• Trick or Truth Essay Contest (2015)

  • The Unreasonable Effectiveness of Mathematics in Physics: the Sixth Hilbert Problem, and the Ultimate Galilean Revolution. For a mathematization of Physics by Giacomo Mauro D\'Ariano

Dear Peter

thank you very much for your compliments, and for your very intriguing comments, which also make me very curious about your own essay (in these days I had too little time for reading essays, which I will doit in these Easter vacations).

I'm happy that you completely share the crucial role of group theory in physics, and consider it as exemplar of the role of math in physics.

You raised a really relevant point about irreversibility, a point that indeed comes out natural in regards of group theory. I have a definite answer for your point. The answer is: the "purification axiom" of quantum theory (Ref. [14]). This means that there is always a theoretical description that is reversible, whereas irreversibility can be always regarded as lack of knowledge, e.g. as the result of partial observation of some systems only. I don't know how this point of view matches your: maybe you can explain this to me.

My fundamental logic is the following. 1) For falsifiability of the theory, we must keep theory and experiment sharply logically distinct (confusing the experiment with its theoretical description is the most common and subtle error in physics research, and it is much more common of what physicists realize). Such a sharp logical distinction is achieved only if: 2) The physical theory is a chapter of pure mathematics, and physics is only interpretation of math (though holding from the very axiomatics up to theorems). Now, epistemological motivations of axioms in terms of logical falsifiability under control of systems (in preparation, transformations, and observations), lead to the axioms of quantum theory, including reversibility (which is part of the purification axiom). Therefore, the falsifiability of the theory under control needs reversibility--not determinism, as supposed by realists.

I would really appreciate further feedback from you.

I strongly believe in the need of a complete mathematization of physics as a crucial methodological step to address and coordinate physics research toward a coherent progress, as it happens in contemporary mathematics.

Thank you again for your interesting post,

and see you on your blog.

Mauro

Dear Mauro,

I just sent a post trying to answer to your questions. It is on my own forum.

Tomorrow I will approach your respond figuring here.

Happy Easter,

All the best

Peter

Dear Mauro

I have carefully read your reply and will try to answer.

First, regarding the "purification axiom", I knew the principle but ignored this denomination; it is a precious information and during Easter vacations - in France there are 3 academic vacation zones and here in Paris it is going from April 16 (I believe) to something like May 2 - I will read "Informational derivation of Quantum Theory"

Personally I agree that quantum reversibility must be posed as an axiom defining experimental quantum irreversibility as lack of information related to micro-macro issues. But regarding information theoretical approaches, there is a significant micro-macro clash: At the macroscopic level, the only lack of information is not sufficient to found irreversibility. By definition, we ignore all from the content of black box, comprising its order state and/or -transitions. For there is irreversibility in the black box, intrinsic factors must play a role.

Contrary to what a great number of authors says, macro-irreversibility is a question of information theoretical entropy AND thermodynamical entropy being certainly linked, but - Brillouin showed it in the fifties - not identical. More precisely, the Boltzmann k having the dimension of an energy is not simply a specification of the dimensionless k figuring in Shannon entropy or in other info theoretical entropy formulations. The links between thermodynamical and info theoretical entropy remain controversial. This is an other argument that macro-irreversible phenomena are not covered by physical theory stricto sensu, whereas, quantum irreversibility being ONLY a question of lack of information favors mathematization.

I agree with all your points 1) to 5), except perhaps a small detail. What do you mean by "realist"? Do you mean "Platonist"? If it is so, I think personally that group theoretically founded reversibility fits well with Platonism, unlike vaguely formulated "determinism". But this will be another debate.

All the best, happy Easter still again

Peter

Dear Mauro--

This is a very significant and thought-provoking article, even though I find some of its claims a bit unqualified. But then, their uncompromising character gives force to your argument, too. One cannot really do the article justice, apart from writing a full-fledge, article-length response to it. Let me, however, venture a few comments of both historical and conceptual nature.

I do agree that your argument has affinities with structural realism, but I shall not address these affinities here, except for noting that your claims are stronger than most made by structural realists (there are other differences as well). Historically, I find your argument close to Heisenberg's later views, when his views become closer to those of Einstein from the late 1920s on (after his work of general relativity, as grounded in Riemann's geometry, and during his work on the unified field theory), rather than to those of Bohr, whom Heisenberg followed initially. At this point, Heisenberg was also influenced by Dirac, whose views were in turn close to that of Einstein in the late 1920s. Cf., such famous pronouncements as "the laws of physics must have mathematical beauty," although this is Dirac's, or Einstein's, deeper thinking on these matters that is important here. I do agree with you that it is not beauty of mathematics, but mathematics itself, sometimes as the mathematics of beauty, that is crucial. Heisenberg also expressly linked his view to Plato: "If we wish to compare the finding of contemporary particle physics with any earlier philosophy, it can only be with the philosophy of Plato; for the particles of present-day physics are representations of symmetry groups, so the quantum theory tells is, and to that extend they resemble the symmetrical bodies of the Platonic view" ("What is an Elementary Particle?" Encounters with Einstein, p. 83). At one point, Heisenberg goes as far as claiming that from the viewpoint of modern quantum physics, Kant's things-in-themselves are mathematical.

That said, however, there is also a continuity with Heisenberg's and, especially, Dirac's earlier views, based in the physical principles. What they maintained, however, was that such principles must necessarily have their mathematical expression, which why Dirac realized a crucial role of noncommutativity so quickly and so deeply (quicker and more deeply than anything else, including Heisenberg, who had his doubt about it). All of Dirac's work may be seen as dedicated to the task of finding the mathematica expression for his principles, and in this sense is deeply mathematical, which is not quite the same as but is a link to the type of argument your advance here. Also, its mathematical nature is not equivalent to doing mathematics (there is an interesting contrast to Von Neumann here). It may indeed be that the disciplinary mathematics is not mathematical enough for physics, which more deeply mathematical than mathematics. Plato argues a similar case for philosophy, which he thought should be more mathematical than mathematics, as a disciplinary practice. In this, it may be less important to be a good mathematician that a truly mathematical physicist, which is to easy, however.

My main point at the moment is that your reversal of Dirac's earlier view, that is, your argument to the effect that these are in fact mathematical principles that must be given physical expressions, is nevertheless both a continuation of and a break from Dirac's program. An important article!

Arkady Plotnitsky

    Dear Arkady

    it is a great pleasure reading from you. Thank you for your very nice compliments, and your insights.

    I agree that my argument is not defining structuralism as such, but I think that it is the logical consequence of taking structuralism seriously. I will enjoy discussing with you in Växjö about this point, since Carnap has been so influential in my work. I understand why you may find my argument closer to Heisenberg's later views rather than to those of Bohr, if we consider that here I'm proposing a complete mathematization of the theory, whereas Bohr never achieved even a minimal mathematization e.g. of his complementarity principle--though his ideas yet not precisely defined were crucial for the progress of the new theory--whereas Heisenberg was somewhat more mathematical in his matrix mechanics, although he was not so in his gedanken experiment. However, you know me personally, whence you also know that I'm very close to Bohr in his operational viewpoint, and I am sure that, giving him an additional life to spend, after his masterpiece discoveries--which, as any act of creation, do not follow from pure logic--Niels would have definitely pursued a mathematization program. Indeed, as you know, the principles that I propose are meant to deriving the largest possible amount of physics as "physics as necessity", from principles of epistemological nature. The mathematization of theory is the only option for logical coherence, whence for falsifiability. If principles contain physical terms with no mathematical definition we are in serious troubles. The fact the the current theory is still not fully mathematized is the reason why physicists overlook circular definitions--e.g. those of inertial mass or force (see the three books devoted to these notions by Max Jammer)--or they forget the issue of the so-called "quantization rule", which is undefined for a general mechanical observable, or they don't care about mathematically undefined tools as the path-integral. Ultimately this has consequences at the science-sociological level, where we are witnessing a review process that is becoming almost utterly opinionated, instead of being a thorough analysis of new proposed theory.

    I love your sentence about the "the mathematics of beauty", in place of "the beauty of the mathematics", and the connection with Plato cited from Heisenberg. I definitely agree with Heisenberg in claiming that from the viewpoint of modern quantum physics, Kant's things-in-themselves are mathematical. Indeed, things-in-themselves are equivalence classes of experiences, and, as such they are mathematical--though not "a priori". What I don't agree with is the "a priory nature" of mathematics. I also agree that my mathematization program is a continuation of Dirac's program, and a break with Dirac's only in not assuming Lorentz invariance as a principle, which instead I derive as a theorem.

    Thank you again for your post

    My best regards

    Mauro

    Dear Mauro--

    Thank you for your kind reply. I entirely agree, and I should have noted this relation to Bohr when I spoke of both the continuity and the break with the preceding views of Heisenberg and Dirac, who were both heavily influenced by and indebted to Bohr, including as concerns the operational aspects of their thinking. And as I said, they certainly continued to maintain the role of (suitably mathematized) physical principles throughout. Indeed, I agree that your own operational framework continues this tradition and contributes to it, and also poses important questions concerning how the mathematical formalism arises from these principles. I also agree on Bohr and mathematics, and I have often argued myself that there are more complex relationsships between mathematics and physics (e.g. Epistemology and Probability, pp. 24-25, 131-136). Indeed, Bohr has an interesting late 1956 essay on the subject, ''Mathematics and natural philosophy,'' in The Philosophical Writings of Niels Bohr, Volume 4: Causality and Complementarity, Supplementary Papers, eds., J. Faye and H. J. Folse (Ox Bow Press; Woodbridge, CT 1994), 164-169. This is only to reiterate that your article raises important historical and philosophical issues concerning the relationships between mathematics and physics, which quantum mechanics made us to rethink. We are far from finished with rethinking, have be rely began it. We are also continuing the debate between Plato and Aristotle concerning the nature of mathematical reality vs. physical reality.

    Regards,

    Arkady

    Dear Giacomo,

    I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

    All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

    Joe Fisher

      Dear Joe,

      I have downloaded you essay, and am now reading it. Please, look at your essay blog.

      My best regards

      Mauro

      Dear Mauro,

      I finally got to read your essay, and I was very pleased. You show very eloquently the difference between eternal and provisional, between mathematics and physics. In the same time, you show with concrete examples how to also make physics eternal, through mathematization. I liked the examples of purely mathematical structures from which physics emerges, in particular the isospin and SU(3) symmetries. And the suggestion that physics should be taken as an interpretation of mathematics, rather than seeing mathematics as a model, an approximation of physics. You said "The reader who considers the proposal of mathematization of Physics preposterously unfeasible has already given up the possibility of acquiring true knowledge in science." I agree, and I said somewhere else "I think that we should admit supermathematical* descriptions as final only if we are sure that we exhausted any hope for a mathematical description. And I don't think this is possible"

      Best wishes,

      Cristi Stoica

      _______________

      *Supermathematical is to mathematical what supernatural is to natural.

        6 days later

        Dear Cristi

        sorry for not replying immediately (I'm in Beijing, and the internet is not so good ...).

        Thank you for your wonderful post. You understood everything of my essay! You got the point!

        I agree perfectly with your sentences:

        "I think that we should admit supermathematical* descriptions as final only if we are sure that we exhausted any hope for a mathematical description. And I don't think this is possible"

        and love the following one:

        "Supermathematical is to mathematical what supernatural is to natural".

        I already wanted to read your essay: I'm doing it right now (if the intended allows it)!

        Hope to know you in person.

        My best wishes to you

        Mauro

        Dear Mauro,

        Maybe we will meet again at TM2015 (I saw you at TM2012, and we've already met last year in February in Bad Honnef, where we discussed a bit about rishons).

        Best wises,

        Cristi

        Dear Cristi

        we met both times very shortly. I hope to see you soon. I'm not sure I will come to TM2015, since I'm teaching two courses in October. Besides I will say that I don't believe on time travels, and I can give good reasons for this--e.g. that time travels would violate quantum theory.

        I read your essay, and I liked it very much. I'm now posting on your blog.

        My best regards

        Mauro

        5 days later

        Hi Mauro,

        Nice essay! I noticed you cited Steven French. He was the external advisor on my PhD thesis. It is somewhat ironic, then, that despite both his influence and Eddington's influence on me, I'm not much of a structuralist, at least in the same sense as you.

        Anyway, speaking of Eddington, I found your approach to be quite reminiscent of Eddington's work in the latter part of his career. In fact he essentially espoused the same exact thing: a purely deductive view of the physical universe. I think this approach, though, only works if mathematics is "discovered" as opposed to "invented." That is a highly debatable point. I tend to think that some mathematics is and some isn't.

        I think the biggest problem I have here is that even a purely mathematical way of looking at things is limited by Gödelian incompleteness, i.e if you want to fully axiomatize physics you will ultimately find that some things are known to be true (perhaps through experiment) but unprovable within that axiomatic framework. There's a great quote from Hawking at the end of Michel Planat's essay that aptly sums up this idea.

        Another issue I had was this: if physics is really just mathematics, but not *all* mathematics as you say (e.g. infinity is just a convenient approximation), and we have to add conditions like decidability and causality to the framework in order to sift out the "right" mathematics, how do we know that the conditions we're using are unique? In other words, how do we know we're not simply projecting our own bias onto the theory? How can we be sure of an objective reality (or don't you believe in an objective reality)?

        Otherwise, it was certainly a stimulating and thought-provoking essay!

        Cheers,

        Ian

          Dear Giacomo,

          I think you may be interested in this paper by Pierre Cartier "A mad day's work : from Grothendieck to Connes and Kontsevich the evolution of concepts of space and symmetry "http://library.msri.org/books/sga/from_grothendieck.pdf

          At the end he mentions that the "cosmic Galois group" acts on the constants of physical theories. But I am not sure that pure mathematicians are able to select the type of mathematics that optimally fit the physics.

          Your essay is excellent and I learned much reading it several times since it appeared. However I do not have the same view than you: I think that it would be dangerous to axiomatize physics, do you remember you learning Bourbaki at school?

          Best,

          Michel

            Dear Giacomo,

            Your ideas are wonderful and I'm sorry I only got to your essay now. You are making a compelling argument for returning to the old and profound purposes that have now been forgotten. Hilbert Sixth problem is equivalent to searching for a Holy Grail of physics, in that it is a quest for finding coherence and meaning. I think such a theory would have an unseen rigorousness and you are doing a very good presentation of the motives. I think your speech about geometrization is raising the important point that geometry helps in finding intuitively a physical interpretation of mathematics because it shows the relational structure. I will end with the beginning and I will say that I was greatly amused by how you (truthfully) characterized Wigner's statement as romantic. Excellent work! Wish you good luck in your work and in the contest!

            Warm regards,

            Alma

              Giacomo,

              Your essay was interesting and well argued, and I found myself in agreement with much of it, importantly in strong agreement with the need for proper physical theories to found the mathematical descriptions. (John Hodge directed me to it as agreeing with my own methodology and I'm glad I managed to read it)

              I see you teach QM, so I hope you may read my essay if only to consider and comment on the physical 'quasi classical' mechanism I identify allowing a mechanical derivation of the results we assign as 'quantum non-locality'.

              You may also be able to understand the highly compressed (experimental) video of a physical dynamic model based on the same foundations, i.e. using your approach to put your theory into practice. To answer the chicken and egg question; here is a viable chicken. It also has an egg inside, which has a viable chicken embryo inside, which has a group of spin states configured to give the ability to produce eggs, etc. The particles clearly came first.

              Possible physical mechanism and implications video

              A comprehensive analysis paper on Bell's 'theorem' etc. and the limits of the D'espagnolet Wigner ineqaulity identified in my essay is available which I hope you may look at after the contest.

              Well done and best of luck in the final dash. But do please comment even after scoring.

              Peter

                Excellent point! Nevertheless, I ask myself if the business of physics is only interpretation of mathematics provided that in quantum mechanics we can analogically extend our labs operations to the external world (assuming that spontaneous physical processes determine outcomes in ways that cannot be totally dissimilar to those controlled by us) and nevertheless we cannot extend our interpretation too far precisely because the latter processes are uncontrolled. In a way it seems to me that you say the same when you speak of physics as "connecting experimental observations", while the "portraying phenomena" can be still true but with a narrower scope.

                Best,

                Gennaro Auletta

                  Dear Ian

                  thank you for your compliments and your interesting comments. You reminded to me to buy the Eddington book on Amazon about the physical law, which can be a great source of inspiration for me. I know from our last discussion (in Boston, am I correct)? that you are not a structuralist. I didn't know that Steven French was your external advisor. How can it be that then you don't completely agree with me?

                  Let me answer to your two pints, very briefly.

                  The power of mathematics in physics is mostly resides in the mathematical induction (not in the physical one, which is logically fallacious, as Hume first noticed). With the mathematical induction, with a couple of modus ponens you prove an infinite large set of true assertions. But as you know, you do not prove the assertion for n=infty! Working with infinity is for mathematical convenience only, for describing analytically asymptotics. If you avoid the "actual infinity" but you work only with the "potential infinity" how can you get an unprovable assertion from 1+1=2, 1+2=3, 1+3=4, ...? Moreover, a theory should be required to be computable, namely allowing a finite-state machine one must be able to describe a finite portion of reality. Clearly you can incur into incompleteness if you want describe the whole universe. But not for a finite portion of reality, as long as you keep it discrete! Again, continuum is a limiting case of discrete ...

                  Second point: About causality, you forget that in my work with Chiribella and Perinotti on the axiomatic derivation of quantum theory we have a precise mathematical definition of causality, which has exactly the physical interpretation that a physicist would agree with.

                  Finally: there are bias that are universally shared. For example: we believe in logic, we believe in the universal validity of a law (or we treated it as such), we believe in isotropy of the low, we believe in locality of interaction. If you don't share one of this biases (which are examples of the principles from which we derive also free QFT), then I would be very interested in knowing which one you don't share. Don't tell me that the law is not universal, as Lee Smolin says, otherwise you are required to provide a higher level law accounting for the variation of the low-level law--otherwise you are giving up the theory. For how long would be a law required to be valid in order to be a law? A century? a second?

                  And, finally: I don't care about objective reality. What I care is that the theory is describing objective experimental data. My daughter still believes that Santa Claus is an objective reality, but the only objective facts are the presents that she gets.

                  Thank you again very much for your compliments and stimulating comments.

                  I will look forward to meeting you soon again. I'll be in Chicago the full August. If you'll pass by, please let me know.

                  My very best regards

                  Mauro

                  Dear Michel

                  thank you for your really nice compliments: you read my essay several times, really? I have been in China with a bad internet, and just back in Italy I find visitors. I will read you essay tonight.

                  There is maybe a misunderstanding about my essay, anyway. A crucial point in my essay is the physical interpretation of the theory, starting from axioms up to all of their consequences. Mathematicians cannot do this work. You need to know physics! Physics is not only of theory (I take the word "theory" very seriously here). Physics is made of laws of limited validity, heuristics, models, etc. or of theories that are not completely mathematical. When I interpret the results or the axioms of a Theory (with capital T) I can either refer to observations, or to heuristics, models or theories (with lowercase t) which just synthesize a huge set of observations, but they don't have the logical coherence of a Theory, whence, are not strictly logically falsifiable (according to Popper, they are a little magical).

                  You know that I am a theoretical physics, but you don't probably know that I started my research work as an experimentalist in my early post graduate years! There I learn one thing from my mentor (prof. Attilio Rigamonti): you want to really understand what's going on, then you must solve the problem at hand with whatever means are needed. If you need to know new mathematics, then learn it! The goal is a logical proof of your thesis, not an exhibition of technical knowledge. But later in my experience I discovered that most theoreticians learn some mathematics and then they seek the problem that may be addressed with the math that they know. I'm not this kind of scientist. And I'm definitely not a mathematician: I don't have the training. I just want to understand. One of the main lessons from Einstein is that if you really pursue the logic of what follows from your principles, you may discover new mathematics previously unknown to physicists. And I find this is marvelous!

                  Thank you again for your feedback and for your nice compliments.

                  My best regards

                  Mauro

                  Dear Alma,

                  I'm delighted that you really share exactly all points I raised! And, let me say that I really love your assertion saying "Hilbert Sixth problem is equivalent to searching for a Holy Grail of physics, in that it is a quest for finding coherence and meaning." Yes! You are right. This is the Holy Grail of physics! And, it maybe the Holy Grail of Popper (Popper said that when you find a theory is like when you find the Holy Grail: you may never know if it is the true one, up to when you falsify the theory, and then you know it wasn't).

                  As I told to Michel, I'm just back from China with an horrible internet. I can now enjoy reading your own essay, of which I'm now very curious.

                  Thank you again for your wonderful and unique comment!

                  My best wishes to you.

                  Mauro