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 Mauro

It was a great pleasure to read your essay, I completely agree with your global development, and fully subscribe to the crucial aspect that you assign to group theory. There may be some philosophical differences between you and me. Your position regarding the choice axiom denotes that you are not exactly a Platonist, and for my part I am an incorrigible Platonist but aware of the difficulties that this approach confronts. Anyway, the profound agreement that I feel with your group theoretical consideration shows me that Platonistic/anti- or non-Platonistic discrepancies are often a question of words, and that there is something more fundamental difficultly to describe by ordinary language, but accessible to mathematical formalization.

If I understand correctly your developments, you say (i) that there is group theory as a (or the) fundamental part of mathematics and (ii) that phenomena belonging to the research field of physics are phenomena we can formalize in terms of mathematical groups. So physics in the bottom of things would not need physics. In other words, (iii) physics can be axiomatized just like any part of mathematics in this sense that physical axioms are experience oriented specifications or "interpretations" (in the model theoretical sense) of group theoretical axioms. Am I right?

Moreover, we can get the same result by heuristic considerations. A physical theory has any chances to be "good" if it permits symmetrically prediction and retro-diction starting from any possible state of the system in question. If this is not the case, the theory has a problem. Now it is obvious that symmetrical prediction and retro-diction presupposes the the underlying system can be formalized in terms of mathematical groups. That is why irreversible processes pose serious epistemological problems within physics. Much has been written about "law like reversibility v/s de facto irreversibility"; this discussion beginning with Boltzmann, Loschmidt, Zermello ...is聽far from reaching聽its end. Anyway, for a physical law to be a law stricto sensu, it must be reversible, so symmetrical in prediction and retro-diction. For this reason, biological issues like theory of evolution or ontogeny BY DEFINITION are not intrinsically mathematizable and represent in relation to physics another world. It is for group theoretical reasons that in matter of biological evolution and ontogeny reductionist approaches - here attempts to reduce biological phenomena to physical laws - have in my opinion no chance of success. Similarly, the Clausius "law" is not a law but a pseudo-mathematical expression, as evidenced by the pseudo-differential without real mathematical signification belonging to it. This is another example of the group theoretical essence of physical laws stricto sensu.

In a semi-technical end note of my own essay I touch briefly a group theoretical consideration which from my standpoint supports Platonism: Contrary to what common sense, intuition, and even simple grammar might suggest, irreversibility is not a direct negation of reversibility. In terms of group theory, these phenomena have nothing in common.

First an intuitive example. Consider an ideal watch without internal frictions etc. whose needles turn by their own inertia at a constant speed. This system, as long as nothing disturbs it, is reversible in terms of the spatial configuration of its needles; it will return to any configuration it occupies at a given moment. Under these conditions, the system (i) is characterized by an entropy variation equal to 0 and (ii) "remains the same" because it conserves its functioning mode. Now let us create an irreversible situation by projecting the system violently to the ground. This time the entropy variation is superior to 0, while the system - reduced to fragments - does not conserve its functioning mode. Nobody would seriously say that the fragments scattered on the ground are the "same" system as the ideal watch in operating condition. So reversibility PRESUPPOSES the conservation of the functioning mode characterizing the considered system, whereas irreversibility CONSISTS ON the transition [conservation of the functioning mode 鈫' non-conservation of the functioning mode].

The intuitive expression "functioning mode of a system" is certainly vague, but it can be formalized by the Klein 4-group where the combination of all the 4 possible transformations gives always the "identity transformation". More details can be found in the end-note of my essay. But briefly speaking, the Klein 4-group formalizes ultimately all systems remaining the same through their transformations. Any physical law is in fine an interpretation I(K4) of the intrinsically reversible Klein 4-group. Irreversibility is the transition I(K4) 鈫' non-I(K4). So "real" physical phenomena are superpositions of IDEAL reversibility and DE FACTO irreversibility. Hence a gas initially in disequilibrium, composed of molecules with their movements dictated by reversible Newtonian mechanics remains ideally reversible but describes de facto an irreversible transition. However no "real" physical phenomenon decriptible by a physical law despite its de facto contamination by irreversible factors could be physically known without this ideal law, this ideally reversible, so ideally eternal law being behind. For this reason, I think that the possibility of discovering physical laws stricto sensu in a "real" world characterized by irreversibility advocates Platonism.

Well, thank you again for your beautiful essay,

Best regards

Peter

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 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 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.

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

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