Okay, the many-worlds interpretation. Multiverse is another sci-fi scenario, fair enough. — Olivier5

Ok.

The Schrödinger equation ? — Olivier5

Is that a question? Make your case that "whatever can happen, will happen." If I flip infinitely many fair coins, it's possible that I never get a tail. Unlikely, but possible. The multiverse (which is what we're talking about in this case) might do something unlikely like that. And since it's a perfectly unfalsifiable notion, it's not science. You already agreed that multiverse theory is sci-fi, yet you claim to be able to make predictions about it. How do you square those two things?

Isn't he just ensuring that what 2 + 2 is equal to is being discussed with respect to a system big enough for the second theorem to apply? — bongo fury

Boolos says that he means proved by "the aid of the whole of math". My guess is that he means ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC. — TonesInDeepFreeze

My point exactly. We have people speculating on what he meant, since what he wrote was wrong.

We don't know what he meant. One must always lie a little in order to simplify, and in this case the wrong judgment was made (IMO) as to what to lie about. Einstein would say he simplified too much.

Okay so, are you going to tell us the difference between the many-worlds and the multiverse, or are you going to keep it for yourself? — Olivier5

I'll keep it between me and Wikipedia.

I never said otherwise. I said "in the multiverse, everything that can happen does happen". — Olivier5

How do you know? And who decides what "can" happen in order to make your claim true?

Thank you, for expanding. PA can't show its own consistency, but PA can be proved consistent outside itself (with other axioms) - and that's a generality that may hold for other arithmetic systems; is that the crux of the argument? — Aryamoy Mitra

Yes, exactly. The proof from ZF is trivial. ZF contains a model of PA; that is, a set in which, with the proper interpretation, every axiom of PA is true. Therefore PA is true if ZF is. And ZF is consistent if we assume the existence of an inaccessible cardinal, which itself is a model of ZF. You just keep adding axioms to push the inconsistency problem up the chain.

And now that I see that the author is the great George Boolos, I am unhappy, because I disagree with what he said. He starts: "First of all, when I say "proved", what I will mean is "proved with the aid of
the whole of math.""

I disagreed with that the first time you wrote it. It's wrong. Proved means, "proved in a given system of axioms." If you take the "whole of math," you lose the entire point of the subject. Given some axiom system we want to know what sentences it can prove. Those are the theorems. But "the whole of math?" I don't even know what that means. Do I get to include the entire hierarchy of the large cardinal axioms? Do I include the continuum hypothesis or not? I was hopelessly annoyed as soon as I read that remark.

Having read Boolos's article, I disagree with it entirely. Incompleteness is NOT about "the whole of math." It's about particular systems of axioms. I believe he's destroyed the essence of the subject in his effort to simplify it. And that's why when you quoted that paragraph, I instinctively said, "No." It's not true that "the whole of math" is either consistent OR inconsistent, nor is it subject to incompleteness. You have to say what the axioms are, then I can say if they're consistent or inconsistent or whether certain statements are independent of the axioms.

But then again I could be wrong. Perhaps he is making the more subtle point that even if I can use ZF to prove the consistency of PA, I still don't know if ZF is consistent. So maybe in the end he's right and I'm wrong. I admit to not being sure whether he's right or wrong. But I still have the right to be annoyed that he ignored the essence of the matter, which is that we are talking about particular axiom systems and not "the whole of math."

In fact if by "the whole of math" we might mean for example the set-theoretic multiverse of Joel David Hamkins, then "the whole of math" includes the models of set theory in which CH is true, and the models of set theory in which it's false; and even perhaps the possibility that PA is consistent, and the possibility that PA is inconsistent [that latter concept is not part of Hamkins's idea, I don't think].

So I just found "the whole of math" to be too imprecise to be correct. Didn't someone (Feynman? Einstein? ) write, Things should be made as simple as possible, but no simpler.

Einstein in fact. https://www.championingscience.com/2019/03/15/everything-should-be-made-as-simple-as-possible-but-no-simpler/

Boolos's article violates the spirit of that saying.

As a clarity, are you refuting the original exposition? This passage, for instance, was word-for-word sourced from another, non-technical resource. — Aryamoy Mitra

I did see the italics but I did not see a link to the source. So I couldn't tell if you were quoting someone else or quoting yourself from some other publication or forum, or just separating out your ideas into quoted form. But in that case I haven't criticized you, I've criticized whoever wrote that passage. Which was:

By the way, in case you'd like to know: yes, it can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five.' — Some Unknown Entity

I believe what they mean is this. For definiteness let's take Peano arithmetic (PA). By Gödel's second incompleteness theorem, PA can not prove its own consistency. That means PA can not prove that it can't prove that 2 + 2 = 5. Agreed so far? Then if PA can prove that it can't prove that 2 + 2 = 5, then PA must have proved its own consistency, which it can only do if it's inconsistent; and if it's inconsistent, then it can prove that 2 + 2 = 5.

Have I got that right?

Now my complaint is that you did not distinguish between "PA can prove ..." and "It can be proved ..."

Because in fact we CAN prove that PA is consistent. The easiest consistency proof is to assume Zermelo-Fraenkel set theory, ZF. In ZF we have the axiom of infinity, which gives us a model of PA. Since there's a model, PA is consistent by Gödel's completeness theorem.

There's another famous proof of the consistency of PA, by Gerhard Gentzen. As Wiki puts it:

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal $\varepsilon_0$", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.

https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

The point is that proof and consistency are relative to given axiom systems. It's true that PA can't prove its own consistency; but we CAN prove the consistency of PA by other means.

So, to sum this all up: Using ZF or Gentzen's proof, I can indeed prove that PA is consistent, and that PA can't prove that 2 + 2 = 5, and that PA can prove that it can't prove that 2 + 2 = 5.

I can always do this as long as I'm willing to go outside PA. And this is true in general. Just because some given system can't prove its own consistency doesn't mean we can't prove its consistency.

Let me know if I've missed your point. And if you'd just say where you got the quote it would be helpful.

I should add for clarity that having used ZF to prove the consistency of PA, I now have the question of whether ZF is consistent. You can push the problem around but never nail it down. Maybe after all that's what the paragraph means, but without the surrounding context it's hard to tell. I admit to being confused as to why Gentzen's proof isn't problematic in exactly the same way.

I think by now you can see why I I originally gave a one word answer. :-)

A while back someone on the site told me the technical names for what I call ontological and epistemological probability but I don't seem to remember them at the moment so I'll just use my terms.

Say you flip a coin. If we believe in Newtonian physics for macroscopic events, the result of the coin flip is absolutely determined by the initial state and exact position of the coin, the flipping force, the air pressure, local gravity (accounting for our local elevation, say), the humidity of the air, and so forth. If we only knew all of those factors, we could predict the outcome of the coin toss with perfect accuracy.

The problem is, we can't know all those things with sufficient accuracy The coin toss is random not because it's inherently random, which it isn't. The coin toss is random because we can't possibly know the factors that go into its determinate outcome. That's what I call epistemological randomness. The event is random because we don't know, or we can't know, enough about the factors that cause it to be one way or the other.

Now a moment's thought reveals that pretty much everything that we regard as probabilistic is actually of this type. It's very difficult to think of anything at all that's truly random -- ontologically random -- which I define as random not because of our ignorance of the deterministic factors, but actually deep down random.

We are hard pressed to give even a single example of an ontologically random event. Most people will fall back on quantum events. The low-order bit of the femtosecond timestamp of the next neutrino to hit your detector is random because QM says it is.

But is it? How do we know that it wasn't determined at the moment of the big bang? And that if we only knew enough, we could predict it? Or that even if we could never know enough, it's still only random by virtue of our lack of knowledge?

This gets into one of the great disputes of twentieth century physics. Is it possible there are "hidden variables," aspects of reality that we just don't know, that, if we knew them, would enable us to see the determinism lurking behind even quantum probabilities? I'm no expert on this stuff, but I understand that Bell's theorem rules out hidden variables. At least up to the limits of our present knowledge of physics.

There's a lot of physics that we don't know. We can't merge quantum physics with general relativity. We don't know what dark matter or dark energy are. There are revolutions waiting to be explicated by geniuses not yet born. It's possible that there is no such thing as randomness; and that probability is simply a measure of our ignorance of the causes of events.

https://en.wikipedia.org/wiki/Hidden-variable_theory

https://en.wikipedia.org/wiki/Bell%27s_theorem

Have I significantly misapprehended the argument — Aryamoy Mitra

Yes.

'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.' — Aryamoy Mitra

No.

You see, in the multiverse, everything that can happen does happen, — Olivier5

One, you (and probably others, I haven't read the entire thread) are confusing the multiverse with the many-worlds interpretation. They're two entirely separate things. And two, your statement is false. Just as you could flip a trillion coins and they all land on heads, there are things that might not happen even in the multiverse (which again, is NOT the many-worlds interpretation).

What if a vicious serial killer tripped on his way back from his most recent depravity and incurred a serious head injury. He is found and taken to the hospital where he lays in a coma for several months. When he awakes he has no memory of his past deeds. He recovers and spends the remainder of his life helping the poor and downtrodden. If evidence arises linking him to the crimes he committed should he be prosecuted. — Steve Leard

None other than Bill Clinton personally executed a guy just like that. Ricky Ray Rector shot a cop in Arkansas then put the gun to his own head and fired. He didn't die, but he lost enough brain function to have no memory of knowledge of what he had done. During the 1992 presidential campaign, Clinton didn't want to appear "soft on crime" so as governor of the state he insisted that Rector be executed. Clinton personally flew back to Arkansas from the campaign trail to witness the execution. Rector had so little mental function that he saved half of his last meal "to eat later."

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

But since what I am looking for is an indication that 2+1 really is the same thing as 3, — Metaphysician Undercover

I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here.

Is it made of something physical or not? — The Opposite

Must be one of those "emergent" properties I keep hearing about. I think emergence is a murky concept and doesn't answer any of the questions people think it does. But wind is emergent. If you look at air molecules you can't see that they'd make wind. If you look at the air in a room you can't see that it would make wind. But when it moves a little, it's a breeze. When it moves a lot, it's wind. When it moves a lot a lot, it's a gale.

Interesting that air has a different name depending on how fast it's moving. For example a car that moves is a moving car. There's not a different name for it.

Well this is a puzzler alright.

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.

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.

Again, what anyone knows or does not know is beside the point. Since it is fact that Henry Fonda is the father of Peter Fonda, by definition (in semeiotic) the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object. — aletheist

Thank you. My granting of @Metaphysician Undercover's point is causing me to waver on 2 + 2 = 4. I need to read more of your posts so I can strengthen my backbone. Henry Fonda IS the father of Peter and that's that. There goes the beer @Meta was going to buy me, which according to @Meta is NOT the same as, "an alcoholic drink made from yeast-fermented malt flavored with hops."
.

I have never denied this, but (in semeiotic) the information conveyed by a sign corresponds to its interpretant, not its object. If we were standing in a room with Henry Fonda--preferably back when he was alive--then we could point at him and truthfully say both "that is Henry Fonda" and "that is the father of Peter Fonda." Therefore, both signs denote the same object, despite signifying different interpretants. — aletheist

As you can see, @Meta has gotten into my head over the years. Must ... stay ... strong ...

The point is that "the father of Peter Fonda" gives different information from "Henry Fonda".
— Metaphysician Undercover
Again, I agree, the two signs signify different interpretants--i.e., convey different information--despite denoting the same object. — aletheist

Thank you for this clarification. Must remember it. Must not weaken. 2 + 2 = 4. Please don't take me to room 101.

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.

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.

Fishfry! Never in a hundred years did I think I'd see this day. Let's go, I'll buy you a beer. — Metaphysician Undercover

Did you mean a beer? Or did you mean an alcoholic drink made from yeast-fermented malt flavored with hops? According to you they're two entirely different things :-)

The point is that "the father of Peter Fonda" gives different information from "Henry Fonda". — Metaphysician Undercover

I agree with you about this point of natural language. But I still disagree regarding 2 + 2 = 4, even thoug I do see the point you're trying to make.

The latter gives nothing, just the name of a person. The first expression also denotes a person, as well as the second expression denotes a person. But the information required to conclude that they are one and the same person is not provided. Even if we add the further premise, "Henry has a son Peter", the condition of reversibility, equality, is fulfilled, but we still cannot conclude that they denote the same person. There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound. — Metaphysician Undercover

That latter is a bit disingenuous. If I say Socrates is a Greek philosopher, someone might object because they think I might have meant Socrates the cat philosopher. That's not really a good objection, if you fully qualified everything there would be no end to it.

... must find that they ought to reduce suffering in the world if they are to feel good with themselves — Shawn

Ah, so the purpose of reducing suffering is so that I can feel better about myself? What if I could feel better about myself by increasing suffering? I think this statement can't be right.

We could say that everyone has a moral duty to reduce the suffering of others. I could easily give lots of counterexamples and corner cases, but at least the basic claim is that I have a duty to reduce suffering. The way you put it, the only reason I would want to reduce suffering is to feel better about myself. That's what virtue signaling is all about. I act in such a way as to convince others that I am a moral person, even if my actions actually make the situation worse. Like opening the border, which only serves to increase human misery and strengthen the cartels, but which allows proponents to feel better about themselves.

I agree that they signify different interpretants, but this does not preclude them from denoting the same object. It is a fact that Henry Fonda is the father of Peter Fonda, so by definition, it is also a fact that the signs "Henry Fonda" and "the father of Peter Fonda" both denote the same object. Someone who does not know the first fact would not know the second fact either, but that is irrelevant to their being facts. — aletheist

I commend you for fighting the good fight against @Metaphysician Undercover. But here I find myself inclined to see his side of it. I might know who Henry Fonda is, but I might not know he's Peter Fonda's father. I can see @Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda."

Just as he claims that 2 + 2 conveys the idea of a process putting two numbers together in some way, which is different information than just referring to the number 4. He's wrong about that mathematically, but he may have a point about Henry and Peter.

A stronger example in natural language is Joe Biden and the president of the US. Joe is always Joe, but being president is contingent and others have held and will hold that office. Joe Biden has always been and will always be Joe Biden; but Joe Biden has not always been president and will not always be president. So these two statements are different somehow.

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?

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.

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.

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.

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.

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.

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.

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.

This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. . — aletheist

If you view real numbers as locations where points might live, then she passes through each location. I don't see how this changes anything. Also I've previously referenced the constructive, hyperreal, and Peircian concepts of the real line, so I don't think I can be accused of narrow mindedness on the topic. Unless you reject the intermediate value theorem, my point stands. And even the constructivists have their own version of IVT, which they make work by restricting functions to being computable ones. Some version of the IVT is always valid regardless of one's model of the real line.

As such, she does not really "pass through" them, we just just use them to track her progress — aletheist

Distinction without a difference. IMO. 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.

think about what it would be like to exist without a physical body, — Jack Cummins

Wouldn't it be like a dream? Every night I have great adventures, run around in semi-familiar worlds, solve problems, interact with people I used to know and some I never knew. Then I wake up and realize that I did all this without the use of my body. Perhaps I'm doing that right now. Maybe I'm a brain in a vat or a program running in the great computer in the sky. In short, existing without a body wouldn't be any different than what we experience now. If you didn't have a body maybe you'd think you did. Maybe you don't have a body and think you do. Just some idle thoughts. I have vivid dreams every night, generally weird but not very disturbing. But they're always realistic. I never have any idea that I'm dreaming. In fact the only way I can tell I'm awake is that when I'm awake, I can wonder if I might be dreaming. When I'm dreaming, I never wonder about that!

Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask? — keystone

No, I'm suggesting that the mathematical concept of the real line doesn't apply to the true nature of physical space; or that if it does, this fact lies far beyond contemporary physics. Even physics doesn't claim it does. Can't we stop here and nail down this point?Why are you deliberately ignoring this point, which I have made to you over and over? Nobody knows if space is infinitely divisible; and everyone knows that we don't know this.

Human beings and even goddesses can not traverse the mathematical real line because the latter is a mathematical abstraction. They can only traverse physical space, and nobody knows if physical space is like the mathematical real line or not. And we DO know that contemporary physics can not address the question.

Since both you and fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. — Ryan O'Connor

@keystone, I'll reply to this one before the other one. I had no trouble @-ing you so at least that worked. I'll continue to quote you as Ryan here but I assume that if you quote me back you'll be on your @keystone account and everything will be clear.

I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? — Ryan O'Connor

As I have asked you several times, is this a mathematical or a physical scenario? If mathematical, the answer is yes. And @jgillI made the same point. By the intermediate value theorem, a continuous function passes through all intervening points between one value and another. And I assume continuity is one of your assumptions here. Is it? Essentially you're asking about the nature of continuity.

If this is a physical scenario, the answer is that one, nobody knows for sure; and two, the question is meaningless with respect to contemporary physics because we cannot reason sensibly below the Planck length. Does she hop from quark to quark? No physicist would regard that as a meaningful question.

By the way I looked up [url=https://www.thoughtco.com/greek-mythology-alanta-1525976[/url], and having been a student of Greek mythology way back in the day I was pleasantly informed. She's the goddess of running. The link I gave, which is better than her Wiki link, is full of good stories.

So: I say again, as I have said several times: Mathematically there is no question that she passes through every point indexed by a real number. Physically, the question is open in general, and meaningless in current theory. Even a goddess can't run between quarks.

If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well. — Ryan O'Connor

You're perfectly well conveying your refusal to read what I'm writing. If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. If this is a physical thought experiment, the question is metaphysically open and physically meaningless. So which question are you asking?

In the Numberphile on Zeno's Paradox, James Grime says the following:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly." — Ryan O'Connor

That's right. Mathematically the sum of an infinite series is defined as the limit of the sequence of partial sums; and the limit of a sequence is defined as the number the sequence gets (and stays) arbitrarily close to. And the Numberphile guy says exactly that.

Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process? — Ryan O'Connor

Depends on whether this is a mathematical or physical thought experiment. The physical question is certainly not resolved by the math, but I've stated that many times already.

I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants. — Ryan O'Connor

If you would stop conflating math and physics, and avoiding answering whether you are asking a mathematical or physical question, all would be clear.

On the other hand, it's perfectly obvious that motion does occur in the real world (unless we live in a block universe or we're all programs running in the great computer in the sky or brains in vats or I'm just dreaming all this.

philosophers need to update their views — Ryan O'Connor

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.

Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term. — Gregory

I found the vid at https://www.youtube.com/watch?v=u7Z9UnWOJNY . I didn't want to watch the whole thing so I skipped to 12:00 and he said that "in the physical world" it's not the same, but I couldn't find the exact quote about brain melting. As I mentioned to @Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series, as the Numberphile guy says; and it's unsolved physically, as he also seems to agree.

In most discussions online people seem to believe the problem is solved by calculus, but that's not true. Only the mathematical problem is solved. The physical problem remains, and even moreso today, because we know about the Planck scale. Below a certain point, our physical theories are not applicable to space or time. The tortoise or the arrow or whatever only have to iterate through 35 or so steps before reaching the point where nothing sensible can be said.

At some point, you can't sensibly divide a distance in half, nor an interval of time. Physics doesn't allow us to get a sensible answer. We don't know whether space and time are arbitrarily divisible; but we DO know that our current understanding of physics doesn't let us sensibly speculate.

fishfry I'm trying to let you off the hook on talking to me about the nature of continua but you keep coming back for more. — Ryan O'Connor

You keep baiting me by posting bad math.

I'm not complaining! — Ryan O'Connor

I like talking to you too even though I don't understand the points you're trying to make

Exactly. And since there is no first nonzero positive real then she cannot take her first step. — Ryan O'Connor

Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that? And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics?

Her journey doesn't begin. Motion is impossible. When I said that 'we cannot answer this question' I should have said 'we cannot answer this question satisfactorily.' — Ryan O'Connor

It's perfectly obvious to anyone who ever took a step -- that's most two year olds -- how the process works. Why do you think the modern theory of the real numbers prevents anyone from walking? "Oh no I can't walk, they're teaching Dedekind cuts to the math majors at the university!"

I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step?

I suspect you would say that you see infinite points there. — Ryan O'Connor

I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted. Like drawing an old guy in a robe with a white beard to symbolize God. Of course the picture itself on my laptop screen is made of pixels, which are discrete, and there are only finitely many of them. Why do you either (a) fail to appreciate both of these points, or (b) think I don't?

I don't. In this picture I see 5 points, each connected by lines. That's it. WYSIWYG. — Ryan O'Connor

Fine. What of it. How would anyone's interpretation of that picture prevent them from walking or allow them to walk?

And if we relate this back to Atalanta's story, I would say that we have still photos of her at 0, 1/3, 1/2, 3/4, and 1. That's all we've got. We are not justified to say that at some time she was at 1/10 or at 7/8 because we don't have the photographs. At one moment we see her at 0, we blink, and when we open our eyes she's at 1/3. — Ryan O'Connor

Yes, motion in the real world is pretty much like that.

Motion happens when we blink, when we are not looking. — Ryan O'Connor

According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation?

Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works?

And I've said it before but perhaps it might sink in this time that this is consistent with QM. — Ryan O'Connor

It's consistent with the Copenhagen interpretation of QM but not with the Many Worlds interpretation. And don't forget that Feynman said we can model reality by assuming that you get from point A to point B by integrating over every possible path, no matter how circuitous. You are just getting yourself tangled up here. You're confusing interpretations of QM, which itself is just a mathematical model that agrees with experiment up to the limit of Congressional funding for particle accelerators; with reality itself. You confuse math with physics, and you confuse physics with ultimate reality. Two series category errors.

If we are continuously observing a quantum system it will not evolve. [/qute]

How do you continuously observe a quantum system? Nobody knows how to do that, you have a false antecedent.
— Ryan O'Connor

It will only evolve when it is not being observed. — Ryan O'Connor

Only in some interpretations. You are making statements about things nobody knows.

Change happens when we blink. Not surprisingly, it's called the Quantum Zeno Effect. — Ryan O'Connor

According to some interpretations, not others. What if a cat blinks? A microbe? A cameral lens? What exactly is a measurement? Nobody knows.

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

How does the mathematical theory of the real numbers prevent me from walking? You're not thinking through your own argument.

Perhaps my point didn't come across. — Ryan O'Connor

That's because you seem to think that the mathematical theory of the real number precludes my getting off my ass and taking a walk. I'll try that one out on my doctor when he tells me to get more exercise. "They're teaching Zermelo-Fraenkel set theory, which leads to Dedekind cuts, so it's all I can do to just change the channel, let alone stand up from the couch."

This is your thesis. Don't you see how silly it is?

All I was trying to say was that 'there is no smallest positive real number' means that Atalanta cannot decide what point to go to first. — Ryan O'Connor

How does the mathematical theory of the real numbers prevent someone from taking a walk? Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense?

In my view, we cannot speak of the 'completely measured continuum' which is what I would call 'the real number line'. We cannot do so because it would require infinite measurements and that's impossible. Instead, we must only speak of systems which we can actually look at (at least in principle), like the one above. And in that case, the smallest positive number is precisely 1/3. — Ryan O'Connor

Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted.

Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this. — Ryan O'Connor

There is no first nonzero positive real. The open unit interval (0,1) does not contain its greatest lower bound. Any math major can tell you that. They learned it in real analysis class.

As long as we think that a line is composed of points we cannot answer this question. — Ryan O'Connor

If we allow that the real line is made of points (which are just real numbers) then the answer is that there is no first nonzero positive real number. That is the answer, so your claim that "we cannot answer this question" is false.

Motion is only possible when she jumps from one point to another. — Ryan O'Connor

Now who's claiming a line is made of points? You are the one doing that!

And what I'm saying is that she doesn't jump over infinite points, she jumps over a continuum — Ryan O'Connor

You may not like the real numbers as a model of the continuum (and after all, some mathematicians agree with you); but that doesn't entitle you to mischaracterize the math of the real numbers. There is no smallest positive real number. Why is that a problem. In your model of a continuum, whatever it is, is there a smallest real number?

I suspect it's the apostrophe...it's always given me computer problems. I've posted a comment in the 'Feedback' section, hopefully a moderator will allow me to change my name. — Ryan O'Connor

Some kind of use-mention problem no doubt.

I deleted the post I wrote. Here's the updated version. First, we must both be frustrated by now. You expressed frustration at trying to tell me something that I thought I had told you three days ago. First you said a still photo stops motion and video shows motion. I pointed out that video is actually a sequence of stills, and that even a still captures motion because it records photons over a nonzero interval of time. Now in your most recent post you are frustrated that you can't explain this to me!

So we're talking past each other. Perhaps we can find agreement at least in that.

Second, it's not the job of math to solve Zeno's paradoxes; and even if it is, it's not an interest of mine. I wish you the best with your efforts in that direction but I can't help. I regard Zeno as a solved problem mathematically via the theory of convergent infinite series; and an unsolved problem physically because our best theories don't allow us to reason below the Planck scale.

And third, dt is a differential form. They don't explain these in calculus so that's why everyone's confused about them. Bottom line is they aren't numbers and they can't be zero OR nonzero. If you use the notation $\Delta t$ that will be accurate. That's an interval of time, zero or nonzero as the case may be.

Your idea about building points from lines or lines from points just went over my head. I don't understand your intention at all. I'm not trying to solve the nature of the continuum here. Mathematically we construct the real numbers and call them a line, but the geometric visualization is incidental and not essential. As far as the true nature of the world, that's above my pay grade but I would personally be very surprised if it's anything at all like the mathematical real numbers.

I haven't much to add beyond this.

Name calling doesn't win debates. — Ryan O'Connor

You should try posting an unpopular opinion in the politics-related threads around here. Namecalling is all they've got.

The second is my own recent paper, — aletheist

Thanks for the info. I was not able to get past the JSTOR paywall even though I have a public JSTOR account (the kind they deign to give to us unwashed non-academics). Do you know why? Usually I can read JSTOR articles on the website even if I can't download them.

I think a much better question to ask — Ryan O'Connor

Ryan, I only quoted this to mention you. I've noticed that when I hit the @ button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues?

Anyway apropos of our convo re video, I thought you might enjoy this. It's a pretty cool video regardless. It's a helicopter taking off with its rotor speed exactly synced to the video frame rate, making the rotors look motionless. So even if things look motionless maybe they're moving. For all we know, still photos are moving like crazy but the lights in our room are blinking. Reality is not what it seems, or something like that. The comments below the video are amusing too.

https://www.youtube.com/watch?v=yr3ngmRuGUc

I ask you questions and challenge your citations and you are completely non-responsive. — tim wood

For the record I won't be responding to you on this issue. "What lockdowns?" Come on, man.

Do you even read your references? — tim wood

This is not a productive conversation. "What lockdowns?" is not a serious response to any of the material in this thread. It'll take a few years before people start to get some perspective and see 2020 as the mass hysteria that it was. If only we could have found some witches to burn, but we'll have to settle for beating up people for not wearing masks.

https://reason.com/2020/05/04/a-new-york-cop-beat-someone-up-over-social-distancing-will-nypd-policing-finally-change-now/

https://www.npr.org/2020/11/27/939499357/french-police-officers-in-custody-after-video-emerges-of-brutal-beating-of-black

https://www.amny.com/new-york/brooklyn/police-violently-arrest-man-in-brooklyn-and-threaten-bystanders-for-not-wearing-masks/

https://timesofindia.indiatimes.com/city/ahmedabad/video-of-cop-beating-man-for-not-wearing-mask-goes-viral/articleshow/79784988.cms