Continua are Impossible To Define Mathematically?
My discussion below is just a scratch in the surface of an old and deep problem. Nevertheless, the point I am making is that I can find no workable mathematical description of continua. This might lend credence to the idea that, like matter, time and space are discrete?
Background
Aristotle held that the continuum was not composed of points. Aristotle’s point (pardon the pun) was that a point lacks extent, whereas a line segment has extent so the second cannot be composed of the first. To Aristotle, the points on a line segment existed only in ‘potentiality’, they do not become ‘actual’ until the line segment is divided. He therefore held that there is a potential but not actual infinity of such points on a line segment.
Aristotle’s position as has been largely subsumed by more modern formulations of the continua. The one we are all taught at school is the Dedekind-Cantor continuum:
- The continuum is constituted by zero-dimensional points (is punctiform).
- A finite line segment is composed of an uncountably infinite set of points.
- Any subdivision of a continuum results in two sub-continua.
- It is possible to continue dividing continua into (sub-)continua forever.
Of particular note, we are all familiar with the ‘Cantor–Dedekind axiom’:
“The real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.”
https://en.wikipedia.org/wiki/Cantor–Dedekind_axiom
As with the teaching of infinity, something which is just an assumption is taught to us as absolute knowledge. I feel our maths teachers are letting us down. :sad:
Attempting to define the continuum
What is a real number? Say in the interval [0,1]? Logically, it must have a ‘width’ that is one of the following:
1. Zero
2. Undefined
3. Greater than zero
Examining each case:
1. Leads to the number of numbers in [0,1] being 1 / 0 = UNDEFINED
2. Leads to the number of numbers in [0,1] being 1 / UNDEFINED = UNDEFINED
3. Leads to the number of numbers in [0,1] being 1 / non-zero = non-zero
Now the prevailing wisdom is that [1] holds - a line segment is composed of an infinite number of zero length points. I cannot make any sense out of this. How can anything zero length (dimensionless) be said to composed something with non-zero length? This is the view Aristotle held.
[2] is clearly unsatisfactory, so I think that we must use definition [3] above. That leads to two further sub-possibilities:
A. A number has a finite, non-infinitesimal width
B. A number has an infinitesimal width
Obviously, option (A) does not result in a continuum - it leads straight away to a discrete model - which could be the way the universe is (Planck length?).
So in search of a viable continuum, we seem to have to take option (B). An infinitesimal is defined as a number x such that 0
As pointed out here:
https://thephilosophyforum.com/discussion/7309/whenhow-does-infinity-become-infinite
It is hard to make any sense of ? as a number, so 1/? seems nonsensical also. But nevertheless, I will press on with the argument:
Is it possible to construct a line segment from infinitesimals? How about if we look at a series that is an infinite sum of all possible infinitesimals:
[math]\sum_{n \to \infty} n/\infty[/math]
Each term in the sum is just 0 so the sum of the series is zero (note that the series does not extend to the limit case of n=? and if it did, that would give ?/? which is UNDEFINED). See Wolfram Alpha for verification:
https://www.wolframalpha.com/input/?i=limit+%28n%2Finfinity%29+as+n-%3Einfinity
So the infinite sum of all possible infinitesimals gives a length of zero. If everything is composed of infinitesimals, then everything therefore has a length of zero. Which is clearly not the case.
So all paths to constructing a mathematical definition of a continuum seem to lead to dead ends?
Background
Aristotle held that the continuum was not composed of points. Aristotle’s point (pardon the pun) was that a point lacks extent, whereas a line segment has extent so the second cannot be composed of the first. To Aristotle, the points on a line segment existed only in ‘potentiality’, they do not become ‘actual’ until the line segment is divided. He therefore held that there is a potential but not actual infinity of such points on a line segment.
Aristotle’s position as has been largely subsumed by more modern formulations of the continua. The one we are all taught at school is the Dedekind-Cantor continuum:
- The continuum is constituted by zero-dimensional points (is punctiform).
- A finite line segment is composed of an uncountably infinite set of points.
- Any subdivision of a continuum results in two sub-continua.
- It is possible to continue dividing continua into (sub-)continua forever.
Of particular note, we are all familiar with the ‘Cantor–Dedekind axiom’:
“The real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.”
https://en.wikipedia.org/wiki/Cantor–Dedekind_axiom
As with the teaching of infinity, something which is just an assumption is taught to us as absolute knowledge. I feel our maths teachers are letting us down. :sad:
Attempting to define the continuum
What is a real number? Say in the interval [0,1]? Logically, it must have a ‘width’ that is one of the following:
1. Zero
2. Undefined
3. Greater than zero
Examining each case:
1. Leads to the number of numbers in [0,1] being 1 / 0 = UNDEFINED
2. Leads to the number of numbers in [0,1] being 1 / UNDEFINED = UNDEFINED
3. Leads to the number of numbers in [0,1] being 1 / non-zero = non-zero
Now the prevailing wisdom is that [1] holds - a line segment is composed of an infinite number of zero length points. I cannot make any sense out of this. How can anything zero length (dimensionless) be said to composed something with non-zero length? This is the view Aristotle held.
[2] is clearly unsatisfactory, so I think that we must use definition [3] above. That leads to two further sub-possibilities:
A. A number has a finite, non-infinitesimal width
B. A number has an infinitesimal width
Obviously, option (A) does not result in a continuum - it leads straight away to a discrete model - which could be the way the universe is (Planck length?).
So in search of a viable continuum, we seem to have to take option (B). An infinitesimal is defined as a number x such that 0
As pointed out here:
https://thephilosophyforum.com/discussion/7309/whenhow-does-infinity-become-infinite
It is hard to make any sense of ? as a number, so 1/? seems nonsensical also. But nevertheless, I will press on with the argument:
Is it possible to construct a line segment from infinitesimals? How about if we look at a series that is an infinite sum of all possible infinitesimals:
[math]\sum_{n \to \infty} n/\infty[/math]
Each term in the sum is just 0 so the sum of the series is zero (note that the series does not extend to the limit case of n=? and if it did, that would give ?/? which is UNDEFINED). See Wolfram Alpha for verification:
https://www.wolframalpha.com/input/?i=limit+%28n%2Finfinity%29+as+n-%3Einfinity
So the infinite sum of all possible infinitesimals gives a length of zero. If everything is composed of infinitesimals, then everything therefore has a length of zero. Which is clearly not the case.
So all paths to constructing a mathematical definition of a continuum seem to lead to dead ends?
Comments (180)
This is a philosophy forum, not a mud slinging contest.
The Cantor-Dedekind axiom is not an axiom in the usual sense. You can construct a complete ordered field over the Euclidean plane from the axioms of synthetic geometry. Since every complete ordered field is essentially the real numbers, this justifies the methods of analytic geometry. This justification required the development of axiomatic methods for geometry and algebra. A classic and highly readable reference is Hartshorne's Geometry: Euclid and Beyond (Chapters 2-3).
The only "explanation" I can offer is:
Consider a line of length 1 unit extending from 0 to 1.It can be repeatedly halved by multiplying with 1/2
Each multiplication will yield a point and a length corresponding to that point. For example the first halving will give us 1/2 which is a point and dimensionless but don't forget the distance from 0 to 1/2 which is a 1 dimensional length. You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.
Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves.
If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/
Or let's say that workable means conformity to your metaphysics. Fine, you haven't found technical and objective treatments that agree with your non-technical home-brewed metaphysics. But have you really shopped around?
https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
https://en.wikipedia.org/wiki/Constructive_analysis
https://en.wikipedia.org/wiki/Computable_analysis
I'm not sure how such options can be intelligible in the first place to someone who hasn't learned at least the basics of mainstream real analysis. Or at the very least a semi-rigorous calculus. As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter. Not so. The black queen on a chess board rules no kingdom. Legal moves are strictly defined and computer checkable. While the queen piece may be called a 'queen' and not a 'pawn' because of her exquisite mobility, what we call her and the intuition associated with that name is not important to the computer that checks whether a move is legal. It's only her place within the structure that matters. Math has to be this 'dry' or machine-like for the vivid objectivity it has in contrast to vague qualms about 'actual infinity' or the 'continuum.'
*I'll extend this point to metaphysics in general by quoting Wittgenstein. What I mean by it in this context is that none of us even have access to what is 'meant' by the 'continuum' or 'actual infinity.' For that matter even your private intuition corresponding to 'finite' cannot play a role. And so on. Only signs are public. In ordinary language, we have comparative freedom. In math no.
[quote=Wittgenstein]
If I say of myself that it is only from my own case that I know what the word "pain" means - must I not say the same of other people too? And how can I generalize the one case so irresponsibly?
Now someone tells me that he knows what pain is only from his own case! --Suppose everyone had a box with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle. --Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing. --But suppose the word "beetle" had a use in these people's language? --If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty. --No, one can 'divide through' by the thing in the box; it cancels out, whatever it is.
That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant.
[/quote]
'The object [the continuum in our 'minds' or 'reality'] drops out of consideration as irrelevant.'
Thanks for the links…
[quote="softwhere;364183”]If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/[/quote]
Wrestled with - and consistently failed to achieve - a sound mathematical description of continua - as I also failed to in the OP.
Weyl was not a believer in the ‘Cantor–Dedekind axiom’. He saw the real number as a discrete concept in contrast to the (alleged) continuous nature of time and space:
“The conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd” - Weyl
So he admits that construction of a valid mathematical model of a continuum is an impossibility - and he never achieves such in his work.
The Cantor–Dedekind axiom is highly questionable to my mind. A real number is a purely imaginary concept. It is like a label so it cannot be said to have any width. So an infinite number of real numbers on a finite line segment is acceptable - in our minds only. A line however represents something that can have objective reality. It must be constituted of something - points or sub-line segments, and the parts must have non-zero, non-infinitimsal width - else something becomes nothing.
Weyl was a supporter of the Brouwerian continuum. As I understand it, the Brouwerian continuum has strange attributes - ‘numbers’ in the Brouwerian continuum are allowed to be dynamical, constantly evolving, quantities in that such a ‘number’ does not have a complete, decimal expansion at any point in time - rather it is in a state of constant evolution as its digits grow with time. This means that in the Brouwerian continuum:
- For real numbers a, b either a < b or a = b or a > b does not hold
- The law of excluded middle: for any real numbers a, b, either a = b or a <> b does not hold
I do not class a system with the above two properties as ‘mathematical’ - in the sense that for me, valid mathematics should be built upon the principles of basic arithmetic and logic. I think once these principles are discarded, then we enter the realm of ‘pure maths’ - maths that does not reliably tells us about the world we actually live in - it may tell us interesting stuff about other realities - virtual worlds with different rules to ours - but it does not describe the universe we live in.
So again, we have mathematics failing to come up with a mathematical description of continua.
[quote="softwhere;364183”]
https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
[/quote]
I am somewhat discouraged that this system denies the law of excluded middle and by the definition: ‘nilpotent infinitesimals are numbers ? where ?² = 0 is true, but ? = 0 need not be true at the same time’ - this is again contrary to basic arithmetic and logic.
It may well have applications, but a reliable description of the nature of our universe is not one of them.
[quote="softwhere;364183”]
https://en.wikipedia.org/wiki/Constructive_analysis
https://en.wikipedia.org/wiki/Computable_analysis
[/quote]
"The continuum as a whole was intuitively given to us by intuition; a construction of the continuum, an act which would create by means of the mathematical intuition "all its points" is inconceivable and impossible" - Brouwer
“Space and time are quanta continua… points and instances mere positions… and out of mere positions views as constituents capable of being given prior to and time neither space nor time can be constructed” - Kant
[quote="softwhere;364183”]
As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter.
[/quote]
Most of the maths you have linked to falls into the category of contrary to basic logic or arithmetic. Now you might call me closed minded, but maths for me has to obey the principles of basic arithmetic. When advanced maths differs from basic arithmetic I feel it is no longer telling us about the real world - it is describing some virtual reality that is not our reality - so it is therefore not helpful in the pursuit of understanding the nature of our universe.
I have tried and failed using my basic maths skills to construct a description of continua. You claim the logic I use is 'fuzzy' but then do not point out any examples of my ‘fuzzy’ thinking - making me think that you are unable to identify any such - please advise.
My research also indicates that no mathematician has ever come up with a sound mathematical description of a continuum - so I would be interested to learn what your favoured mathematical prescription for a continua is?
I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero.
I think the core of the issue is in in assuming that a line can be constructed from points on the line.
If we are talking about pure and abstract maths, and I think we are,, then there is no reason for this.
A line can be defined by an equation such as X=Y, where all numbers that satisfy that equation lie on the line. It can be considered to be a continuum as for any two points on the line another point can be identified that is between those two points.
There is no need to consider that the line is made up of points.
If you assume that the Cantor–Dedekind axiom is true - that the real numbers correspond to points on a line - then we can take the purely mental model we have of real numbers as being infinitely divisible - and apply it to a mathematical line segment - resulting in a mental model of the segment as containing an infinite number of zero-length points (something than does not make physical sense - but we can do it purely in our minds).
But I'm not sure that the Cantor–Dedekind axiom can be said to apply to 'real life' lines - when I wrote the OP, it was buried deep in my consciousness under the category of 'unquestionably the way the world works'. I have since changed my mind about this.
A real life line segment (as opposed to a purely imaginary, mathematical line segment) is something substantial - and it must be composed of something - and the components must have non-zero length - else it is nothing. So I do not see how a physical line segment could ever correspond to our mental model of the real number line (obviously here I assuming some sort of continuous substance in reality to construct the real line segment with - time and space are the only two candidates - no form of continuous matter is known).
Real numbers I think are more like logical/mental labels - they have zero width - they are just a way to label parts of a line rather than the constituents of a real line.
So I think there is probably an equivalence between the real number line and an imaginary/mathematical line segment - both exist in our minds only - so the impossible stuff like them being composed of an infinite number of zero-length points happens in our minds only (where the impossible is possible), but I doubt the equivalence holds to any line with real life existence.
There is obviously also the question of whether time and space qualify as 'something' rather than 'nothing' (Relationism Versus Substantivalism). I am in the 2nd camp on that question.
Wiki:[i]"Other mathematical systems exist which include infinitesimals, including non-standard analysis and the surreal numbers. Smooth infinitesimal analysis is like non-standard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/?, where ? is a von Neumann ordinal). However, smooth infinitesimal analysis differs from non-standard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and non-standard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in non-standard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.
Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points."[/i]
It IS possible to learn something on this forum! Thanks.
I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error.
Points are locations. Location are distances. Distances are lengths. Lengths are spaces between two points. As you can see there's a definitional circularity: points defined in terms of lengths and lengths in terms of points and I think this is the reason why a point belongs to the list of undefined terms in geometry.
Anyway...
When we mark off a point, say C, on a line AB= 5cm, it needs an identifying label and that's its distance from one of the endpoints of the line. Suppose C is 3cm from A. We label point C = 3cm. C, the point, is zero-dimensional but remember it divides AB into two lines viz. AC = 3cm and BC = 2cm. AC + BC = 3cm + 2cm = 5cm. The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. Can you now see that, since a line can be divided into an infinite number of non-zero lengths and each length has a point associated with it, there'll be an infinite number of points in any line. In some sense, a point is just a label/name for a length and while a length is a building block/structural component of a line the point is nothing more than the name for lengths.
Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question.
Quoting TheMadFool
That does not appear to be the case - see the argument in the OP that sums an infinity of infinitesimals to zero (length).
Now you might disagree with the maths in the OP and say an infinity of infinitesimals sums to ? * 1/? = 1. But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1.
Then I suppose one could further argue that the infinitesimals on the segment are somehow longer than the infinitesimals on the subsegment - but that's contrary to the definition of continua as all having identical structure (and identical cardinality - uncountably infinite).
What were Aristotle's objections to points being zero-dimensional?
Quoting Devans99
I don't get this part.
Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length?
- He felt that something with zero extent like a point could not constitute something with non-zero extent, like a line.
- He also regarded points on a line as a 'potential infinity' in that the points are not 'actualised' until the line is divided. In this way, he justified his believe that 'actual infinity' is impossible.
There maybe others too, I'm no expert on Aristotle.
Quoting TheMadFool
The mainstream definition of continua (Dedekind-Cantor) does seem to lead to this conclusion:
1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length
So point 4 is expressing the fact that:
(size of a point) * (cardinality of the continua) = (size of the continuum in question)
0 * ? = 0
Equivalently, if we assume that points have 'infinitesimal' length, then, depending on how you do the math, you get:
1/? * ? = 0
or
1/? * ? = 1
And you always calculate the same length for all different lengths of continua.
I think 4 doesn't follow.
Consider two lines AB = 2 cm and CD = 4 cm
Divide these lines i.e. find points on these lines by dividing them with k
1. k = 1; for AB, 2/1 = 2; for CD, 4/1 = 4
The point on AB is 2 and the point on CD is 4
2. k = 2; for AB, 2/2 = 1; for CD 4/2 = 2
The point on AB is 1 and the point on CD is 2
3. k =3; for AB, 2/3; for CD, 4/3
The point on AB is 2/3 and the point on CD is 4/3
.
.
.
n. k = n; for AB, 2/n; for CD, 4/n
The point on AB is 2/n and the point on CD is 4/n
As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite.
However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths.
Agreed.
Quoting TheMadFool
That numerical value does not tell us the width of a point and length is the sum of widths of the constituent points / sub-line segments.
https://en.wikipedia.org/wiki/Integral
Hopefully that will answer your questions.
In order to understand the justification for analytic geometry, you need to study synthetic geometry. In particular, Hilbert's work on the foundations of geometry. Your questions were resolved more than a century ago. From the axioms of geometry we can construct models of the real numbers that satisfy our intuitive notion of a coordinate system. There isn't much interesting to see here.
I'm saying that your objections are more than a century out of date. In order to understand why, you need to learn a bit of synthetic geometry and abstract algebra. I don't see any way around this.
- Yet the Dedekind-Cantor continuum is taught in school along with the fact that a point has zero width. So my objections are bang upto date, as far as I can see.
- No-one has yet pointed out any logic/math error in my OP.
- If I have something wrong, then someone should set me straight, rather than vague hand waving
- At least a link to your preferred definition of the continuum would be nice
How is the date of the objection relevant? If it's a reasonable objection then it's a reasonable objection, regardless of the date.
And, as Devans99 indicates, the issues have not been "resolved more than a century ago". They've simply been ignored.
The notion of a continuum is relative to the notion of a 'hole', which is describable in terms of the absence of a topology-preserving surjection between two topologically structured sets.
"Gaps" only exist within the line of countably computable reals for platonists who think that non-computable numbers exist. Even if they are granted the existence of an uncountable number of non-computable reals to occupy those "gaps", the resulting model of the reals cannot control their cardinality, suggesting to the Platonist further holes, forcing him into constructing hyperreals and so on, without ever being able to fill the gaps in the resulting continuum.
This is another good reason for rejecting the non-constructive parts of mathematical logic.
There is a distinction between:
1. The process of division - we cannot go on dividing an object forever because we would never finish the division process - so we can not take an actual object and make an actual infinity of pieces out of it
2. Something that exists and was never created could exist in a form that is already divided infinitely
So it seems it is impossible to create/manufacture the actually infinite, but actually infinite things could exist - uncreated things - as some hold spacetime to be uncreated (not my view).
I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century.
https://download.tuxfamily.org/openmathdep/euclid/Euclid_and_Beyond-Hartshorne.pdf
It is 500 pages long and the words continua and continuum are never mentioned. It talks about 'points' a lot without even defining the term as far as I can see.
?
If you think that the objections have been resolved, then you're simply wrong. And pointing to some "highly online" reference (whatever that means) does not make you right. If you would take the time to produce the supposed resolution we could show you how it simply covers up the problem rather than solving it.
The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. It merely reflects the fact that distance is something that exists between points.
You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other".
Quoting A Seagull
But it is made out of points. It's just that the length of a line is not measured the way @Devans99 thinks it is measured. You don't measure the length of a line by summing the lengths of its smallest parts (which are points.) Points have no length. They do not have such a property. Length is something that exists between two points. In order to measure the length of a line you must count the number of pairs of points-at-a-certain-distance that constitute it.
Quoting Devans99
Points have no zero-length. They have no length at all. I think this is part of the problem. A lot of people do not understand that the statement "Points have no length" does not mean "Points have zero-length".
continuous series of points in a continuous series of instants.”; Bertrand
Russell, I Principi della Matematica, (Milano: Einaudi, 1963), 637.
Yet he also said points were non-sensical.
“It is just as impossible for anything to break forth from it [the One] as to break into it; with Parmenides as with Spinoza, there is no progress from being or absolute substance to the negative, to the finite.”; Hegel, Science of Logic, 94-95.
Math cannot solve this. Maybe philosophy can
It depends on how you define the word "existence". You can define it any way you like. You can define it in a way that implies length. That which exists has length. If we accept that definition then it follows that points (among many other things, such as sounds, colors and other sensations) do not exist. And since definitions have no truth-value, you can't argue against such a definition by saying "That's not true definition of existence!" Unless, of course, you're arguing against it on the basis of use-value. And that would be my response. Such a definition of existence has a limited (even questionable) use-value. It's certainly not how most people define existence. Most people define existence (not necessarily verbally but certainly intuitively) in such a way that even things that have no length (such as points, colors, sounds, etc) can be said to exist.
Hegel had interesting things to say on limits. Google it.
"The linear series that in its movement marks the retrogressive steps in it by knots, but thence goes forward again in one linear stretch, is now, as it were, broken at these knots, these universal moments, and fall asunder into many lines, which, being bound together into a single bundle, combine at the same time symmetrically, so that the similar distinctions, in which each separately took shape within a sphere [Parmenides's One], meet again." Phenomenology of Mind
According to Nietzsche, Hegel "systematized the riddle" of being and nothingness thru teaching that all is "obscure, evolving, crepuscular, damp, and shrouded".
Marx wrote:
"First making the differentiation and then removing it therefore leads literally to nothing. The whole difficulty in understanding the differential operation (as in the negation of the negation generally) lies precisely in seeing how it differs from such a simple procedure and therefore leads to real results."
Negations of nothing!
"Marx recognized the differential equation as an ‘operative formula’ — ‘a strategy of action’ which, when it arises, constitutes a reversal of the differential process, since the ‘real’ algebraic processes then arise out of the symbolic operational equation, which originally itself arose out of a ‘real’ algebraic process... The German mathematician Gumbel led a team to decipher them [Marx's 1000 pages of mathematical manuscripts] and published a report in 1927 listing the wide range of subjects dealt with." marxist dot com
This might be relevant, since against we understand math through the world:
http://www.hawking.org.uk/godel-and-the-end-of-physics.html?fbclid=IwAR297vm3qpeViCnrXcGXBuRo-PXCEXIOcUiQxlFQWk1e20Xvu-e90P_OhrA
Russell talked about Zeno's paradoxes in lectures and in books. He said that they had extreme subtlety. Ironically, you have to have an asymmetry between the hemispheres in order to see this
If you check the OP, I did consider the possibility that points have no size. That leads to the size of a point being UNDEFINED and all line segments having an UNDEFINED length.
Quoting Magnus Anderson
You establish above that your line segment is made out of 10, 1cm sub-segments. IE if we switch to discreetly/finitely sized sub-segments (/points), suddenly we can come up with a meaningful definition of length. That should tell you something - with discrete/finite sized points/sub-segments, lo and behold, the maths suddenly starts making sense. It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions.
Quoting Magnus Anderson
You cannot simultaneously hold that line segments (which have length) are made out of points and points have no length - that’s a plain contradiction.
Quoting Magnus Anderson
Colours have a wavelength, so do sounds so they can be said to have existence. Points however, defined as having no extent, clearly cannot not exist. A line segment has extent so can be said to have existence. Maths, claiming that non-existent things can be the constituents of existing things, is tying itself in a logical knot.
A point has zero length, a line has zero area, a plane has zero volume, a unit sphere has ...
What's so odd about that? (How many of them do you want to banish anyway?)
Volume of an n-ball (Wikipedia)
Would the Archimedean properties be worth mentioning here (since they seem to be ignored all over the place)?
Infinites and infinitesimals aren't real numbers (nor rationals etc):
[math]\infty \notin \mathbb{R}[/math]
[math]1/\infty \notin \mathbb{R}[/math] (archaic Wallis notation)
Don't use them as if they were (outside of convenience in specific contexts).
Quoting Magnus Anderson
Related to dense sets:
Dense order (Wikipedia)
But that's not what it leads to. If points have undefined length it does not follow that line segments have undefined length.
Quoting Devans99
What's non-sensical about the statement that points have no size?
Quoting Devans99
How's that a logical contradiction? How does "Line segments have length" contradict "Line segments are made out of points that have no defined length"?
Quoting Devans99
Colors do not have wavelength. Rather, it is light waves that have wavelength and light waves are not colors, they are the cause of colors.
OK. Please describe those contradictions. I've not encountered them in my many years. :roll:
I'm focusing on your claim that mathematicians assume but have not justified the methods of analytic geometry.
Quoting Devans99
I'm not a mathematician, but you're using mathematical language in ways that seems to betray a misunderstanding of fundamental mathematical concepts rather than problems with mathematics. Of course, I could always be mistaken.
Quoting Devans99
Would you be satisfied with a proof that one can construct a complete ordered field from the axioms of geometry such that the methods of analytic geometry work out? In other words, would you be satisfied with a construction of such a field from purely geometric premises? My hand-waving is an attempt to communicate these ideas without simultaneously writing a textbook on algebra and geometry. You, on the other hand, refuse to understand basic mathematical concepts.
Quoting Devans99
I'm satisfied with the real numbers (up to isomorphism) for the purposes of this discussion.
OK, let's say that points have no length whatsoever, they have no size, so it is incorrect to classify them in the category of things with lengths, such as line segments. Therefore we cannot say that they have zero length, just like you say.
Quoting Magnus Anderson
How can you determine where a point is on a line, such that you could use this as an instrument, a tool in the act of measurement? We've already placed the no-length point as right out of the category of things to be measured, so how can a point appear on a line to be measured? How can you find and identify any point?
Quoting Magnus Anderson
A line can be measured. A point cannot be. There is a fundamental incommensurability between a line and a point, such that it is impossible that a line is made of points. The claim that a line is made of points is illogical and irrational.
I accept that a point as defined is zero-dimensional and has neither width nor length.
Your contention is that if the above statement is taken to true then a line segment can't exist for they're a collection of points and adding nothings (points in this case) together can never yield something/line segment.
I admit that most of the information I have on the topic of lines and points can be summarized in the following claim: there are an infinite number of points on any line segment. This can be easily proven, right? Afterall if a/b and c/d are any two points on a line segment, the point (a+c)/(b+d) will lie in between a/b and c/d: a/b < (a+c)/(b+d) < c/d. That out of the way we can now focus on the original statement: there are an infinite number of points on any line segment. Now, supposes, in deference to your concerns, we change the meaning of a point from a zero-dimensional object to something of a fixed, non-zero length. What do we notice?
1. There are an infinite number of points on a line segment (proven above)
2. Giving due respect to your objection, instead of a point being zero-length we now define a point is a non-zero length
So,
3. There are now an infinite number of non-zero length points.
So
4. Any and all lines would have to be of infinite length because we proved there are is an infinity of points AND if points are a non-zero quantity then doing the math we get infinity
5. It is false that any and all lines are of infinite length e.g. a line of 6 or 8.5 or pi centimeters is a finite line.
Ergo
6. We have to reject one of our premises
7. That there are an infinite number of points on a line is true and proven so has to stay
Ergo
8. We must reject that a point has a non-zero length
So,
9. A point is zero-dimensional and has neither width nor length and there are an infinite number of them on any line.
Doesn't it make sense? A point necessarily is zero-dimensional or else we come to the preposterous conclusion that all lines are of infinite length.
Compare the above argument with yours. You've swung the other way by attempting to show a line segment is impossible since they're composed of zero-dimensional points and no matter how many nothings there are we simply can't get something.
From here let's use some prepositional logic.
S = a line segment is structurally dependent on points
Z = a point is zero-dimensional
P = it is possible for a line segment to exist. For example a 5 cm line segment
I = all line segments are infinite
My argument is as follows
1. (S & Z) > ~P...(this is your claim) premise
2. (S & ~Z) > I...(this is part of my argument above) premise
3. ~I...premise - there are finite line segments
4. P...premise
5. ~S > (Z & P)...premise
6. S...assume for reductio ad absurdum
7. ~I > ~(S & ~Z)...2 contra
8. ~(S & ~Z)...3, 7 MP
9. ~S v ~~Z...8 DeM
10. ~S v Z...9 DN
11. ~~S...6 DN
12. Z...10, 11 DS
13. ~~P > ~(S & Z)...1 contra
14. P > ~(S & Z)...13 DN
15. ~(S & Z)...4, 14 MP
16. ~S v ~Z...15 DeM
17. ~~Z...12 DN
18. ~S...16, 17 DS
19. S & ~S...6, 18 conj (CONTRADICTION)
20. ~S...6 to 19 Reductio ad absurdum
21. Z & P...5, 20 MP
QED
Basically, the assumption that points are structural components of lines is false and their dimension being zero has absolutely no relevance to the length of a line segment and so lines are possible geometric objects even if points are zero-dimensional.
2) If points are not zero dimensional, you can make a triangle out of it and half the hypotenuse would be smaller than a point!
3) Banach-Tarski paradox assumes that every object is the same size. Does not Cantor?
The debate in the late Middle Ages about how many angels can dance on the head of a pin was about this question, because angels are dimension-less. Nominalism was popular in those days too btw
How about the real plane where a point is denoted by (x,y) ? Still have a problem?
Complex analysis may be hard to justify for those who know little mathematics, but ask a QM physicist about path integrals and their predictive values.
Here is a fundamental question: Is it reasonable (however defined) for philosophers who have not studied mathematics to argue basic principles of the subject? This is a far reaching question that can be applied to sciences in general. Remember, math came first, then its set theory foundations, PA and ZFC, were postulated in the 1800s and 1900s. Those mathematicians knew mathematics.
The Axiom of Choice sounds so very benign (one can pick an element out of each non-empty set in a collection of sets), but look what it leads to! And you think the Axiom of Infinity is bad!!! :scream:
Though you cannot measure how long a point is (since it has no length, as per definition) you can identify a point. And we do so through a complex process that involves the movement of our bodies (if we're talking about identifying points in physical space, that is.)
There are many things that have no size but that nonetheless exist (and are not logically contradictory, illogical or otherwise irrational.) The word "existence" does not imply size. For example, colors and feelings exist, and yet, they have no size. A typical counter-argument is that colors are light waves and that light waves have size (their wavelength.) But light waves are not colors. Rather, light waves are things that cause colors. (This is evident in the fact that light waves can exist without conscious beings whereas colors can't.)
It's just a line. It signifies a spatial dimension. Why does it have to be "made of" something? You may as well be asking me what a dimension is made of. It's an idea.
Quoting John Gill
It may be the case, that a person was discouraged from entering the field of mathematics because its fundamental principles appeared to be very difficult to understand. So this future-to-be philosopher could not understand mathematics and went on to become a philosopher instead. Then the philosopher goes back to revisit the fundamental math principles and finds that they appear to be totally irrational and this is why they cannot be understood. In this case it is reasonable for a philosopher who has not studied mathematics to ask the mathematicians to justify their basic principles.
Quoting Magnus Anderson
Points don't exist in physical space. According to the description they are non-spatial. So I don't see how you can identify something which has absolutely no spatial extension by moving your body. Care to explain how you think you might do that?
Quoting Magnus Anderson
I agree that there are many things we can talk about, which do not have spatial existence (God being one of them). The problem is with the claim that a line, which is supposed to have spatial existence, is made up of non-spatial things.
Right, now the problem is the inconsistency between the definition of a point, and the definition of a line. A "point" is completely dimensionless. A "line" has dimension, yet the "line" was said to be "made up" of points. Do you see the inconsistency? No matter how many dimensionless things you add together you will not produce a thing with dimension.
Quoting tim wood
So what are you saying? We ought to just get rid of definitions altogether, and mathematicians will use these terms however they please? Then we can produce some sort of idea of what each mathematician means by looking at each one's use of the terms. That sounds very difficult, because there would be nothing to encourage consistency of use.
Quoting tim wood
What is trivial to one person is important to another. So for instance, I think that it is very significant that a "point" is incompatible with a "line", and that the ratio between the circumference and diameter of a circle is irrational, and that the relation between two perpendicular sides of a square is irrational. You, as well as many others, might think that these matters are "trivial" and uninteresting.
But then why do you partake in threads like this? I've seen the same tactic from atheists who say that whether or not God exists is really unimportant, yet they will argue incessantly that god does not exist. So, what is really important to them is whether or not a person believes that God exists. But how could this be important if whether or not God exists is unimportant?
Now, I'll ask you how is it possible that the "true" nature of the "point", and the "line", and the fact that there is inconsistency between these conceptions, is trivial, when the mathematician will claim that consistency is of the utmost importance?
I am not sure why you think so, Points do exist (both in time and space.) Consider that at any point in time, you occupy certain point in space. So there exist at least some of the points that we can imagine. Points that do not exist cannot be occupied by anything under any set of circumstances.
Physical space is made out of points. The fact that physical space is made out of things that have no size (points) does not mean that it has no size itself. Not sure why you think so.
No, this is not true. But it's such a common misunderstanding that a bit of exposition is in order.
One of the first things to know about the philosophy of math is the distinction between a number, on the one hand, and a representation of a number, on the other. 5, five, fünf, cinco, 2 + 2 + 1, 4.999..., the number of points on a mystical pentagram, and "half of ten" are all distinct representations of the same number.
So what's a number? If one is a Platonist, the number 5 is the abstract thingie out in the Platonic world "pointed to" or represented by each of those representations that I listed. If one rejects Platonism, I suppose we could get by saying that the number 5 is the collection of all possible finite-length strings of symbols that represent that number. (Although when it comes to numbers, an anti-Platonist has a tough row to hoe, suggesting that a Martian mathematician would not necessarily know the concept of five, or that there weren't five things before there were people to count them).
But whether one is a Platonist or not, one must distinguish between the abstract concept of the number 5 and any of its many representations.
Now, a real number is a real number. The exact definition of a real number is a technical matter for math majors and isn't too important at the moment. What is important is to understand that a real number HAS a decimal representation, sometimes two! But it is wrong to say that a real number IS a decimal representation.
One can be forgiven for wrongly believing the latter, since it's what we tell high school students. And even in technical disciplines, the distinction's not important. A working professional engineer, even a working professional research physicist, does not need to know or care about the distinction between a real number and its decimal representation.
There are only two classes of people who need to carefully make this distinction: mathematicians, who are trained on this topic in their undergrad years; and philosophers, who SHOULD BE but often aren't cognizant of the distinction.
Now decimal notation happens to be broken. Some perfectly sensible and familiar rational numbers, such as 1/3 = .3333333..., have infinitely-long decimal representations. That's not because 1/3 is broken; it's because decimal representation is.
Likewise some perfectly familiar integers, like 5 = 4.9999..., have TWO distinct decimal representations. Again, this isn't because real numbers are broken [though for the record, in this post I'm not necessarily arguing that they're not!]; it's because decimal representation has these well-known flaws.
Quoting Gregory
So what about a real number like [math]\pi[/math]? Does [math]\pi[/math] represent or encode an infinite amount of information? NO it does not! I will now show three distinct finite length descriptions of [math]\pi[/math] that uniquely characterize that particular real number:
* The first is the famous Leibniz series for [math]\pi[/math] :
[math]\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots[/math]
This is a finite string of symbols in a formal language that unambiguously characterizes the real number [math]\pi[/math], and that, Platonism or not, would without doubt be recognized as such by a Martian mathematician.
* But ok suppose that you don't like those dots. We can compactify the notation as follows:
[math]\frac{\pi}{4} = \sum_{k=0}^\infty \frac{(-1)^n} {2k + 1}[/math]
[NOTE: Typo in formula corrected].
* But ok, perhaps you don't like the notion of infinitary processes, even in compact notation that only uses finitely many symbols in a formal language. In this case:
[math]\pi[/math] is the smallest positive zero of the sine function.
Our Martian mathematician would have no trouble agreeing with that. And it's a finite string of symbols written in plain English, with no infinitary process in sight, that uniquely and unambiguously characterizes the real number [math]\pi[/math].
So in fact the real number [math]\pi[/math] encodes only a finite amount of information. This is true also of every real number you can name, such as the square root of 2, the base of the natural logarithm [math]e[/math], the golden ratio [math]\varphi[/math], and so forth.
Are there real numbers that can't possibly be expressed with a finite amount of information? Most definitely. These are the noncomputable real numbers as defined by Alan Turing in his landmark 1936 paper that defined the nature of computation. In constructive math, one doubts the existence of such numbers; but that discussion is for another time.
Takeaway: Every real number that you know encodes only a finite amount of information and can be expressed using only a finite number of symbols. Decimal representation is defective to the extent that some reals require infinitely many symbols and others have two distinct representations. But a real number is not any of its particular representations, so this is not a philosophical problem.
Quoting Gregory
Math proves no such thing, since your examples don't hold up to mathematical scrutiny. But more strongly, as many posters have already noted (hundreds if not thousands of times on this site over the years), math can prove nothing at all about the world. At best, math is a super-handy (and as Putnam and Quine might put it, indispensable) tool for building mathematical models of physical theories. Newton and Einstein both used math to express their respective theories of gravity; even though neither, strictly speaking, is true. They're both just approximations that describe the experiments and observations we're able to do with our current level of technology. I hope this point is clear not only here, but once and for all on this forum. Science isn't true. Science is our best mathematical model that fits the observations we're able to make with the equipment we're able to build subject to technology and funding. (As the government bureaucrat pointed out in the film, The Right Stuff: "No bucks, no Buck Rogers!")
Quoting Gregory
Way cool, but perhaps a little off the mark in this particular conversation.
Think about what you're saying tim, "a location" without any size is nonsensical. What could possibly identify that location unless there was something there with size? If there is a dot, to show the location, there is something there with size. If there is no dot, then there is no identified location. But "location" implies particularity and particularity is identifiable. How can the mentioned "location" be a location without something to show that location.? Otherwise it is just imaginary. And an imaginary location is not a real location, therefore not a location at all. It seems very clear that it is contradictory to say that a point has no size and it is also a location.
Quoting tim wood
It's no wonder I appear confused, you totally misuse words. There is no such thing as "objects of definitions", unless you are using "object" in the sense of "goal". And you are clearly not talking about the goals of definitions. These are subjects, not objects. Points, lines and other defined principles, ideas, are subjects. We are taught the subjects of knowledge, not the objects of knowledge. If you call them "objects" you imply the existence of some unjustified Platonic realm full of non-spatial objects.
So if you want me to agree with you on any definition of "point", we'd have to start with a principle to avoid such a category mistakes. We cannot say that a point is a location because locations are marked by objects with spatial extension, not subjects. Therefore we must disambiguate. Is a point a non-spatial idea (subject of knowledge), or is it a particular spatial location (object)? We can't have both without contradiction.
Quoting Magnus Anderson
You said points have no size. I do not see how any part of time could have no size. If it has no size, then no time is passing at that "point", therefore it is not part of time. The same principle holds for space. If it has no size, then it cannot be part of spatial existence, because there is no space there. It is very clear to me, that if points have no size, then they are excluded from space and time, because things existing in space and time have size. Having size is what makes them spatial-temporal. Do you not understand this?
Depends on what you mean by marking off distance on a ruler. If you mean real rulers and real markings, then it's kind of hard to even talk about exact distances. If you have an idealized model in mind - putting a point somewhere on a segment of a straight line, then why can't we mark off an irrational distance? If you make a mark somewhere at random, the distance it marks off is pretty much guaranteed to be an irrational number.
I don't know about that, never having gone through the proof of B-T. However . . .
Wiki: Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.
Sure, I'm fine with this, but it's more a case of pointing out a very common and simple mistake, rather than a case of "mocking".
Quoting tim wood
No, I'm not prepared to argue that an idea has a size, but as I've demonstrated, I'm prepared to argue that a location has a size. Therefore, if a point is an idea, and has no size, it is impossible that a point is a location.
Quoting tim wood
What's this supposed to mean, locations have size for reasons unrelated to size? Let me remind you of my argument. All locations have size. That's why a point cannot be a location.
Quoting tim wood
I think so. You seem to think that I was arguing that a point must have size. That's not the case, I was arguing that a location has size, but a point does not. I thought I made it clear that what I was arguing is that if a point has no size it is not a location.
Quoting tim wood
There clearly is an issue. You said that a point has no size, and that a point is a location. As I pointed out, the two are contradictory. So which do you think is the case? Does a point have no size, or is a point a location? Since the two definitions are contradictory we cannot proceed until we resolve this matter.
To suggest that a point has no size in theory, yet has size in practise is an unacceptable solution, because this would mean that the definitions of the theory contradict the definitions of practise, leaving the theory with all its definitions absolutely useless. To have a useful theory requires consistency between the definitions of the theory, and the definitions employed in application. That's what an acceptable theory is, one which is actually applicable in practise.
Quoting tim wood
I am not proposing any definitions. I am just arguing that the ones proposed here are not good, due to their contradictions involved with how they relate a point to a line..
So, I am happy to start with the assumption that a point has no size, and is immaterial, as an idea, like you suggest. Now the question is how does a point relate to a line. If we can agree that a line also has no size, and is immaterial, as an idea, then we might be on our way to making some progress toward describing the relationship between a point and a line. Let's consider a problem together then. Some people might say that two points, and a line between the two points creates something measurable. I don't think so. I think that this is just an idea, an idea which exists as a tool for measuring. So the line segment spoken of here is actually not something which could be measured.
:)
So the AC is a logical consequence of infinite divisibility and thus is not an axiom? :gasp:
So how does Zeno produce a non-measurable set? :chin:
What's ridiculous is that people like you refuse to accept the obvious, and keep touting your contradictory definitions.
Thanks jorndoe, I've never seen that circle paradox before. If you look, you'll see that the centre of the circle moves just as far relative to the horizontal surface as the point on the circumference does, and it's not rolling at all. So the rolling wheel creates a movement which is other than just the circumference moving around the centre, it "carries" the centre along in a horizontal motion relative to the ground. That's why the wheel's a good mode of transport, you put an axle in the centre and you can "carry" stuff. So we have carriages and cars.
In the depiction therefore, all horizontal movement is carriage. No point in the wheel, accept the centre, actually makes that horizontal line, so the line represents only the movement of the centre. The movement of any point in the wheel would make an arc, not a horizontal line. Therefore the horizontal line only represent carriage. The point part way between the circumference and the centre, is not rolling along the ground, it is being carried, And since it is not in the centre it would be carried in an arc; a bumpy ride..
Now, take the point on the circumference and follow the arc that it makes. Is that arc the same length as the horizontal line? No, obviously it's much longer. Why is it so much longer? It's longer because that point on the wheel, like any other point, is being carried, along with also moving in that circular motion. So the distance it actually traverses is much longer than the horizontal line. The horizontal line on the ground represents the length of the circumference, but it doesn't represent the movement of any part of the wheel. It is a fictional line of movement which represents the movement of the centre, but transposed to a parallel, on the ground. And the other line is also such a transposed parallel, not representing the movement of any point on the wheel. .
I do not understand why you think that things that exist in space and/or time must have size. Why is it impossible for something to exist (in space and/or time) and not have size?
This depends on the meaning of the symbol "0.333~". According to the way most people define "0.333~", it is not true that "1/3 = 0.333~". By standard definition, 0.333~ is a number smaller than 1/3 (in the same way that 0.999~ is a number smaller than 1.) It's not a decimal representation of it. "0.333~" does not mean "the limit of 0.333~".
Bertrand Russell on the Axiom Of Choice: "At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it."
Just as I described, if it takes up no time, it is not "in time". Things which exist in time, have temporal extension, that's what existing in time means. Likewise things which exist "in space" have spatial extension, that's what existing in space means.
If you're having difficulty understanding what this means, then try to imagine a point in time which has no temporal length. At this point, no time is passing. When no time is passing, this is not "in time" because time is always passing, that's what time is; and if time were not passing, we would no longer be "in time". Likewise, think about a thing which occupies no space, All the space around us is occupied by objects, air, etc. A thing which occupies no space could be anywhere, and everywhere, or even nowhere, all at the same time. Bit "existing in space" means that the thing has a particular spatial location, so a thing with no spatial extension cannot be "in space"..
Quoting tim wood
Contradictory definitions are rejected on the basis of the law of non-contradiction. So it is pointless, and rather ridiculous to put forward a definition which is contradictory, and expect that someone ought to accept that definition.
I avoid the "..." notation because it looks ugly when used in forums without LaTeX support. But yes, that's what I mean.
"0.333~" represents the infinite sum 3 x 1 / 10^1 + 3 x 1 / 10^2 + 3 x 1 / 10^3 + ... + 3 x 1 / 10^inf. It does not represent its limit.
[math]
\displaystyle\frac{1}{3} = \displaystyle\sum_{n \in \mathbb{N}} \frac{3}{10^n} = 0.333\cdots
[/math]
where [math]\mathbb{N}[/math] does not include [math]0[/math]
To render:
Anyway, to 's point, the three expressions around the [math]=[/math] symbols are just different ways of writing the same number, out of any number of ways.
Whether they can be said to exist or not, these are abstract objects, not my sandals. :)
The formalisms, theorems, etc, is how you treat them, you don't wear them on your feet.
Say, there's 10 meters over to the neighbor's front door.
That's a distance between two places here in the world.
Maybe some prefer saying "there's roughly 10 or 11 meters over there"; doesn't really matter much.
Unless you walk the wrong way, then it's almost 40,000 km longer.
The mathematical treatment (or modeling) of these things hold up just fine.
Mathematicians? Not necessarily. "Flaws" . . . not necessarily. Incidentally, your compact form of Leibnitz expansion has a simple error. And 1/3 =.333... = limit of a geometric series, well defined. You may be talking about mathematicians who labour in foundations. Making such fine distinctions is unnecessary in most math careers, IMHO.
Once again, I can't argue with this. :brow:
What do you think an "infinite sum" is then if not the limit (if it exists) of the partial sums?
The standard view of the positional notation is that it is a representation of a number as a series, with digits serving as coefficients in front of the base, and their position designating the power of the base (positive before the dot, negative after the dot). But I still have no idea what you think "most people" think of it.
Furthermore, the difference between the two concepts is bigger than that. For example, two infinite sums that approach the same value can represent different quantities. For example, [math]0.9 + 0.99 + 0.999 + \dots[/math] represents a number greater than [math]0.5 + 0.25 + 0.125 + \dots[/math] even though they both aproach but never reach [math]1[/math]. Indeed, there are numbers greater than [math]0.999\dots[/math] but lower than [math]1[/math]. Hexadecimal [math]0.FFF\dots[/math], for example, lies somewhere between the two numbers.
I've never seen an infinite object. In fact, I really don't see how an object could be infinite. What we sense and apprehend are the boundaries of an object. Without such boundaries what we'd be perceiving could not be apprehended as an object. And since we perceive boundaries it's questionable that we could even apprehend the infinite. If "infinite" is to even make sense as a concept, it cannot refer to an object
You should be aware that the mind is image oriented. It creates, analyzes, and stores images. Vision is the dominate sensory input from the external world. It's the nature of the mind to form the simplest images possible to represent things outside the mind.
The purpose of abstraction is to eliminate detail irrelevant for our purpose. A simple example, children use this form of abstract representation with their 'stick' figures for people. Thus we use ideal lines, circles, cubes, etc. to convey information to others. A social benefit is realized via storytelling.
This is most obvious in numbers used for counting, assessing the multiplicity of a collection of things. The numbers exclude all attributes of the things being counted.
As for the 'point', it too is an abstraction to serve as a location/coordinate. A surveyor places a stake as a marker/point for a property line. Being dimensionless, you can't see it, but an object is provided in the form of a marker, blob of medium on a surface, pixel on a screen. The point is somewhere within that marker/blob/pixel. This is not a problem since any calculations requiring the location will not vary to any significant degree. The same situation for the 'line' having no width. In graphics it's a continuous marker, in surveying it might be a laser. The line formed from points is a contradiction of terms since a point has no extent. How many zeros are added to a register to accumulate 1? We don't see trajectories or orbits either, but they still serve a purpose.
The continuum is another story.
An interesting quote by Poincare, The Measure of Time, 1898
"We helped ourselves with certain rules, which we usually use without giving us account over it [...] We choose these rules therefore, not because they are true, but because they are the most convenient,...
In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism.“
If the survey stake is the marker point, then the point is not dimensionless, and clearly can be seen. But if the survey stake simply represents a point, and the point represented is dimensionless, then the dimensionless point is not a location at all, because the survey stake marks the location.
What you say here, that the survey stake is the location point, and also that it is dimensionless and can't be seen, doesn't make sense, because clearly the survey stake can be seen.
Quoting Gregory
This is nonsense. It's actually very obvious that an orange cannot be divided infinitely.
I'm so happy that Cauchy, Weierstrass, and others settled this issue for mathematical analysis long ago. It made my career so much easier. :cool:
Depends on what you mean by mark. Every point on the real line "marks" that point. All the irrationals and even the noncomputable reals. Every real number is the location of some point on the real number line.
Of course if you mean construct as in compass and straightedge, that's different of course. But every point is at some exact real number location to the left or right of the origin. All the points are marked, the way I see it. Perhaps we're talking semantics.
I happened on this remark which you made a while ago. I wanted to note a correction.
Consider the rational numbers. They have what's called a dense linear ordering, which means that between any two rationals there exists another rational. Just take their midpoint, for example, which must be rational.
But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line.
That is not a continuum, mathematically or morally.
What characterizes a continuum is the least upper bound property. This says that every nonempty set of reals that is bounded above, has a least upper bound.
The least upper bound property is false for the rationals. For example the set
[math]S = \{x \in \mathbb Q : x^2 < 2\}[/math]
where [math]\mathbb Q[/math] is the set of rationals, does not have a least upper bound in the rationals. But it DOES have a least upper bound in the reals.
That's exactly why the reals are regarded as a mathematical continuum, and the rationals aren't.
Note please that I'm only saying what a mathematical continuum is. I'm not addressing any of the many philosophical objections there could be to calling the real numbers a continuum, But mathematically, the reals are a continuum and the rationals are not.
https://en.wikipedia.org/wiki/Least-upper-bound_property
Oops thanks I left out the exponent of the -1. Fixed it.
Not sure I followed the rest of your remark. My post is intended to clarify the thinking of all who say that "irrational numbers introduce infinity into mathematics." This is actually false. It's noncomputable numbers that introduce infinity into mathematics. A far more subtle and philosophically interesting point.
In any event I'm not arguing that working mathematicians care about these fine points in their daily work. I'm simply noting that a number is not any one of its representations; and that having an infinite decimal expression is an artifact of radix notation and not the real numbers themselves. After all pi and the square root of 2 have perfectly finite Turing machines that generate their decimal representations. That's the key point. Not who thinks about what during the work day.
Naming conventions are a matter of historical accident. Banach-Tarski is a theorem. It's not actually a paradox. It is however a veridical paradox, which is a true result that is contrary to intuition. If you want to call it the Banach-Tarski theorem that would be both accurate and less confusing. It's not really a paradox. It does show the distinction between abstract math and physics. The kinds of set-theoretic manipulations required in Banach-Tarski can't be done in the real world ... as far as we know, anyway.
That's right, the existence of irrationals really throws a wrench into the rational number line. Where do those irrationals exist in relation to that line?
Quoting fishfry
The real numbers cannot fulfill the conditions of a proper definition of "continuity". Real numbers produce a sequence of contiguous units. Contiguity implies a boundary of separation between one and another. This boundary must produce an actual separation between one number and the next, to allow that each has a separate value. This is contrary to "continuity" which is the consistency of the same thing.
So mathematicians have created a term, "continuum", which applies to a succession of separate units, allowing that each is different, so there is something missing in between them, and that "something", which is the difference in value, is unaccounted for. Therefore "continuum" means something completely different from "continuity".
The rational numbers are an attempt to account for this "something", the difference in value, which exists between the reals. This an attempt to create a true continuity. However, the irrationals appear, and foil this attempt. So mathematics still does not have a continuity.
We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful.
Right, but the point is only a location when there is a physical object to mark the location. The point, as dimensionless, cannot be a location because there needs to be something physical to mark a location. Therefore, what I've been explaining, is that the point cannot be both dimensionless and a location at the same time.
2) For rigor, we can use Cantor's arguments for taking infinities out of infinities
3) So Banach-Tarski comes right from Cantor
4) Therefore you can make a universe out of a pebble
The conclusion is that the limits of calculus go out the window!
I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion
Assuming God has laid forth unto us the Axiom of Choice.
Quoting Gregory
This is quite entertaining. :smile:
The point though is that #1 is wrong, things are really composed of finite parts. Things are finite. However, we can dream up an imaginary thing, a set of numbers or something like that, and stipulate that this thing is composed of infinite parts. But what good is that? It's just an imaginary thing, and the description of that thing, 'composed of infinite parts', is not consistent with any real thing which are all composed of finite parts, so it's all just a fiction.
I think modern science has demonstrated that there is a smallest unit of space. That's what quantum physics is all about. It's definitely not zero though, because these quanta of space contain our entire physical reality.
"In reality, space is therefore amorphous, a flaccid form, without rigidity, which is adaptable to everything, it has NO properties of its own. To geometrize is to study the properties of our instruments, that is, of solid bodies." Poincare
This is a common misconception. What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself.
As for the thread title and OP, Cantor's analytical definition provides an adequate model for most mathematical and practical purposes. However, it is a bottom-up construction that wrongly attempts to assemble a continuum from distinct parts corresponding to the real numbers. In other words, it presupposes that any line, surface, or solid of any length, area, or volume is composed of all such parts--namely, points--which is why the Banach-Tarski theorem follows from it.
By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. Those parts are indefinite unless and until we arbitrarily mark them off for a particular purpose, such as measurement, by inserting limits of lower dimensionality, which can hypothetically be of any multitude or even exceed all multitude. A line is composed of lines that are contiguous at points, a surface is composed of surfaces that are contiguous at lines, and a solid is composed of solids that are contiguous at surfaces.
Peirce's rather poetic analogy between semeiosis and music in "How to Make Our Ideas Clear" (1878) seems relevant here: "In a piece of music there are the separate notes, and there is the air ... Thought is a thread of melody running through the succession of our sensations ... [Belief] is the demi-cadence which closes a musical phrase in the symphony of our intellectual life."
No, Spinoza was not holding us back. Spinoza was not fully understood in the shadow of Descartes. On my YouTube channel I have a playlist that I call 'My Freedom from Nominalism Worldview'. Most people believe they have to choose an either this or that from what is conventionally taught in western culture academia and theology. There is a path to understanding that didn't make the popular cut due to the lure of materialism and the ontological individualism of 'I think therefore I am'. Do a search for this ... "Shaken by Nominalism: The Theological Origins of Modernity" to get some insight, or visit my YouTube playlist for some excellent learning videos. I recommend watching them in order to best understand them. Make sure you click on the playlist tab.
Space and time are concepts derived from our observations. Therefore it makes no sense to say that either space or time extends beyond what is observable. What we call "space" and what we call "time", are abstract ideas created to account for what we observe. If some aspects of reality are beyond what is observable, then they are neither spatial nor temporal because these are empirically based concepts.
I fully agree that there are aspects of reality which are not observable, but these are non-spatial, and non-temporal aspects of reality, as "space and "time" apply to observations.. That there are such non-observable aspects of reality, is what validates the concept of the "immaterial", and "non-physical".
Quoting aletheist
The problem here is the difference between a continuum, and a continuity, which I explained already. "Continuum" as implied by common usage means a collection of contiguous, but separate individual units. "Continuity" implies one uninterrupted whole. So if we assume "a whole", it could be composed of separate parts, like a continuum, or it could be a continuous whole, a continuity, but it cannot be both, because of contradiction. So to speak of a whole, as a continuity, is to speak of one thing, and to speak of a collection of parts, as a continuum, is to speak of a completely different thing. "Continuum" and "continuity" have very different meanings.
This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves.
Quoting Metaphysician Undercover
That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. As George Box famously put it, "All models are wrong, but some are useful."
Quoting Metaphysician Undercover
Only in the sense that continuity is a property, while a continuum is anything possessing that property.
There's no such thing as real space and times. We see things, we create things, distinct objects, and we see how things move around and change, and we make these concepts of space and time to account for these perceptions. What would real space, or real time even be like? We have no such concept, of real space or time.
What the concepts of space and time represent, is our understanding of the existence and movement of objects. There are not any things represented by "space" and "time". Kant covered this thoroughly, these are fundamental "intuitions" he called them, which provide for us the possibility of understanding the physical existence of things.
Quoting aletheist
This is the commonly accepted definition of "continuum" in mathematics. You can't say that the mathematical definition is wrong, because it's a mathematical term. It's not used anywhere else, so when mathematicians use the term, this is what they are talking about. What we have to respect is that "continuum" is not the same thing as a "continuity". If you think that "continuum" ought to be used to refer to a continuity, then it's you who is wrong, because this is not how the word is used in mathematics, which is its normal place of usage.
Quoting aletheist
This is not true, and that's the point I've been trying to make. A "continuum" consists of a series of distinct elements, whereas "continuity", or "continuous" refers to one uninterrupted whole. Therefore a "continuum" as the word is commonly used in mathematics cannot be continuous, nor can it have the property of continuity. If it had the property of continuity it would be one continuous thing, and not a series of distinct things, as "continuum" implies.
If you adamantly deny the reality of space and time, then there is nothing more for us to discuss on that front.
Quoting Metaphysician Undercover
The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not.
https://arxiv.org/ftp/physics/papers/0310/0310055.pdf
This paper inspired considerable controversy. You be the judge. :chin:
Is a line segment a "thing?" As opposed to a wooden rod, say? Does a line segment actually exist as a physical thing?
This is exactly why it is a mistake for us to try to discuss "real space", and "real time". If physicists produce a concept of space and time as a continuum, in the mathematical sense, yet we want to say that "real" space and time are continuous, and this is different from the mathematical continuum, to talk of "real" space and time is a mistaken approach. It is mistaken because there is no such thing as real space and time, these concepts are derived from the observations of objects which are assumed to be real. So the correct approach would be to say that physicists incorrectly model the existence and movement of objects, and that they ought to be modeled as continuous rather than as a mathematical continuum.
Notice, that what is being modeled is the movement of objects, and these are what are assumed to have real existence, not space and time. "Space" and "time" are produced from these models, as logical conclusions. Model A shows objects to move in such and such a way, therefore there must be such and such "real space and time", to substantiate that model. But model B shows objects to move in a slightly different way, so there must be a slightly different "real space and time" to substantiate that model. The "real space and time" is just something created by the model, as an incidental conclusion. This is why model-dependent realism is popular. But if the model is the "correct" one, then there is a real space and time of such and such nature to support that model. If there is no "correct" model, as model-dependent realism proposes, there is no real space and time. Space and time are not themselves being modeled. You can see this in the difference between Newtonian principles and Einsteinian principles. Movement of objects is modeled, and a "real" space and time of such and such a nature is required if the model is supposed to be true.
Ultimately, we might produce a model of moving objects which required no space or time, but our present conceptual structure falls back onto the reliance of space and time. However, the fact that our conceptual structures of the movement of objects requires that space and time are real, does not mean that space and time actually are real. If you and I agree that there is misconception, related to the use of "continuum", and "continuous", then the conceptual structures are inaccurate, and therefore a conception without any real space or time might be the correct one.
Wow that nutball Peter Lynds is still around? I heard of him about ten years ago ... maybe fifteen or twenty, now that I think about it. It was on Usenet. So it must have been at least twenty years. Tempus fugit.
I remember that he had solved Zeno, or some such ... I actually don't remember the particulars. Everyone on sci.math was talking about him with varying opinions. Lynds had written two papers. I downloaded and read them both line by line with close attention. I concluded that Mr. Lynds needed a good course in basic calculus. I regarded him at that time as a minor crank.
I have not heard his name since then. If in the intervening decades he has managed to acquire mindshare among ... well anyone, frankly ... I wish to register my dissent.
No need to disagree, this isn't a hill I'm going to die on. And maybe he's reformed his ways and is now writing articles of actual substance. If so I'd appreciate a link to anything recent that he's done. If there's new evidence I'll change my opinion. As it is. the mere mention of his name annoys me. As Dennis Miller used to say back when he used to be funny, That's just my opinion, I could be wrong.
Quoting Metaphysician Undercover
In my post that you were replying to, I took pains to explicitly mention the following:
Quoting fishfry
Since I wrote that, I wonder why you replied as you did. I'm perfectly well aware of the philosophical objections to calling the real numbers a continuum.
It's also a fact about the world that mathematicians regard the real numbers as the (mathematical) continuum.
My disclaimer made it perfectly clear that I understand the distinction between the mathematical continuum and the various philosophical approaches. In recent years I've studied the classical intuitionist continuum, (Brouwer et. al.), the modern constructivist continuum, the hyperreal line of nonstandard analysis (that's the one with the reals plus infinitesimals), and I've even seen a little Peirce. I have sympathy towards these points of view.
I wonder why you feel compelled to respond to me so irrelevantly, with such trite philosophy and inaccurate mathematics. Your remarks regarding the rationals are particularly unenlightened.
I'd welcome substantive engagement with you but I told you in the other thread that I find your style unpleasant. To me this is more of the same. You saw my disclaimer but couldn't help yourself piling on with the usual ... usual.
I am in the process of writing a reply to some of the things you read in the bijection thread. I read everything you wrote. I have a better understanding of your ideas. This is not the place and I'll try to finish that post up soon. I'll put my thoughts into a larger context. Meanwhile what is your point in telling me what I already know?
Quoting Metaphysician Undercover
You have asked a mathematical answer. I'd be happy to give you a mathematical answer.
My concern is that no matter what mathematical points I make, you're just going to pile on with the usual nihilism and simplistic philosophy that math must be wrong because it's not true. As if everyone doesn't already know that math isn't literally true. You're the only one flailing at this strawman.
The irrationals fill in all the holes in the rationals. I already illustrated this with a sequence of rationals that approaches the point sqrt(2) but there's a hole there instead of a point. The irrationals fill in those holes.
We can formalize the process of filling in the holes with various technical constructions of the reals. There are several, the two best known being Dedekind cuts and Cauchy sequences. The details aren't of interest. The point is that it can be done within set theory and it allows us to found calculus in a logically rigorous way, something that escaped Newton and Leibniz. We can also axiomatically define the reals as "the unique Cauchy-complete totally ordered infinite field." When you unpack the technical terms, you end up with an axiomatic system that's satisfied within set theory by the Dedekind cuts or Cauchy sequences. It's all very neat. One need not believe in it or care. It must be frustrating to you to both not believe in it, yet care so much!
I could drill the math down a lot more but should probably wait for encouragement, and if none is forthcoming I should leave it be. I don't think you're curious about the math at all. You just want to throw rocks. But why? People uninterested in chess don't spend their lives hating on chess. They just ignore it. You think math is bullshit? Maybe you're right. Maybe it is all bullshit. The thing is why do you keep repeating the point over and over as if we haven't all heard you already? And as if we all don't already understand the point?
Am I being too harsh? Maybe. I don't know. You misunderstand me. Perhaps I misunderstand you. Why would you reply to my post at all? You don't think I know anything. Why'd you even bother?
Ah. But I'd call that a strawman. The "widespread misconception." There isn't ANYBODY out there beating the drum for the proposition that the mathematical real line is the "true" continuum, whatever true might mean in this context. The overwhelming majority of mathematicians give the matter no thought at all. If they're studying differential equations or abstract algebra or proving Fermat's last theorem, they're simply not concerned with such matters.
And among those mathematicians who have taken the time and trouble to think about the nature of the continuum, they'd know about Brouwer and the modern constructivists and the hyperreals and if they studied some philosophy they'd get some context for the conceptual objections to the real numbers as the "true" continuum (again, whatever that might mean), and they'd most likely agree that the standard mathematical real numbers leave a lot to be desired in the philosophical realm,
I just don't think there are that many people who have thought about the matter for five minutes and don't have some sympathy for the alternative point of view. I don't think there is a widespread misconception. I think there's a widespread lack of interest in the question; and among those who are interested, some degree of agreement that the real numbers don't express everything we think must be true about a continuum.
There are no holes in the rational numbers, just like there are no holes in the reals. The numbers are produced from a different set of rules. This is what your illustration illustrates, it does not illustrate a hole:
Quoting fishfry
What you illustrate here is that an irrational number cannot be expressed in the rational system. If we did not already know what the sqrt(2) is, as an irrational ratio, we would not know that your proposed rational sequence does not converge to that. You could make up any random, fictional, irrational number (not grounded in a true irrational ratio like sqrt(2) or pi), and show that a sequence of rationals does not converge. But this does not illustrate a "hole". it just illustrates some sort of incompleteness, like 1.5 (3/2) illustrates an incompleteness in the reals.
The reals fail to account for division. The rationals are an attempt to address this failing, but to the extent that there are irrational ratios, the rationals fail in the attempt to account for division.
So you could make another demonstration to show that there are rational numbers which cannot be expressed in the real system, but this does not indicate "holes" in the system. Such demonstrations, which demonstrate incompleteness do not illustrate "holes", they just show that the different systems follow different rules. There is not one set of rules to cover all mathematical operations. If there is incompatibility in the rules, it will appear as what you call a "hole".
The existence of irrational numbers is easily accounted for by the fact that in the rational system (rules), there is infinite possibility of numbers between any two rationals. "Infinite possibilities" is not within the bounds of any logical system, it refers to where the system fails. Therefore illogical divisions (irrational ratios) are allowed to exist within that system, which allows for infinite divisibility. The illogical divisions are within the rules of the rational numbers, system. So the irrationals do not represent holes, as if they are outside the rational numbers system, they are within the system, a manifestation of the illogical proposition of "infinite divisibility". "Infinite possibility" is itself illogical. As a rule, "infinite divisibility" allows for the illogical, i.e., for the irrationals to exist within the rational system.
Quoting fishfry
That's right, I'm not curious about that math because I think it's the wrong approach. approaching the irrational numbers as if they are holes in the rational numbers, and trying to fill those holes with the reals, is completely backward. If these are actually "holes", then the reals are like a sieve. There's a massive "hole" whenever an odd number is divided by two for example.. Why would you try to patch holes with a sieve?
To resolve a problem requires a clear, and complete analysis of the problem itself, to understand its true nature. This is what I offer, a more thorough analysis of the problem, as an aid, to assist in resolution of the problem. What I do is not a case of throwing rocks, or hurling insults, it's a case of examining the foundations for weak points. If your attitude is that these foundations were built by the greats, therefore there are no weak points, (appeal to authority resolves fundamental problems), then I think you are in need of God's help. Hopefully, God as the ultimate authority, will show you how his principles differ from those authorities which you appeal to.
Fair enough, thanks.
A length that is irrational comes into play when you have a length that is the "smallest" length as the right sides of the triangle. The irrationals are not imaginary numbers. They simply go on forever, within a limit. But that infinity within the spatial limit is still an infinity of points. And half that hypotenuse would be the new true smallest length. It goes on forever, hence Cantor and then Banach and Tarski. I think it's also fishy that with finding the length of the hypotenuse with a length of One on the ride sides, you multiply one times one and get no further than one. I could be wrong about that minor point
:lol: Not likely. I thought the thread needed some comic relief.
Good, I thought it was just me. In fact after I posted I looked up his name and found that when he submitted his 2005 papers (so fifteen years ago) one reviewer rejected him saying that he "lacks fundamental understanding of basic calculus." Exactly the same criticism I noted.
Then we have nothing to talk about. I still owe you my thoughts on the bijection thread, where you made a couple of remarks that I can use as a starting point.
It's curious, though, that you are so interested in the philosophy of math yet so uninterested in math. Your math is wrong. You have some ill-formed ideas that are leading you astray. And you have no interest in clarifying your own errors. Hard to know what to make of that.
Quoting Metaphysician Undercover
Not something I've ever said. If I have said such a thing, please be so kind as to quote my words directly. You can't because I never said any such thing. It's pathetic that your only means of discourse is to lie.
Since I explicitly mentioned all the alternative models of the continuum that I'm familiar with, your remark has no basis in reality; other than your own reality of incoherent bullshit.
The irrationals are relations which cannot be resolved, like the ratio between the circumference and the diameter of a circle, or the distance between two points of equal distance from a point at a right angle. That such relations are irrational indicate that the two things being related to one another are incommensurable.
Quoting fishfry
Yes, sorry, I misunderstood.
I think they are commesurable only in the sense that there is infinity contained within the finite in geometry and in our world
I think that "infinity contained within the finite" is contradiction pure and simple. That's like saying that a finite thing is inherently infinite. Contradiction.
In mathematics, to "truly understand" is to accept the axioms without question. This allows that contradiction within an axiom is acceptable as understood. Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem.
So your argument on this thread is that there is not a contradiction in math, but that it's incomplete?
It's mostly a metaphysical problem. Most mathematicians and physicists do quite well without contemplating such issues. But that's not to say that "infinity" and "objects" are not concerns, as a physics person might tell you in reference to renormalization procedures and quantum entities, for instance.
Yes it is incomplete, but I think what I meant was that there is contradiction in math but it's acceptable because it's unavoidable. It's unavoidable because mathematics is a reflection of our understanding of reality, and our understanding of reality is limited by our capacities as human beings. So contradiction just reflects our imperfections. If mathematics were without contradiction, it would be perfect, and our understanding of reality would be perfect. The incompleteness is therefore our inability to completely rid the system of contradictions.
Quoting jgill
I agree, these issues are simply accepted, taken for granted, perhaps almost subconsciously, and not worth thinking about for most mathematicians and physicists. They work with the tools they have. Likewise, that our knowledge of the physical universe is incomplete is also taken for granted. If we put two and two together, we might conclude that a better system would provide us with a more complete knowledge.
I do not believe that could be true according to the known laws of physics. A sufficiently high frequency would eventually require back-and-forth movement of the bow in space smaller than the Planck length.
To be clear, since this is a common misunderstanding: The Planck scale is the point at which our theories of physics break down and may no longer be applied. It does NOT mean the world itself is quantized. Below the Planck length we simply do not know and can't even speculate, because our physics no longer works.
With that understanding, if there is a length below which we can not analyze or understand; there is a highest possible frequency. And within them, to discriminate one from another must likewise become impossible. There are not infinitely many gradations, to the best of my understanding of things.
After all quantum theory says that light is not infinitely variable. Pretty reasonable that sound isn't either.
Your eyes tell you there is a continuous gradation of light. Physics proves otherwise.
For that matter your senses tell you the world is flat. Science shows otherwise.
Even your ears have physical limitations. There's no way you could distinguish an infinite number of different notes. And if you just assume that there is so many different notes that they must be infinite, that's an unsound premise.
Thanks for this. I would add that the same is true of the Planck time, since it is defined as the duration required for light to travel the Planck length in a vacuum.