She passes through each point or she passes through each location that is the address of a point. Not sure what you are getting at. — fishfry

Put another way, there are no real points or locations that she passes through, only the ones that we invent to describe her motion. Again, space does not consist of such discrete points or locations, we introduce them for our own purposes. Not sure I can convey what I mean any more clearly than that.

Again, space does not consist of such discrete points or locations, we introduce them for our own purposes — aletheist

In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now.

Not sure I can convey what I mean any more clearly than that. — aletheist

How many times must I say the same thing? I don't get it. To be fair you may not be reading my replies to @keystone aka @Ryan, but rest assured that I have made the point you are making many times over.

There is no true space that cannot be divided by the mind. And this applies to the spaces that are divided from each other. And again and again. Understanding the contradiction is a type of wisdom on the subject. It's not scary. "A healthy mind can accept a paradox" (Chesterton)

The "Internet Encyclopedia of Philosophy" has a superb article on "Inconsistent Mathematics", which is a field of study also called paraconsistent mathematics. It mentions Zeno's paradox

(I believe all of Zeno's arguments are reduced to one single insight: spatial objects are finite with infinite parts. If this does not give you a headache one day of your life then you never understood it. The argument that motion through a point is really "rest" I think is false. Zeno had one great insight and it was an insight into mathematics, not physics)

Unless you reject the intermediate value theorem, my point stands. ... Some version of the IVT is always valid regardless of one's model of the real line. — fishfry

Well, the IVT is not valid in smooth infinitesimal analysis. As Bell states in his book that I suggested a while back, "the classical intermediate value theorem, often taken as expressing an 'intuitively obvious' property of continuous functions, is false in smooth worlds." The Wikipedia article on SIA explains:

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.

It is presumably less surprising that the Banach-Tarski paradox also does not arise in SIA.

In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now. — fishfry

So we agree, then? The mathematical real line is an extremely useful model of a continuous line, but like all representations, it does not capture every aspect of its object--in this case, a true continuum.

So we agree, then? The mathematical real line is an extremely useful model of a continuous line, but like all representations, it does not capture every aspect of its object--in this case, a true continuum. — aletheist

I have never claimed otherwise. If I've somehow failed to communicate that, I'll try to do better. But I would disagree with one aspect of what you said. The mathematical real line fails to describe what's physically provable about space, in terms of people walking and "passing through every point." But why should space be a continuum at all? That's an open question. How would anyone know what a "true continuum" even is? The idea of a continuum, let a lone a "true" one, is a conceptual abstraction. It's like arguing about which mathematical model of the tooth fairy is correct.

I would ask you: What is a "true continuum," and how would you know one if you saw it? Are you claiming there is such a thing in the world? Or if not, and if it's only a conceptual abstraction, how can anything be an accurate model of it?

ps -- SIA denies excluded middle and so falls into the general category of constructivism. No wonder many common standard theorems are false in such a framework. I've already mentioned that IVT is false in constructivist math but they patch that up by demanding that all functions are computable. In SIA, all functions are continuous.

But why should space be a continuum at all? That's an open question. — fishfry

I agree, it is a hypothesis--one that I happen to find much more plausible than space consisting of discrete parts. I would say the same about time, which Peirce considered to be "the continuum par excellence, through the spectacles of which we envisage every other continuum."

How would anyone know what a "true continuum" even is? — fishfry

I guess it comes down to definitions. Modern mathematicians stipulate that the real numbers constitute the (analytical) continuum, but (at least arguably) that approach is not entirely consistent with the common-sense notion of what it means to be continuous.

I agree, it is a hypothesis--one that I happen to find much more plausible than space consisting of discrete parts. I would say the same about time, which Peirce considered to be "the continuum par excellence, through the spectacles of which we envisage every other continuum." — aletheist

Your intuition is seriously at odds with modern physics. Do you think physics is wrong? How do you square this.

Secondly I wanted to repeat in case you missed the ps to my last post, that SIA denies excluded middle. So it's a flavor of constructivism. No wonder that IVT is false, it's false in constructive math. And no wonder Banach-Tarski is false, constructivism denies the axiom of choice.

I guess it comes down to definitions. Modern mathematicians stipulate that the real numbers constitute the (analytical) continuum, but (at least arguably) that approach is not entirely consistent with the common-sense notion of what it means to be continuous. — aletheist

Your common sense notion is incompatible with modern physics. And you are failing to distinguish between physical space, which we think is "really there," and the idea of the continuum, which is just a philosophical concept with no actual referent in the real world.

Your intuition is seriously at odds with modern physics. Do you think physics is wrong? — fishfry

No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement, or as Wikipedia puts it, "the minimum distance that can be explored. ... The Planck length is sometimes misconceived as the minimum length of space-time, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry."

Secondly I wanted to repeat in case you missed the ps to my last post, that SIA denies excluded middle. — fishfry

I did not see your PS until now, but I am well aware that the logic of SIA is what has come to be known as constructive or intuitionistic. Peirce was skeptical of excluded middle, but for very different philosophical reasons than Brouwer and Heyting--he believed that reality itself does not conform to it, because it is fundamentally continuous and general, rather than discrete and individual. He stated this in slightly different ways in alternate drafts of the same text.

To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it. — Peirce (NEM 3:758, 1893)

No doubt there is an assumption involved in speaking of the actual state of things ... namely, the assumption that reality is so determinate as to verify or falsify every possible proposition. This is called the principle of excluded middle. ... I do not believe it is strictly true. — Peirce (NEM 3:759-760, 1893)

In itself, yes; but we can still "divide" it at will to suit our purposes. — aletheist

To identify a point on a line is not to divide that line. So it's not really a matter of dividing which you are talking about.

For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space. — aletheist

The proposed units would be arbitrary, but I do not think you could call them discrete. And, according to the issues brought forward by special relativity, the supposed "same unit" would be different depending on the frame of reference, or more precisely, we could not determine the "same unit" from different frames of reference. This makes the whole idea of measuring the continuum through the means of units rather difficult. I suggest that if the application of units works well for measuring space and time, they are probably not actually continuous.

The speculation which is the reverse of yours is that continuity is what is artificial. The continuum is something created by human minds, and physical existence contains no such continuity. Problems such as Zeno's paradoxes arise because we apply principles of continuity to a physical world which is discontinuous.

"Pierre Gassendi in the early 17th century mentioned Zeno’s paradoxes as the reason to claim that the world’s atoms must not be infinitely divisible. Pierre Bayle’s 1696 article on Zeno drew the skeptical conclusion that, for the reasons given by Zeno, the concept of space is contradictory." Internet Encyclopedia of Philosophy

I think Bayle's conclusion is the only one that is eventually reached by logic on this. The great mathematician Alfred Whitehead write on the subject of "wholes and parts" and yet Wikipedia said some of his results appear to be wrong. This subject is a tangle which however allows you to view it with intuition instead of logic and get something out of it instead of eternal frustration. I have no doubt no one will find "THE answer" to this riddle. Yet it can be accepted as a trick played on us by the fathomless

As I have asked you several times, is this a mathematical or a physical scenario? — fishfry

I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness.

In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now. — fishfry

Are you referring to mathematical space or physical space? I assume in this context you're referring to physical space, right? Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?

Mathematically there is no question that she passes through every point indexed by a real number....If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. — fishfry

A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]?

One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas?

Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too. — fishfry

Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.)

That is what I mean when I say that the whole is real and the parts are entia rationis, creations of thought. — aletheist

No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement — aletheist

Yay!

The parts are always there! — Gregory

If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing?

If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing? — keystone

No. The zeros (points) is approached by infinite space that is also finite. That is the only way to put this question honestly. "In the early 19th century, Hegel suggested that Zeno’s paradoxes supported his
view that reality is inherently contradictory." Internet Encyclopedia of Philosophy

It's said everywhere that modern math found the answer to this paradox, to which I use words of Heraclitus "Hearing they do not understand, like the deaf. Of them does the saying bear witness: 'present, they are absent'" I say this only because most people don't understand what the paradox means so there "explanations" completely miss the mark

Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed. Not so long ago some writer came out with his "solution" and got a lot of praise until everyone realized that it was Henri Bergson's answer that had plagiarized and it was not a true solution to the problem to begin with

I mean no offense

I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness. — keystone

Ok. Then for clarity let me add another important condition. The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy.

Are you referring to mathematical space or physical space? — keystone

By space I mean physical space. And by physical space I sometimes mean "true" physical space as in ultimate reality, if there even is such a thing; and other times I mean the theories of contemporary physics.

I assume in this context you're referring to physical space, right? — keystone

Yes.

Is it fair to assume that you believe that mathematical space can be modelled with the real numbers? — keystone

No, I'd be very surprised if this turns out to be true. The mathematical real numbers are far too strange to be real in the sense of physical reality.

A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]? — keystone

There are no computers involved in this. Computers are far too limited, since Turing machines are restreicte to finite sequences of instructions and finitely many steps. A Turing machine already has an unbounded tape (memory) and speed of execution is not a factor at all. So this is a bad model of mathematics. For example, no computer program can approximate or generate the decimal digits of a noncomputable number. So I reject this idea totally. The set of computable numbers is a countably infinite proper subset of all real numbers, so they are missing a lot.

A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point. If one is a constructivist this becomes false, but the constructivists patch the problem by restricting attention to computable functions to make it true again.

One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas? — keystone

Your idea is totally invalid. Computers, even theoretical abstract ones, are not relevant. What can be computed is a tiny subset of what can be mathematically proved to exist. As I noted, Turing machines can only generate the countably infinite set of computable real numbers. The noncomputable reals, of which there are far far more, are entirely beyond the range of computation.

The concept of computable real numbers was first elucidated by Turing himself. His great 1936 paper is called, "On Computable Numbers, with an Application to the Entscheidungsproblem." Never mind what the Entscheidungsproblem is. What's important is that it was Turing himself who first pointed out this profound limitation in the ability of computers to represent or characterize real numbers. The exact limitation your simulation or computation idea is bumping into. If we restrict ourselves to only numbers that are computable, then you CAN drive a truck through the real number line, because the computable real line is full of holes. [A skinny truck of course, the width of a point].

In fact the computable numbers, just like the rationals, are infinitely divisible but not complete.

Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.) — keystone

Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were.

No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. — aletheist

I have never done that, so I'm happy to see that I am therefore not the target of that particular criticism.

I interpret it instead as a limitation on the precision of measurement, or as Wikipedia puts it, "the minimum distance that can be explored. ... — aletheist

It is not exactly a measurement problem. The fact that I can't measure exactly one meter with a meter stick is the problem of the approximation of measurement.

The Planck scale represents the point at which contemporary physics breaks down and is not applicable; so that we can not rationally discuss or compute what happens below those lengths, durations, and energies. That's subtly different than just the approximateness of measurement.

The Planck length is sometimes misconceived as the minimum length of space-time, — aletheist

Not by me and frankly never by me.

but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry." — aletheist

We're perfectly in agreement on this point except that my physics is not strong enough to catch the reference to Lorentz symmetry. I found a physics.SE thread linking the terms but didn't read it. Should I?

I did not see your PS until now, but I am well aware that the logic of SIA is what has come to be known as constructive or intuitionistic. — aletheist

Ok. Then it's fair for me to note that I already mentioned constructivism in replies to @keystone and can't personally do anything about the legion of neo-intuitionists running about these days, though I would if I could :-)

Peirce was skeptical of excluded middle, but for very different philosophical reasons than Brouwer and Heyting--he believed that reality itself does not conform to it, because it is fundamentally continuous and general, rather than discrete and individual. He stated this in slightly different ways in alternate drafts of the same text. — aletheist

That's interesting. The differences among the LEM-deniers are too subtle for me to appreciate. But then again I don't have much interest in reality when it comes to math. I don't feel a need to justify math in the name of reality. And I can always fall back on history: complex numbers, non-Euclidean geometry etc., to show that no matter how weird math gets, people often find a use for it. I am not defending math as any kind of model of reality any more than I would defend the game of chess on that basis. I'm immune to criticisms based on anyone's idea of what's real because I don't think math is particularly real. On my formalist days at least. The rest of the time I take a Platonist view. I have no hard convictions in this regard.

To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it.
— Peirce (NEM 3:758, 1893) — aletheist

Yes but does he deny the way the knight moves in chess? LEM can be taken as a rule in a formal game. I would never defend LEM by saying the world works that way. How could I pretend to know such a thing? Peirce could be 100% right yet LEM-based math is still an entertaining pastime. I never speak of the absolute state of things. You see you are trying to get me to take the other side of a question I reject entirely. I don't say, "LEM-based math is right because the world works that way." I would never say such a thing. I'm not making any claims about the world, nor about LEM-based math's applicability to the world. I say, "LEM-based math is interesting in its own right, and if the arc of history is turning against it via the neo-intuitionists, so be it." Sort of like how Cubs fans must have felt about night games [Chicago's Wrigley Field only got lights as recently as 1988].

You say that standard math doesn't apply to the "true continuum." I say to you, "Yes I certainly agree. And by the way, what do you mean by true continuum." That's the conversation I believe I'm trying to have. I'm not claiming knowledge of the world or staking out any metaphysical position.

No doubt there is an assumption involved in speaking of the actual state of things ... namely, the assumption that reality is so determinate as to verify or falsify every possible proposition. This is called the principle of excluded middle. ... I do not believe it is strictly true.
— Peirce (NEM 3:759-760, 1893) — aletheist

I have no argument or complaint. He may be right for all I know. That doesn't bear on any point I'm making. You are arguing against multiple strawmen positions I'm not taking.

The Planck scale represents the point at which contemporary physics breaks down and is not applicable — fishfry

I agree, this is a better concise summary of what it means.

We're perfectly in agreement on this point except that my physics is not strong enough to catch the reference to Lorentz symmetry. — fishfry

Heh, we are in the same boat on that, I was just quoting Wikipedia. :cool:

I don't feel a need to justify math in the name of reality. — fishfry

Oh, I completely agree. Again, mathematics is the science of drawing necessary conclusions about strictly hypothetical states of things. Whether those premisses match up with reality is a matter of metaphysics, not mathematics. They can be just about anything imaginable, although some of the most interesting cases come about when we remove a previously taken-for-granted axiom like the parallel postulate in geometry or excluded middle in logic, but still manage to come up with a consistent and useful system.

You see you are trying to get me to take the other side of a question I reject entirely. — fishfry

No, that was not my intention, and I am sorry that I came across that way. I was just trying to provide more background about my own position.

You say that standard math doesn't apply to the "true continuum." I say to you, "Yes I certainly agree. And by the way, what do you mean by true continuum." That's the conversation I believe I'm trying to have. — fishfry

I can only cover so much ground in this format. My long answer is the paper that I provided.

You are arguing against multiple strawmen positions I'm not taking. — fishfry

Again, I apologize for giving you that impression.

There is no absolute infinity. Note the equation --

infinite = 1/infinity

Try to cross multiply

There is only "can increase without limit" but the limit is never reached.

The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy. — fishfry

I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions.

No, I'd be very surprised if this turns out to be true. The mathematical real numbers are far too strange to be real in the sense of physical reality. — fishfry

Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?" By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question.

There are no computers involved in this. Computers are far too limited...So I reject this idea totally. — fishfry

I wasn't granting you a Turing machine, I was granting you an infinite computer with no restrictions. But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1? As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it?

A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point. — fishfry

Your statement/position implies that all that exist between a and b are points. The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: $(0,\frac{1}{3}), (\frac{1}{3},\frac{1}{2}), (\frac{1}{2},\frac{3}{4}), \text{and} (\frac{3}{4},1)$. We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions.

Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were. — fishfry

Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon.

The zeros (points) is approached by infinite space that is also finite. — Gregory

I don't understand this statement.

Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed. — Gregory

I am probably wrong, but your statement isn't proof of that. Like I said to fryfish, my view has neither been formalized or proven correct. If I provided a formal theory and proved it correct then that would certainly be newsworthy, but as it stands I'm just looking for flaws in my ideas.

I agree, this is a better concise summary of what it means. — aletheist

Ok. My point, which I sadly forgot to make, is that even with the clarified fact that the Planck scale is the scale below which contemporary physics can't be applied, your position -- that the physical world embodies or instantiates or contains or is a "true continuum" -- is not supported by contemporary physics. And I asked you what a true continuum is, and how you'd know one if you saw it. And how exactly would you see it? Good questions all.

Heh, we are in the same boat on that, I was just quoting Wikipedia. — aletheist

Yes but even with the clarified definition of Planck scale my point still applies. That your belief that there exists a "true continuum" is incompatible with contemporary physics. If I said inconsistent that was too strong. I should have said, "not supported by." That would be more accurate.

Oh, I completely agree. Again, mathematics is the science of drawing necessary conclusions about strictly hypothetical states of things. Whether those premisses match up with reality is a matter of metaphysics, not mathematics. They can be just about anything imaginable, although some of the most interesting cases come about when we remove a previously taken-for-granted axiom like the parallel postulate in geometry or excluded middle in logic, but still manage to come up with a consistent and useful system. — aletheist

Ok. I still want to know, and think there could be an interesting discussion around, your claim that there exists a "true continuum." You said math doesn't model it, as if there even is any such thing to be modeled.

No, that was not my intention, and I am sorry that I came across that way. I was just trying to provide more background about my own position. — aletheist

Ok, but I still don't understand your position. You claimed that math doesn't model the "true continuum." I have two questions. What is a true continuum, and what makes you think any such thing exists in the physical world?

I can only cover so much ground in this format. My long answer is the paper that I provided. — aletheist

Can I get a short answer? What is a true continuum and what makes you think such a thing exists in the physical world, such that the question of whether it's accurately modeled by math is a meaningful question? After all, we can't ask if math models the tooth fairy. And I claim the tooth fair is in the same category of things as the "true continuum" -- a fairytale. Except that I can't trade the continuum a tooth for a quarter

Again, I apologize for giving you that impression. — aletheist

No prob. I'm still curious about what's a true continuum and why you think there is such a thing.

I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions. — keystone

I'm just clarifying that we often take "infinitely divisible" as a mathematical continuum, but this is very loose speech. The actual condition required is completeness in the metric sense.

Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?" — keystone

Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification.

By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question. — keystone

What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc.

I wasn't granting you a Turing machine, — keystone

If you use the word computation it's a Turing machine by default unless you explicitly say otherwise.

I was granting you an infinite computer with no restrictions. — keystone

I wish you'd take the trouble to read what you yourself wrote. You said a computer (or simulation) with infinite memory. A Turing machine already has unbounded memory so you haven't added anything without supplying further qualification. Which you didn't supply.

But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1? — keystone

In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as @aletheist noted.

As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it? — keystone

f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this.

Your statement/position implies that all that exist between a and b are points. — keystone

On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific.

The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: (0,13),(13,12),(12,34),and(34,1)(0,13),(13,12),(12,34),and(34,1). We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions. — keystone

If I go out to the nearby highway and remove all the mile markers and road signs does that make them disappear? You're confusing labeling with existence. If I remove the label from a can of soup the can still contains soup. You're making a very disingenuous point.

I find your claim silly and not at all a serious argument or position.

Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon. — keystone

Argumentum ad mulletus.

... your position -- that the physical world embodies or instantiates or contains or is a "true continuum" -- is not supported by contemporary physics. — fishfry

That is not an accurate statement of my position. I hold that space is a true continuum, but not that it is something physical; rather, it is the real medium within which everything physical exists. Ditto for time, albeit in a different respect (obviously).

And I asked you what a true continuum is, and how you'd know one if you saw it. And how exactly would you see it? — fishfry

We all perceive space and time, and some of us formulate the hypothesis that each is truly continuous in a way that no collection of numbers, even an uncountably infinite one, could ever fully capture because of their intrinsic discreteness. Nevertheless, this does not preclude the real numbers (for example) from serving as an extremely useful model of continuity for the vast majority of practical purposes.

You said math doesn't model it, as if there even is any such thing to be modeled. — fishfry

I have said before, and I just said again, that the real numbers do very successfully model a continuum. They just do not constitute a true continuum. That requires a different mathematical conceptualization, and smooth infinitesimal analysis turns out to be a promising candidate.

Can I get a short answer? — fishfry

I gave it a shot, hope it helps.

That is not an accurate statement of my position. I hold that space is a true continuum, but not that it is something physical; rather, it is the real medium within which everything physical exists. Ditto for time, albeit in a different respect (obviously). — aletheist

Hmmmm. The way I'm hearing this, and correct me if I'm misunderstanding you, is that there's an absolute space against which everything else happens. A fixed, universal frame of reference. I hope you can see the problem there. And if that's not what you mean, then I don't understand what you're saying.

After all in general relativity spacetime is a manifold. It's like a twisted space without the ambient space. Like a globe, the earth say, embedded in Euclidean 3-space ... except there is no ambient space. There's only the globe, with its intrinsic curvature.

And again you have used the phrase true continuum, which you haven't defined.

I do appreciate that you've put something on the table that I can at least begin to ask questions about.

We all perceive space and time, and some of us formulate the hypothesis that each is truly continuous in a way that no collection of numbers, even an uncountably infinite one, could ever fully capture because of their intrinsic discreteness. — aletheist

I understand that you mean that the real numbers are composed of individuals, namely the real numbers. But the real numbers are not a discrete set. The integers are a discrete set, because around each integer you can draw a little circle that doesn't contain any other integers. You can't do that with the real numbers. So I get that Peircians don't like the fact that the reals are the union of their singleton points. But I don't see how you can label some alternative model "true" without evidence.

Nevertheless, this does not preclude the real numbers (for example) from serving as an extremely useful model of continuity for the vast majority of practical purposes. — aletheist

Yes of course, we agree on that. And for my part I go farther. The reals are fun to study and if the physicists find them useful that's great because it means the universities will continue to fund the math department. Else they'd fire the whole lot. But the reals are far too weird to represent anything real. If nothing else, the reals as we understand them depend on the vagaries of which set-theoretic axioms we choose. If anyone thought the real numbers were "really real" then physics postdocs would get grants to count the number of points in a meter in order to determine whether the continuum hypothesis is true. Since nobody has applied for any such grant, I take that as evidence that nobody takes the real numbers seriously as an accurate model of anything physical. They're only an abstract mathematical model, one with deeply strange properties.

I have said before, and I just said again, that the real numbers do very successfully model a continuum. — aletheist

Ok.

They just do not constitute a true continuum. — aletheist

LOL. As Reagan said to Jimmy Carter, "There you go again." I wish you would define a true continuum.

That requires a different mathematical conceptualization, and smooth infinitesimal analysis turns out to be a promising candidate. — aletheist

Brouwer's revenge. The intuitionists are back with a vengeance. I don't doubt the historical momentum. It's the influence of the computers.

I gave it a shot, hope it helps. — aletheist

Yes, you did give me specific things to be unclear about and to push back on. Is my end of the conversation hopeless unless I read Bell and Peirce?

But I do think you need to explain yourself about this mythical background space in your first paragraph, which sounds suspiciously like the luminiferous aether, whose existence was disproved by the famous Michelson-Morly experiment, leading Einstein to special and then general relativity. There is no preferred frame of reference in the universe according to modern physics. There is no fixed background against which all other physical things can be measured.

pd -- Reading your paper, with much eye-glazing I'm ashamed to say. I did come across this:

"...he restates the second and third properties of Time as a continuum: any lapse can be made up of two lapses that have a common instant between them, although it need not "have a final and an initial instant." For example, the present is "assignable" as the "limiting instant" between the past, which has no initial instant, and the future, which has no final instant."

This sounds suspiciously like the idea of a Dedekind cut. Except that there's a point missing, as in the union of the open intervals (0,1) and (1,2). Am I understanding that right?

The way I'm hearing this, and correct me if I'm misunderstanding you, is that there's an absolute space against which everything else happens. A fixed, universal frame of reference. — fishfry

No, that is not what I am saying. I am not really talking about physics at all, just a hypothetical/mathematical conceptualization that might have phenomenological and metaphysical applications.

Brouwer's revenge. The intuitionists are back with a vengeance. I don't doubt the historical momentum. It's the influence of the computers. — fishfry

Peirce came before Brouwer, and my interest in SIA/SDG has nothing to do with intuitionism or computers. If Peirce had followed through on his skepticism of excluded middle and omitted what we now (ironically) call "Peirce's Law" from his 1885 axiomatization of classical logic, then he would have effectively invented what we now (unfortunately) call "intuitionistic logic" and it might be known instead as "synechistic logic"; i.e., the logic of continuity.

Is my end of the conversation hopeless unless I read Bell and Peirce? — fishfry

Maybe not hopeless, but I suspect that there is a "curse of knowledge" aspect here on my part, given my immersion over the last few years in Peirce's writings and the secondary literature that they have prompted.

Reading your paper, with much eye-glazing I'm ashamed to say. — fishfry

Thanks for the attempt, sorry for the resulting effect.

This sounds suspiciously like the idea of a Dedekind cut. Except that there's a point missing, as in the union of the open intervals (0,1) and (1,2). Am I understanding that right? — fishfry

Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end."

I'm wondering just what this thread is about. It appears to be about systems of ideas and their rules, seeming conflicts between systems, and a great confusion about any contact these systems have with the world itself. Some being useful, seemingly even necessary, for understanding the world; and some mistakes and mistaken about that applicability. The former being mathematics, the latter a collection of eccentricities or mistakes about mathematical ideas.

Zeno's paradoxes being an example of such mistakes. Atalanta gets started, Achilleus gets from A to B and passes the tortoise on the way, and the arrow flies; and these the incontrovertible facts of the matter. Any description that fails to acknowledge these is just at least by that amount nonsense. In my opinion Zeno was merely demonstrating how a lack of care and even rigor in description could lead to nonsense. I imagine he would be astonished that anyone would be confused by his paradoxes and mistakenly take them for other than the academic exercises that they are.

In this thread I find @jgill and @fishfry, who I believe are or were professional mathematicians, with patience and forbearance returning again and again to the math to hold off and correct error and misunderstanding. For that they get a lot of points. And the folks who poke at them might take a moment to consider were they would be if they did not have them to poke at.

I'll end this rant with an observation on Pierce's views on the law of the excluded middle (LEM) as presented and interpreted above:

To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it.
— Peirce (NEM 3:758, 1893)
No doubt there is an assumption involved in speaking of the actual state of things ... namely, the assumption that reality is so determinate as to verify or falsify every possible proposition. This is called the principle of excluded middle. ... I do not believe it is strictly true.
— Peirce (NEM 3:759-760, 1893) — aletheist

"
What Piece questions is not the LEM, but instead the applicability of it as referenced. To be sure, he calls it the "principle of the excluded middle," and in my opinion the substitution of "principle" for "law" makes all the difference.

In sum, mathematics and my conception of Wanda the purple six-headed hippopotamus in tutu and ballet slippers have exactly the same ontological status: they're both ideas, mental constructs. Neither having anything whatever to do with reality, the world, except as mediated by applicability, and Wanda very little of that.

Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification. — fishfry

Okay, that makes sense.

What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc. — fishfry

I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system.

If you use the word computation it's a Turing machine by default unless you explicitly say otherwise. — fishfry

Okay, my mistake.

In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as aletheist noted. — fishfry

I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence.

f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this. — fishfry

What I'm trying to convey is that no matter where Atalanta's mathematical universe lives (whether in an infinite computer or the mind of God) it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. The only way to build her universe is to deconstruct it from a continuum, working your way down from the big picture to specific instants.

When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant. And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible.

I'm sure you agree with the above paragraph (and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) but my point is that knowing a function doesn't imply that we can describe it completely using points. Any attempt to do so would be akin to listing the real numbers.
 

On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific. — fishfry

I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points. If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum. So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.

You're confusing labeling with existence....I find your claim silly and not at all a serious argument or position. — fishfry

If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them). Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument. My view is only silly when seen from a point-based view because you assume that all we can talk about are actual objects...an infinite number of them.

Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end." — aletheist

We don't have two adjacent portions before the marker. The marker is a pointer that demarcates but does not split the line thereby making it discontinuous. The marker is not part of the line. A materialist overlay is unnecessary in math. It's cheaper to think of it as an abstract pointer anywhere to an abstract endless line in one dimensional space.