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TOPIC: 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 [refresh]

TOPIC: 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 [refresh]

Wigner called the effectiveness of Mathematics in Physics (and in natural sciences) "unreasonable". Against the widespread romantic position, I argue that what is unreasonable is the use of physical principles for founding physical theories. Physics without physics? This may seem an oxymoron. But 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. In contrast to the present call for mathematization, the current major physical theories either have mathematical axioms that lack physical interpretation, or have physical postulates. I therefore call for the construction of a theory that, though with limited (but relentlessly growing) domain of applicability, yet will have the eternal validity of mathematics. A theory on which natural sciences can firmly rely. This is what I consider should be the answer to the Hilbert’s call contained in his Sixth Problem.

Im am professor of theoretical physics at the University of Pavia, where I teach "Quantum Mechanics" and "Foundations of Quantum Theory", and enjoy research with a marvelous group of young collaborators.

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...

view entire post

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...

view entire post

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Dear John,

thank you for reading my essay.

Just few words (writing is very time consuming).

The probabilistic theory is more general than the deterministic one, since it includes it a a special case. Causality should not be confused with determinism. Falsifiability of propositions in a theory are guaranteed by an axiom, such as the perfect discriminability axiom of ref [14]. Probability is an extension of logic, and, this is the right spirit with which we should take probabilism. The Einsteinian preference for determinism has a merely ontological motivation, not a logical one.

Regarding the notion of particle, to which you seem mostly interested in, I only stress again that the notion that survives quantum field theory has nothing to do with that of a localizable object: the particle is an irreducible representation of the Poincarè group. What I tried to convey in my essay is that ultimately physics should be only an interpretation of a purely mathematical construction.

Regarding an underlying structure for particles, that's what my ref. [18] is about.

My best regards

Giacomo Mauro D'Ariano

thank you for reading my essay.

Just few words (writing is very time consuming).

The probabilistic theory is more general than the deterministic one, since it includes it a a special case. Causality should not be confused with determinism. Falsifiability of propositions in a theory are guaranteed by an axiom, such as the perfect discriminability axiom of ref [14]. Probability is an extension of logic, and, this is the right spirit with which we should take probabilism. The Einsteinian preference for determinism has a merely ontological motivation, not a logical one.

Regarding the notion of particle, to which you seem mostly interested in, I only stress again that the notion that survives quantum field theory has nothing to do with that of a localizable object: the particle is an irreducible representation of the Poincarè group. What I tried to convey in my essay is that ultimately physics should be only an interpretation of a purely mathematical construction.

Regarding an underlying structure for particles, that's what my ref. [18] is about.

My best regards

Giacomo Mauro D'Ariano

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

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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

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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...

view entire post

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...

view entire post

report post as inappropriate

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

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

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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

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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...

view entire post

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...

view entire post

report post as inappropriate

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

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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

report post as inappropriate

Dear Jacek

I'm in Nagoya, and is 0.30AM now. I need to prepare my talk for friday. I will read your essay definitely before sunday. It seems that we share the idea of a geometry as exemplar of a theory with physical interpretation. And I also believe that mathematization of Physics is not premature. The previous axiomatization attempts that you are mentioning failed because they were not geometrical, they included physical notions within the axioms. As I said, physics should stay only at the interpretational level of pure mathematics.

Thank you also for your nice compliments.

Until next on your blog.

My best

Mauro

I'm in Nagoya, and is 0.30AM now. I need to prepare my talk for friday. I will read your essay definitely before sunday. It seems that we share the idea of a geometry as exemplar of a theory with physical interpretation. And I also believe that mathematization of Physics is not premature. The previous axiomatization attempts that you are mentioning failed because they were not geometrical, they included physical notions within the axioms. As I said, physics should stay only at the interpretational level of pure mathematics.

Thank you also for your nice compliments.

Until next on your blog.

My best

Mauro

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

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Great essay that I have strong agreement with. Starting with your comments "

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

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Dear Ed,

I'm happy that you completely share my point of view, since it is very different from the common one, but it is crucial for driving theoretical physics toward the right direction.

You made me interested in your own essay, which I definitely will read soon. Consider only that I just came back from two intense weeks in Japan, and I am typing my answer in bed on an iPad.

Until next on your blog.

My best

Mauro

I'm happy that you completely share my point of view, since it is very different from the common one, but it is crucial for driving theoretical physics toward the right direction.

You made me interested in your own essay, which I definitely will read soon. Consider only that I just came back from two intense weeks in Japan, and I am typing my answer in bed on an iPad.

Until next on your blog.

My best

Mauro

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...

view entire post

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...

view entire post

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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...

view entire post

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...

view entire post

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

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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

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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...

view entire post

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...

view entire post

report post as inappropriate

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...

view entire post

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...

view entire post

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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...

view entire post

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...

view entire post

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

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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

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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

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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

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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.

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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

Best wishes,

Cristi Stoica

_______________

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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

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

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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

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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

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

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

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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

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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...

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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...

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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

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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

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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...

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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...

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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

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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

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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

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

Dear Mauro,

I am very glad I summarized all your points properly, for it means I understood them well. It was my pleasure doing so!

I hope your trip to China was as fruitful as you planned it to be. Welcome back!

Warmest regards,

Alma

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I am very glad I summarized all your points properly, for it means I understood them well. It was my pleasure doing so!

I hope your trip to China was as fruitful as you planned it to be. Welcome back!

Warmest regards,

Alma

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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

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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

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Peter

thank you for your post, and nice compliments, and a pleasure to meet you again in this competition. As I wrote in the last entries of this blog, I'm just back from China with an horrible internet. It is 1 AM here, and I am coming back from a dinner with a visitor. But I have still some hours to read your essay in time for voting it, looking at your video, and writing something on your blog.

Thank you again

My best regards

mauro

thank you for your post, and nice compliments, and a pleasure to meet you again in this competition. As I wrote in the last entries of this blog, I'm just back from China with an horrible internet. It is 1 AM here, and I am coming back from a dinner with a visitor. But I have still some hours to read your essay in time for voting it, looking at your video, and writing something on your blog.

Thank you again

My best regards

mauro

Mauro

Thanks for your kind comments on my blog. I responded there, reproduced below; (though I'm a little confused which is your christian name!)

Giacomo,

Interesting response. I see that view as rather 'dumbing down' the problem to that of the chicken and egg, suggesting the two views of 'which came first' are irreconcilable. In my first lines I accept your case, then reconcile it coherently with reality.

I've derived the chicken/egg solution elsewhere nearby; to simplify, It's a chicken, which contains an egg, which contains a chicken embryo, which contains the physical constituents required to form an egg, with a chicken inside, etc etc, chicken soup all the way down!

So at the bottom, when our microscopes get enough resolution do you think we'll find a mathematical formula written out? Will it be in Arabic numerals, perhaps Roman? Babylonian? or even Mayan? Or do you not agree that it's more likely to be some fundamental relative motion, spin or OAM state of motion DESCRIBABLE or representable by a simple equation (in Arabic or whatever system the observer wishes)?

In the same way SR and QM are probably only irreconcilable due to inadequate understanding I feel our approaches are Ying and Yang, both essential and more "inseperable" than irreconcilable, except perhaps in blinkered or 'tunnel' vision.

Do you really not agree? If so what would you expect to 'observe' as 'information' being processed at the smallest (sub-planck?) scale? A micro computer?

I'm not convinced we generally think things through thoroughly enough before pinning our colours to them them. Can you convince me?

Best wishes

Peter

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Thanks for your kind comments on my blog. I responded there, reproduced below; (though I'm a little confused which is your christian name!)

Giacomo,

Interesting response. I see that view as rather 'dumbing down' the problem to that of the chicken and egg, suggesting the two views of 'which came first' are irreconcilable. In my first lines I accept your case, then reconcile it coherently with reality.

I've derived the chicken/egg solution elsewhere nearby; to simplify, It's a chicken, which contains an egg, which contains a chicken embryo, which contains the physical constituents required to form an egg, with a chicken inside, etc etc, chicken soup all the way down!

So at the bottom, when our microscopes get enough resolution do you think we'll find a mathematical formula written out? Will it be in Arabic numerals, perhaps Roman? Babylonian? or even Mayan? Or do you not agree that it's more likely to be some fundamental relative motion, spin or OAM state of motion DESCRIBABLE or representable by a simple equation (in Arabic or whatever system the observer wishes)?

In the same way SR and QM are probably only irreconcilable due to inadequate understanding I feel our approaches are Ying and Yang, both essential and more "inseperable" than irreconcilable, except perhaps in blinkered or 'tunnel' vision.

Do you really not agree? If so what would you expect to 'observe' as 'information' being processed at the smallest (sub-planck?) scale? A micro computer?

I'm not convinced we generally think things through thoroughly enough before pinning our colours to them them. Can you convince me?

Best wishes

Peter

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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

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Best,

Gennaro Auletta

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Dear Gennaro

thank you for your compliments and your point raised! The essay would indeed need a little expansion to be complete. In short, the role of physics is not just the interpretation, but to build also the whole pragmatic knowledge to build up the interpretation, namely as I wrote in my answer to Michel Planat:

Physics is not only made 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).

I hope this answers your legitimately raised point.

See you in july!

My very best

Mauro

thank you for your compliments and your point raised! The essay would indeed need a little expansion to be complete. In short, the role of physics is not just the interpretation, but to build also the whole pragmatic knowledge to build up the interpretation, namely as I wrote in my answer to Michel Planat:

Physics is not only made 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).

I hope this answers your legitimately raised point.

See you in july!

My very best

Mauro

Dear Giacomo,

It is perfectly true that I red your essay several times from the beginning. I did the same for some essays that I did not comment and did not rate. It is interesting that our scientific evolution can be compared. I started as an engineer and did my "state thesis" in applied physics (nonlinear wave propagation), I did a lot of experiments at that time and also later. I arrived at quantum physics only twelve years ago and one training was to organize the ICSSUR'05 conference that you know about because you were in the scientific committee. As you I was not trained as a mathematician but always was fascinated by the tricks it provides. I knew about dessins d'enfants but it took 20 years before I found the opportunity to use them in quantum contexts. The Grail for me would be to use them in large scale physics or in biology.

Thanks for your comment on my page.

My best regards,

Michel

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It is perfectly true that I red your essay several times from the beginning. I did the same for some essays that I did not comment and did not rate. It is interesting that our scientific evolution can be compared. I started as an engineer and did my "state thesis" in applied physics (nonlinear wave propagation), I did a lot of experiments at that time and also later. I arrived at quantum physics only twelve years ago and one training was to organize the ICSSUR'05 conference that you know about because you were in the scientific committee. As you I was not trained as a mathematician but always was fascinated by the tricks it provides. I knew about dessins d'enfants but it took 20 years before I found the opportunity to use them in quantum contexts. The Grail for me would be to use them in large scale physics or in biology.

Thanks for your comment on my page.

My best regards,

Michel

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