aleph2.6 is the one for RxR.
I redid the post.
There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject.
Indeed. So N can't be mapped 1-1 onto. You can at most map N to infinite points, infinite times. Say 1 into 2, 2 into 4, 3 into 6, etc. But into still. Then you still need to map into all inbetween intervals. On each interval an infinity of infinities of naturals is needed. So you need inf^3 times to map N into R. Pffffff.... I'm done! Thanks for the resistence! :smile:
There are a lot of good books indeed. Thanks for the references. I prefer the math use in physics though. And the alephity of the continuum has implications for particles moving in it. That's why I think the aleph of the volume is different from the line and plane. I have booked a hotel for us... Just kidding! Gnight!
You still demonstrate that you don't understand these basic concepts.
And you desperately need one if you are not to remain mired in your terrible confusions.
Well... I don't take it too seriously... You are probably right. Still, I can't see how R and RxR can have the same cardinality. There are just inf^3 times as many points in RxR as there are in R.
I resist misinformation.
The problem is not so much that you don't take it seriously, but that you take it seriously enough to stubbornly persist in claims that are false or just ersatz gibberish from your own mind uninformed about anything other than itself.
Quoting AgentTangarine
You could ask for more details about the proof mentioned by jgill and about the proof in the Quora thread.
For a proof in greater generality for any infinite S, as I said, it requires learning set theory.
The point is that the proof in quora is incorrect. It's making use of decimal expansions also but overlooks the majority of them.
Name one.
You should edit Wikipedia. We need Wikipedia and Wikipedia needs you i.e. we need you!
No offense intended AgentTangarine. I don't think your're guilty of misinformation. TonesInDeepFreeze is conflating facts with opinions.
Quoting TonesInDeepFreeze
Well, the point made is that a pair of numbers (x,y) say (0.678567..., 0,98678...) is contained in a single number 0.65456456.... The infinite number behind the 0 should contain both the infinites behind the 0 of x and y. This is not so.
You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.
Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.
No way. One can offer alternative systems; I enjoy reading about them if they are rigorous. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion.
Saying in a case like this "Oh, it's all opinion anyway" is intellectual dereliction.
That is not a real number, you understand, right?
I'm currently reading a book on mathematical philosophy. I'm no good at math although I'm fascinated by the subject. I noticed that math, its various branches, start life more as vague intuitions rather than crystal-clear concepts/ideas. Rigor comes much, much later if I'm not mistaken.
Too, the definitions in math give me the impression that true understanding is being sacrificed for logical formalism.
To know, we would have to have access to the mental states of mathematicians. We would have to know how long was the time between their first pre-formal musings and then putting them down in concrete formulations. There is no reason to believe that for many mathematicians that time might be very brief.
Anyway, if cranks said, "Here are my pre-formal musings, maybe something could come of them", then that would be one thing, but instead cranks insist that their view and only their view is correct; that ordinary mathematics (and even the alternative systems that the crank is ignorant of) are wrong. It is the crank, not the mathematician, who is dogmatic and exclusionary.
I have never had any such impression. Very much to the contrary.
Example?
Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many. (same for (.0329576914..., .0925318623...).
Quoting TonesInDeepFreeze
It's a pair of numbers. You must quote the whole line I wrote.
Give me an example of a number you think is left out. I bet it's not. Avoid the .999... =1.0 thing. Have you figured out what the algorithm is?
Quoting Agent Smith
The definition you are seeing is the formal aspect. It's a kind of final touch to an idea that began as an interesting notion.
Yes, a pair of numbers. Not a number as you wrote.
You keep resorting to saying that I must consider the rest of what you posted. But each time it turns out that the rest of what you posted doesn't actually qualify into correctness the initially incorrect statements you make.
But funny,
Whatever you mean by an ordered pair being "contained" in a number, what we have is each number mapped to an ordered pair. The claim of the prover is that the whole mapping is 1-1 and onto RxR. All it takes then is to see that no ordered pair is mapped to by two different numbers, and that each ordered pair is mapped to by a number.
The diagonal proof shows that any map from N to R is not onto R. That is, there are real numbers not mapped to.
I wrote:
Well, the point made is that a pair of numbers (x,y)...
Quoting TonesInDeepFreeze
Where did I do that?
Quoting TonesInDeepFreeze
It's you who is the crank. You are exclusionary and dogmatic. And you have no sense of humor. Sense of rigor, maybe. Jgill knew to convince me (well, almost...) in one comment. But he's a real mathematician.
Cantor diagonal setQuoting TonesInDeepFreeze
That's not what the proof is about. It just shows that [0-1]is uncountable. Every time you think you counted a new number shows up. After infinity.
Just now, and in the other thread that was deleted yesterday.
You don't need the diagonal proof to realize that. Every real number can be mapped from N^3. Every real number can be reached from N^3. N^3 can be mapped onto R.
You have not shown any dogmatism by me. Nor any exclusion other than of ignorant confusion and misinformation.
Quoting AgentTangarine
It shows that [0 1] is uncountable by showing that any map from N to [0 1] is not onto [0 1]..
On the contrary. I even told you I contradict myself in previous posts. I never told you to consider other posts. Anyhow... I'm truly tired and my beloved has awoken. Damned! 7 hours about infinities.. I'm off to bed. Gonna contemplate about mr. Gill's procedure. It was fun! :smile:
True, there are other proofs of the uncountability of [0 1]. Cantor gave one of those other proofs.
Quoting AgentTangarine
There is no map from N^3 onto R. And even if there were, it would prove the countability of R not the uncountability. You are again completely backwards and confused.
Yeah, you are right about that! Sorry that I called you dogmatic and exclusionary! You certainly got me interested in this aleph topic! As a physicist I find it difficult to believe that the number of points on a line is the same as on a plane or in a volume. The number of directions are different though. Or not even that?
By "the rest of what you posted" I meant the rest of what you posted in that post, just as I was responding exactly to your complaint that I hadn't quoted more of your post.
Then I bet you really would not like Banach-Tarski.
Anyway, stepping back, do you understand the proof of the equinumerosity of N and NxN?
You can map N to all reals between 0.1 and 0.999999...
You can do this N times (for all smaller decimals). You can do this for all N size one intervals. Where am I wrong?
In fact, I like that theorem!
Off the top of my head, infinitesimals.
Quoting jgill
Yep, the formalism is necessary to ground what started off as an intuition, in logic; a necessity no doubt; after all, in math not having a crisp definition trying to find a missing person without either having a good description or photo of that person.
My point is that making a definition precise (usually) means losing some/all of the feelings that go with the intuition. The transition from just a vague notion to a clear-cut, well-defined idea is, for me, a heart-to-brain relocation of an idea and that's what bothers me. Math's link to the heart pops up occasionally though - I've heard of mathematicians being moved by the elegance and beauty of some formulae for example. I have a book titled The Heart of Mathematics which is on my reading list.
You keep saying that. It's dogmatism.
Quoting AgentTangarine
In thinking that the fact that in your own mind you imagine that it must be so implies a mathematical proof. And in thinking that your disconnected and mathematically unsyntactical dribblings are too mathematical proof.
Yet you reject other theorems from the same axioms.
I think it's the continuum that confuses me, and its break-up into (onto?) its parts. I think the break-up of a square into lines is the same as a line into points. But maybe the break-up of a square into points is the same as a line into points.
I mean, where am I wrong if I say that N^3 can be mapped on R?
That's almost a good example. But it's better described as the centuries-ago formulations being more than vague intuitions yet not adequately formalized. Then, centuries later, it was discovered with mathematical logic and model theory how to vindicate the notion rigorously.
Of course, one can look back centuries, even to the ancients, to see that their mathematics has seen been formalized.
What I thought you had in mind though are cases where mathematicians had vague notions and then they or their contemporaries formalized those notions themselves.
(1) How do you know that unless you've interviewed mathematicians about it?
(2) I highly doubt that mathematicians very much regret whatever such loss of feelings there might be, as I would think mathematicians are primarily eager to communicate their notions clearly and objectively to other mathematicians and to prove their results.
I cherish the mysteries of mathematics. That is not impaired by formalization. On the contrary, formalization leads to even deeper mysteries for me.
I told you. You don't have proof of it. You only think you do.
Also, you have a serious problem when it's pointed out to you that you are in contradiction with yourself and your response is "So?".
Start by adopting a specific definition of 'the continuum'. The term is often used flexibly, but I would settle on the continuum understood to be the pair of the set of reals with the standard ordering on the reals:
c =
Or, if you prefer to think of it as merely the x-axis:
{
Or when we refer to "the cardinality of the continuum" we are thinking of the continuum as merely the set of real numbers.
But where am I wrong in my proof? Cannot N be mapped onto 0.1-1? You ñeed N numbers for that: 1-99999999.... What number do I leave out here? Or do I leave numbers out between 0.1-0.9999999....?
Do you mean to suggest that there is a 1-1 function from N onto 0?
By the way, you claimed that I have no sense of humor. Well, you haven't said anything funny. Neither have I very much, since I don't find this context with you to motivate me to make jokes. That is not a lack of sense of humor. You don't know me.
I was a bit angry when I wrote that! Forget it! I don't even know you!
There is a 1-1 function from N onto [0.1-1] (the interval from 0.1 to 1). Isn't there?
Given any function f from N to [0 1], the diagonal proof constructs a member of [0 1] that is not in the range of f.
I feel pretty safe in thinking that you don't know the diagonal proof.
Moreover, even if the diagonal proof were found to be incorrect (it won't be) then that would not constitute a proof of its negation.
I've told you a couple of times already: To prove your claim, you must prove that there is a function whose domain is N and whose range is R and that it is 1-1. And I won't even ask that your proof be constructive by showing a particular such function, only that one exists, even though Cantor's proof is constructive: showing, for any given function from N to R, a particular real number not in the range of that function.
The diagonol proof doesn’t apply here. I was thinkiñg that too. The interval I talk about is 0.1 to 1. Not zero to 1.
You say it doesn't apply, yet you mentioned it directly in connection.
I might think you are trolling me, but even worst trolls don't usually have your endurance.
It's a 1-1 map!
Strangely enough, you need more numbers between 0 and 0.1.
Quoting TonesInDeepFreeze
No. I mean: why can't N be mapped onto 0.1-1 (or 0.1-0.9999999999.....). I use every member of N one time.
So maybe what you mean by "0.1-1" is [0.1 1]?
If you're going to ask me to explain why you have no proof, then it would help for you to use recognizable notation, and would be more courteous too.
Now you say Quoting AgentTangarine
What does "use the numbers mean"?
Do you mean they are all in the range of a function from N to some set? WHAT function? You have not adduced any function. For any given k in N, what is f(k)?
Quoting AgentTangarine
So maybe you just mean that all the natural numbers are in the domain? Well, duh, yeah. So what? The domain is not at issue. It is the range that is at issue. You need to prove that there is a function whose domain is N and whose range is R (i.e. every real number is in the range).
So I ask:
Are you clear that your task is to prove that there is a function whose domain is N and such that every real number is in the range of the function?
Okay:
0.1 connects with 1
0.2 connects with 2
0.3 connects with 3
.
.
.
0.14 with 14
0.15 with 15
.
.
0.53 with 53
0.54 with 54
.
.
0.768 with 768
0.769 with 769
.
.
0.99998 with 99998
.
.
ad 0.9999999999......
Do I have to list all N numbers?
The bijection between R and RxR is not continuous.
This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology.
Yes. I get that. Still... something is nagging. If I map all naturals on [0.1-0.999999] (there you go...) what do I leave out? [0-0.1] seems to contain more numbers than [0.1-0.99999...]:
0.01-0,0999999...
0.001-0.009999...
0.0001-0.00099999...
.
.
.
Which is absurd. Still... On each of the intervals (including [0.1-0.99999...]) you can map the set N directly. Can you break up a continuous interval, like [0.1-1], up in real points? Like 0.1, 0.2, 0.3,...,0.91, 0.92,...,0.110, 0.111,...,0.1222, 0.1223,..., 0.2111, 0.2112,..., 0.24444, 0.24445, ...0,2023432, 0.2023433, ..., 0.655555, 0.655556,.......,0.999999999999....
Every cardinal is contained once. Or is one left out? The diagonal? But how can that be? Say that number is 1000000023432500876.... Isn't that contained in N? But not in R?
If so, can you represent x by 0.5878900... and y by 0.197867....(to name two arbitraries) to map on r?
Or is continuum just continuum, no matter the dimension? No points attached?
Can you knock it off please? The thread title is "Infinites Outside of Math". You've turned it into a factual discussion about math. That Agent has... spicy takes of dubious correctness... on how infinite sets work is besides the point. Please feel free to take it to personal messages, though.
Agent - would it take that much effort to try to fact check what you've written if you're genuinely interested in it?
Tones - horse is dead now.
Quoting AgentTangarine
EG: you'd find out that those repeating 9s after the decimal point become the number above in the limit. So if you write 0.999... that already equals 1, or if you write 0.0999... that already equals 0.1.
Quoting AgentTangarine
There you've provided at most a countable set of countable sequences, which together turn out to at most countable.
If you're interesting in in these things, I'm sure someone involved would be happy to provide you with study materials.
That's exactly what I have done. And didn't agree with. But it's clear now. Thanks to my opponent. I don't agree with him though.
That's exactly what I not do. I question it. That's all. I have been given no satisfactory answer though. Jgill came close. But the bijection he prescribes is discontinuous and suffers from the same problem as the number of points on the interval [0.1, 0.999999....). You just can't make points touch, or break the continuum up in points. How many points lay between 0.1 and 0.999999...? Do all these numbers constitute the continuous interval [0.1-0.99999...)? (What is continuous? Undivided.) You can assign natural number to each of these numbers. Is a number left out, by the diagonal argument? If so, isn't that a new natural number, contradicting that you left one out?
What if I asked the question, as a new thread, if the continuum can be broken in parts? It's maybe a math question. Maybe not. It will not alter the essence though of this thread, which is life-related.
I think that would be fine. There's been plenty of discussions before about the nature of the continuum. Just try and keep it away from the mathematical equivalent of pseudoscience.
The "point" is that constructing a continuum out of points seems like a pseudoscience to me. The fact that the line, plane or volume have the same cardinality is because of the attempt to reduce them to points. I'll give it a try later.
I'm well aware of the situation. :cool: Having communicated with him I believe he has a graduate degree in physics. However, his concerns are with the "physical" or interpretive aspects of the science rather than mathematical descriptions. Even advanced math in QT seems not to have reached axiomatic set theory to a noticeable extent. Kenosha Kid is another genuine physicist. To the best of my knowledge neither of these gentlemen work in the profession at present.
Ignoring the references to cardinality and accepted theory , it does seem like the unit cube has more of "something" than one of its edges.
Quoting AgentTangarine
Really?
(1) A bijection between a proper subset, or even the entire set, of terminating decimal expansions (which reduces to the set of finite sequences of natural numbers) and the set of natural numbers is not a bijection between [0 1] (which is represented by the set of denumerable (not terminating) expansions) and the set of natural numbers.
(2) Among the members of [0 1] not represented by terminating expansions are both some of the rational numbers in [0 1] and all of the irrational numbers in [0 1]. Those numbers are not represented in the domain of the bijection.
(3) To answer "What are examples of members of [0 1] that are left out of the domain of the bijection?", we could mention for example: .333.. (i.e. 1/3) and .14159265358979323846264338327950288419716939937510... (i.e. the decimal portion of pi).
Also, the domain of the bijection is only a proper subset of the set of terminating expansions, as for example, not even .01 is a member of the domain. However with a more careful construction, of course, we can show a bijection between the entire set of terminating decimal expansions and the set of natural numbers.
(4) To answer "What natural number is not in the range?", we say that no natural number is not in the range. But so what? Uncountably many members of [0 1] are not in the domain, so it is not a bijection between [0 1] and the set of natural numbers. Of course, any mathematician knows that there is a bijection between the set of terminating expansions and the set of natural numbers. Indeed it is a basic tool of formal languages and computing that the set of finite sequences on a countable set is countable. And, indeed, Cantor proved that there is a bijection between the set of rational numbers and the set of natural numbers, thus, a fortiori, there is a bijection from the set of terminating decimal expansions into the set of natural numbers. But Cantor also proved that there is no bijection between the set of denumerable (non-terminating) expansions and the set of natural numbers. Versions of the proof may be found in any good textbook on set theory.
Suggested study materials (studied in this order).
Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar. I consider this to be the best textbook for learning symbolic logic, which is extremely helpful (for me, crucial) for understanding set theory.
Introduction To Logic - Suppes. This has the best explanation (it is superb) of formal definitions that I have found.
Elements Of Set Theory - Enderton. My favorite textbook on set theory. It is beautifully written.
Axiomatic Set Theory - Suppes. Good as a backup to Enderton.
Philosophy Of Set Theory - Tiles. An overview of the intuitions and philosophy behind various views on set theory and mathematics.
However, a question has been asked and our task is to answer it as best as we can. Where do we begin?
[quote=Brazil nightclub fire survivor (2013)]I was only there for seconds. It felt like an eternity.[/quote]
A "few seconds" isn't infinity but this hapless burn victim experienced eternity (infinity). This can't be a mathematical infinity for only a "few seconds" had gone by. Ergo, this must be nonmathematical infinity. We're not out of the woods yet for eternity can only be described mathematically. I'm only offering a possible starting point for an inquiry.
'is infinite', as a set theoretic notion, is a 1-place predicate defined:
S is finite iff there is a bijection between S and a natural number
S is infinite iff S is not finite
S is Dedekind-infinite iff there is a bijection between S and a proper subset of S
Salient about infinite sets is that every Dedekind-infinite set is infinite (this is the other side of the coin of the "pigeonhole principle"), and, with an appropriate choice axiom, every infinite set is Dedekind-infinite.
And mathematical infinity is not just magnitude or number. Yes, there are infinite ordinals and infinite cardinals, which are called "numbers" (there is not, as far as I know, a mathematical definition of 'is a number') but there are infinite sets that are not ordinals or cardinals.
(I'm not a physicist, but I recall reading of this. Could be wrong)
Does this kind of reasoning really work? Since at the same time,
r=.xyzabc... <-> p=(.xyzabc..., 0, 0)
and
r=.xyzabc... <-> p=(.xyzabc..., 0, 0.0...01)
and so on?
And moreover,
r1=.xyzabc... <-> r2=(.0....1xyzabc...)
r1=.xyzabc... <-> r2=(.0....2xyzabc...)
...
My point is, every point on the line can be mapped to an edge of the cube. What about all the points in the rest of the cube?
The interleaving algorithm looks good to me. But then, you can map those interleaved points onto a single edge as well. This can go on and on in a cycle. Why is there not therefore a paradox?
Probably because points have no dimension. The real numbers are complicated and fascinating.
Look, suppose I pick at random a point in the interior of the cube: (.251956... , .629435..., .194735...) Then that point corresponds to a unique point on the x-axis edge: .261529194947...
You just alternate digits. Here, you do it by going the other way: find the point inside the cube corresponding to the point on the x-axis: .381402258639...
The process is continuous in the epsilon-delta sense.
English is a real language. The range of sentences expressible in English is infinite. Is that an infinity in the world?
Of course, these sentences can not be enumerated, at least in the world.
I guess whether there are enumerated infinities in the world depends on the nature of the world. Is space-time closed or open? Is there really a unit length, the plank length? Or do lengths truly map to real numbers?
Well put.
Now that I think about it some more, the argument is convincing.
Outside of mathematics, I believe that some people do experience infinity in mediation, other spiritual practices, and with psychedelics. Also, one can experience the law of non-contradiction transcended, so that all dualities are one.
Here's another, albeit mathematical, way of experiencing [math]\infty[/math]: Live! If one lives for even just 5 minites, [math]\infty[/math] moments from t = 0 to t = 5 mins would have to flow by. The same can be done with distance, mutatis mutandis.
Given a given line segment has exactly the same number of points as any other line segment, no matter their difference in lengths, and given lifespans can be mapped onto lines, we could say that
1. Everyone one of us - infant, child, adult, man, woman, young, old - has the same lifespan ([math]\infty[/math], measured in terms of moments/instants). We're kinda immortal mortals. The Greeks, it seems, refused to touch [math]\infty[/math] with a barge pole for a very good reason: paradox galore!
2.
This blaze survivor was experiencing instants/moments ([math]\infty[/math]) when the fire was raging around him and not the timespan ("seconds").
As a kindergarten child, I couldn't count beyond 10; the numbers greater than 10 are beyond my ken. This, is relative infinity and since it's in fact not an absolute infinity (more on this in a while), it can be treated as a nonmathematical/qualitative infinity.
Now, as an adult, I could go on counting (the natural numbers), but of course I'll never complete the task; this, is absolute infinity and it's mathematical/quantitative.
Qualitative infinity has to do with limits (our own, our tools'); Quantitative infinity is, at the end of the day, a concept to which limits don't apply.
When I see your alias, I always wonder how or why you have chosen it ... Thinking should not be tiresome! :smile: Not if directed correctly and it is controlled, and one doesn't "torture" one's mind by chosing to talk about such subjects as infiniteness! :grin:
Quoting TiredThinker
I'm not sure what you exactly mean by "going right to number lines", but if you imply that "infinity" and "infiniteness" are normally connected to numbers (Math), I think that this is too much restricting. There are a lot of things we call "infinite", even if actually they are not, but only very huge in size, from simple to complex: a line, a circle, love, time, space, the Universe (although its infiniteness is still debatable), ... That is, anything the limits of which cannot be determined.
“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”
? Albert Einstein
An abstraction with utility, but not an actuality. All things are finite, including the universe. There has never been any reason to suggest otherwise apart from our inablity to formulate thoughts on the subject.
Unbounded spaces e.g. Earth's surface, etc.
You mean because we can travel around a circle forever? I mean infinite more like if absolute coordinates existed we could go straight in one direction without reaching a limit.
If anyone persists in this madness, s/he would need a nonquantitative definition of limit.
The obvious question: is [math]\infty[/math] a number?
Have a dekko at the following:
6 and 2 are numbers
[math]6 + 2 = 8[/math]
[math]6 - 2 = 4[/math]
[math]6 \times 2 = 12[/math]
[math]6 \div 2 = 3[/math]
Now [math]\infty[/math]
[math]\infty + \infty = ?[/math]
[math]\infty - \infty = ?[/math]
[math]\infty \times \infty = ?[/math]
[math]\infty \div \infty = ?[/math]
[math]\infty[/math] isn't a number like 2, 3, 4, 6, 8, 12.
Don't get me started on [math]0[/math]!
Potential infinity, as the intuitionists keep stressing and as programmers demonstrate practically, is the only concept that is needed, both inside and outside of mathematics, that refers to finite entities of a priori indefinite size.
The myth of absolute infinity is what give the illusion of mathematics as being an a priori true activity that transcends Earthly contingencies.
* There are rigorous definitions of 'limit', in various contexts, in mathematics.
* To my knowledge, there is no general mathematical definition of 'is a number'. However 'ordinal number', 'cardinal number' and the predicate 'is infinite' have rigorous mathematical definitions, and there are proofs that there are ordinal numbers and cardinal numbers that are infinite.
* The lemniscate [here I'll use 'inf'] does not ordinarily denote a particular object. Probably its two most salient uses are for (1) points on the extended number line and (2) in expressions such as "the SUM[n = 1 to inf] 1/(2^n). With (1), inf and -inf can be any arbitrary objects (they don't even have to be infinite) that are not real numbers, serving as points for the purpose of a system. With (2), 'inf' is eliminable as it is merely convenient verbiage that can be reduced to notation in which it does not occur.
So your examples of arithmetic involving inf are not meaningful. However, there are rigorously defined operations of ordinal addition, subtraction, multiplication, and division.
* I agree that you in particular are better off not speaking on 0!, which also has a rigorous mathematical treatment.
* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.
* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1.
* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
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* One may take mathematics as a priori true without commitment to infinite sets.
Do straight lines exist? And even if you traveled the earth forever you will see the same places more than once.
Take another sip, my friend, then off to bed. :yawn:
Thanks.
You just went right past what I wrote.
I'm just puzzled/intrigued by the fact that you can't do math with nihilism and also with [math]\infty[/math].
I merely stated the needed corrections to your uninformed argument.
What's so uninformed about:
Quoting Agent Smith
?
You're trolling. What was uninformed is the post::
Buh bye.
AFAIK, there are "straight lines" only in the abstract Euclidean space. However, the shortest path between any two points is a geodesic ...
The circumferential path does not end (i.e. it's in-finite); no point ("place") on the path is a boundary, therefore the path is unbounded.
In which case, you surely agree that absolute infinity isn't a semantically meaningful assignment to a mathematical entity, for any semantic interpretation of the symbol of infinity as referring to extensional infinity, is question begging.
Quoting TonesInDeepFreeze
The semantic notion of absolute infinity (whatever that is supposed to mean) isn't identifiable with the unbounded quantifiers used in classical mathematics, logic and set theory, due to the existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas, which usually cannot be fully formulated or solved in those notations due to the inconvenient truths of software implementation and physical reality. Most software engineers don't regard themselves to be mathematicians or logicians, due to historical reasons concerning how mathematics and logic were initially conceived and developed.
Quoting TonesInDeepFreeze
SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics, i.e. mathematics with lazy evaluation, that rejects both formalism and platonism, in which unbounded universal quantification is understood to refer to potential infinity, which leads to his formulation of non-classical continuity axioms. In a similar vein, Edward Nelson's Internal Set Theory adds tenses to Set Theory, by distinguishing the elements of a set that have so far been constructed that have definite properties, from those that will potentially be constructed in the future, that have indefinite properties.
Perhaps one might say that to view the groom as contradicting his earlier vows amounts to a definition of 'negated absolute infinity' - but this interpretation is unnecessarily problematic in asserting the negation of a statement that isn't a verifiable proposition with verifiable meaning.
On a curved surface. Straight lines otherwise. We're talking 3D, not 4D space-time.
I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me.
Quoting sime
In the complex plane "infinity" is called "the point at infinity" and correlates directly with the north pole of the Riemann sphere - a specific point. But I've never used this concept.
sure, its an overstatement born of frustration with somewhat outdated formal traditions that still remain dominant in the education system.
I am not interested in an infinite trip. I am asking about and infinite landscape. Only new information all the time.
:chin: Such as unknown unknowns which necessarily encompass "knowns":
[quote=Karl Popper]The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know; our knowledge of our ignorance. For this indeed, is the main source of our ignorance - the fact that our knowledge can be only finite, while our ignorance must necessarily be infinite.[/quote]
That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".
Moreover, most remarkably, in another thread:
https://thephilosophyforum.com/discussion/comment/653236
you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that.
/
Quoting sime
Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?
Moreover, my earlier remark stands and should not be overlooked:
Quoting TonesInDeepFreeze
You simply skipped the point that mathematical definitions are never circular or tautologies.
Quoting sime
Whatever what is supposed to mean? "The semantic notion of absolute infinity" or "absolute infinity"? Anyway, in either case, who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?
Quoting sime
What specific non-standard models and specific "axioms and equations" do you have in mind?
What is your rigorous mathematical (or even non-rigorous philosophical) definition of "extensionally infinite"?
Quoting sime
Formal classical mathematics is exact in the sense that there is an algorithm to check whether a given finite sequence of formulas is or is not a proof. Any alternative to classical mathematics that also has that attribute would need to be also evaluated for simplicity of the formulation of the system.
Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets?
You mean without absolutely infinite sets, presumably. The overall approach would be to stress that the mathematical notion of a derivative approximates the real-life practice of directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function. This is opposite to the conventional way of thinking, which construes the practice of measuring a slope as a means of approximating an ideal, abstract and causally inert mathematical derivative.
With this in mind, the classical definition of df/dx with respect to the (?, ?)-definition of a limit, can be practically interpreted by interpreting ? to be a potential infinitesimal, and ? as representing a random position on the x axis given the value of ? , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded.
dx may be arbitrary, but dy depends on dx and is not arbitrary. That's where the observed function comes into play. Maybe you are talking about something else. You seem confused about these things. Or maybe I misinterpret.
Quoting sime
Non-standard analysis is what you are referencing here in a somewhat befuddled fashion. What is a potential infinitesimal?
Fractals are generated by simply iterating certain complex functions. They are not "infinite spaces" but images on computer screens. The iteration process is finite, say n=1000. There is no abstract infinity other than implied replications of patterns like turtles all the way down. You never really get there.
It is the reciprocal of a potentially infinite number, e.g. a random value taken from the codomain of the rational valued function 1/x.
'non-standard analysis' is the correct umbrella term, but it is already befuddled by the various alternatives that fall under it, some of which receive rightful criticism for obscuring matters even further, e.g the hyperreals .
Why in the world would I couch anything in terms of "absolutely infinite"? The notion of "absolute infinity" does not occur in classical mathematics since Cantor was superseded by Z set theory, as I had already alluded to:
Quoting TonesInDeepFreeze
And you skipped that question. Would you please answer it?
You keep talking about "absolute infinity" but in absence of any reference to it in mathematics after Cantor, and in absence even of your own definition of it.
Quoting sime
Mathematical objects are in general abstract. And knowing what you mean by "causally inert" awaits knowing your definition.
Quoting sime
A philosophical view, such that we are already familiar with, on the notion of 'potential infinity' is of course welcomed. And I have little doubt that it has been formalized somewhere. But you again skipped my point that it is only by seeing a particular formulization that we can compare its ease of use with classical mathematics. From the (admittedly not deep) reading I have done, generally alternatives to classical mathematics are much more complicated to formulate.
Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one).
Then, aside from the mathematical and philosophical subject, we now are talking about heuristics in the form of pedagogy. Nothing you have mentioned so far appears to be any more pedagogically promising than the classical definition of 'the derivative'. Indeed, there is no reason to think that a high school or college student would not have a much harder time grasping non-standard analysis combined some way with "potential infinity" than grasping the usual notion of a derivative. Nor have you pointed out any instance in which scientific or engineering calculations would be better enabled by your seemingly personal graft of non-standard analysis with a notion of 'potential infinity'.
Since hyperreals are formalizable in set theory, they are formalizable in internal set theory too.
You skipped it again.
Quoting TonesInDeepFreeze
You skipped that point twice now. Maybe you'll address it later, but as it stands, you fail to substantiate your claim as to what Brouwer or any intuitionist or any constructivist or any mathematician or philosopher ever said.
Overall, good observation.
To my knowledge, it isn't possible to point to a complete formalisation of potentially infinite logic, because it doesn't yet exist. All we have are fragments or incomplete axiomatizations of the concept, that have been invented by different logicians over the years with respect to different systems for purposes other than the current discussion. This is somewhat similar to the proliferation of different programming languages. There isn't any inter-subjective agreement as to how to formulate open-world reasoning.
- The I, S and T axioms that Edward Nelson introduced are useful for formulating what it means to reason with as-of-yet unconstructed elements of an unfinished set, in spite of the fact he proposed the axioms in the context of ZFC as an alternative to model-theoretic non-standard analysis in ZFC. ZFC is of course inadmissible for the purposes of this discussion, and is the reason why his formalisation doesn't tend to be associated with formalizing potential infinity. But the I,S and T axioms, divorced from the problematic axioms of ZFC appear to be a relevant fragment of some formalization of potentially infinite logic.
-Brouwer's notion of choice sequences, i.e. unfinished sequences, serve as the template for potentially infinite sequences but his formulation doesn't to my understanding provide what I,S and T does.
Choice sequences allow the expression of unfinished sequences, e.g
{1,2,3,...}, where the dots "..." are understood to mean "to be continued"
Brouwer introduces continuity axioms that define what it means to prove a universal proposition over a domain that consists of such unfinished sequences. However, his concepts, at least to my understanding, doesn't permit direct talk about numbers that we have presently declared, but cannot currently quantify, e.g. "The height of the tallest human being who will ever live"
- Linear Logic, as opposed to intuitionistic logic, is the logic i would associate with Intuitionism and potential infinity, because it is a resource conscious logic. Again, as you might say, it is "not evidently associated with p.i", especially in view of it's exponential fragment. Squinting at axioms to see their practical significance is still an unfortunate necessity.
For potential infinity. Though, in ZFC, we can describe sequences of finite sets.
Quoting sime
Problematic for some people, not for others. And I am pretty sure that if you take ZFC out from under IST you're left with a theory that accomplishes quite little.
Quoting sime
Has anyone tried?
Quoting sime
It looks very interesting. But you couldn't seriously propose it as a way for college freshmen to learn calculus.
Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer. And I still don't know why in the world you go on about "absolute infinity", which is a notion pretty much entirely unused since Cantor was superseded by axiomatic set theory.
.
Nope. You need to reread the article.
There are no passages in the article that support your misrepresentations. If you think there are passages that support you, then you can cite them specifically.
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
You fabricated those.
And what was your point arguing about "absolute infinity", which is a notion that has virtually been out of play since axiomatic set theory? For that matter, as I've already asked, what is your definition of "absolute infinity"?
:up:
Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity, relative to which the existence of said entity cannot be independently verified, empirically evaluated, or constructed with respect to a finite amount of data.
Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. Quine's famous example "All Bachelors are unmarried men" can be held as being true by definition in the mind of a particular speaker, but in doing so it can no longer be regarded as being representative of how a community of speakers might use the words "bachelor" and "unmarried man", given the limitless potential of them using the words non-equivalently.
Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension, whereupon it is inconsistently alleged that a finite description of a function can somehow represent a limitless amount of information that is also exact.
The alternative interpretation corresponding to "potential infinity" is to consider the definition of such entities as being vague and verifiable, as opposed to being semantically precise but unverifiable.
Quoting TonesInDeepFreeze
It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals.
Actually my phrasing is a slightly weaker statement, considering that potential infinity is usually used in the context of monotonic sequences, as in infinitesimals or infinitely large numbers. The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. This constitutes a random stopping event, in the sense that the time of the pause is not defined a priori at the time when dx is declared to be infinitely small or x to be infinitely large.
Defining potential infinity in terms of a "non-terminating" process is problematic however, given the fact that 'non-termination' isn't a verifiable proposition if interpreted literally, which is a concept belonging to absolute infinity. What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. Then consider the fact that any of the finite extensions generated by pausing a process are countable and isomorphic to the integers. These are my considerations when thinking of "potential infinity" in terms of a priori unbounded finite numbers instead of in terms of a "non-terminating" process. However, my definition might cause confusion due the fact it is more general and includes random variables with unbounded values.
Apart from Cantor's speculations I never came across this idea. It can't be applied to sets so it sits out there in an unattainable splendor, ignored by most in the profession.
Instead of defining 'absolute infinity' you give various notes on your ideas about it.
'absolute infinity' is a noun. For mathematics, the definition of a noun requires the definiendum (a noun phrase for defining a noun) on the left and a definiens (a noun phrase for defining a noun) on the right.
So a definition of 'absolute infinity' would be of this form:
absolute infinity = [insert a mathematical noun phrase here]
/
Quoting sime
There you still don't provide a definition, but you take 'absolute infinity' to refer to a certain kind of "interpretation". Here are the problems for you:
(1) You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it.
Accepting that there is a common notion of an intensional definition as opposed to an extensional definition, we still see that 'an extensional definition' just boils down to the definiens being a finite list. But a set is infinite iff it has no finite listing. So there is nothing gained from saying 'absolutely infinite' rather than merely saying 'infinite'. The definitions:
x is finite iff x is 1-1 with a natural number*
x is infinite iff x is not finite
Then your sense of 'absolutely infinite' is just 'infinite'.
Quoting sime
Then syntactic infinitistic mathematics is not itself liable.
Quoting sime
What example do you offer of a mathematician or philosopher of mathematics making that mistake?
Quoting sime
What total function do you offer as an example? 'total' is better thought of as a 2-place relation. A function is total on a set iff the domain of the function is that set. So what exactly do you have in mind regarding totality? And, for example, as to the the infinitude of the set of natural numbers, what total function do you think is improperly employed?
/
Quoting sime
No it is not "practically equivalent". You ADDED "finite", "a priori indefinite", "finite entities" and especially "eventually finitely bounded". Those are not mentioned, neither literally or practically, in the article's explanation of the notion of potential infinity (except 'finite' is mentioned regarding the finitude of each natural number). Generally, the article does not say that potentially infinite sequences are finite sequences. Even for an intuitionist, a finite sequence is not a potentially infinite sequence, notwithstanding that only a finite portion of a potentially infinite sequence is constructed at any given point. To really stress the point, especially, the article says nothing like "eventually finitely bounded".
Quoting sime
Please cite a source where that is mentioned as part of the "common definition".
Quoting sime
You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off from them.
/
* That is circular with my definition of 'is a natural number' in another thread, but that can be overcome by using a different definition of 'is a natural number' such as 'an ordinal that is well ordered both by membership and the inverse of membership').
Simply:
Where do you find "eventually finitely bounded" in Brower, or even any secondary source, on potential infinity? Please cite a specific passage.
Again, I'm not asking about your own notions; I'm asking where you got your idea as to what Brower said about it.
And you might hold off snidely telling me to "reread the article(s)" when you still don't cite specific passages in an article and the article you mentioned does not at all support your claim.
I think you may have misinterpreted the Wikipedia article:
There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition.
You have a very sophisticated writing style that I find a bit hard to process, but others may not. And when a person writes well there is a temptation to put aside statements that might give one pause. I'm guilty, but fortunately TonesInDeepFreeze is sharper in this regard.
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.
The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future.
Quoting TonesInDeepFreeze
It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject, and I have already presented my arguments in enough detail as to why it is better to think of potential infinity in that way.
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Quoting TonesInDeepFreeze.
Intuitionism is partially aligned with constructively acceptable versions of non-standard analysis. If you want an more authoritative but easy-read sketch, Read Martin Lof's "The Mathematics of Infinity" to see the influence Choice sequences have had on non standard extensions of type-theory (which still cannot fully characterise potential infinity due to relying exclusively on inductive, i.e. well-founded types.
Quoting TonesInDeepFreeze
Classical mathematics and Set theory conflate the notions of absolute with potential infinity, hence only the term "infinity" is required there. Not so in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk.
Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory:
1) The Law of Excluded Middle is not only invalid, but false with respect to intuitionism.
2) The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity.
For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..."
3) Choice Axioms obscure the distinction between intension and extension, whereupon no honest mathematician knows what is being asserted beyond fiat syntax when confronted with an unbounded quantifier.
3) The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. Not so according to potential infinity, since it cannot be determined that two functions are the same given a potentially infinite amount of data .
We also have Markov's Principle: according to absolute infinity, an infinite binary process S must contain a 1 if it is contradictory that S is constantly zero, and hence MP is accepted. Not so according to potential infinity, due to the fact that 1 might never be realised. This principle is especially relevant with respect to Proof theory, since any proof by refutation must eventually terminate at some point, before knowing for certain whether an unrefuted statement is refutable. So unless we are a platonist who accepts absolute infinity, Markov's principle isn't admissible.
We seem to be looking at a common notion from different perspectives. I assume you are a CS person (?). First of all a student in a math class is more likely to ask, What is an infinite sequence?, whereupon the definition as a function of the positive integers is provided. Then comes the distinction between a sequence in which each term is defined by a formula and one that self-generates by recursion. Examples would make these ideas clear:
(1) [math]{{S}_{n}}=S(n)=3n+1,\text{ }n\in N[/math]
(2) [math]{{S}_{n}}={{S}_{n-1}}+{{S}_{n-2}},\text{ }{{S}_{1}}=1,{{S}_{2}}=1,\text{ }n>2[/math]
Then the idea of a finite expansion (largest value of n) or an infinite expansion (non-terminating).
[math]<{{S}_{1}},{{S}_{2}},\cdots ,{{S}_{n}}>\text{ , }<{{S}_{1}},{{S}_{2}},\cdots >[/math]
Then, of course, there are special and/or more complicated cases, like a sequence which is non-terminating but assumes the same value after a certain point, or a sequence that is convergent in a metric space, etc.
In all my years I don't recall using the expression "potential infinity".
Look in virtually any introductory textbook or set of lecture notes on computability. And we don't need a notion of 'potential infinity' to explain the notion of non-termination. The classical treatment of computability is replete with the notion of non-halting. For example, it is a well known simple fact that a program to list the natural numbers does not halt.
Moreover, it is the responsibility of proponents of the notion of 'potential infinity' to provide the needed definitions to support the notion; not the responsibility of people who don't rely on the notion. The fact that you are mixed up about this subject shouldn't entail that you try to patch that up by supplying incorrect and incoherent claims and attribute them to Brouwer and the intuitionists. You continue to say that Brouwer and intuitionists understand potentially infinite sequences to be "eventually finitely bounded". Yet, after multiple requests, you fail to provide a source where Brouwer or anyone said that.
Quoting sime
Classically, it can be explained as not one finite sequence but as a sequence of finite sequences. Or, non-classically, to avoid having an infinite sequence of finite sequences, as anyone can read in virtually any article about 'potential infinity', even at the most basic level: For any finite sequence, there is a finite sequence of greater length, but there is not an upper bound to the lengths of such finite sequences. That is the OPPOSITE of saying that there is a finite bound on the lengths.
Quoting sime
One can posit that no physical process continues without termination. But, as I've asked you again and again, please cite where Brouwer said that a potentially infinite sequence is "eventually finitely bounded". More generally, if you have no Brouwer source to point to, then you should not conflate your claim that an ideal process is not realized physically so a non-terminating process is "eventually finitely bounded" with Brouwer who, as far as we know, never advocated that a potentially infinite sequence is "eventually finitely bounded", especially as the notion of potential infinity is the OPPOSITE.
Quoting sime
You use terminology in such a sloppy yet grandiose way:
(1) Logical equivalence is a special notion. You haven't shown any "logical equivalence".
(2) By mere fiat you declare a logical equivalence. Still, you do not cite anything Brouwer said that even suggests (let alone is "logically equivalent") to "eventually finitely bounded", especially as it is the OPPOSITE of the notion of potential infinity.
Quoting sime
I will look at it. I am not well versed in type theory and category theory.
Quoting sime
ZFC could not possibly conflate the notions since set theory doesn't even have a notion of 'potential infinity' nor does set theory mention 'absolute infinity'.
Again, 'absolute infinity' is a notion of Cantor that is not used in ZFC. You persist to use 'absolute infinity' in your own personal sense (for you, 'absolute infinity' is the notion that an intensional definition can specify a set that cannot be finitely listed), which is very different from Cantor's use of the term.
And even with your sense, though ZFC does define sets that are not finite, that is not "conflating" with some other notion ('potential infinity') that ZFC does not even address.
Quoting sime
A rigorous definition would be good for those who use the concept. But ZFC does not use it. And ZFC does provide a rigorous axiomatization for classical computer theory. If you want a rigorous non-classical computer theory, then it's your job to make it rigorous; your lack of doing that is not a fault of ZFC.
And I guess there are some people in computer science who regard ZFC as junk, but that is not at all any kind of consensus or, as far as I know, even a large contingent. You don't legitimately get to speak on behalf of "computer science".
Quoting sime
Wow. You are so wrong, and so obviously so. A combination perhaps of ignorance and intellectual dishonesty.
The very purpose of Z is to not include axioms that would allow an absolute infinity. Why don't you get a book on introductory set theory and inform yourself on this subject?
Quoting sime
Yes, LEM is not a logical validity in intuitionism. That doesn't contribute to claiming that Cantor's sense of 'absolute infinity' is the same as your own personal sense of it. (Also, there is a technical question about what 'false' in a semantic sense means for intuitionism. In finite domains, LEM is TRUE in intuitionism. Of course, we would have to look at specific intuitionistic set theories to see whether LEM is false in any given model.)
Quoting sime
I have no idea what you have in mind with the notion of the axiom of regularity "preventing unfinishable sets". ZFC has no predicate "unfinishable sets", so the axiom of regularity couldn't allow nor "prevent" anything about. It is incorrect to say that ZFC makes determinations on notions that are not even expressed in ZFC.
And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.
Quoting sime
I don't know what that is supposed to mean. But, to be clear "..." is not in the language of set theory, not even as extended by definition, but rather it is informal notation that can be eliminated with actually rigorous notation.
Quoting sime
The axiom of choice is intuitive. One is free to reject it, but you are incorrect to say that mathematicians don't know what the axiom asserts.
And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.
Quoting sime
No, according to what YOU mean by 'absolute infinity'; it's not what Cantor meant.
And the axiom of extensionality doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.
Even more basically, even though ZFC does capture a great deal of Cantorian set theory (but not including Cantor's 'absolute infinity'), that does not entail that what Cantor meant by 'absolute infinity' is what you mean by it.
1. sime's claim of what Brouwer meant is the OPPOSITE of the basic notion of potential infinity, and sime has shown no source that supports his claim.
2. sime is plainly incorrect that Cantor's notion of absolute infinity is the same as sime's, and sime's argument about ZFC in this context is incorrect since ZFC dramatically DIFFERS with Cantor on the matter.
We can measure things and we can make linear statements about those things with mathematics but beyond that we have to use linear algebra. There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics.
Traditionally the first axiom of mathematics is commutative principle. Increasingly, commutative principle is seen as the first derived axiom with the first being the axiom that given any observable set we can assign whole numbers to that set as a form of measurement.
Take it how you will, the simple truth is that it's the psychology of individuals and their personality cults that have had the most influence on how everyday people understand the universe, not the ideas of finite and infinite. Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything.
Lots of non-linear material out there. That's where the study of differential equations gets interesting. So much more.
1 infinity + 1 infinity = 1 infinity :chin:
Infinity broke math. Leopold Kronecker 1, Georg Cantor 0. Sorry Cantor old chap, I'm a big fan, but Kronecker's got a point!
Necessary to allow the goodness to pour forth. :cool:
so ?+1!=?
All it takes is shifting the axioms in the philosophy of mathematics down by one. Seems reasonable to me.
It's interesting you have come upon a well-kept secret within the mathematics community. I, myself, have witnessed such rants at the top echelons of the profession. It's an embarrassing spectacle and rather unexpected within a discipline so rigorous and demanding calm reasoning. Sad. :cry:
People gifted in mathematics tend to be (very much a generalization) very judgemental, love symmetry, almost obsessively orderly, emotionally distant, become easily obsessed with problems, stoic in their self image, etc...
Of course there are often exceptions to this as a person can be a brilliant mathematician but not see themselves as naturally gifted. Einstein comes to mind as he forced himself to do math in order to explain his theories but preferred to visualize stuff in his head and then labor to write it down as equations.
Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be.
One of my favorite examples of the difference between linear calculus and set theory elegance is to compare Euler Angles with Quaternions. In linear algebra the quaternion equation iterates over vectors and translates (rotates) their position in 3 coordinate planes. It tells you the new location of each member of a vector. With Euler angles, however, the description of the matrix is what gets rotated, not the matrix.
Quoting SkyLeach
I can write very concise and precise definitions and descriptions, then use them in sheer babble.
I'm not familiar with that. Where can I read about it?
Quoting SkyLeach
I've never seen such a "first axiom of mathematics" as you describe it. Is it something you've read or just an idea of your own?
Quoting SkyLeach
What is a "derived axiom"? Ordinarily, sentences derived from axioms are called 'theorems'.
Quoting SkyLeach
Where is that stated as an "axiom"?
Quoting SkyLeach
What famous mathematicians in particular do you have in mind? Where have you witnessed these "meltdowns"?
The use of sets is ubiquitous in mathematics. I didn't know that famous mathematicians were having "meltdowns" about sets.
Again, there is no object that is infinity (other than such things as points of infinity on extended numbrer lines). There are sets that have the property of being infinite.
That cardinal arithmetic is idempotent for infinite sets (especially for the set of natural numbers) is not really not problematic if you bother to read the proof.
What is your basis for that claim?
[note]Developmental, evolutionary and neuronal modeling inferred explanation for how and why the brain needs math for modern communication (and its limits)[/note]
I see that as: mathematics is missing constructs for allegorical analogy which are an essential base component for practical daily use languages.
Language is a serialization of 4D reality as it is "understood" by the speaker that uses shared context and perspective to allow the listener to reconstruct the serialization in the prefrontal cortex as a cognitive narrative (series of causal set functions applied to models).
That process relies very heavily on allegorical analogy to compare properties and functions of sets to completely different sets and infer meaning from the exercise.
Returning to mathematics; the purpose of mathematics is to validate the properties and functions applied to the sets. It makes no assertions, descriptions or assumptions about the nature of the sets. It's intended to strictly regulate validation of function and derived outtcome only.
The entire idea is that if you have applied mathematics to the properties and functions (as many as makes sense at least) and if two individual's precision of definition of a set in question is sufficiently detailed they can be relatively certain of reaching the same cognitive result and thus certain of objective agreement.
- Me
I quoted myself because you seemed to have missed my detailed comparison when you asked where Quaternions came into things.
You refer repeatedly to "sets". Care to expand on that ?
So in this context a set is the boundary of associations encapsulating a concept as distinct from others. It is best defined given current technology by an fMRI showing active regions experiencing state change during specific cognition. Typically it includes the Parietal, Occipital, Temporal and Prefrontal for real-world objects and their allegorical associations. Distinctions between regions are associated with how concepts are introduced (visual, written and spoken instigation as well as direct vs. representational association with cognitive burn-in (reinforced mapping)).
Math is an abstraction of determinism such that you can apply the same series of functions to apples, oranges and the loves of your life because even though two of the former are physical sets of atoms that can be seen and touched while the latter is an allegorical association of evaluated relationship qualifications between a concept of self and a concept of other; all three are treated by the human brain identically. That's because they are cognitively identical processes.
Obviously, just saying "demographics" is not a basis for claiming:
Quoting SkyLeach
Obviously, just saying "ZFC" is not a meaningful answer to the questions I asked.
https://thephilosophyforum.com/discussion/comment/659140
And no, I'm not going to ignore you but I'm also not going to argue (as in fight) with your ego instead of your rational mind. It's a waste of my time.
EDIT: I mixed up two replies in one. Glad I caught it on the editorial re-read. That's what I get for typing in the middle of a family in full Saturday boogy.
:ok:
Were (are) you an aeronautical engineer or Naval person? I know little of rotation theory, but the gimbal lock problem can occur in one of these schemes but not the other. The short paper I linked is as close to this as I get. I almost always work in the complex plane.
I was joking about the behavior of fellow mathematicians. In all my days I never saw a rant. And your description of a math person's personality is valid sometimes, but more often they are social animals - the practice of mathematics is a very social activity. I recall being at an autumn meeting at the Luminy campus of the University of Marseilles in 1989 at which there was communal dining and quite a jovial atmosphere. And a summer meeting at the University of Trondheim in 1997 where a member of the royal family attended a convivial banquet overlooking a ski jump where their Olympic team put on a performance. Other international meetings displayed similar atmospheres.
Sorry about the ramble above, but many assume math talent means the sort of personality you have described. Which, incidentally, omitted the fact that many in the profession have musical talent.
I am an engineer but software, not mechanical/chemical/structural/etc...
[hr/]
As for Quoting jgill
Please allow me to clarify using my own quote:
Quoting SkyLeach
I am very aware of the problems of bias in generalization and that's why I pointed out it was a generalization. Any time you combine multiple data points in a demographic you wind up with a much flatter distribution curve. "... tend to be ..." drops to 12% instead of the 50:50 median split of a single data point. Most people conceptualize bias generalizations at 50+ during conversation because... well I'll avoid going into why right now...
As soon as you add any other variables you can completely invalidate the metric. I was using my knowledge of Jungian and the MMPI which are the most widely used and thus easiest to draw inferences from. They have their limits, however, so I try to make it clear that I'm generalizing and try never to use them to evaluate any individuals (or prejudice my behavior). The operative word being try, I'm human too.
EDIT: one further thing to add. When it comes to a discipline, generalizations become much more appropriate because we are very much social animals and when we make choices about our own bias which is essential to cognitive function we must choose between social consensus and personal time investment for validation (hopefully those, not the bad ones like bigotry, faith or inanity). Academics is, at its core, an appeal to authority.
I have read some of what the others said and my good question is, what kind of infinite are you looking for? As long as you live this mortal life, you will not possess any knowledge of the True Infinite, aka God, at least in the sense that you will have no real evidence of this mysterious Infinity. Any belief in the Infinite must be accepted with faith. And to an extent, even lesser infinities are unknown, until you actually encounter them. Before that, we can only speculate on the nature or existence of infinities.
Well, all I meant to ask was whether something uncountable (an example of an uncountable infinity is the set of real numbers R) can be considered mathematical? After all, math is, bottom line, about countability (0, 1, 2, 3,...).
In a sense. An appeal to the authority of consensus.
Then again, as I said, there is a very strong and growing debate about that since the argument for sets being the the first self-evident truth has a lot to offer and, of course, can't (or hasn't been) invalidated.
The essential argument is that numbers are simply a form of measurement and mathematics is a precise language for making falsifiable linear statements about those measurements. Numbers are, after all, assigned to representations of sets. Apples/fruit/kilometers/distance/seconds/time... you get the idea.
True that the variety of numbers has expanded over history. Yet we seem distinctly more comfortable with the category of natural numbers than with any other I can think of.
Quoting SkyLeach
Mathematics is what a consensus of mathematicians says it is. That's the "authority".
I know this wasn't a question for me, but have to say: Yes, it is a mathematical object.
Just as is even Cantor's Absolute infinity. (Which he didn't talk much about, but still...)
Assuming what? I infer that you don't have any actual studies to cite, since you continue to reply without mentioning them.
And I know a little about the notion of studies in psychology and sociology. Asking you to cite an actual study that justifies your claims is not a fault from my own understanding.
With emphasis on 'you're', I guess you mean that you have a better response but you're holding it close to vest because I'm the one asking?
Anyway, my point stands that just saying 'ZFC' is not a plausible response to the questions I asked.
Quoting SkyLeach
What's a limbrain?
Quoting SkyLeach
You don't know that I am an ISFJ. And even if I were, being any one of the types does not contribute to disqualifying a person from having asked pertinent questions.
Quoting SkyLeach
I would guess that most people who post do so with motivated to enhance their sense of self-worth by exercising their prerogative to express their ideas, and many people often with criticisms of the ideas or claims of others. You have no evidence that my motivation is any more for gratification of ego than average. Especially, I come nowhere close to the kind of egotism found in a forum such as this that is displayed by people with terrible grandiosity when they bloviate their personally devised philosophies, with irrational arguments denying any weight to, and insulting other views by people who know something about the subject, and doing that by way of egregious distortions of even the rudiments of the subject.
And even if my motivation were entirely egotistic, that would not disqualify my points themselves.
Quoting SkyLeach
I am interested in mathematics and philosophy of mathematics. I find that it is worthwhile not to distort or misrepresent the mathematics itself when philosophizing about it. Clearing up misinformation and misunderstanding that is posted about mathematics is a worthy first step toward discussion of philosophy about it.
Quoting SkyLeach
You have not mentioned anything I've said that is irrational.
On what data points do you base your characterization of the personalities of mathematicians?
You have test Briggs-Meyer and MMPI test results of sets of mathematicians that you draw inferences from? How would you obtain such test results?
Quoting SkyLeach
But you haven't given any basis even for the generalization you stated.
If there is one field that is the least based on appeal to authority, I don't know what would be a better candidate than mathematics.
I won't argue against the notion that what is the study of mathematics is based on what professionals in departments called 'mathematics' do. But as to whether a purported proof is correct or not (unless it is extraordinarily complicated) is not a matter of consensus. Whatever consensus there might be, if one shows an incorrect inference in a purported proof, then the proof is disqualified from being deemed an actual proof.
I don't know of a common definition of 'mathematical object'.
I said that (except in special contexts) in ordinary mathematics, there is no object named by the word 'infinity'. I mean that there is no constant symbol added to the language of ZFC where the constant symbol is rendered in English as 'infinity'.
Instead, there is a predicate symbol that is rendered in English as 'is infinite'. And there are particular constant symbols that are defined as particular sets and we have theorems that those sets are infinite. For example, let 'w' [read as omega] stand for the set of natural numbers. It is a theorem that w is infinite, and it is a theorem that w is countable. Or, let 'R' stand for the set of real numbers. It is a theorem that R is infinite, and it is a theorem that R is uncountable
It's still a matter of consensus to determine whether the proof is valid. Yes, in some abstract realm a proof is valid or not according to logical principles, but humans have to agree before it becomes an accepted piece of mathematics. Occasionally a proof is so long and so complicated the verification process is difficult.
Quoting TonesInDeepFreeze
Guess there's not a consensus, then.
The set of real numbers is defined in set theory, which is a mathematical theory. Set theory makes reference only to pure sets. Not sets of apples or nations or thoughts, but only sets provided by the purely abstract axioms. Indeed, the empty set itself doesn't have to be taken as given but is derivable from purely abstract axioms.
Quoting Agent Smith
The natural numbers are foundational. That doesn't entail that mathematics must be limited to the natural numbers. And set theory takes not the natural numbers as primitive, but merely the relation 'is a member of', i.e, membership. And with axioms about membership, the natural numbers are constructed, and from the natural numbers, then the integers, then the rationals, the reals are constructed,.
Only when the participants haven't themselves checked the purported proof. They may take the word of the referees that the purported proof is correct. My point is that in principle, it is objective to check whether a purported proof is correct, and if an incorrect inference is clearly shown, then no consensus can alter that the purported proof is not correct. (Again, I'm setting aside situations that are so terribly complicated that there is real debate.)
Quoting jgill
Granted. But they will ordinarily agree if they check the proof. It's not based on consensus in the same sense as other kinds of questions.
Quoting jgill
No, there is a consensus that the purported proof is not correct.
I understand. :ok:
Counting lies at the heart of mathematics. An uncountable object (e.g. the set of reals), therefore, must be, in some way, nonmathematical, oui?
Non.
What do you think of the following problem?
A man invests $7,127 in selling shirts. His selling price is 3 shirts for the price of $200. How many shirts must he sell to break even?
Where n = total sets of 3 shirts each he sells.
The money from selling n sets = 200n
[math]200n = 7127[/math]
[math]n = 35.635[/math]
The number of shirts he must sell to break even = [math]3 \times 35.635 = 106.905[/math]
106.905 shirts!
In addition, the economic and sociopolitical spectrum tends to limit entry into academics as part of it's statistical weighting towards wealth and privilege.
Finally, the social authority encourages ego which, like the other factors mentioned, retards change and adoption of new ideas.
To remind you, in case you forgot, it's a waste of my time. Don't care what you infer, you've already demonstrated that isn't going to be affected by anything I say or do.
Of course I did. It's on record as a post.
Meanwhile, your characterization of mathematicians, as a generalization, is unsupported by any data whatsoever.
Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry!
Perhaps this will help.
Starting Premises:
Look around you (allegorically speaking) and point at anything that is a 1 or a 2 ... or any other number.
Numbers aren't things, they're representations.
Q: What do they represent?
A: Cognitively distinct concepts.
Q: What is a cognitively distinct concept?
A: Anything the rational mind considers a discrete set.
Q: Is there any limit on what the mind considers a distinct concept?
A: Yes, the mind is only able to process allegorical comparisons as set theory and has increasing trouble with concepts sufficiently distant from experience to create one-off pathways (dangling pathways or singular references or [in the case of damage] Disconnection Syndrom
Q: Can you put that in terms I can understand?
A: Maybe. Concepts like massive measurements are very hard for the mind to put into comparative concept and since that's how the mind actually works a great many mistakes are made when doing it until the mind has compensated.
Q: Can you give an example?
A: Yup. Timescales in galactic terms. Distances in interplanetary terms. Infinities. Distance between two points along a curved path like a planet's surface instead of a straight line. Asymmetric periodicity.
Right now, traditionally and with the same consensus as JGill mentioned, numbers are referred to as measurements.
Counting (commutative principle) is just an axiom. Numbers aren't counting except in the sense of combining any two measurements.
When we think of counting apples it can be hard to think it's a measurement. In effect, however, you're measuring volume (just not being very precise). If you're counting apples to fill a pie then the correct answer can vary because the size of the apples definitely will vary based on what kind of apple you are counting. Granny smith vs. red delicious (for example) since granny smith are small green apples and red delicious are quite large red apples.
Ok given that all numbers are just relative measurements between two points in a conceptual context of the mind we can begin to argue for set theory being the first axiom:
I know that's a lot to absorb but it's my best attempt at simplifying the entire argument.
Do you mean differentiable manifolds? A cylinder created by moving a circle through space is not curved? A sphere in 3-D is not composed of points?
The sequence <2,7,9> can be seen as a vector. The set {2,7,9} cannot, since the positions of the elements is arbitrary.
Quoting SkyLeach
Not my experience at all. After my PhD and getting a tenured position I belonged to a small international group of academic mathematicians, all of whom eagerly sought new ideas, novel ways of looking at things, unusual results, etc.
What you might be referring to is when a grad student has to choose a project (dissertation) and is not capable of making that choice, their advisor will guide them onto a path he thinks they are able to follow and hopefully do original research - the fundamental requirement for the degree. And, sad to say, sometimes the advisor will in effect do much or most of the original research and give credit to the student.
However, it is true that some departments are "governed" by a cliche that exerts pressure to push ahead toward certain research goals, having little patience with deviations. I've seen this also. This can occur when the cliche forms around a prominent, celebrated academic. Especially when there are grants to be captured.
Quoting SkyLeach
What's that? So a set of random integers is defined that way? Do you speak of a set or a sequence?
Lots of material here for TIDF.
:lol:
Jokes aside, my example is still in the rational domain. Perhaps I could give an example from construction/engineering: a circular [math](\pi)[/math] dome like the one that tops the Hagia Sophia (Turkey) for an irrational number in the calculation.
Quoting jgill
When is something curved? If the value of the tangent differs from place to place? A circle seems curved. So does a cylinder. So does a torus. You can move a circle through space so it becomes a torus. Maybe the way you move points, lines, surfaces, etc. through the higher dimensional space determines if they are curved (apart from Ricci or Riemann tensors). If I move a point wildly through space the ensuing line gets curvature. A circle hasn't though. A sphere does. But the value of the derivative on it is the same everywhere (how do you define a derivative on a manifold?). A torus has Gaussian curvature but it can be defined such that it has zero curvature. There's more to it...
Can you built a line with points? You can move it through space. Or stack lines to form a surface? Planes to build a volume? What if we take a point from a line? Can you still move on it continuously? Math addresses these questions, and confirms, but still. A line made of points? How you glue them together? How are strings kept on 3d while the 3d soars in 4d?
Quoting jgill
It's strange that you read those statements as "all sets are" instead of "any set can be". Maybe my word choice was poor? I tend to think in terms of software, not geometry, so there really doesn't need to be a pattern in the numbers in a set just a relationship between them that associates them, even if they're pseudo-random numbers.
Typically a vector is rationally linked with some spatial coordinate system but in software you just can't limit them by guessing the relationship between them from their values.
Quoting jgill
This isn't about personal experience or a single academic discipline. When I think of the problems in academia I just don't think about math as one of those. It's not that it isn't one, it's just that the only time the problems ever came up was over the rants against infinity and set theory (which are actually a bit funny to read at times). Mathematics is quite possibly the most empirical of all the sciences.
When I talk about many of the problems in academia I tend to be thinking of cosmology, astronomy, paleontology, the humanities (psych, anthro, socio, etc...) The more empirical and rigid a discipline is the less they seem to get into academic problems.
I double majored but since I graduated I don't think I've read more than a half dozen research papers in mathematics. It's just not one I need to follow closely because new algorithms or solutions sets are rare.
not if you're using set theory. Calc 3 FTMFL
EDIT: damnit I just realized that my perspective will probably get me argued with again...
Set theory isn't just sets of points, it can also be a scene described as a space (hilbert, sobolev, etc...) with objects described functionally instead of sets of points. I deal far more with scenes described rather than sets of points except when rendering a solution set.
Sorry to hear that. Good day.
Can a surface be counted? Area can but the surface itself?
Are you arguing this way?:
The set of real numbers can't be counted.
Consciousness can't be counted.
Consciousness is not mathematical.
Therefore, the set of real numbers is not mathematical.
Your perspective of "set theory" is not the normal math perspective. If it works for you, fine.
Quoting SkyLeach
OK. Not a topic I have an opinion about.
differentiable manifolds
The point was that I have both perspectives. I double majored in mathematics. This being a philsophy forum I default to the more general view (and certainly in this context discussing the mathematics-only perspective would make little sense).
I'm also curious why you just skipped over the whole part about a vecor being defined by element association since that was the mathematics perspective and the practical application perspective.
The answer is, of course, yes that's the definition of commutative principle we just use a different word with a root modifier for plurality instead of the singular.
That's why I said it didn't make any sense. You can have multiple surfaces, consciousnesses, etc...
These are vectors as I know them. I guess I don't understand what exactly you are talking about. That could be me. But if you point out your description in the article that would be good.
Yes, it's literally the first sentence.
Trouble is, then we have to get into what a vector space is...
And from there we have to get into specific allegorical definitions (while trying to avoid jargon) in order to limit the definition of 'element association'.
Really though, at this point the specifics are specious since the bolded part of the definition of vector space pretty much covers it.
My word "association" is indistinguishable from the definition of mathematic field:
Maybe a real-life example would clear it up better though
In radiology the system that stores images is called PACS. Most of the time, those systems deal with DICOM images. Those images can be regular computer graphics, but not for things like PET. PET is stored as a matrix defined in the sagittal, axial and transaxial planes. The value of each element in the matrix is called a Housfield scalar. It's a relative absorbtion rate (defined by the precision of the PET hardware) between 0-255 (or bigger).
You can't do anything with the matrix in that form, however. You must either convert it to a 2D vector space or a 3D vector space. You can do this directly or by scaling it in a process called fusion with MRI or CT data which is generally stored as a more traditional JPEG2k or MPEG.
Thus, the resulting vector space is either a 3 element vector of (x,y,h) or 4 element vector of (x,y,z,h) with x,y and z mapped to the coordinate space and h mapped to the housfield absorbtion.
It's a whole other dimension, but it's still valid.
OK, take the set {5,2,4} and generate a vector space since this set can be defined as a vector.
Anyhow, here:
Edit the ray stops at 20, but that was to keep the post short.
Edit2: it should be pointed out that since you didn't define the association, the possibilities are infinite. The only way to make it useful is to define how they're associated.
More or less. I'm sucker for (good) analogies.
It's a ridiculous analogy.
This distinction between, on the one hand, a set in and of itself without specifying an order and, on the other hand, an ordered tuple is crucially basic to mathematics, especially linear algebra.
Curly braces are used for sets in general. Either by set abstraction such as
{x | x is a natural number less than 4}
or for specifying the members individually such as
{0 1 2 3 }
where the order in which the members are listed, or redundancies are irrelevant, so:
{0 1 2 3} = {3 1 0 2} = {3 3 1 0 2} etc.
Angle brackets [alternatively, parenthesis] are used for order tuples such as
<0 1 2 3 > [alternatively (0 1 2 3)]
and order and redundancy do matter, so none of the below are equal to one another:
<0 1 2 3>
<3 1 0 2>
<3 3 1 0 2>
etc.
/
This is not just a matter of notation, but is a crucial concept, especially in linear algebra. I don't know why someone would be posting such bold claims as yours about mathematics and linear algebra while not even knowing that there is a distinction between merely a set and an ordered tuple.
Saying that linear algebra is the foundation of mathematics while not knowing the basic notion of an ordered tuple is like saying benzene rings are the foundation of chemistry while not knowing what an atom is.
Of course it is, but I am not a mathematician and you're literally talking about writing style. I pointed that out already. You're talking about notation which is literally how you write it down by definition so that there isn't a bunch of words needed to explain things.
I literally said exactly that:
Quoting SkyLeach
See, in the real world people can't specialize in everything. There are limits to how much jargon and special notation and career details a generalist can cover because there just isn't any possible way to read and remember every possible detail of every discipline, even if you spent 100% of your time doing nothing but learning and no time at all in practical application.
And you're just plain wrong about it not being "just a matter of notation". That's bloody well exactly what it is and it's specialized for your particular field because it makes sense in a field where you write down equations all the time. In fields where writing equations is a complete waste of time except for the rare instance when it's going into the documentation it's mostly worthless trivia.
As for knowing it there would be absolutely no difference at all in me saying you don't know linear algebra because you didn't know that what I pasted into my post was set notation for a set of tuples (immutable ordered sets).
If I had thought you'd try such a specious and obvious Ad Hominem attack merely because you don't like my argumet I would have taken the time to run the output through SymPy. I'd have done that because it's a tool for turning functional code notation into symbolic mathematics notation.
The tool's purpose is to ease the grunt work required to turn functional logic into symbolic mathematics for publication because ain't nobody got time for that shit but a mathematician.
Quoting TonesInDeepFreeze
Blah blah blah "HAHA! I found you out! Have at the pretender!"
Seriously what the hell is your problem? I don't care if you like the argument for set theory based axioms or not. It's a core concept of the philosophy of mathematics and I think it makes a lot of sense but I'm not on a crusade to make you accept it and you had no call whatever to try to trick me so you could try to shame me.
It doesn't really matter that you failed completely (and made yourself look like a jackass) so much as the fact that I'm a real person. I have real feelings. I like people and genuinely enjoy talking about philosophy and so far all I've gotten since I came here is a bunch of angry people trying to take out their personal issues on random strangers.
Seriously, go see a professional and deal with your issues, not my fault and I don't deserve your shit.
EDIT: I also didn't now that Python borrowed the term "Tuple" from mathematics. Cool. I learned a new etymology. So yeah, I guess I could have answered your question easily if I had known we shared that.
I'm not angry about you.
Quoting SkyLeach
No, that's the point, I'm not. I'm talking about the very concept, as it was mentioned to you by even another poster. Moreover, even if it were merely (which it is not) notation, then still it is hard to imagine anyone has studied linear algebra without having seen this utterly ubiquitous notation.
Anyway, now that I've given you an explicit explanation, you may understand the difference between a set taken in and of itself and an ordered tuple, and you now have the common notation you may use going forward to contribute to communicating clearly about mathematics with people who have actually studied it and know about linear algebra..
I just feel sad that yet again my hopes of having a good conversation has met a disaster of self loathing instead of a peer.
I didn't get the joke!
As much as I can't stand that guy, he's right. Your entire argument is semantic and rooted in an incomplete understanding of the philosophy of mathematics and semantic meaning of the words you are using.
Pray tell, how is what I wrote/said ridiculous? :smile:
I want to run something by you since you seem to know your way around the philosophy of math.
I recall a video im which Michio Kaku says (paraphrasing) "In blackholes, the (relevant) equations when used result in this: [math]\infty = \infty[/math] and then physicists can't make head or tail of it. Math breaks down"
If [math]\infty[/math] does that to math, is [math]\infty[/math] mathematical?
Quoting Michio Kaku
Quoting Michio Kaku
If I completely avoid theoretical physics questions and stick to mathematics do you notice something here?
I'm specifically talking about MK's approach to theories not his own...
Mathematics is a declarative language. You can make statements. You can make incomplete statements if you provide variables or equations with variables. You cannot, however, speculate or expound on theory. Mathematics is also linear. It makes clearly defined logical statements in a linear series. You can make a linear set of those linear statements. You can stack up as many linear statements describing as many dimensions or universes as you like too.
MK is a theoretical physicist specializing in cosmology and grand unified theory.
The problem MK (and all other theoretical physicists) are running into with their theories is the difficulty of constructing an analogy of sufficiently granular specificity (enough details/variables) to describe the function of the universe linearly and in an unbroken form from quantum theory into relativity (Einstein space-time).
Add to that MK's problem accepting any theory not his own and he says some truly whacked out things on a regular basis. I couldn't find your quote, but just from the ones I did find we already know he's going to utterly disagree with QMRE (quantum-modified Rocharch Equation) and infinite universe because any times infinite anything comes up (or the math showing that super-blackholes just can't form - based on Radjaharma's sp? research) he just says "bullshit" and ignores it.
He doesn't debate. He doesn't offer explanations. He doesn't refute. He just says it's not worth his valuable time. It's 100% ego-driven logic.
Quoting Agent Smith
Nope. It doesn't do that. That's a linear assumption, NOT an axiom. It's not a proof. It isn't part of mathematics at all. It's just a belief held by some people with big egos and funding to defend.
Can we do math with [math]\infty[/math]? Try some basic operations (+ × ÷ -) on it and check what happens.
(I tried bbcode math and mathml and texzilla but this forum doesn't support it :-( )
No work
UTF-8 but no solve 'cause not MathML
The Fourier Transform (signals math) - edit: this looks horrid. (appended what it should look like)
edit: and here is what it's doing:
You see what I mean now? You can't sum up infinity so you can't solve the answer to ?*2 but you can work with it like any other variable and to do many modern forms of equation you have to.
Great imagery! Somewhat similar:
Epicycles spring to mind.