create ambiguity in the definition — Metaphysician Undercover

Disambiguation is what mentioned definitions do.

Cantor did nothing to help our understanding of infinity IMO; he has lead us down the wrong path entirely. — Devans99

To me this frames infinity as an object that already exists, already has a nature. Philosophers can compare their intuitions in natural language, but mathematicians have got to make some rules.

So perhaps the burden is on Cantor's critics to offer a mathematical substitute.

My (and Galileo's) point exactly - you fundamentally cannot measure something that is
uncountable/infinite - you would never finish measuring it - it is impossible to measure and claiming that bijection can provide a sound measure is ignoring the evidence (of paradoxes). — Devans99

You misunderstand me. The measure of a set is different than its cadinality. https://en.wikipedia.org/wiki/Lebesgue_measure

Some of the confusion about Cantor seems to involve not realizing that mathematicians also have other ways of comparing sets. Measure is more intuitive than cardinality. [0,2] has twice the measure of [0,1] and yet the same cardinality. And then also have homeomorphisms in topology. These concepts each treat of a different quality that sets can have in common. With people we can talk about height, eye color, shoe size, etc. It's the same with sets.

Yes, we can class mathematics as "normal discourse", but to characterize "normal discourse", as working with finite objects of meaning, is what Wittgenstein demonstrates as wrong. This is why we must work to purge the axioms of mathematics from the scourge of Platonism, To consider proofs as finite objects is a false premise. — Metaphysician Undercover

I find it hard to make sense of this. Proofs are obviously finite objects, or we could never finish reading or writing one.

Wittgenstein can't demonstrate that this or that as wrong mathematically. He comments ultimately about interpretations of the calculus (game of symbols). I like some of his critiques. And I also like intuitionism, finitism, constructivism. The philosophy of mathematics is deep and complicated. A person can have philosophical doubts about mainstream mathematics and still be good at it.

And then most people never learn pure math. They learn algebra, trig, calculus, applied linear algebra. This stuff is fairly intuitive and incredibly useful. To me anti-Cantorian passion suggests a love for pure math in that it wants to get the infinite 'right.' The door is always open. A person could construct a system. To be math, it would need rules. This guy actually tried to deliver a replacement for the foundations he objected to: https://en.wikipedia.org/wiki/L._E._J._Brouwer

A few people work on systems like that. But most people who use math don't care at all about philosophical disputes. Math is just a tool that they use according to conventions.

Therefore it is very unlikely that we actually have any truly finite proofs, because definitions are produced with words, which themselves need to be defined, etc., ad infinitum. — Metaphysician Undercover

It can and has all been done with symbols. It's like a game of chess. In practice words are used to abbreviate formal proofs and aid the intuition.

A formal proof is a finite sequence of formulas, each member of which is either an axiom or the result of applying a rule of inference to previous members of the sequence. Typical rules of inference are modus ponens and the substitution of equals for equals. A grammar for formulas, a collection of axioms, and a collection of rules of inference together define a logical theory.

For the usual theories of mathematics, e.g. set theory or number theory, it is a relatively modest exercise to write a program called a proof checker that will check, in a reasonable amount of time, whether a given sequence of formulas is a proof. — link

http://www.cs.utexas.edu/users/boyer/ftp/ics-reports/cmp35.pdf

This isn't surprising. Unless proofs could be verified, math wouldn't be a normal discourse. It would just be a quasi-literary metaphysics with mathematical themes.

To me this frames infinity as an object that already exists, already has a nature. Philosophers can compare their intuitions in natural language, but mathematicians have got to make some rules. — softwhere

But as I understand it, maths frames infinity as an object that already exists (axiom of infinity). I believe that axioms should be more than assumptions - they should be self-evident truths - and what is self-evident about the existence of an actually infinite set? Parallel lines not meeting I can swallow, but an actually infinite set?

So perhaps the burden is on Cantor's critics to offer a mathematical substitute. — softwhere

I am not a mathematician but I would imagine that the vital areas of mathematics (IE calculus) could function well enough using just the concept of potential infinity. I do not see why the concept of actual infinity is needed - it just leads to paradoxes.

You misunderstand me. The measure of a set is different than its cadinality. — softwhere

I am not disputing it is possible to measure intervals, I am disputing the common mathematical claim that there is an actually infinity of points on a line segment length 1.

How many points do you claim there are on a line segment length 1? The answer must logically be one of the following:

1. Infinite number
2. Finite number
3. Undefined
(there are no other possibilities)

If it is [1], that means 1/0=∞ which is nonsense
If it is [2], then a point must have non-zero length which is not the definition used in maths.
So I contend it must be [3].

But as I understand it, maths frames infinity as an object that already exists (axiom of infinity). I believe that axioms should be more than assumptions - they should be self-evident truths - and what is self-evident about the existence of an actually infinite set? — Devans99

I think that's a good question. To me it's fair indeed to operate at this level and engage in a philosophical debate about the rules of the game that human beings freely decide upon. I'm not a specialist in set theory, but certainly a set theorist wants to construct more familiar mathematical objects like natural numbers.

the other axioms are insufficient to prove the existence of the set of all natural numbers. — Wiki

Basically they had to have it if they wanted the natural numbers, and they had to have the natural numbers. But others have wanted to take the natural numbers as fundamental.

The primary concern of mathematics is number, and this means the positive integers. . . . In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself. — Erret Bishop

https://en.wikipedia.org/wiki/Errett_Bishop

'God created the positive integers' is just a metaphor for their obviousness to intuition. Other comments involve constructive/intuitionistic logic. There's a fascinating ideological purity in this that appeals to me. But the mainstream chose otherwise. At the same time, most mathematicians don't worry themselves about this stuff in my experience. (I stand out by questioning the meaning and value of the game, but then I should have studied philosophy instead.) They learned a certain set of rules that they are happy with. The rest is disreputable 'philosophy,' inferior because it's just 'opinions.'

I am not disputing it is possible to measure intervals, I am disputing the common mathematical claim that there is an actually infinite of points on a line segment length 1. — Devans99

I still think you are confusing mathematical and metaphysical claims. What do you mean by 'actually infinite'? Math isn't done with ambiguous philosophical terms. You can complain that the axiom of infinity violates your intuition (fair enough), but it's trivial to show that [0,1] is an infinite set using 'the rules.' What that means philosophically or metaphysically is another question entirely. It is essential for math that it function independently of metaphysics. Once the rules are chosen, there is no more room for confusion or ambiguity. It is a dead machine.

How many points do you claim there are on a line segment length 1? The answer must logically be one of the following:

1. Infinite number
2. Finite number
3. Undefined
(there are no other possibilities)

If it is [1], that means 1/0=∞ which is nonsense
If it is [2], then a point must have non-zero length which is not the definition used in maths.
So I contend it must be [3]. — Devans99

What you neglect here is the ambiguity of 'infinite number.' This is pre-mathematical metaphysical ambiguity. In truth, I think you are guilty of the very think you accuse mathematicians of. If one wants to be strictly and even mechanically logical, then one needs strictly and even mechanically defined terms. This is precisely why notions of cardinality and measure were painstakingly developed within a 'realm of law.'

Cantor's most famous breakthrough was showing that one notion of infinity was not enough. His work has consequences for theoretical computer science. It's not only artistically charming. Whatever limitations or blemishes one finds in mainstream math, it's a spectacular structure.

Also, choice #1 does not imply that 1/0 = infinity. Saying so is pseudo-math. Since your attracted to this stuff, why not study some math? Even if you just want to criticize it, your criticism will only be plausible if you can project a basic competence --that you actually know what math is and how it operates from a position within math. Fascinating criticisms of math can be made, but all the criticisms I've seen online by the untrained have so far been just projected misunderstandings. I don't mean to be offensive. I like the critical philosophy of mathematics. It just has to know its target in order to hit it.

Basically they had to have it if they wanted the natural numbers, and they had to have the natural numbers. But others have wanted to take the natural numbers as fundamental. — softwhere

The problem I see is that (applied) mathematics forms the basis for our understanding of reality. So scientists pick up definitions and theories from maths and apply them to the physical sciences.

Now the set of natural number exists purely in our minds - my believe is there is nothing in reality akin to it. So there is this impossible concept which is taken from maths and is being applied in the physical sciences - producing erroneous results - cosmology is the biggest offender.

I feel a way of defining the natural numbers without resorting to questionable concepts like actual infinity would be the way to go.

Also, choice #1 does not imply that 1/0 = infinity. Saying so is pseudo-math. — softwhere

It does imply that 1/0 = ∞, we need only pre-school maths to arrive at such a conclusion:

1. Maths claims an infinite/uncountable number of points on a line segment
2. A point is defined by maths to have length 0
3. The line segment in question is length 1
4. So we must divide the interval into 0 equal pieces to find the number of points in it
5. Hence the number of points is 1/0 = ∞

Where is your dispute with the above reasoning?

It does imply that 1/0 = ∞, we need only pre-school maths to arrive at such a conclusion: — Devans99

The problem is that you think you can do philosophy of math with only pre-school mathematics. If you don't know how to read and write proofs, then you don't really know what math is.

Where is your dispute with the above reasoning? — Devans99

I've already discussed that. Measures are countably additive set functions. Cardinality is actually important here. But measures will make no sense without a mastery of basic real analysis. Usually one learns this over a course of years. It's like learning to become fluent in a language. Since inexpensive Dover books are easily available, I won't go into great detail that's likely to be ignored anyway.

If you actually want to resolve your confusion rather than install it as a work of genius, you'll just have to learn some math.

The infinite sum concept in maths has definite problems, see here for an example: https://en.wikipedia.org/wiki/Thomson%27s_lamp — Devans99

It should please you to know that you are at the point mathematicians were at two and a half centuries ago as they pondered what infinite sums meant. Does the sum S=1 -1 +1 -1 + 1 ... make any sense? Indeed, the partial sums are 1,0,1,0,1,... like that pesky light switch (which is seen as pretty silly these days - not a "definite problem"). After Cauchy and Weierstrass and others formalized convergence criteria there was still the amusing question of series that oscillated like the one above. Then other mathematicians developed summation processes (SP) that had the following features: If a series converged in the normal sense, it must converge to the same value in the SP , but some series that did not converge in the normal sense might "converge" in this new way.

For instance, one such SP is to add the first n partial sums and divide by n. If the given series converges as n gets larger without bound, this new process will converge to the same value. But in the above conundrum, note that this process yields a limiting value of 1/2. And some mathematicians long ago stipulated that value for the series, before Cauchy and Weierstrass had their says.

As for measures, although the simplest is the length of an interval on the real axis, used in the Riemann integral, they get much, much more complicated and abstract and are used in what are called functional integrals. I suggest you don't go there. If you are curious, go to my page in researchgate and pull up the note on functional integrals.

A point has length 0, say the line segment is length 1, then the number of points on it is 1/0=UNDEFINED. It is not infinite or unbounded, it is just UNDEFINED. It's not surprising considering a point is defined to have length 0 - so cannot exist - something with all dimensions set to zero clearly does not exist - so the question can be rephrased as 'how many non-existent things can you fit on a line segment' - an answer of UNDEFINED is exactly what you'd expect. — Devans99

This has bothered me since you first brought it up not a while ago. I'm not a mathematician but 1 here is a length and when you divide a length you don't get a point. What you get is another length.

Also, a point isn't defined in terms of how big/small it is i.e. it isn't dependent for its existence on its own dimensions which as you rightly pointed out is zero. A point is actually defined in terms of its distance from the origin (0,0) or some other reference point.

Dividing a length by a point doesn't make sense in the same way as dividing Tom, Dick, Harry and John by Dick or Harry doesn't make sense.

The BB suggests that space maybe finite - space has been expanding at a finite rate for a finite time since the BB - so that suggests finite space (finite spacetime too). What lies beyond is pure nothing - there is no space and no time beyond the boundaries so nothing can exist. — Devans99

What lies beyond the boundary of "finite" space? Can an infinite space not expand?

Imagine three galaxies in infinite space A, B, and C. Suppose the distance between them is 4,000 lightyears. Can't the space between these galaxies increase, not because they're moving but because space is being created between them. In other words I see a possibility of an infinite and expanding space.

But nothing can exist forever in time, so it must have a start. See for example the argument I gave in this OP: — Devans99

What about time itself? Did it have a beginning? If space can be infinite and time is "just another" dimension, and if space can be infinite can't time be too?

I feel you are avoiding addressing my point and attempting to blind me with references to advanced math. If advanced math disagrees with basic math / basic logic then advanced math must contain logic errors.

The measure of the interval [0,1] is 1 and the measure of the interval [0,2] is 2. This way of classifying size also leads to the conclusion that a point must have non-zero length:

length of a interval = pointsize * pointnumber

Neither of 'pointsize' and 'pointnumber' can be zero because then the measure of the two intervals would be equal (zero in both cases). So a point cannot have zero length.

?

I still feel maths does not currently have a complete understanding of infinite series:

- At any intermediate point in the evaluation of Grandi's series, it always has a sum of 1 or 0. Therefore logically the final sum can only be 1 or 0 - there are no other possibilities
- But mathematical methods for evaluating series yield the sum of 1/2
- Maths calls the series divergent as the individual terms do not approach zero
- But if we knew whether ∞ is odd/even we could evaluate the series
- But my contention is there is no such thing as actual ∞
- So IMO the final sum of a series (taken to ∞) is a meaningless concept (in some instances)

This has bothered me since you first brought it up not a while ago. I'm not a mathematician but 1 here is a length and when you divide a length you don't get a point. What you get is another length.

Also, a point isn't defined in terms of how big/small it is i.e. it isn't dependent for its existence on its own dimensions which as you rightly pointed out is zero. A point is actually defined in terms of its distance from the origin (0,0) or some other reference point. — TheMadFool

OK, so your interpretation is (as I understand it) that that a line segment is not composed of infinite points, but is composed of sub-lengths. I am in agreement. I would point out that the length of a sub-length cannot be zero else all line segments would have the same size.

Imagine three galaxies in infinite space A, B, and C. Suppose the distance between them is 4,000 lightyears. Can't the space between these galaxies increase, not because they're moving but because space is being created between them. In other words I see a possibility of an infinite and expanding space. — TheMadFool

If the distance between them is currently 4000 ly and the universe is expanding. then there must have been a time in the past when the distance between them was 3000 ly, 2000 ly, 1000 ly, 0 ly. At the final point, when the galaxies are co-located, the universe cannot be expanding. So I think that infinite expansion is impossible. I believe there are some cosmologists who disagree with me on this.

What about time itself? Did it have a beginning? If space can be infinite and time is "just another" dimension, and if space can be infinite can't time be too? — TheMadFool

It comes to a question of origins. I believe that there must have been a first cause for everything in time (the cause of the BB probably). Then the obvious question is what caused the 'first cause'. We could answer that by introducing another cause to cause the 'first cause', but then we would need another cause, and another, so we end up with an infinite regress with no ultimate first cause of everything - which is impossible.

So there seems to be a need for a first cause and there cannot be an empty stretch of infinite time preceding the first cause - then there would be nothing but emptiness to cause the first cause - which is impossible.

So there seems to be a need for an 'uncaused first cause'. How do you get an 'uncaused cause'? Well causality is a feature of time, so placing the first cause beyond time seems to be the only way to have an 'uncaused cause' - then there is nothing logically or sequentially 'before' the first cause - the first cause has permanent uncaused existence. The first cause is then the cause of / creator of time (time must have a start).

I can sum up the argument with an altered version of the PSR:

- Everything in time must have a cause

Which leads naturally to a timeless first cause.

This also leads to, incidentally, an answer to 'why is there something rather than nothing?' - there has always been something - uncaused and beyond time - there is simply no 'why?' for something that is uncaused.

OK, so your interpretation is (as I understand it) that that a line segment is not composed of infinite points, but is composed of sub-lengths. I am in agreement. I would point out that the length of a sub-length cannot be zero else all line segments would have the same size. — Devans99

The issue here is that there is an incommensurability between distinct spatial dimensions. Pythagoras demonstrated that the ratio between two perpendicular sides of a square is irrational. The same type of irrationality arises from other two dimensional figures, like the circle, with the irrational pi.

This incommensurability is extremely evident in the relationship between the non-dimensional point, and the one dimensional line, according to TheMadFool's explanation.

Whenever we add another dimension to our spatial representations we add a new layer of complexity to this fundamental incommensurability, such that by the time we get to a four dimensional space-time the irrationality involved is extremely complex. What is indicated by this fact, is that our representation of spatial existence, in the form of distinct dimensions, is fundamentally flawed.

Agreed. A 0-dimensional point can in no way be the constituent of a 1-dimensional line segment - the point has zero length and the line segment has non-zero length. So it is incorrect to say, as mathematicians often do, that a line segment contains an actual infinity of points.

Likewise, a 1-dimensional line cannot be the constituent of a 2-dimensional plain - the line has length but zero width so it cannot be the 'parts' of a plain (which has non-zero length and non-zero width).

As you are both mathematicians, I hope you don’t mind if I take the opportunity to explain my concern a bit further. I feel there are some deep problems with infinity in the foundations of mathematics and it seems to me that many mathematicians are very invested in their discipline and their hard-won knowledge - so that they are not usually eager to confront such issues.

For example, take the extended complex numbers - the set of complex numbers plus ∞. The definition used for ∞ is z/0=∞. Now you can call that ‘an assumption’ if you like (and a pseudo-justification in terms of limits can be given) but it is plainly a wrong assumption. I believe there are then fields of maths (like complex analysis) which build on the idea of the extended complex numbers. Then people in the physical sciences build further theories based on these ideas. The net result is whole vertical slices of human ‘knowledge’ which are based on wrong assumptions and are therefore not valid knowledge.

Similar bad assumptions to the above example can be found in the hyperreal numbers and the projectively extended real line. Another example, already discussed above, is the axiom of infinity from set theory - the assumption of the existence of actually infinite sets of objects. It is a bad assumption to make and set theory is based on that bad assumption. Many things in maths and science are then built upon the foundation of set theory. Again we have whole swaths of knowledge based on bad assumptions - all that ‘knowledge’ is therefore not valid.

I am not a mathematician so I cannot call out more examples than this, but I’d be surprised if there are not more. It is therefore not surprising (I hope) that laymen such as myself are disinclined to try learning more of advanced mathematics - my (admittedly limited) experience of such is that (what can happen) is early on, in the foundations, a bad assumption is made and then a body of interesting, complex but ultimately invalid results are derived based on that bad assumption.

I feel mathematics has a responsibility to the rest of the physical sciences to keep the assumptions reasonable. By reasonable, I include assumptions that are provisional - they may lead to interesting, but provisional results (eg non-euclidean geometry). I also class as reasonable assumptions that widen the scope of traditional mathematics, such as the introduction of new types of numbers (eg complex numbers). But assumptions that are plain wrong/bad (counter logical) lead nowhere useful, lead other folks (in the physical sciences) astray, and result in lots of clever folk wasting huge amounts of time on wild goose chases (eg a good portion of modern cosmology is like this IMO).

Your thoughts?

Pythagoras demonstrated that the ratio between two perpendicular sides of a square is irrational. — Metaphysician Undercover

What do you mean? The ratio "of"? Take a square with sides =1. The 1/1=1. Are you talking about the hypotenuse of a right triangle? Like each side = square root of two?

The definition used for ∞ is z/0=∞. — Devans99

This is wrong, and shows how difficult it is to debate with you. Apparently you know so little of math you cannot even frame your beliefs correctly.
https://math.stackexchange.com/questions/2424005/what-does-infinity-in-complex-analysis-even-mean/2424111

Software, what is your math background?

“Nonetheless, it is customary to define division on C ∪ {∞} by
z/0 = ∞ and z/∞ =0”

https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers

"There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^*!=0 in the extended complex plane C-* is defined to be a quantity known as complex infinity.”

http://mathworld.wolfram.com/DivisionbyZero.html

I've never known a mathematician who actually used z/0 = infinity. It's simply a symbol that ultimately refers to the north pole of the Riemann sphere.

Here's a comment from the talk page on Wikipedia. Not mine.

"The Riemann sphere is just the complex plane with an extra point added in, called the point at infinity. For analogy, look at the real line. There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit.

In the same way, one can pretend that all rays in the complex plane originating from 0 actually have an endpoint, and they all eventually meet at infinity, a point far-far away (not accurate as Elroch mentions above, but helpful in imagining things).

The Riemann sphere is not the same as the usual sphere, but they are topologically equivalent. Imagine a normal sphere, remove the north pole, and make the obtained hole there larger and larger (assume the sphere is made of very flexible rubber). Eventually, that sphere without a point can be flattened in a plane, the complex plane. The original north pole corresponds to the point at infinity in the complex plane.

There is only one Riemann sphere, as the point at infinity is just a symbol, its actual nature is not relevant. "

You need to move away from your fixation on the symbol z/0, which you seem to believe has some quality that disrupts reasoning in mathematics. I suggest you look into the Axiom of Choice, about which there is genuine relevance to mathematical thought.

But you did score a point here! Congrats. :grin:

There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit. — John Gill

We are talking about the nature of time, whether it has a beginning or end specifically. Such a conversation is intimately linked to the existence or non-existence of Actual Infinity. Maths treatment of the subject could hardly be described as definitive - a set of non-sensical assumptions IMO. Notice I have highlighted the phases 'pretend', 'without worrying much'... such words hardly inspire confidence...

There is no largest number - numbers go on for ever in each direction - there is no such number as ∞ - so a mapping of non-existent points at actual infinity to the north pole of a sphere is a nonsense procedure.

And yes z/0=∞ appears to have 'valid' (!) applications in maths:

"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=∞ well-behaved."

- https://en.wikipedia.org/wiki/Riemann_sphere

Carry on in your crusade. It's above my pay grade and below my interest. (that's not my quote, someone on Wiki TALK). The link to math stackexchange is more reliable than the Wiki sites IMO. Even there things get out of hand. For me and my colleagues, |z| getting larger and larger without bound means z -> infinity. An actual point at infinity is irrelevant in practice. If I think of time going to infinity, I mean it in this sense. If you look at the projective plane sitting below the Riemann sphere, you can see z moving further and further out, without bound, and as it does so its projection on the sphere moves closer and closer to the north pole, but never reaches it.

@Devans99, you should know that the extended reals is not a semigroup, a group, a ring or a field.
There are certain contexts where such (sort of "artificial") constructs are convenient, as long as that's understood.
I don't think you can simply lean on this sort of convenience here and call it a day.

Many things in maths and science are then built upon the foundation of set theory. Again we have whole swaths of knowledge based on bad assumptions - all that ‘knowledge’ is therefore not valid. — Devans99

'Foundations' is a misleading metaphor. In general, applied math comes first. Calculus was invented and successfully applied long before a careful definition of real number system was given. Pure math isn't, in my view, the genuine foundation of applied math. Instead human beings just trust tools that tend to give them what they want, reliably.

If you live in the city, consider the tall structures of concrete and steel. Why is it that they don't tumble down? Aren't they based on the axiom of infinity? No. Formal set theory is arguably more of an aesthetic enterprise. Engineers don't need it. Pure math is its own beast, and I suggest that its prestige is parasitic upon that of the technological enterprise.

But assumptions that are plain wrong/bad (counter logical) lead nowhere useful, lead other folks (in the physical sciences) astray, and result in lots of clever folk wasting huge amounts of time on wild goose chases (eg a good portion of modern cosmology is like this IMO). — Devans99

I think you'd have to make a case that physical scientists are being led astray, a claim I find less than plausible. I know very little about modern cosmology, but some of your other comments make me think that you are wrestling with Kantian themes. Is a beginning of time thinkable? No. Is beginningless time thinkable? Also no. I'm sympathetic to that kind of thing. I like instrumentalism as a way to make sense of science. To me neither math nor physical science offers 'metaphysical' truth (replaces philosophy).

We are talking about the nature of time, whether it has a beginning or end specifically. Such a conversation is intimately linked to the existence or non-existence of Actual Infinity. Maths treatment of the subject could hardly be described as definitive - a set of non-sensical assumptions IMO. Notice I have highlighted the phases 'pretend', 'without worrying much'... such words hardly inspire confidence... — Devans99

You are asking math to do what it cannot do and does not claim to do, namely metaphysical philosophy. If you want a deep investigation of time, look into Hegel or Heidegger or Kant. The results of math being so certain comes at the cost of their significance being indeterminate.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. — Russel

I think this is an illuminating exaggeration. Intuition is important in math, but math is protected from the endless misunderstandings found on philosophy forums by rules and exact definitions.

The measure of the interval [0,1] is 1 and the measure of the interval [0,2] is 2. This way of classifying size also leads to the conclusion that a point must have non-zero length:

length of a interval = pointsize * pointnumber — Devans99

I will humor you. The number of points in [0,1] is uncountably infinite. Measure, however, is only countably additive.
http://mathworld.wolfram.com/CountableAdditivity.html

That means you can't split [0,1] into points and then add the measure of the points to get the measure of the interval. You can, however, split [0,1] into [0, 1/2) and [1/2,1]. In this case all is well. One can even measure Q, because Q is countably infinite. And the measure of Q is 0. Note that the two most famous kinds of infinity (that of N and that of R) actually do come into play right away in measure theory.

Keep in mind that I might as well be talking about the rules of chess. You can say that these are all fictional entities. It is true that theoretically this is the foundation of statistics. But I suggest that human beings would still apply math without careful justifications that few of them ever study in the first place. We trust tools that work. Consider Hume's problem of induction. All of our technology is based on a gut-level faith in the uniformity of nature.

Are you talking about the hypotenuse of a right triangle? — John Gill

Yes.

A point has length 0 — Devans99

So a point cannot have zero length. — Devans99

:chin:

Such a conversation is intimately linked to the existence or non-existence of Actual Infinity. — Devans99

At some point I think this leads us into the philosophy of language. How do the signs 'actual infinity' function in our community? Is there ever some sharp meaning in our head? Maybe the question isn't binary, one of clear existence or non-existence. For why should 'exist' be any less complicated semantically than 'actual infinite'? I

Well causality is a feature of time, so placing the first cause beyond time seems to be the only way to have an 'uncaused cause' - then there is nothing logically or sequentially 'before' the first cause - the first cause has permanent uncaused existence. — Devans99

I understand where you are going. But how sensible is this time before time? I find it as questionable as intuitions of actual infinity. Personally I think human cognition runs aground on issues like this. It's as if we just weren't equipped for such questions. If time and space are automatic intuitions, then we run the risk of talking non-sense. At the same time, we can create mathematical models that defy intuition that nevertheless have practical power.

This is why I like instrumentalism as an interpretation of science. And I also think that reality as a whole is inexplicable on principle. Some principles always must remain 'true for no reason.' They are patterns that are just there. Later we may derive them from still larger patterns, but this just expands the circle whose outermost ring is contingently true.

If you live in the city, consider the tall structures of concrete and steel. Why is it that they don't tumble down? Aren't they based on the axiom of infinity? No. Formal set theory is arguably more of an aesthetic enterprise. — softwhere

I would contend that set theory gives an unwarranted legitimacy to actual infinity which influences the physical sciences. Some cosmologists are obsessed with actually infinite time and space and they lean on concepts borrowed from pure maths to justify such infinite models.

I will humor you. The number of points in [0,1] is uncountably infinite. — softwhere

Measure theory does not seems to provide any justification for the above claim - neither do I see any justification anywhere else in maths.

I understand where you are going. But how sensible is this time before time? I find it as questionable as intuitions of actual infinity. Personally I think human cognition runs aground on issues like this. — softwhere

There are a number of good arguments that all point to a start of time (one example: perpetual motion is impossible, things are currently in motion -> a start of time). If there is another type of 'time' 'before' our time, the same arguments apply to that 2nd type of time - it must also have a start. Obviously there cannot be an infinite regress of such times, therefore we seem to be left only with the possibility that something must exist that is timeless /atemporal.

You maybe right stating 'human cognition runs aground on issues like this'. Philosophers have grappled with the nature of timelessness for centuries and I have not yet encountered a satisfactory explanation of what it could be.

I imagine our universe as a 2d space time diagram - a plane of finite dimensions. Then I imagine a point off to the side of this plane that represents a timeless thing. Then I imagine there is a mapping from the point off to the side to each point in the plane - the timeless thing experiences everything in time in one 'eternal now' and can interact with anything in spacetime. Of course this does not really shed any light on the nature of the point - the timeless thing.

Sorry I should of said:

- A point has zero length according to maths definition
- But according to my intuition, a point must have non-zero length