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. — aletheist

Ok not physics. Thanks for the clarification, I'm sure you can see that this idea would not hold up as physics.

But if it's a "hypothetical conceptualization," how can you claim with a straight face that the standard real line doesn't model the "true continuum?" I do understand Peirce's point that the real line isn't a continuum because it's made up of individual points. But I am objecting to your claim that it's meaningful to say whether something is or isn't a good model of an abstract idea. What if my true continuum isn't the same as yours? Just as you can't say whether set theory is a good model of the tooth fairy. One conceptual fiction's like another. You have a vague idea (ok perhaps not vague to you) of a true continuum, and you're saying the real line isn't it. How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum?

Peirce came before Brouwer, and my interest in SIA/SDG has nothing to do with intuitionism or computers. — aletheist

Ok, yes you pointed that out to me earlier. Was Brouwer familiar with Peirce? It's interesting that these ideas were already floating around.

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. — aletheist

I agree with you that Peirce should be more famous. Didn't realize he pioneered LEM rejection.

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. — aletheist

I wish I could dispatch a clone to read up on Peirce. It's on my to-do list, none of which ever gets done.

Thanks for the attempt, sorry for the resulting effect. — aletheist

Most philosophical papers glaze my eyes. Some I find very clear, but many confuse me. Not just yours. I'll take another run at it in light of this interesting conversation. But I must admit that given a finite block of time, I'd be more likely to spend it trying to learn some new standard math rather than philosophy. I find philosophy very hard to grab on to.

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. — aletheist

Is this @keystone's point about points not existing on the number line till we mark them with labels? Sounds similar. Surely Peirce must have been familiar with Dedekind. Dedekind cuts are a clever idea, because we can logically construct the reals given only the rationals. With Peirce's description, I don't know what we've got. Philosophy versus math again. Perhaps I shouldn't even try to talk about philosophy.

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

Grrrr. That makes no sense. If we divide the rationals into two classes, those whose square is less than 2 and those whose square is greater than 2, we can define sqrt(2) as that exact pair of classes of rationals. That's Dedekind's idea. Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set? Having trouble with this. How can one point become two points? That's a more mysterious trick than Banach-Tarski.

In this thread I find jgill and @fishfry, who I believe are or were professional mathematicians, — tim wood

@jgill is a professional mathematician. I don't say was, because it's not the kind of status one loses by virtue of being retired.

I am a failed math grad student. My own zeal is that of the fallen priest (I'm thinking Richard Burton in Night of the Iguana and probably some other roles too). Nobody has missionary zeal as strong as one who has been cast out of the order they strove to join.

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. — keystone

I've taken and taught calculus and have no idea what the "geometrical objects of extension" are. The objects of the Cartesian coordinate system are ordered pairs of real numbers; or in the general case, ordered n-tuples. The pictures are for intuition and visualization. The actual objects of study are analytic. Just real numbers and ordered lists (in computer parlance) of same.

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. — keystone

Distinction without a difference. You can think of the real line as a set of points or as a set of locations or addresses. Just like a street is a collection of houses or it's a collection of addresses where there might be houses or there might be empty lots. Except that by the completeness of the real numbers, there are no empty lots. But if you want to think of real numbers as locations on a line, that's perfectly ok.

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) — keystone

Computers are too limited and the mind of God is too expansive. The mathematical universe lives in the world of symbolic math.

it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. — keystone

I don't have to name all the people in China to know there are a billion of them.

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. — keystone

That's just not true mathematically.

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. — keystone

See, you DO believe in the intermediate value theorem.

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. — keystone

Right. It's unnecessary.

I'm sure you agree with the above paragraph — keystone

Yes I do, but it's trivial and doesn't support your point.

(and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) — keystone

Why? Believe me when I get offended around here I let the offender know about it.

but my point is that knowing a function doesn't imply that we can describe it completely using points. — keystone

On the contrary. A function is a collection of individual ordered pairs. That's the set-theoretic definition of a function. It's not the same as the path of a moving point as it was for Newton. But he was doing physics when he had that viewpoint.

Any attempt to do so would be akin to listing the real numbers. — keystone

No. Not true. To say that a function is a collection of ordered pairs does not mean we are required to explicitly list them.
 

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. — keystone

You can think of real numbers as locations on the real line if you like. Locations or addresses. But the completeness of the real numbers means there is a point at every location.

But more to the point, "point" is just another name for a real number. The set of real numbers is the collection of all the real numbers and vice versa. You are thinking "points" are things separate from real numbers, but mathematically they are not. Or in n-space, points are n-tuples of real numbers.

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. — keystone

There are no such things as an actual or potential point. There are only real numbers and n-tuples of real numbers.

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.]/quote]

Nonsense. You keep repeating this and I keep calling it nonsense (last time I called it silly) but I'll soon run out of adjectives and also of patience. This isn't going anywhere. I disagree with your view and don't find there to be any meaningful content in it.


— keystone

If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. — keystone

No no no no no I did not. @aletheist said that this was Peirce's point of view and I noted that this seemed similar to yours. I admit that when Peirce says it, I say, "Peirce, interesting guy. I wish I knew more about him." And when you say it, I say, "Nonsense." But in either case I do think the idea is mathematical nonsense. Philosophically I have no idea what you or Peirce could possibly mean by this and I emphatically deny ever giving the slightest consideration to the idea. It's wrong.

I do know that the intuitionists try to tie existence to human cognition, and this point I also disagree with despite there being many smart people on board with the idea.

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). — keystone

Ok. I can't keep disagreeing with the thing you keep repeating over and over. I disagree strongly with what you said here.

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. — keystone

I understand that. And I am totally serious when I say this idea is nonsense utterly devoid of content. Although when @aletheist tells me Peirce said it, I scratch my chin and go, "Hmmm, that Peirce sure was an interesting guy." But what I think to myself is, "Nonsense."

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. — keystone

I don't see that either of us has said anything new in a long time.

I don't see that either of us has said anything new in a long time. — fishfry

I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks!

I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks! — keystone

No prob, likewise. After all when Peirce says the same thing I go, "Hmmmm I need to learn more." So maybe there's something to it. Who's to say.

The solution to Zeno that has been proposed here is that parts appear on the geometric item only when noticed. Did not Parmenides say thought is being? Is his philosophy not early Greek idealism? Are not these new QM ideas just modern idealism? I have accepted that unbounded space is identical to bounded space not just because its a bigger idea than these other opinions, but because it's true

I do understand Peirce's point that the real line isn't a continuum because it's made up of individual points. — fishfry

I appreciate this and would be content to leave it at that.

How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum? — fishfry

I guess it comes down to the meaning of the concept of continuity. Someone immersed in modern mathematics, where the real numbers are routinely called a continuum, is understandably satisfied with that definition. Someone like Peirce who objects to finding any discrete parts whatsoever in something that is supposed to be continuous can never accept it. He was motivated primarily by logical considerations rather than mathematical ones.

Was Brouwer familiar with Peirce? — fishfry

According to a paper by Conor Mayo-Wilson, "Peirce and Brouwer seemed to have no knowledge of each other's work." However, they were indirectly linked through Lady Victoria Welby, with whom Peirce exchanged a fair amount of correspondence including some of his most important writings about semeiotic, and whose ideas about significs were later adopted by a group of Dutch thinkers that eventually included Brouwer.

Surely Peirce must have been familiar with Dedekind. — fishfry

Yes, Dedekind's name appears in a bunch of his writings, and his most fundamental disagreement with him was about the relationship between mathematics and logic. For Peirce, logic (generalized as semeiotic) depends on mathematics, as does every other positive science; while for Dedekind, mathematics is a branch of logic.

Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set? — fishfry

Peirce would not talk about "sets"--or "collections," his usual term--when referring to a continuum at all. By definition, a collection consists of discrete parts, which are ontologically prior to the whole ("bottom-up"); while in a continuum, the whole is ontologically prior to the parts ("top-down").

How can one point become two points? — fishfry

Because the only points at all are the ones that we create by marking them. When we mark a line without separating it, we create one point. When we separate the line, we create two points, one at the discontinuous end of each resulting portion. When we put them back together, we have only one point again. The points are never parts of the line itself, because they are of lower dimensionality. Every part of a line is one-dimensional, but a point is dimensionless. SIA seeks to capture this with its infinitesimal segments that are long enough to have "direction" but too short to be curved.

What Peirce 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. — tim wood

Indeed, I believe that his use of "principle" rather than "law" for excluded middle is very deliberate. As he wrote elsewhere, "Logic requires us, with reference to each question we have in hand, to hope some definite answer to it may be true. That hope with reference to each case as it comes up is, by a saltus [leap], stated by logicians as a law concerning all cases, namely, the law of excluded middle. This law amounts to saying that the universe has a perfect reality."

Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is definite (so that no assertion can be both true and false of it), individual (so that any assertion is either true or false of it), and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)."

and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)." — aletheist

This is strange. Do you understand it? Because I do not. Try reading it closely and see if it doesn't begin to seem to you that the writer is confused about his subject.

classical logic is strictly applicable only — aletheist

in the land of Ps and Qs, that being the place where the game of logic is played by certain rules. Therein truth or falsity determined exactly by those rules, that is to say, recognized by people who follow the rules. Even God can play if he likes, but even he must obey the rules. But what (exactly) does this have to do with the world?

Edit: And I see "logic requires us." Logic does no such thing, nor can. Hmm.

In philosophical terms, since all objects are spatial and subject to one or another type of geometry, we have to say that objects are finite in form and infinite in content. To be perfectly honest when contemplating a ball or a cup will lead to this conclusion. There are, I admit, many types of geometry, and if someone finds a way to explain "the spatial" in a way that is comprehensive and avoids paradox, I am all ears. (I like how non-Euclidean geometry is on an infinite curve that revolves back into compactness. The weirdness of it gives me a faint hope that Zeno's paradox could be solved, but the final result might be way over my head)

gives me a faint hope that Zeno's paradox could be solved, — Gregory

Just for the heck of it, what, exactly and precisely, is your difficulty with Zeno? In a well-crafted sentence or two or three. I ask because I find no problem therein. Not because I'm clever, but because I've encountered the reasoning of others.

Infinite subdivisions imply an infinite within the finite. How these opposites coincide is the issue

Good, and ty! Now, where's the infinity?

If you can forever take a part from a solid and there always be a part remaining, it is not fully finite but instead has an aspect of infinity. Common sense says something should be either finite or infinite but not both

The ancient Chinese knew about this: "One of the few surviving lines from the school, 'a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted,' resembles Zeno's paradoxes."
https://en.wikipedia.org/wiki/School_of_Names

My point is that the infinity is a matter mind and not of substance. Zeno's paradoxes at least make clear the hazards of not clearly distinguishing the two. It's a species of map and territory. The arrow flies. Zeno says it does not and cannot. Now you answer: does it fly? Then, where, exactly, is the error in Zeno's description that says it does not/cannot? Plainly somewhere in Zeno's description. And it can live there because descriptions aren't real in the sense that arrows are real. Descriptions enjoy the freedom of the mind that can conceive of things that in the real world of arrows cannot be. And do you think Zeno was confused about any of this?

The ancient Chinese new about this: "One of the few surviving lines from the school, 'a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted,' resembles Zeno's paradoxes." — Gregory

A critical difference: in Zeno everything is in motion. With the stick, not so. As it happens, there is a physical limit to how many times the stick can be halved. But were it possible to take a half per day without any limit - that being a mathematical and thereby not real sense - then the stick would indeed last forever.

Zeno added motion to the question of infinite divisibility to make the question even more difficult. I think mathematics does apply to objects in that two haves will have the same volume when united. I see no reason why these processes of division can't go on forever in real objects, although I recognize that objects are finite. Hence the paradox: "the infinite" WITHIN "the finite". Infinite content

I think mathematics does apply to objects — Gregory

Can apply to objects. Not to be confused with being an object.

The infinite (as idea) is not within the finite (as object). And in fact has nothing directly to do with any object.

Can God divide a soccer ball into infinite parts?

I recognize that our thoughts are imperfect in this regard. But I think they are interesting because they lead to either Parmenides's speculations or to the ever-changing "fire" of Heraclitus. All three thinkers spoke of very ancient ideas, and Zeno leads to one of the other two

Can God divide a soccer ball into infinite parts? — Gregory

My idea of God can divide my idea of a soccer ball into my idea of lots of parts. What does that have to do with real soccer balls.

Further, it seems to me that many problems arise from, not trying to model reality in ideas, but trying to find reality in ideas.

Matter is clunky and awkward. Abstractions, not so.

If matter does not perfectly conform to geometry, then this alone is the answer to infinite divisibility problem. I am more likely to accept a contradiction than accept that objects only exist when perceived, as others have said on this thread. Anyway, maybe I should read more Wittgenstein

I've been scanning and reading Bell's book that you kindly linked. Although I've been a math person for many years I've been concerned only with certain areas of classical complex analysis and a few abstract vector spaces. So, when I read that in smooth infinitesimal analysis (SIA) all functions from the reals to the reals are continuous I immediately think, what of a step function, like Heaviside function, how can that be interpreted as continuous? However, I gather that in accordance with the axioms of SIA such functions may not be considered in the first place, as they are not defined on the (augmented) reals. In other words, all functions defined in the system are continuous by definition. Am I correct? If so, then the claim that all functions from the reals to the reals are continuous is misleading. But perhaps I'm not interpreting things properly. :chin:

I like the infinitesimal line segment approach to continuity. It's how I program many of the functions I graph on the computer, although I'm using short real line segments.

Perhaps you and/or fishfry would comment on these ideas and elaborate on them to make SIA clearer. Sometimes in math very arcane subjects arise from elementary observations. Like reading a research paper in which simple motivation is left out, replaced by a series of odd looking lemmas leading to the proof of a theorem. :cool:

Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is ... real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)." — aletheist

This is strange. Do you understand it? Because I do not. Try reading it closely and see if it doesn't begin to seem to you that the writer is confused about his subject. — tim wood

It makes perfect sense to me. Again, the basic definition of real is being such as it is regardless of what anyone thinks about it. If we are talking about a fictional universe, then what is true or false of it depends entirely on what its creator decides about it, but not on what anyone else thinks about it. In Shakespeare's "Hamlet," the title character is the prince of Denmark and kills Claudius because Shakespeare says so; but no one can now truthfully claim that within the universe of that play, Hamlet is the king of Spain and spares Claudius. That is why there are objectively right and wrong answers on tests that students of English literature have to take after reading it.

And I see "logic requires us." Logic does no such thing, nor can. Hmm. — tim wood

This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. In the case of classical logic, one of those rules is excluded middle--every constituent of the universe of discourse must be treated as "individual (so that any assertion is either true or false of it)," which "amounts to saying that the universe [of discourse] has a perfect reality."

https://www.amazon.com/Towards-definitive-solution-Zenos-paradoxes/dp/B0006CXZPY/ref=sr_1_43?s=books&ie=UTF8&qid=1549513691&sr=1-43&keywords=zeno+paradox

Too bad they don't have this is book form for sale anymore

Full disclosure, I am not a mathematician, so my ability to address the details of SIA is admittedly limited.

In other words, all functions defined in the system are continuous by definition. Am I correct? — jgill

I believe so, as this seems to be simply what it means for a world to be smooth. As Bell says on p. 276, "If we think of a smooth world as a model of the natural world, then the Principle of Microstraightness guarantees not just the Principle of Continuity--that natural processes occur continuously, but also the Principle of Microuniformity, namely, the assertion that any such process may be considered as taking place at a constant rate over any sufficiently small period of time." For me this is reminiscent of the following passage.

Accepting the common-sense notion [of time], then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. — C. S. Peirce

A step function obviously has a discontinuity that violates this requirement. Of course, the "false continuity which the calculus recognizes" is that of the real numbers, which Peirce elsewhere calls a "pseudo-continuum."

so that what is true or false of it is independent of any judgment of man or men, — aletheist

How can this be when the "true or false of it" is exactly dependent on people and not otherwise?

This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. — aletheist

How do you characterize classical logic? Are you caviling about "game"? It's a way of doing certain well-defined things according to well-defined rules. That's not "my characterization." It's a fair statement of what classical logic is. Accept it, refine it, or correct it.

How can this be when the "true or false of it" is exactly dependent on people and not otherwise? — tim wood

I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it.

How do you characterize classical logic? — tim wood

It codifies how we can properly draw necessary conclusions about states of things that are definite, thus conforming to non-contradiction; individual, thus conforming to excluded middle; and real, in the sense that they are such as they are regardless of what we think about them.

What is true or false of a fictional universe is only dependent on what its creator thinks about it. — aletheist

No. What he says about it.

so that what is true or false of it is independent of any judgment of man or men, — aletheist

How can this be when the "true or false of it" is exactly dependent on people and not otherwise?
— tim wood
I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it. — aletheist

I do not know why you resort to fictional universes. Indeed, if it's a fictional universe, then no proposition about it is true - except the proposition that it is a fictional universe. But you have evaded the point. Truth and falsity are assigned to propositions. If no propositions, or, if no one to assign truth or falsity, then no truth or falsity. You need the assigner.