Can "Infinity" ever be more than a mathematical abstraction?

Discussion in 'Physics & Math' started by Seattle, Jun 24, 2018.

  1. someguy1 Registered Senior Member

    I think I get your point, except that nobody ever does this. Nobody goes, "Here's the Pythagorean theorem, but when we're doing physics we don't allow noncomputable numbers as its inputs or outputs."


    Why are you fixated on noncomputable numbers? I agree they're interesting, but they have no relevance to physics unless someone is claiming the world is a computer or a computation. In which case computability is the least of the issues on the table. But you're not claiming that. So why do you think anyone either is or should be deleting noncomputable numbers from the inputs and outputs of mathematical functions and equations used in physics? Why do you think this?

    If all you care about is "accuracy and precision" as you put it, then rational numbers are fine. That's what all physical measurements are. Rational numbers with error bars.
    Last edited: Aug 7, 2018
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  3. iceaura Valued Senior Member

    Nobody has to. They never come up. They are meaningless, physically.
    Just as when we use the Pythagorean Theorem, we allow the negative inputs and outputs - we just don't use them for calculating distances. We discard the meaningless.
    Not my idea. They came up, complete with various claims of significance.
    You'll have to ask them - those physicists you mentioned. I'm just noting that it works, so far. My guess is that they had no way to include them, in the first place, and didn't need them, in the second.
    But if I also care about modeling physical reality, they aren't. I need logarithms, spiral and circular geometries, that kind of stuff.
    The rationals aren't enough to provide formulas and solutions for our models.
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  5. someguy1 Registered Senior Member

    I can't help thinking we're in violent agreement. If we're having a disagreement or a discussion, I honestly no longer know what it's about.

    Neither are the computables, if you care about continuity. You've already stated you don't care about continuity but you're going to call non-continuity continuity. That's what I'm understanding. But we're way past any substantive conversation IMO.
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  7. iceaura Valued Senior Member

    If I only care about physical continuity, they appear to be sufficient.
    I only care about successful modeling of physical continuity, in the question of whether there exist infinities that are "more than" mathematical abstractions.
  8. Write4U Valued Senior Member

    It's like saying; "they had to commit him, he was more ....ummmmm...., more than happy"!

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  9. someguy1 Registered Senior Member

    You don't know that's true. What if the world is accurately modeled as the mathematical real numbers? Then all the noncomputable points would still be there. And remember, a real number can be viewed as the address of a point. Just because a point lives at a noncomputable address, it's still a first-class point in good standing, a valid part of the universe (under the assumption that the universe actually looks like the real numbers). So I don't think you have an argument to support your thesis. Noncomputable numbers MIGHT turn out to be important in understanding the world.

    Just as, by analogy, the negative numbers, and zero, and rationals, and irrationals, and complex numbers started out as seeming strange, and eventually turned out to be a part of nature. Math has a way of going from "weird and useless" to "essential for physics" within a few generations. From Riemann's non-Euclidean geometry to Einstein's theory of relativity.

    You only care about modeling reality and no longer about reality itself? You could get drummed out of the physics club that way. That's something a math major would say!
  10. Write4U Valued Senior Member

    IMO, that is a bad assumtion. The universe does not look like numbers, it looks like values which are translatable into numbers. But that does not automatically mean that all possible numbers are represented as a specific (computable) physical values.
    And some values like Pi cannot be numerically represented with 100% accuracy. But then does it need to be?

    Evolution is a continuous process except for "mutation" which skips intermediate evolutionary states. Perhaps the mathematical nature of the universe just skips certain numbers because they are not significant to the mathematical function at a given time.

    The Fibonacci Sequence skips non-Fibonacci numbers (1-1-2-3-5-8-13-21-etc), yet is eminently functional as demonstrated throughout the universe, from flowers to galaxies.

    Even as I believe the universe to be mathematicall deterministic, I recognize also the probabilistic nature of specific local events. It started in Chaos and ordered mathematical patterns emerge with time.

    It would be useless to reduce the FS into smaller numbers. The logarithmic ratio remains the same at any value.
    Last edited: Aug 7, 2018
  11. arfa brane call me arf Valued Senior Member

    The fact that physics apparently ignores the uncomputable numbers, or that apparent motion is continuous and 'rational' although the rationals have an empty interior, to me implies that nature does compute real numbers. It doesn't compute uncomputable numbers, they're only mathematical after all. So are all the holes in the rationals.

    Physics and physical numbers like distances have what we call "physical units". What really are they except heuristics? What does mathematics say about these units? Nothing really.
  12. arfa brane call me arf Valued Senior Member

    No, I think that assumption isn't sound. The universe we live in has computable numbers in it.

    So (perhaps) numbers like Avogadro's or the ratio of mass to charge of the electron, the fine structure constant etc, must all be computable. So far we have a handful of "significant" digits, we have a good enough approximation for these numbers; it's good enough for our current technological reach.

    What of the nonzero uncomputable numbers in Robinson's schema where you have h is infinitesimal if h < 1/n? n is a finite number so must be computable. Is h also computable? I can't see how you could compute it, it's defined to be beyond that, in a sense, a "hyper-number", since it also can't be zero. This excludes the naive solution of a TM printing a zero as an approximation of any h.

    On the other hand, that's what the TM that corresponds to a continuous path through space and time does, it outputs zeros as approximations for the uncomputables. Sounds like an idea for a SF novel.
  13. someguy1 Registered Senior Member

    Says who? What makes you think this? What makes you say this? What does it even mean? A glance at any physics book falsifies your claim. Newton: "F = ma as along as F, m, and a are computable" No he didn't say that. Wave equation, based on t a real number. I can't find a single physics equation that says, "noncomputable numbers excluded." You're just making this up.
  14. arfa brane call me arf Valued Senior Member

    Says who? Can you justify this beyond a conjecture? Why should the universe look like the real numbers, why not like a subset of the real numbers--those with a computable value?

    If the universe does compute. what does: "compute an uncomputable" mean? What could it possibly signify?
  15. someguy1 Registered Senior Member

    There are two meanings of "physics." If by physics you mean the historically contingent theories of physicists, then my previous post stands. F = ma is not qualified by throwing out noncomputable numbers. You must be making a bad joke if you think it is.

    On the other hand if by physics you mean "ultimate reality," if there even is such a thing, then any claims you make about it are metaphysical speculations.

    I have no idea. You're the one using the phrase. You'll have to explain it.
    Last edited: Aug 8, 2018
  16. arfa brane call me arf Valued Senior Member

    In fact, F = ma says nothing at all about computable or noncomputable numbers, it doesn't even say anything about measurement. That's something we have to deal with.

    When you throw an object with mass so it describes a continuous path through space, the distance it travels along this path is not approximate, measurement is. You observe that it stops moving upward and starts moving downward, so it must have zero vertical motion at some point, but when and where is this point? It must have an exact location in space and time, except we can't do better than approximate that.
    Except that you yourself claim this: "Just because a point lives at a noncomputable address, it's still a first-class point in good standing, a valid part of the universe (under the assumption that the universe actually looks like the real numbers)."

    This isn't your own metaphysical speculation? Please explain.
    Last edited: Aug 8, 2018
  17. someguy1 Registered Senior Member

    So when you said, "The fact that physics apparently ignores the uncomputable numbers ..." you were wrong. Glad we got that straight. Unless by physics you mean "ultimate reality," in which case you're simply making an unprovable metaphysical claim.

    QM disagrees with you. Or more accurately, some interpretations of QM disagree with you. As far as what reality "really" does, that's a metaphysical assumption on your part. You are not God and you don't know what reality "really" does. "It must have an exact location in space and time" is something that you can never know.

    By the way can you please read Tiassa's reply in the following thread and stop doing whatever you're doing? I don't get alerts when you reply to my posts, and if someone posts to a different thread in the Physics/Math section I can't see your post unless I dive into this thread to check.
  18. arfa brane call me arf Valued Senior Member

    . Not if the laws of physics are exact. As for QM, the wavefunction of a rock you just threw up in the air seems to have no noticeable part in its motion.

    Motion is 'assumed' to be continuous, we never observe classical objects, like rocks, jumping around discontinuously.

    Physics has observation, mathematics doesn't. Nonetheless, to first order in any equations of motion I invoke, rocks fall back to earth so must have zero vertical velocity at some point (more exactly the centre of mass does).

    The problem is that although this inflection point must exist, detecting where and when it is involves observation and measurement--accuracy. Saying there is no way to prove the existence of inflections of vertical (or even continuous) motion because of QM is just wrong.
    That was a bit TIC, what I meant there was you and iceaura appeared to agree that in physics, measurement gets you rational values. Apparently we don't have to consider the holes in the real line, that doesn't really seem all that meaningful. Nonetheless, mathematically there are indeed, numbers with nonzero but meaningless(ly small) values.

    BTW, let me quote from my actual 1st year calculus text:
    Last edited: Aug 9, 2018
  19. iceaura Valued Senior Member

    In physical science, you never do.
    It agrees perfectly with experiment. It has support in all - all - of the evidence, so far. It makes sense. It's useful.
    But they didn't get rid of each other.
    There are infinities in a world modeled by functions evaluated over the computables only. That will be the case if and when any use is found for noncomputables.
    In the matter of infinities that are "more" than abstractions, I don't even care about the math in any rigorous sense.
    Guilty as charged.
  20. someguy1 Registered Senior Member

    You're still equivocating "the laws of physics."

    Do you mean:

    * The historically contingent laws of physics as published in physics journals? Or

    * The "true ultimate laws," if such laws exist, of the "external world," if such a thing exists.

    If you mean the true laws of the world, then I suppose they're exact. Whatever that means, since you don't know what the true laws of the world are. But if you mean the historically contingent theories of physicists, those are of course not exact at all. At best you get a few decimal places of accuracy and a renewal of your research grant.

    The word "noticeable" totally undermines your point. When you watch a movie at 24 frames per second there is no "noticeable" frame-by-frame effect, even though that is exactly what's going on. You go back and forth between what's absolutely true and the highly imperfect observations we can make. In my opinion you are not stating any coherent point because you keep going back and forth between these two points of view.

    Then you entirely concede my point. The assumption of continuity and that an object has an exact position at a given time is a metaphysical assumption. The exact opposite of an established scientific fact. You ASSUME things then you argue that they are true because you assumed them.

    Math has tons of observation. You see number patterns then you figure out the underlying reason. Mathematics is very empirical. It's the presentation of math that gets cleaned up into definition-theorem-proof. But the actual work of math is based on observation. Please make a note of that.

    You are implicitly invoking the intermediate value theorem (IVT) that states that if a continuous function is negative here and positive there, it must be zero somewhere in between. But that is a consequence of the continuous nature of the real numbers. In the computable real line, for example, the IVT is false and so is your claim.

    You are assuming what you claim to be proving or believing.

    No, that's two problems. The first problem is that there is no particular reason to believe that the inflection point exists. That's a metaphysical assumption about the continuous nature of the real numbers. It might be true, it might be false. You are not God and you simply can not know.

    If such a point exists we can approximate it. In fact we can approximate it even if it DOESN'T exist. That is the point!!!!!! The approximation may be all we can ever know, and it may be all that's true!

    QM says a particle has no exact position till we measure it. So go complain to Max Born and all those guys.

    Sorry I don't know that Three Letter Acronym (TLA).

    Me, iceaura, and everyone else who ever spent more than five minutes thinking about the problem. If you work in a lab you get a bunch of decimal places. A finite number of them, and a VERY SMALL finite number at that, like six or eight. That's a rational number. And you know the error tolerance of your apparatus. So you have a rational number with error bars. This is science 101. I have no idea why you would waste keystrokes arguing otherwise.

    For those who claim the universe is computable or rational or fails to be topologically complete in any way, it's a problem. For me it's not a problem, I don't think the world's a computer and I have no idea if it's continuous. And even if it is, why should it be the mathematical real numbers? There are alternative ideas about the continuum.

    What do you mean by that? There are no infinitesimals in the standard reals. There are infinitesimals in the hyperreals but the hyperreals are not topologically complete. They have holes! I would like you to be much more explicit in what you mean by this.

    Surely that is a very low standard of evidence. First year calculus? A famous math professor once said to me, "Freshman calculus is a futile exercise in mindfucking." This is exactly the kind of thing he was talking about. But no matter, you have a quote about Leibniz which has absolutely nothing to do with anything we're talking about, and absolutely nothing to do with modern math. Newton, for example, did NOT rely on the infinitesimal concept. His idea of a limit was much closer to our modern point of view. And in any event, limits have replaced infinitesimals in modern thinking. Quoting a freshman freaking calculus textbook about what Leibniz thought over 300 years ago has absolutely no value whatsoever. I'm glad if you find comfort or meaning in it, but I found that quote incredibly off the mark at every level.

    My opinion of course. I already had a power outage and a visit to the dentist today in case I went overboard disparaging your freshman calculus book and/or Leibniz.
    Last edited: Aug 9, 2018
  21. someguy1 Registered Senior Member

    Sorry I don't remember what the context was and didn't go back to check. But nothing can "agree perfectly with experiment" unless it's a discrete measurement like counting the eggs in a dozen. The world's best experiment gets 12 decimal places of accuracy. That's a great theory, it's not exact reality. Science is never "perfect." We're slipping into scientism again.

    This was in reference to my saying that zero, negative numbers, complex numbers, etc., used to be thought weird then became indispensable to physical science. In that context I do not understand your comment.

    What model is that? Some guy wrote a book on computable physics but he's the only guy. It is a VERY obscure pastime to be trying to redo physics on computable math.

    I am wondering about your interest in noncomputable numbers. It's my understanding, PLEASE correct me if I'm wrong, that I brought up noncomputable numbers to make a point about something entirely different (someone claiming that ALL real number frequencies actually occur in the world). To the best of my knowledge, noncomputable numbers are somewhat of a niche interest, and most people here never thought about them much before.

    Did you see me talk about them then decide they are important enough to write this many posts about with respect to physics? Or is this an interest of yours that preceded my mentioning the topic? I'm just wondering if I brought all this on myself. There are a lot of weird intermediate fields [mathematical fields, not physical ones -- two totally different meanings] between the rationals and the reals. For example there are numbers that are definable but not computable. Chaitin's Omega is one famous example. I'm just wondering why you are into the computables so much in your recent posts.

    Oh but there are uses for noncomputables. Let me tell you two of them.

    * The noncomputables plug all the holes in the computable real line, so that you can have a continuum!

    * Say you have the world of Turing machines (TM) that define the notion of computability. There is a noncomputable problem, the Halting problem. In computer science you can add an "oracle" which is a black box that solves the Halting problem. By adding this oracle to the set of conventional TMs you get an augmented, more powerful notion of computation, call it "TMs + halting problem oracle." But now there will be a NEW noncomputable problem, so you can add ANOTHER oracle. This process keeps going on forever.

    The structure you get is equivalent to the transfinite ordinal numbers. Turing studied this in a paper called "Ordinal Models of Computation." He was a very smart guy, he was decades ahead of everyone else on this stuff.

    Another way to look at this is, what if I took the computable real numbers, and I added a SINGLE noncomputable real number. I'd get a stronger and slightly more interesting system than just the computables. It could "do more" in the sense of the oracles. So adding an oracle is like adding a single noncomputable. Remember you don't have to add ALL the noncomputables, you could just add them one at a time and study the hierarchy of ever slightly more powerful logical systems you'd get.

    So noncomputable reals have application in theoretical computer science, in studying infinitary models of computation. I hope this wasn't all too vague, I pretty much said everything I know about it.

    Don't care about math in a rigorous sense. No crime against that but then participate in a discussion about the relation between math and physics? The whole point of this thread is to discuss whether mathematical infinity has physical importance. And of course mathematical infinity is a rigorous, technical subject. You can't ignore it and then say you're still on topic. Right. We are talking about the rigorous theory of infinite sets, and its relation to physics. Not that we've actually talked about that, to be fair. We haven't even gotten there yet.

    LOL I forgot what this was in reference to, too!
  22. arfa brane call me arf Valued Senior Member

    If you say so. The assumption of continuity--continuous motion--and that derivatives exist which correspond to exact local times and positions does however form the basis of Newtonian physics (that we can't measure them exactly has nothing to do with the calculus, nothing).
    No I don't think that's how it goes at all. F = ma isn't an equation that says "At best, you get a few decimal places of accuracy". It says nothing about measurement, remember?
    So does physics. Except physics doesn't say anything about an "absolute" reality or truth.
    Really? Can you give an example of a mathematical object with "tons of observation"? The integers under addition? A set of numbers?
    Actually I'm invoking the observational fact that if objects with mass move up, they come back down. Since there's a derivative at each point, it must exist where the motion inflects (in the gravitational field), the tangent to this point is exactly horizontal.
    What you're saying here is Newton and Leibniz developed a metaphysical theory . . .? If we launch a rocket into space, it might be where we think it is, or it might not?
    And yet an object such as a rock you can throw is a large number of particles, all of which have exact positions relative to say, a central point. We don't perform quantum experiments when we throw rocks around. We do kinematics experiments with rigid bodies (i.e. with fixed positions for all the particles in the rock). The centre of mass of a rigid body does have a definite position at all times.
    I disagree. I think Leibniz has quite a bit to do with modern math.
    Says you. What Leibniz thought 300 years ago has no value, but what Newton thought does?
    So any opportunity to denigrate Leibniz's ideas is one we should take (that is, follow your lead), to point out that . . . something goes here, I just know it.

    Oh well. Here's some following text from that textbook (ironically titled "Freshman Calculus"):
    If it's true that infinitesimals are logically unsound, why is there nonstandard analysis? Why does the idea persist, and even seem to be useful? Can nonstandard numbers resolve Zeno's paradoxes (you betcha)?
    Last edited: Aug 10, 2018
  23. someguy1 Registered Senior Member

    I prefer not to argue point by point about the issues you raised, Leibniz and infinitesimals. I'll pass here. FWIW the hyperreals don't solve any problems the standard reals don't. At best some people claim they offer pedagogical benefits in freshman calculus. Studies are inconclusive. Students come out confused regardless.

    For those interested in the subject I strongly recommend Googling around and reading some articles on the hyperreals, not just depending on the Wiki page.

    I never said that. Nor did the quotes you posted. You're just engaging in the "Are you saying ..." game where someone says, "I think that in general, governments should control their borders," and someone else says, "Oh you hate Canadians." That's exactly how I'm reading your most recent post, one sentence after another. You no longer seem to be arguing in good faith.

    Another example of the same. I won't play. Are you saying YOU hate Canadians?
    Last edited: Aug 10, 2018

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