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

Are models based on probabilities `physical theories'? I suggest `physical models' necessarily require a cause - effect model. I think probability models suggest a cause - effect model. The probability models are measurement models.

For example:

Perhaps I should expand on one of the examples about a conceptual mystery such as the double slit experiment that math may show some insight to a better model. That the math of quantum mechanics (QM) works has been shown. However, the mystery is why? Thus, several concepts (interpretations) such as wave--particle--duality and the Bohm Interpretations have been conceived. But both (all) start from concepts and try to derive the Schrodinger equation (the math part of QM that works). But suppose we start with the observation that the math works and try to conceptually model why it works. Ask ``what is the math doing?'' Schrodinger equation defines the total energy as the sum of the potential and kinetic energy. The kinetic energy is the inertial mass energy. The potential energy derives from the potential field (whatever a field is) that imparts (somehow - by contact or action at a distance?) energy onto the (inertial) mass. These energies seem to have some relation to wave dynamics. Our scale (mm to km) observes waves in mediums that have unbounded differentiability (continuous). So I suggest the wave and its medium is real (yes I know there is argument here.) But then general relativity also mathematically suggests a gravitational ether. A gravitational ether (called ``space'' today) is influenced by matter and influences matter through a gravitational field that exerts a force by contact through its divergence. Just what is need for the photon inducing potential energy Photon diffraction and interference .

Would you classify the group models of particle classification the same as you classify statistical analysis? The periodic table was developed first by noting common characteristics of elements. A few holes were filled (predicted) by where the hole was in the classification scheme. Later, the causal underlying structure of atoms explained the periodic table. Indeed, the position of an element indicated something about the atomic structure. The same type of classification is true for the group models. Holes in the group model have been used to predict particles that were found. Can this be used to imply an underlying structure of particles? How would such a study proceed? Is anyone working on the structure of particles (papers I see seem to stop with the group description with no indication of an underlying structure)?

Dear Mauro,

Thanks for a lovely essay. I didn't know you were a structural realist. Are you fully against ontology, meaning that physical theory does not tell us anything about reality?

Another question. You're arguing for a purely mathematical set of axioms, but your own example contains notions like "our control" or "what we observe". These aren't necessarily physical; one may take them as instrumental or computational, as you rightly suggest. But they are surely not mathematical! Isn't it a problem for you?

Hope to see you around soon,

Alexei

Dear Alexei

it is a pleasure to hear from you. Thank you for reading my assay, and for kind your compliments.

It seems that you noticed in my essay a transition from operationalism toward structural realism. Indeed, in the past I stressed the operational vision against the ontological one, since, as I still maintain, ontological believes precludes new more effective viewpoints. I believe in a "reality without realism", as Arkady Plotnitski says, and this is well synthesized by a structuralist viewpoint a la Carnap, in which what should be the concern of science are not the "objects", but the relations among them. Also, my position is quite close to the internal realism of Hilary Putnam, as Gregg Jaeger pointed me out. In the past have been mis-identified with an operationalist a la Bridgman. What I maintain is the idea that we cannot "describe reality as it is" as Nino Zanghi says, since otherwise we should provide also initial conditions. We can only connect objectively known portions of reality with objective observations, but, due to possible existence of complementarity, we need also to choose which observation to perform. Therefore, we connect preparations with observations.

As regards the mathematical nature of our axioms for quantum theory [14], the framework is indeed monoidal category theory, and all notions are purely probabilistic, hence mathematical. What we "control" and what we "observe" is interpretation of the math. Finally, the algorithm is a strict mathematical notion (see Gödel) and computer science (the good one) is a branch of mathematics.

I will read your essay, and ask you questions.

I'm at your disposal for further clarifications.

My best regards

Mauro

Dear Mauro,

Another great essay from you, well done !

But next time, don't forget to put an accent on the "à" when you write "a la Carnap". I am French, so it was easy for me to spot ;-)

All the best,

Patrick

Dear Professor D'ariano,

You ended your essay with this insult: "The reader who considers the proposal of mathematization of Physics preposterously unfeasible has already given up the possibility of acquiring true knowledge in science."

Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

You have never had any understanding of reality.

Warm regards,

Joe Fisher

Dear Joe

I never had a post removed, in particular yours, which indeed seems to appear as duplicate here in the following. With your comments you are proving that there are different personal visions of "reality".

Thank you for your compliments about the "exceptionally well written" essay. My best regards.

Hi Giacomo,

Thank you for very interesting essay. As you have noticed there were many attempts to formulate axioms in physics (D. Hilbert, J. von Neumann, L. Nordheim, H. Weyl, E. Schrödinger, P. Dirac, E. P. Wigner and others). All these efforts failed . However a deductive system can consist of axioms or other, already established theorems. As far theorems were reserved exclusively for mathematics. That means that we can use theorems only if we accept that the reality is isomorphic to mathematical structures. (Not necessarily vice versa).

I propose to use the geometrization conjecture, proved by Perelman (so it is a theorem). We have the set of 8 Thurston geometries. We can treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime... and get all interactions and matter.

"Is mathematization of Physics premature?" No way.

If you are interested you can find details in my essay

http://fqxi.org/community/forum/topic/2452

I would appreciate your comments. Thank you.

Jacek

Dear Giacomo Mauro D'Ariano,

Great essay that I have strong agreement with. Starting with your comments ".. the point made here is that the theory should be a purely mathematical construction, whereas its physical connotation should pertain only the interpretation of the mathematics. An exemplary case is that of group theory and physical symmetries." and arguing "the Dirac equation is the archetype of a mathematical theorem with physical interpretation" you start with a strong base of provable mathematical models, but leave open the ability of overlay physical interpretations and models over this base.

I would be very interested in your comments on my essay here where I start with group theory and symmetries to build physical models of particles that match the mathematical properties of particles of the standard model. Finding objects that match S(3) and SO(3) groupings starts the quest and leads to objects that behave as SU(2) symmetries for electrons and SU(3) symmetries for quarks and hadrons.

Very insightful essay, enjoyed it a lot.

Regards and best of luck in the contest.

Ed Unverricht

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