Physics & Math: Certainly related thanks to the uncertain nature of them both. by Nikolaos Pappas

Nikolaos Pappas

Essay Abstract

Physics and Mathematics share a deep, dialectic yet not at all trivial relationship. From the ancient Greek natural philosophers to Newton and to 1990 Fields medals this relationship has been ever-present with all its tensions and contradictions yet repeatedly producing impressive results. Its many successes made the physics-math connection seem absolutely natural and self-evident. However, the basis of it is rather ill-understood and to these days we conceive it primarily in a phenomenological way. In this essay a different approach is proposed. Examining the analogy between Heisenberg's Uncertainty Principle and Godel's Incompleteness Theorems we argue that both sciences, being much more than human inventions, are obliged to incorporate the same fundamental restrictions and the tendency to deal with the same issues as reflections of their common origin, Nature, thus, in this context, the striking effectiveness of mathematics in fundamental physics comes as no surprise at all.

Author Bio

2013: PhD in Theoretical Physics. Thesis title "Black hole properties in the context of 4- and higher-dimensional gravity theories". 2006: degree in Physics (first in class during both my under- and postgraduate studies). 2002: degree in Medicine. 9 published papers with 50 citations. 1 chapter in collective volume. Invited speaker in 3 international conferences. Peer-reviewer for 5 international scientific journals. Speak 5 languages: greek,english, french,german, spanish.

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Gary Simpson

Nikolaos,

This was a nice essay ... very easy to read. Many of the essays present similar arguments, so you are clearly in the mainstream of thinking. Godel features prominently in many of the essays.

Best Regards and Good Luck,

Gary Simpson

Nikolaos Pappas

Thank you for your kind comment, Gary.

I feel that Godel's theorems were expected to be used a lot in this conversation since, while producing a groundbreaking result, they are rigorously proven and this is a feauture that most (if not all) of the ideas concerning the math-physics connection do not have!

Allow me to point out, however, that the approach of the incompleteness theorem as an uncertainty principle in math is an original idea that goes around in my head for some time now which I hope could shed new light in our understanding of the math-physics connection.

Michel Planat

Dear Nikolaos,

Very well written and convincing short essay. You arrive at S. Hawking conclusion (quoted at the end of my essay) in a sligthly different way. My remark is that if one is able to incorporate several aspects of the reality in the same description (e.g. group, numbers, topology, geometry and so on, as Grothendieck suggest) then the uncertainty vanishes somehow. It is just that the 'projections" are "orthogonal", as in the Heisenberg's sense.

Good luck,

Michel

Nikolaos Pappas

Dear Michel,

thank you for your kindness.To be honest, I was not aware of the talk by S. Hawking that you mention until now. I am excited nevertheless that the views of this great physicist come so closed to the idea presented in my essay.

As far as your remark is concerned, I am stand in the oposite side. I think that the limitations expressed by the uncertainty principle and the incompleteness theorems are absolutely foundamental and cannot be lifted no matter how many parameters we take into account. Furthermore,I am not quite sure about the legitimacy of argument about "orthogonal projections". The way I understand things, Heisenberg's picture of quantum mechanics is not to be comfused with the uncertainty principle.

Thank you for your time.

Good luck to you too,

Nikos

George Gantz

Nikolaos -

Thank you for the very nice essay! The connection between incompleteness and uncertainty is interesting and novel, but I'm not sure they are quite cut from the same cloth. There are also other dialectic problems, for example wave/particle duality in physics and non computability in math that also, according to many, cannot be reconciled.

I fully agree with you as to the indeterminacy of "Nature" - including both math and physics. But I'm wondering how you respond to that. This is the key topic in my essay, and I would appreciate your thoughts when you have a chance to read it.

Sincerely - George Gantz

Nikolaos Pappas

Dear George,

I am very pleased you enjoyed my essay. The way I see it, both uncertainty and incopleteness actually state that a system cannot always have fully determined properties at the same time thus expressing a fundamental limitation posed By Nature and not by human (in)ability or any technological limitations so in your words I actually do think that they are cut from the same cloth. DUalities like the one you mention on the other hand do not cause any problems since one can usually switch smoothly from one picture to its dual.Like in the case of quantum mechanics which at the proper limit smoothly goes to classic mechanics (as a matter of fact the same also hold true for the transition of discrete to continuous mathematics - a nice example that phys and math share common qualities).

I promise to find the time to read your essay soon and share any thoughts.

All the best,

Nikos

Edwin Klingman

Dear Nikolaos Pappas,

I too found your essay elegant, almost poetic. As you do not attribute your statement at the top of your essay, I assume it is your own. As obvious as it is, I do not believe I have ever seen it so clearly stated, that our most profound and elegant theories state:

"...our knowledge of Nature is, and shall forever remain, uncertain (Heisenberg) and incomplete (Godel)..."

Certainly appears to leave room for free will, does it not?

I see you are a particle physicist, and I note that Marni Dee Sheppeard, in her essay, notes of the Standard Model that

"...enormous effort went into maintaining locality, while quantum physics would abandon it."

As another comment on my essay states, the problem with Bell's theorem has a "self-concealing nature", which I have tried to unveil. It is not a math error, but essentially a 'map' error, that derives from an over-simplified local model. I hope that you will read my essay and welcome any comments you might have.

Thank you for your inspiring essay concerning, as you say, the "flame inside the mind and soul" of the physicist.

With best regards,

Edwin Eugene Klingman

Nikolaos Pappas

Dear Edwin,

First of all please excuse me for taking so long to reply to your post. I'd like to thank you for your thoughts and comments. Even more I'd like to thank you for recognizing the poetic mood I was in when writing this essay! The opening statement you mention is indeed mine and as a matter of fact it serves as the opening statement of my PhD thesis.

It certainly leaves room for modesty since this kind of limitations are valuable reminders not to be too arrogant as far as to what is our place in Nature is concerned. Free will together with consciousness are far beyond the reach of Physics and/or Math to my mind.

Being a gravity-ist myself, I have made appeal to the EPR non-locality in one of my papers and if you ask me I feel quite comfortable with the so-it-seems non-local nature of quantum physics even though I have many unanswered questions about it running in my head.

I shall read your essay and get back to you on that.

Once again, thank you and good luck.

Best,

Nikos

Chidi Idika

Dear Nikolaos,

Your conclusion that: "... there is a fundamental indeterminacy in Nature that is directly reflected upon the sciences derived by its study." is exactly what I adopt as a postulate in my essay Observer as the Mathematician's "Constant" and the Physicist's "Quantum".

But I must admit that what is most odd to connect with is my stance that this fundamental indeterminacy is the very state we all know as "the observer". Yet exactly this stance is the required leap. Think about it, "observer" is strictly speaking not a physics concept nor is it a mathematics concept but it captures in an intuitive sense man's place in the scheme of things.

Now if you can picture observer rather in the non-local sense of a sensory threshold/initial condition, akin to Newton's infinitesimal or Boltzmann's entropy, namely as at once the effective unit and limit of physical information (observables) the picture begins to emerge: observer can indeed be the same state physics calls the quantum of observables (the natural unit). And hence the connection between Godel and Heisenberg.

I invite you to read my essay and leave your frank comment.

Joe Fisher

Dear Nikolaos,

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

James Hoover

Nikolaos,

We all seem to struggle over the connection of the micro and the macro worlds.

Your pointing out the analogy between Heisenberg's Uncertainty Principle and Godel's Incompleteness Theorems imposing the same fundamental restrictions and are more than human inventions makes me wonder about the quantum-classical connection. Heisenberg is looking at the quantum world and math mostly at the classical world.

My essay mentions connections of mind, math and physics citing studies that look at macro mysteries like the navigation of the European robin and the process of photosynthesis, pointing out quantum connections scientists have found.

I agree with your mention of the "unreasonable effectiveness" of math in physics but that triad connection of math, mind and physics -- I think -- makes it all work: http://fqxi.org/community/forum/topic/2345.

Jim

Sujatha Jagannathan

You have commemorated well from the beginning till the end with hopeful inputs.

- Sincerely

Miss. Sujatha Jagannathan

Jason Edwards

Dear Nikolaos,

You found a connection where there is none.

"There is a straightforward and rather shocking similarity between this theorem and Heisenberg's uncertainty principle."

The latter is about measurements, the former is about axiomatic foundations.

Jason

Nikolaos Pappas

Dear Jason,

"You found a connection where there is none...The latter is about measurements, the former is about axiomatic foundations."

This is exactly how I do not see things. Allow me 2 comments.

1. The uncertainty principle is not about measurements. It lies at the foundations of Physics as core property of Nature. In other words it is as profound for Physics as incompleteness theorems are for Mathematics.

2. Things in many cases might appear to be of different nature but more often than not in science we just see them this way because we lack a deeper understanding or their connections. And when we get the right ideas we find it totally natural to think of them as different aspects of the same entity. Electric and magnetic phenomena used to be considered distinct until the work of J.C.Maxwell proved that they all stem from the same source and today we always say "electromagnetism". Particles used to be considered as different from waves. Quantum mechanics introduced the wave-particle duality and the "dispute" was over. In the same sense I believe that the similarities of uncertainty and incompleteness indicate that there is an underlying connection between them which is also projected and manifested by the efficiency of math in describing physical processes.

All the best,

Nikos