DOING AWAY WITH INFINITY SOLVES MUCH MATH & PHYSICS

Particle physics: Fundamental physics is frustrating physicists: No GUTs, no glory, intones the Economist, January 11, 2018. Is this caused by a fundamental flaw in logic? That’s what I long suggested.

Says The Economist:“Persistence in the face of adversity is a virtue… physicists have been nothing if not persistent. Yet it is an uncomfortable fact that the relentless pursuit of ever bigger and better experiments in their field is driven as much by belief as by evidence. The core of this belief is that Nature’s rules should be mathematically elegant. So far, they have been, so it is not a belief without foundation. But the conviction that the truth must be mathematically elegant can easily lead to a false obverse: that what is mathematically elegant must be true. Hence the unwillingness to give up on GUTs and supersymmetry.”

Mathematical elegance? What is mathematics already? What maybe at fault is the logic brought to bear in present day theoretical physics. And I will say even more: all of today logic may be at fault. It’s not just physics which should tremble. The Economist gives a good description of the developing situation, arguably the greatest standstill in physics in four centuries:

“In the dark

GUTs are among several long-established theories that remain stubbornly unsupported by the big, costly experiments testing them. Supersymmetry, which posits that all known fundamental particles have a heavier supersymmetric partner, called a sparticle, is another creature of the seventies that remains in limbo. ADD, a relative newcomer (it is barely 20 years old), proposes the existence of extra dimensions beyond the familiar four: the three of space and the one of time. These other dimensions, if they exist, remain hidden from those searching for them.

Finally, theories that touch on the composition of dark matter (of which supersymmetry is one, but not the only one) have also suffered blows in the past few years. The existence of this mysterious stuff, which is thought to make up almost 85% of the matter in the universe, can be inferred from its gravitational effects on the motion of galaxies. Yet no experiment has glimpsed any of the menagerie of hypothetical particles physicists have speculated might compose it.

Despite the dearth of data, the answers that all these theories offer to some of the most vexing questions in physics are so elegant that they populate postgraduate textbooks. As Peter Woit of Columbia University observes, “Over time, these ideas became institutionalised. People stopped thinking of them as speculative.” That is understandable, for they appear to have great explanatory power.”

A lot of the theories found in theoretical physics “go to infinity”, and a lot of their properties depend upon infinity computations (for example “renormalization”). Also a lot of problems which appear and that, say, “supersymmetry” tries to “solve”, have to do with turning around infinite computations which go mad for all to see. For example, plethora of virtual particles make Quantum Field Theory miss reality by a factor of 10^120. Thus curiously, Quantum Field Theory is both the most precise, and most false theory ever devised. Confronted to all this, physicists have tried to do what has worked in the past, liked finding the keys below the same lighted lamp post, and counting the same angels on the same pinhead.

A radical way out presents itself. It is simple. And it is global, clearing out much of logic, mathematics and physics, of a dreadful madness which has seized those fields: INFINITY. Observe that infinity itself is not just a mathematical hypothesis, it is a mathematically impossible hypothesis: infinity is not an object. Infinity has been used as a device (for computations in mathematics). But what if that device is not an object, is not constructible?

Then lots of the problems theoretical physics try to solve, a lot of these “infinities“, simply disappear. 

Colliding Galaxies In the X Ray Spectrum (Spitzer Telescope, NASA). Very very very big is not infinity! We have no time for infinity!

The conventional way is to cancel particles with particles: “as a Higgs boson moves through space, it encounters “virtual” versions of Standard Model particles (like photons and electrons) that are constantly popping in and out of existence. According to the Standard Model, these interactions drive the mass of the Higgs up to improbable values. In supersymmetry, however, they are cancelled out by interactions with their sparticle equivalents.” Having a finite cut-off would do the same.

A related logic creates the difficulty with Dark Matter, in my opinion. Here is why. Usual Quantum Mechanics assumes the existence of infinity in the basic formalism of Quantum Mechanics. This brings the non-prediction of Dark Matter. Some physicists will scoff: infinity? In Quantum Mechanics? However, the Hilbert spaces which Quantum Mechanical formalism uses are often infinite in extent. Crucial to Quantum Mechanics formalism, but still extraneous to it, festers an ubiquitous instantaneous collapse (semantically partly erased as “decoherence” nowadays). “Instantaneous” is the inverse of “infinity” (in perverse infinity logic). If the later has got to go, so does the former. As it is Quantum Mechanics depends upon infinity. Removing the latter requires us to change the former.

Laplace did exactly this with gravity around 1800 CE. Laplace removed the infinity in gravitation, which had aggravated Isaac Newton, a century earlier. Laplace made gravity into a field theory, with gravitation propagating at finite speed, and thus predicted gravitational waves (relativized by Poincaré in 1905).

Thus, doing away with infinity makes GUTS’ logic faulty, and predicts Dark Matter, and even Dark Energy, in one blow.

If one new hypothesis puts in a new light, and explains, much of physics in one blow, it has got to be right.

Besides doing away with infinity would clean out a lot of hopelessly all-too-sophisticated mathematics, which shouldn’t even exist, IMHO. By the way, computers don’t use infinity (as I said, infinity can’t be defined, let alone constructed).

Sometimes one has to let go of the past, drastically. Theories of infinity should go the way of those crystal balls theories which were supposed to explain the universe: silly stuff, collective madness.

Patrice Aymé

Notes: What do I mean by infinity not constructible? There are two approaches to mathematics:1) counting on one’s digits, out of which comes all of arithmetics. If one counts on one’s digits, one runs of digits after a while, as any computer knows, and I have made into a global objection, by observing that, de facto, there is a largest number (contrarily to what fake, yet time-honored, 25 centuries old “proofs” pretend to demonstrate; basically the “proof” assumes what it pretends to demonstrate, by claiming that, once one has “N”, there is always “N + 1”).

2) Set theory. Set theory is about sets. An example of “set” could be the set of all atoms in the universe. That may, or may not, be “infinite”. In any case, it is not “constructible”, not even to be extended consideration, precisely because it is so considerable (conventional Special Relativity, let alone basic practicality prevent that; Axiomatic Set Theory a la Bertrand Russell has tried to turn around infinity with the notion of  a proper class…)

In both 1) and 2), infinite can’t be considered, precisely, because it doesn’t finish.

Some will scoff, that I am going back to Zeno’s paradox, being baffled by what baffled Zeno. But I know Zeno, he is a friend of mine. My own theory explains Zeno’s paradox. And, in any case, so does Cauchy’s theory of limits (which depends upon infinity only superficially; even infinitesimal theory, aka non-standard analysis, from Leibnitz + Model Theory survives my scathing refounding of all of logics, math, physics).  

By the way, this is all so true that mathematicians have developed still another notion, which makes, de facto, logic local, and spurn infinity, namely Category Theory. Category Theory is very practical, but also an implicit admission that mathematicians don’t need infinity to make mathematics. Category Theory has now become fashionable in some corners of theoretical physics.

3) The famous mathematician Brouwer threw out some of the famous mathematical results he had himself established, on grounds somewhat similar to those evoked above, when he promoted “Intuitionism”. The latter field was started by Émile Borel and Henri Lebesgue (of the Lebesgue integral), two important French analysts, viewed as  semi-intuitionists. They elaborated a constructive treatment of the continuum (the real line, R), leading to the definition of the Borel hierarchy. For Borel and Lebesgue considering the set of all sets of real numbers is meaningless, and therefore has to be replaced by a hierarchy of subsets that do have a clear description. My own position is much more radical, and can be described as ultra-finitism: it does away even with so-called “potential infinity” (this is how I get rid of many infinities in physics, which truly are artefacts from mathematical infinity).  I expect no sympathy: thousands of mathematicians live off infinity.

4) Let me help those who want to cling to infinity. I would propose two sort of mathematical problems: 1) those who can be solved when considered in Ultra Finite mathematics  (“UF”). 2) Those which stay hard, not yet solved, even in UF mathematics.

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11 Responses to “DOING AWAY WITH INFINITY SOLVES MUCH MATH & PHYSICS”

  1. SoundEagle Says:

    Hi Patrice,

    Happy New Year to you! Thank you for writing this excellent post to put infinity back in its place, if it has any place at all but in the dark recess of a black hole. 😉

    I would like to point out that there are typos in your sentence “In any case, it is not “constructible”, not even to be to be extended consideration,” where “to be” has been repeated.

    I wonder what your opinions are on the following articles:

    http://nautil.us/issue/45/power/this-man-is-about-to-blow-up-mathematics

    http://nautil.us/issue/46/balance/the-fifth-force-of-physics-is-hanging-by-a-thread

    http://nautil.us/issue/2/Uncertainty/the-deepest-uncertainty

    • Patrice Ayme Says:

      Hi SoundEagle! I thought I had answered you, but something went wrong, I just discovered. And happy new year to you too, and thanks for the appreciation and for pointing out the typo (which I removed). I am unaware of all these articles. I will look at them ASAP. (I tend to be severely busy!!!!!!!!!!!!!!!) Keep them coming. Except whe there are multiple links, your comments should appear immediately!

      • SoundEagle Says:

        I did not receive any answer or reply from you other than the one that you just made about two minutes ago.

        • Patrice Ayme Says:

          OK, I read the articles. Thanks. I knew of Friedman. His adviser, Putnam was also famous. One could say that his “Emulation Theory” is the opposite of what I am doing.
          There is an agenda for him, one for me: he tries to make relevant 150 years of infinity craze. I try, instead, to wipe it out, so mathematicians could concentrate on useful math (that’s a lot of math, including all differential equation theory, geometry, topology, etc.)

          The fifth force? The only seriously 5th force proposed has been MOND, and now I view that ugly, ad hoc theory as demolished
          https://patriceayme.wordpress.com/2017/03/31/dark-matter-emergence-if-so-is-a-new-quantum-revolution-at-hand/

          The French “Microscope” experiment was the torsion balance writ large.

          Cantor became mad. Need I say more? I studied the infinity stuff for decades, in the same mood as Friedman… Before realizing my mistake… Suddenly I realized the most basic proofs of math had logical gaps, trying to prove what they had assumed to start with… But then, when confronting mathematicians it’s often like visiting cardinals in 1600 CE, and telling them god is all in their heads… 😉

  2. benign Says:

    Patrice,

    What about Zero? Wouldn’t Zero be lonely without Infinity? And why should there be only one Zero? Isn’t a bound implied? Is Zero a “proper zero” or an imposter? If there is no Infinity can there also exist a Zero?

    I have just finished reading Claude Swanson’s “Life Force: the Scientific Basis,” all 750 pages of it, which argues that the Russians have a theory due to Nicolai Kozyrev (debunked as pseudo-science by Wikipedia) of left and right-handed “torsion fields” that accounts for dark matter and quite a few other anomalous gravitational and paranormal-type things, that has allegedly accumulated quite a bit of experimental confirmation behind the “propaganda curtain” (see http://journals.sfu.ca/seemj/index.php/seemj/article/download/425/386 for intro).

    I believe that this is really the forefront of physics research, as it also addresses the issue of consciousness.

    Infinity poses a problem for some
    the problem is really ’bout something or none
    you can’t have it all
    so go on and bawl
    ’cause scrazy is always how Big Things are done

    cheers (just having fun),
    benign

    PS scrazy = strange + crazy

    • Patrice Ayme Says:

      “Zero” is not a problem, it can be defined in many ways, all equivalent. “Zero” is so to speak a physical object, in particular a lack of action, or operation. I could have zeroed you, by considering you a big zero, not taking any action to reply to your comment, for example.
      There is definitively a zero operation. Not taking action is zeroing it.
      There is no infinity operation. One can’t just add infinity to a normal (definable, constructible) operation and get a normal, definable, constructible operation.
      The whole gymnastic about the Incompleteness Theorems is all about that, making refined distinctions between notions of infinity. It made big news at the time, but, in the end, Godel didn’t present us with a constructible Godel number. Precisely because his reasoning uses infinity! So Goedel says: there are Goedel numbers… Yes, as long as you believe in the Cantor Diagonalization process as capable of producing a number (it can’t!) Each time infinity appears, real numbers disappear!

      Physicists with renormalization knowledge may contest what I just said. But actually renormalization processes amount to taking limits, and substracting them from each other. I have no problem with that, as the arithmetic cut-off a la Patrice would apply to them.

      “Torsion” is strictly defined in manifold theory, it’s something beyond (so to speak) curvature, and there is none so far observed in the physical world. Entanglement, on the other hand, which is infinitely (hahaha) stranger, is observed, and there is no space for it in manifold, or supermanifold, or varifold theories…
      Entanglement generates space, but how it does is not clear (Cal Tech’s Sean Carrol had a paper on this, a few days ago). This is my main area of scientific research, and my viewpoint is gaining audience… I am happy to report…
      Just grim fun here too.
      Happy new year…

  3. benign Says:

    🙂 well done, good answer. however, Kozyrev’s “torsion” is a whole different animal, allegedly capable of affecting gravity and time. Swanson has surveyed the Russian experimental literature and finds it credible; it goes back a ways. he’s a Princeton physics Ph.D. in gravitation theory. you might look into it

    given the paranormal experimental results, eg Jahn and Dunn, which I believe, I conceptualize of the physical vacuum as a “singularity” in which everything is connected to everything else across all space and time, out of which our local reality is spun. psychics see into it through a glass darkly, to coin a phrase.

    Godel was right, IMHO. every theory is incomplete.

    grim cheers and happy new year to you too

    b

    • Patrice Ayme Says:

      Glad you like it. That every theory is “incomplete” has a very precise meaning in Godel case, and a meaning that rests on infinity. In a trivial way, every theory, in the sense of every 2nd order logic, being limited in its universe, is incomplete… The man in the street doesn’t need to no more than that. Decades ago, I was discussing the subject with top notch mathematician friend of mine, and we stumbled on the fact that the “Godel numbers” are NOT constructible.
      The average research mathematician doesn’t give a hoot about incompleteness. My guess is that most of them don’t know the Godel proof(s). Nor does it matter from what I said. Actually some of professional math friends, after talking to me checked that their math didn’t depend upon infinity. BTW, as I said, Category Theory eschews the problem of infinity (under the rug!)

      My beef with physics is that THERE, in Quantum Field Theory, computations are pushed to infinity, raw and direct.
      I don’t believe in “paranormal”, UFO, and winged horses carrying Muhammad to bother the Jews in Jerusalem… BUT, I do believe evolution is intelligent, and can see how it works, and that’s plenty enough to have many conventional people suspecting me of weirdness, so I am fully satisfied…

  4. Gmax Says:

    If you are right, this will put math on its head. Physics aside, what’s the interest of your vision for pure math?

  5. ianmillerblog Says:

    Hello Patrice – happy new year! Needless to say, being something of a rebel, I am happy enough to include infinity, but as a warning sign – “Don’t go there!” To me, quantum field theory has to be wrong – you can’t go around cancelling infinities because they don’t have precise values by definition. What is infinity squared? INFINITY! Also, any theory that gets a prediction wrong by a factor of 10^120 has to be wrong. And it manages this even after cancelling out some infinities. Think of what it could do if it left them in.

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