PROOF IS PHYSICAL

”Information is physical”. Always. Of course. What else?

Yet, the mystery is far from dispelled, as we don’t know what “physical” is. We don’t know, what physics is, for sure. Some roll out the Quantum, and say:”here is physics: it from bit”. However, we are not certain of what the Quantum is (= we don’t know whether quantum theory is “complete” or not; ultimately it’s a Physical Problem, experimentally determined; Von Neumann thought he had a “formal” proof, but he was wrong).

Are there Physical Problems that are not Mathematical Problems? Or Physics Proofs that have not Mathematical Proofs? Well, at this point, there are. Take general fluid flow. Be it water inside a fluid, or a meteor going hypersonic, these Physical Problems exist, and have solutions, that the physical objects themselves are Physical Proofs. It is not clear that they have Mathematical Solutions, let alone Mathematical Proofs.

Theorems From Physics? claims that:

“mathematical theorems are not supposed to be contingent. This is a fancy philosophical term for propositions that are “true in some possible worlds and false in others.” In particular, the truth of a mathematical proposition is not supposed to depend on any empirical fact about our particular world.”

With all due respect, that’s theology. Conventional theology, so called “Platonism”, but still theology. For me Plato, and his modern parrots are seriously obsolete, and “an embarrassment, for these people are friends”, as Aristotle put it.

I can show that the proof that square root of two is irrational contains assumptions made on an empirical basis (along the lines of mn = nm, actually; similarly, the choice between Presburger arithmetic and Robinson, or Peano, or Ayme arithmetics, can be viewed as empirically driven.)

However, what is an achieved mathematical proof? Just a neural arrangement. Similar neural arrangements in the minds of noble primates called mathematicians. Thus, a mathematical proof is a physical object constructed similarly in the minds of many. So a mathematical Proof is a Physical Proof, just as the fluid in a tube is a Proof of a Physical Problem, the flow problem. And similar tubes have similar “proofs”, once similar fluids similarly flow.

So any Mathematical Proof is a Physical Proof.

***

Patrice Ayme

***

Notes:

1) Could Quantum Theory be Wrong?

(Meaning not as perfect as it is taken to be.) Actually the main objection I have against the Quantum-as-it-is is exactly the same as the objection Isaac Newton had against his own theory of gravitation: instantaneous interaction at a distance with nothing between made no sense, said Newton.

(Einstein remedied this partly by proposing that gravitation was a field propagating at the speed of light.)

2) The preceding was a comment of mine on the “Gödel Lost Letter and P=NP” site in Theorems From Physics?

And most notably the following passages: “The philosopher in us recoils dogmatically at the notion of such a “physical proof”…  Imagine that someone shows the following: If P is not NP, then some physical principle is violated. Most likely this would be in the form of a Gedankenexperiment, but nevertheless it would be quite interesting. Yet I am at a loss to say what it would mean. Indeed the question is: “Is this a proof or not?”

Actually this is exactly the general method I used to prove there is a largest number. Basically, I said, if there is infinity, there is a violation of the conservation of energy principle. Oh, by the way, if you want to know, in my system, the proof of P = NP is trivial (as everything is polynomial; four words proof, so I should the Clay Prize, hahaha)…

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30 Responses to “PROOF IS PHYSICAL”

  1. Dominique Deux Says:

    True, the support or medium of mathematical proof is material, or physical.

    Yet a given proof can be formulated by minds which never had knowledge of each other, or even mutually held knowledge. Pascal was kept away from mathematics as a child; yet he proceeded (at ten) to formulate anew, from scratch, all the major tenets of Euclidian geometry, using his own words (“rounds” for circles, “bars” for straight lines, and so on). To the Platonician herd, this is “proof” of the independent “existence” of mathematical proofs and concepts in a separate plane. To non-Platonicians, myself included, this merely means that mathematics go a step further than physics in dissociating themselves from sensory experience. I’ll leave this at that – There are salaried metaphysicists to sift through that muck – us laymen should only take care to eradicate theological contamination on sight.

    • Patrice Ayme Says:

      Thanks Dominique! Well, that’s roughly my position. The physics of neural networks of calls mathematics. That’s why dogs know calculus, without having gone to school, but for the school of hard knocks.

      I did not know this about Pascal. I am dubious that he was truly isolated. People in his family (uncle) taught Einstein calculus when he was in his early teens. (Usually smarties say Einstein was no mathematician, and Einstein said that himself about himself; but few 13 year olds, even now, know calculus, and without it, no Einstein physics… Although Special Relativity he copied from Henri Poincare’ requires little calculus, if any, the reasonings are in common.)
      PA

  2. Paul Handover Says:

    Sorry, I have been avoiding this place. The ‘book’ is down to the last 2,000 words and all other aspects of my life have been put on hold!

    • Patrice Ayme Says:

      Very good, Paul, I will read your book when finished! I have been sort of maximally busy, and now I’m engaged in traditional Thanksgiving travel… Will see what happens to the price of fuel, when fracking has exploded…
      PA

      • Paul Handover Says:

        And it is now finished; all 53,400 words. Would love you to read it but not!! until it has been edited! A very Happy Thanksgiving from Jean and me to you and all your family.

        • Patrice Ayme Says:

          Very happy thanksgiving to you and Jean, and your overseas brood… I am totally swamped by all I have to do. Just on dogs + Neanderthals, I have three no less, systems of thought in different dimensions (N invented dogs; Y the racism against Ns?; what does it mean to “invent” dogs?)

          Plus my ideas in math (math from physics) seem to be gaining traction, finally!
          PA

  3. Serge Says:

    Serge

    November 27, 2013 9:18 am

    “What is an achieved mathematical proof? Just a neural arrangement. Similar neural arrangements in the minds of noble primates called mathematicians. Thus, a mathematical proof is a physical object constructed similarly in the minds of many. So a mathematical Proof is a Physical Proof.”

    Well, not exactly. It’s rather the duplication of those neural arrangements from one brain to another that deserves to be called a proof. This is usually achieved by means of a text written in some human-readable language. But whenever you replace the neurons with transistors, you get what’s usually called a program. I suspect the difficulty of P?=NP to reside in what programmers call a deadlock, that is two concurrent processes which prevent each other from running. The novelty being here that the deadlock occurs in the head of the computer scientist!

    • Patrice Ayme Says:

      Thank you for commenting, Serge. I don’t see any difference between what you said, and what I said. But maybe I miss something, and you can instruct me.
      It’s true that the duplication is achieved by text. That is by neuronal, or more exactly, axonal communications. And that, one can in turn duplicate that with transistors. So far, so good.

      However, one should pay attention to this: there is much more to the brain than axons. There are dendrites, but also astrocytes. This introduces a TOPOLOGICAL character to thinking absent in transistor networks. That’s also why there is more to human mathematics than just programming in a computer.

      More to say, but I have to travel.
      PA

  4. Patrice Ayme Says:

    Gowers: The best experiments give us the best kind of direct contact with reality that we have. It would be naïve to think otherwise (a naivety some philosophers have been culprit of). Such experiments are precisely designed by trying to eliminate all what could go wrong and all what could act as a screen between us and reality.

    If a philosopher of science thinks otherwise, she/he ought to come up with a better thought experiment. To start with. Otherwise she/he would be just striking a pose.
    PA

  5. Alexi Helligar Says:

    “The P = NP problem has been solved, I must admit, in my humble opinion, very elegantly…”
    Very interested! Tell more.

  6. Alexi Helligar Says:

    Patrice Ayme: “Any Mathematical Proof is a Physical Proof.”

    Alexi Helligar: I’m getting the feeling that all of physics is geometry on a non-Euclidean space.

  7. EugenR Says:

    Dear Patrice, I watched recently some of the videos of Richard Feynman about the subject mathematics-physics and find it fascinating and funny. It is pleasure to listen to this great intellectual, who made the most abstract thoughts understandable to almost everyone and one of the best standup comedies as well;

    here is the link to the first of the serial of many short videos;

    • Patrice Ayme Says:

      Dear Eugen: I did not get the link. Yet. Pls persist. be it only for other readers. Feynman was a great physicist, and very smart.
      One of my earliest book was a full edition of the Feynman lectures on physics… Years, if not a decade or more, later, I noticed a mistake, a pretty bad one, in his exposition of the 2-slit. Ironical, as Feynman viewed it certainly as the greatest riddle in physics, as I do. (of course Feynman knew the physics, but the way he exposed it would have led the naive into error; I don’t know if the mistake has been corrected in more recent editions.)

      And I have my a suggestion about the 2-slit!

      I had the priviledge to talk with him one on one about physics, and more precisely about that… My main idea pertaining to the Quantum and the Big Bang. “It would change everything“, he had the kindness to say.

      My theory predicted actually something akin to Dark matter and even Dark Energy. Enormous amounts of it. As observed. Feynman would have been totally fascinated by these recent developments… That I had predicted to him, and we convened that finding such would be an indication of the validity of my flight of fancy…

      Funny that there are strong indications that I am right, whereas the SUSY bandwagon where zillions of the best and brightest worked to great fanfare and celebration are, so far, coming completely empty, said the LHC (I did work on the math side of it).
      PA

  8. EugenR Says:

    An other one,

  9. Paul Pieter Kruijmer Says:

    “Any Mathematical Proof is a Physical Proof”?
    It is normal, physics are based on mathematiques on a lot of situations.

    • Patrice Ayme Says:

      Paul Pieter: Let’s say that mathematics presents physics with all-ready-made programs of computation. But a question has always been: where does mathematics come from?

      Plato (who was not a mathematician, although he understood their importance as a discipline of the mind) said: in another world, taylor made for math. Most mathematicians, to this day, follow that dubious, not to say absurd, theology. I don’t.

      I say instead: mathematics comes from physics. Consequence: physics direct can demonstrate mathematics. My point of view has been gaining, very recently. It has/will have tremendous consequences on mathematics.
      PA

  10. Serge Says:

    November 27, 2013 9:39 am

    I agree that the physical universe can prove its own consistency, but that seems to be all it can do from a mathematical viewpoint, doesn’t it? Because, as long as the universe hasn’t delivered the totality of its axioms to us poor humans, no physical proof will be rigorously convertible into a convincing mathematical proof…

    • Patrice Ayme Says:

      Serge: The universe has been perceived to present naturally with some axioms. However, that was with yesterday’s physics, I claim there are new, stronger axioms. Just ignoring them does not mean they can be ignored while preserving veracity.

      For example, one is free to do mathematics with (A and Non A) viewed as true. One can deduce lots of things from that. Actually anything whatsoever. But it does not make those deductions true.

      I use energy conservation. That axiom from physics is relatively recent, and has had not been integrated in mathematics. Until very recently (by me).
      PA

  11. gmax Says:

    How do you feel that famous academics are adopting your ideas without quoting you? And then attributing them to each others? As if they had invented them?

    • Patrice Ayme Says:

      Human beings are, in general, nothing, but for the tribal structures within which they grow. Krugman clinging to Summers at this point is examplary. It’s the same in math, physics, philosophy, banking, etc.
      PA

  12. Serge Says:

    I agree that the physical universe can prove its own consistency, but that seems to be all it can do from a mathematical viewpoint, doesn’t it? Because, as long as the universe hasn’t delivered the totality of its axioms to us poor humans, no physical proof will be rigorously convertible into a convincing mathematical proof…

    • Patrice Ayme Says:

      Serge: The universe has been perceived to present naturally with some axioms. However, that was with yesterday’s physics, I claim there are new, stronger axioms. Just ignoring them does not mean they can be ignored while preserving veracity.

      For example, one is free to do mathematics with (A and Non A) viewed as true. One can deduce lots of things from that. Actually anything whatsoever. But it does not make those deductions true.
      PA

  13. Serge Says:

    November 27, 2013 3:26 pm
    Patrice,

    There are certainly a few other things more to the brain! Some people even conjecture that the conscious part of the mind be a quantum computer… I find this hypothesis quite seducing, since it would imply that classical computers can only have a “subconscious” way of thinking. I don’t know where it leads, but I like it.

    The difference with what you said in the first place lies in the necessary distinction between processes and programs. Neural arrangements aren’t proofs (programs), they’re thoughts (processes). Making this distinction’s vital to understanding what’s expected of a proof: it must be efficient as a communicating device.

    My theory of mental deadlock seems likely – at first sight – to explain a lot of impossibilities in complexity theory, but I have to make sure it does work.

    • Patrice Ayme Says:

      Serge: A lots of meta characteristics of the quantum and of consciousness are similar. And, moreover, the quantum makes biology possible (non locality itself is used for computing/finding the lowest energy solutions; see chlorophyll).

      What I was pointing at was much more prosaic: there is a whole topological computing going in the brain… on top of, but entangled with, the axonal system. Without going quantum, that makes human brains completely different from transistor networks.

      You tell me a proof is “expected” to be “efficient as a communicating device”. Well, efficient long range communication is precisely what axons do in the brain.

      It is true that brain regions can shut down other brain regions (even without epilepsy; it actually happens routinely, everyday.)

      If all brains are similarly limited, one obviously constrains what one can talk about. Including “proofs”.
      PA

      • Alexandre de Castro Says:

        Patrice,
        today, I posted a preprint showing that connection. P =? NP is intrinsically linked to the entropy constraint.

        http://vixra.org/abs/1403.0058

        Dick and Ken in the entry entitled “Theorems from Physics” also discuss some ideas of my theoretical work on one-wayness.

        See the central idea in my current preprint:

        “In computer science, the P =?NP is to determine whether every language accepted by some nondeterministic algorithm in polynomial-time is also accepted by some deterministic algorithm in polynomial-time. Currently, such a complexity-class problem is proving the existence of one-way functions: functions that are computationally ‘easy’, while their inverses are computationally ‘hard’. In this paper, I show that this outstanding issue can be resolved by an appeal to physics proofs about the thermodynamic cost of computation. The physics proof presented here involves Bennett’s reversible algorithm, an input-saving Turing machine used to reconcile Szilárd’s one-molecule engine (a variant of Maxwell’s demon gedankenexperiment) and Landauer’s principle, which asserts a minimal thermodynamic cost to performing logical operations, and which can be derived solely on the basis of the properties of the logical operation itself. In what follows, I will prove that running the Bennett’s algorithm in reverse leads to a physical impossibility and that this demonstrates the existence of a non-invertible function in polynomial-time, which would otherwise be invertible.”

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