Peter Lynds simply rehashed Bergson on Zeno. Time is not a thing, so dividing it is rather arbitrary. The paradox of Zeno is about the line segment itself, not just about the motion. Maybe Bergson answers the motion part. But he doesn't show how it is not a contradiction for a line segment to be both infinite and finite in the same respect

Time is not a thing, so dividing it is rather arbitrary — Gregory

Is a line segment a "thing?" As opposed to a wooden rod, say? Does a line segment actually exist as a physical thing?

A rod is real and can be divided infinitely. Lines are just our way of mulling over the world in our brains

The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not. — aletheist

This is exactly why it is a mistake for us to try to discuss "real space", and "real time". If physicists produce a concept of space and time as a continuum, in the mathematical sense, yet we want to say that "real" space and time are continuous, and this is different from the mathematical continuum, to talk of "real" space and time is a mistaken approach. It is mistaken because there is no such thing as real space and time, these concepts are derived from the observations of objects which are assumed to be real. So the correct approach would be to say that physicists incorrectly model the existence and movement of objects, and that they ought to be modeled as continuous rather than as a mathematical continuum.

Notice, that what is being modeled is the movement of objects, and these are what are assumed to have real existence, not space and time. "Space" and "time" are produced from these models, as logical conclusions. Model A shows objects to move in such and such a way, therefore there must be such and such "real space and time", to substantiate that model. But model B shows objects to move in a slightly different way, so there must be a slightly different "real space and time" to substantiate that model. The "real space and time" is just something created by the model, as an incidental conclusion. This is why model-dependent realism is popular. But if the model is the "correct" one, then there is a real space and time of such and such nature to support that model. If there is no "correct" model, as model-dependent realism proposes, there is no real space and time. Space and time are not themselves being modeled. You can see this in the difference between Newtonian principles and Einsteinian principles. Movement of objects is modeled, and a "real" space and time of such and such a nature is required if the model is supposed to be true.

Ultimately, we might produce a model of moving objects which required no space or time, but our present conceptual structure falls back onto the reliance of space and time. However, the fact that our conceptual structures of the movement of objects requires that space and time are real, does not mean that space and time actually are real. If you and I agree that there is misconception, related to the use of "continuum", and "continuous", then the conceptual structures are inaccurate, and therefore a conception without any real space or time might be the correct one.

Absolute space seems obvious. Absolute time, no

Perhaps Peter Lynds' essay — jgill

Wow that nutball Peter Lynds is still around? I heard of him about ten years ago ... maybe fifteen or twenty, now that I think about it. It was on Usenet. So it must have been at least twenty years. Tempus fugit.

I remember that he had solved Zeno, or some such ... I actually don't remember the particulars. Everyone on sci.math was talking about him with varying opinions. Lynds had written two papers. I downloaded and read them both line by line with close attention. I concluded that Mr. Lynds needed a good course in basic calculus. I regarded him at that time as a minor crank.

I have not heard his name since then. If in the intervening decades he has managed to acquire mindshare among ... well anyone, frankly ... I wish to register my dissent.

No need to disagree, this isn't a hill I'm going to die on. And maybe he's reformed his ways and is now writing articles of actual substance. If so I'd appreciate a link to anything recent that he's done. If there's new evidence I'll change my opinion. As it is. the mere mention of his name annoys me. As Dennis Miller used to say back when he used to be funny, That's just my opinion, I could be wrong.

The real numbers cannot fulfill the conditions of a proper definition of "continuity". Real numbers produce a sequence of contiguous units. Contiguity implies a boundary of separation between one and another. This boundary must produce an actual separation between one number and the next, to allow that each has a separate value. This is contrary to "continuity" which is the consistency of the same thing.

So mathematicians have created a term, "continuum", which applies to a succession of separate units, allowing that each is different, so there is something missing in between them, and that "something", which is the difference in value, is unaccounted for. Therefore "continuum" means something completely different from "continuity".

The rational numbers are an attempt to account for this "something", the difference in value, which exists between the reals. This an attempt to create a true continuity. However, the irrationals appear, and foil this attempt. So mathematics still does not have a continuity. — Metaphysician Undercover

In my post that you were replying to, I took pains to explicitly mention the following:

Note please that I'm only saying what a mathematical continuum is. I'm not addressing any of the many philosophical objections there could be to calling the real numbers a continuum, But mathematically, the reals are a continuum and the rationals are not. — fishfry

Since I wrote that, I wonder why you replied as you did. I'm perfectly well aware of the philosophical objections to calling the real numbers a continuum.

It's also a fact about the world that mathematicians regard the real numbers as the (mathematical) continuum.

My disclaimer made it perfectly clear that I understand the distinction between the mathematical continuum and the various philosophical approaches. In recent years I've studied the classical intuitionist continuum, (Brouwer et. al.), the modern constructivist continuum, the hyperreal line of nonstandard analysis (that's the one with the reals plus infinitesimals), and I've even seen a little Peirce. I have sympathy towards these points of view.

I wonder why you feel compelled to respond to me so irrelevantly, with such trite philosophy and inaccurate mathematics. Your remarks regarding the rationals are particularly unenlightened.

I'd welcome substantive engagement with you but I told you in the other thread that I find your style unpleasant. To me this is more of the same. You saw my disclaimer but couldn't help yourself piling on with the usual ... usual.

I am in the process of writing a reply to some of the things you read in the bijection thread. I read everything you wrote. I have a better understanding of your ideas. This is not the place and I'll try to finish that post up soon. I'll put my thoughts into a larger context. Meanwhile what is your point in telling me what I already know?

That's right, the existence of irrationals really throws a wrench into the rational number line. Where do those irrationals exist in relation to that line? — Metaphysician Undercover

You have asked a mathematical answer. I'd be happy to give you a mathematical answer.

My concern is that no matter what mathematical points I make, you're just going to pile on with the usual nihilism and simplistic philosophy that math must be wrong because it's not true. As if everyone doesn't already know that math isn't literally true. You're the only one flailing at this strawman.

The irrationals fill in all the holes in the rationals. I already illustrated this with a sequence of rationals that approaches the point sqrt(2) but there's a hole there instead of a point. The irrationals fill in those holes.

We can formalize the process of filling in the holes with various technical constructions of the reals. There are several, the two best known being Dedekind cuts and Cauchy sequences. The details aren't of interest. The point is that it can be done within set theory and it allows us to found calculus in a logically rigorous way, something that escaped Newton and Leibniz. We can also axiomatically define the reals as "the unique Cauchy-complete totally ordered infinite field." When you unpack the technical terms, you end up with an axiomatic system that's satisfied within set theory by the Dedekind cuts or Cauchy sequences. It's all very neat. One need not believe in it or care. It must be frustrating to you to both not believe in it, yet care so much!

I could drill the math down a lot more but should probably wait for encouragement, and if none is forthcoming I should leave it be. I don't think you're curious about the math at all. You just want to throw rocks. But why? People uninterested in chess don't spend their lives hating on chess. They just ignore it. You think math is bullshit? Maybe you're right. Maybe it is all bullshit. The thing is why do you keep repeating the point over and over as if we haven't all heard you already? And as if we all don't already understand the point?

Am I being too harsh? Maybe. I don't know. You misunderstand me. Perhaps I misunderstand you. Why would you reply to my post at all? You don't think I know anything. Why'd you even bother?

The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not. — aletheist

Ah. But I'd call that a strawman. The "widespread misconception." There isn't ANYBODY out there beating the drum for the proposition that the mathematical real line is the "true" continuum, whatever true might mean in this context. The overwhelming majority of mathematicians give the matter no thought at all. If they're studying differential equations or abstract algebra or proving Fermat's last theorem, they're simply not concerned with such matters.

And among those mathematicians who have taken the time and trouble to think about the nature of the continuum, they'd know about Brouwer and the modern constructivists and the hyperreals and if they studied some philosophy they'd get some context for the conceptual objections to the real numbers as the "true" continuum (again, whatever that might mean), and they'd most likely agree that the standard mathematical real numbers leave a lot to be desired in the philosophical realm,

I just don't think there are that many people who have thought about the matter for five minutes and don't have some sympathy for the alternative point of view. I don't think there is a widespread misconception. I think there's a widespread lack of interest in the question; and among those who are interested, some degree of agreement that the real numbers don't express everything we think must be true about a continuum.

The irrationals fill in all the holes in the rationals. I already illustrated this with a sequence of rationals that approaches the point sqrt(2) but there's a hole there instead of a point. The irrationals fill in those holes. — fishfry

There are no holes in the rational numbers, just like there are no holes in the reals. The numbers are produced from a different set of rules. This is what your illustration illustrates, it does not illustrate a hole:

But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry

What you illustrate here is that an irrational number cannot be expressed in the rational system. If we did not already know what the sqrt(2) is, as an irrational ratio, we would not know that your proposed rational sequence does not converge to that. You could make up any random, fictional, irrational number (not grounded in a true irrational ratio like sqrt(2) or pi), and show that a sequence of rationals does not converge. But this does not illustrate a "hole". it just illustrates some sort of incompleteness, like 1.5 (3/2) illustrates an incompleteness in the reals.

The reals fail to account for division. The rationals are an attempt to address this failing, but to the extent that there are irrational ratios, the rationals fail in the attempt to account for division.

So you could make another demonstration to show that there are rational numbers which cannot be expressed in the real system, but this does not indicate "holes" in the system. Such demonstrations, which demonstrate incompleteness do not illustrate "holes", they just show that the different systems follow different rules. There is not one set of rules to cover all mathematical operations. If there is incompatibility in the rules, it will appear as what you call a "hole".

The existence of irrational numbers is easily accounted for by the fact that in the rational system (rules), there is infinite possibility of numbers between any two rationals. "Infinite possibilities" is not within the bounds of any logical system, it refers to where the system fails. Therefore illogical divisions (irrational ratios) are allowed to exist within that system, which allows for infinite divisibility. The illogical divisions are within the rules of the rational numbers, system. So the irrationals do not represent holes, as if they are outside the rational numbers system, they are within the system, a manifestation of the illogical proposition of "infinite divisibility". "Infinite possibility" is itself illogical. As a rule, "infinite divisibility" allows for the illogical, i.e., for the irrationals to exist within the rational system.

I could drill the math down a lot more but should probably wait for encouragement, and if none is forthcoming I should leave it be. I don't think you're curious about the math at all. You just want to throw rocks. But why? People uninterested in chess don't spend their lives hating on chess. They just ignore it. You think math is bullshit? Maybe you're right. Maybe it is all bullshit. The thing is why do you keep repeating the point over and over as if we haven't all heard you already? And as if we all don't already understand the point? — fishfry

That's right, I'm not curious about that math because I think it's the wrong approach. approaching the irrational numbers as if they are holes in the rational numbers, and trying to fill those holes with the reals, is completely backward. If these are actually "holes", then the reals are like a sieve. There's a massive "hole" whenever an odd number is divided by two for example.. Why would you try to patch holes with a sieve?

To resolve a problem requires a clear, and complete analysis of the problem itself, to understand its true nature. This is what I offer, a more thorough analysis of the problem, as an aid, to assist in resolution of the problem. What I do is not a case of throwing rocks, or hurling insults, it's a case of examining the foundations for weak points. If your attitude is that these foundations were built by the greats, therefore there are no weak points, (appeal to authority resolves fundamental problems), then I think you are in need of God's help. Hopefully, God as the ultimate authority, will show you how his principles differ from those authorities which you appeal to.

I don't think there is a widespread misconception. I think there's a widespread lack of interest in the question; and among those who are interested, some degree of agreement that the real numbers don't express everything we think must be true about a continuum. — fishfry

Fair enough, thanks.

A length that is irrational comes into play when you have a length that is the "smallest" length as the right sides of the triangle. The irrationals are not imaginary numbers. They simply go on forever, within a limit. But that infinity within the spatial limit is still an infinity of points. And half that hypotenuse would be the new true smallest length. It goes on forever, hence Cantor and then Banach and Tarski. I think it's also fishy that with finding the length of the hypotenuse with a length of One on the ride sides, you multiply one times one and get no further than one. I could be wrong about that minor point

And maybe he's reformed his ways and is now writing articles of actual substance. — fishfry

:lol: Not likely. I thought the thread needed some comic relief.

:lol: Not likely. I thought the thread needed some comic relief. — jgill

Good, I thought it was just me. In fact after I posted I looked up his name and found that when he submitted his 2005 papers (so fifteen years ago) one reviewer rejected him saying that he "lacks fundamental understanding of basic calculus." Exactly the same criticism I noted.

That's right, I'm not curious about that math — Metaphysician Undercover

Then we have nothing to talk about. I still owe you my thoughts on the bijection thread, where you made a couple of remarks that I can use as a starting point.

It's curious, though, that you are so interested in the philosophy of math yet so uninterested in math. Your math is wrong. You have some ill-formed ideas that are leading you astray. And you have no interest in clarifying your own errors. Hard to know what to make of that.

If your attitude is that these foundations were built by the greats, therefore there are no weak points, (appeal to authority resolves fundamental problems), then I think you are in need of God's help. — Metaphysician Undercover

Not something I've ever said. If I have said such a thing, please be so kind as to quote my words directly. You can't because I never said any such thing. It's pathetic that your only means of discourse is to lie.

Since I explicitly mentioned all the alternative models of the continuum that I'm familiar with, your remark has no basis in reality; other than your own reality of incoherent bullshit.

A length that is irrational comes into play when you have a length that is the "smallest" length as the right sides of the triangle. The irrationals are not imaginary numbers. They simply go on forever, within a limit. — Gregory

The irrationals are relations which cannot be resolved, like the ratio between the circumference and the diameter of a circle, or the distance between two points of equal distance from a point at a right angle. That such relations are irrational indicate that the two things being related to one another are incommensurable.

Your math is wrong. — fishfry

Yes, sorry, I misunderstood.

I think they are commesurable only in the sense that there is infinity contained within the finite in geometry and in our world

I think that "infinity contained within the finite" is contradiction pure and simple. That's like saying that a finite thing is inherently infinite. Contradiction.

Exactly mr. undercover. The world is an Esher painting

I've been told that if I truly understood calculus, I would see how there is no contradiction in something spatially being finite and infinite at the same time. I suck at math so I could be the stupid one in the conversation :(

Levi-Srauss, who's "mythic discourse" is very Jungian, said "since the purpose of myth is to provide a logical model capable of overcoming a contradiction... a theoretical infinite number of slates will be generated, each one slightly different from the others." Hence i, tongue in cheek , said above that God created the world as a poet, not a mathematician

I've been told that if I truly understood calculus, I would see how there is no contradiction in something spatially being finite and infinite at the same time. I suck at math so I could be the stupid one in the conversation :( — Gregory

In mathematics, to "truly understand" is to accept the axioms without question. This allows that contradiction within an axiom is acceptable as understood. Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem.

So your argument on this thread is that there is not a contradiction in math, but that it's incomplete?

Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem. — Metaphysician Undercover

It's mostly a metaphysical problem. Most mathematicians and physicists do quite well without contemplating such issues. But that's not to say that "infinity" and "objects" are not concerns, as a physics person might tell you in reference to renormalization procedures and quantum entities, for instance.

So your argument on this thread is that there is not a contradiction in math, but that it's incomplete? — Gregory

Yes it is incomplete, but I think what I meant was that there is contradiction in math but it's acceptable because it's unavoidable. It's unavoidable because mathematics is a reflection of our understanding of reality, and our understanding of reality is limited by our capacities as human beings. So contradiction just reflects our imperfections. If mathematics were without contradiction, it would be perfect, and our understanding of reality would be perfect. The incompleteness is therefore our inability to completely rid the system of contradictions.

Most mathematicians and physicists do quite well without contemplating such issues. — jgill

I agree, these issues are simply accepted, taken for granted, perhaps almost subconsciously, and not worth thinking about for most mathematicians and physicists. They work with the tools they have. Likewise, that our knowledge of the physical universe is incomplete is also taken for granted. If we put two and two together, we might conclude that a better system would provide us with a more complete knowledge.

Violins can create an infinity of sounds with infinitesimal changes. Aristotle's finite universe is a myth

Violins can create an infinity of sounds with infinitesimal changes. — Gregory

I do not believe that could be true according to the known laws of physics. A sufficiently high frequency would eventually require back-and-forth movement of the bow in space smaller than the Planck length.

To be clear, since this is a common misunderstanding: The Planck scale is the point at which our theories of physics break down and may no longer be applied. It does NOT mean the world itself is quantized. Below the Planck length we simply do not know and can't even speculate, because our physics no longer works.

With that understanding, if there is a length below which we can not analyze or understand; there is a highest possible frequency. And within them, to discriminate one from another must likewise become impossible. There are not infinitely many gradations, to the best of my understanding of things.

After all quantum theory says that light is not infinitely variable. Pretty reasonable that sound isn't either.

My ears tell me sound is infinite, when I study music. There is an infinity within the limit of the highest and lowest frequency.

My ears tell me sound is infinite, when I study music. There is an infinity within the limit of the highest and lowest frequency. — Gregory

Your eyes tell you there is a continuous gradation of light. Physics proves otherwise.

For that matter your senses tell you the world is flat. Science shows otherwise.

I don't see an infinity of quality in light personally. Maybe if I finally tried LSD

My ears tell me sound is infinite, when I study music. — Gregory

Even your ears have physical limitations. There's no way you could distinguish an infinite number of different notes. And if you just assume that there is so many different notes that they must be infinite, that's an unsound premise.

To be clear, since this is a common misunderstanding: The Planck scale is the point at which our theories of physics break down and may no longer be applied. It does NOT mean the world itself is quantized. Below the Planck length we simply do not know and can't even speculate, because our physics no longer works. — fishfry

Thanks for this. I would add that the same is true of the Planck time, since it is defined as the duration required for light to travel the Planck length in a vacuum.