The Experimental Verification of Mathematics via an Axiom of Measurement by Christopher Levi Duston

Armin Nikkhah Shirazi

Dear Chris,

Having myself first learned to think more like a physicist and only in the last 1.5 years or so more like a mathematician (though I find thinking like a physicist still more intuitive, or, I should say, easy to come by), I think I can well appreciate where you are coming from.

Perhaps it would be worthwhile to step back for a moment and think about exactly what role you want this to play in the connection between physics and mathematics. The spectrum ranges on one end from something like a philosophical or physical principle to the other as a rigorously formulated mathematical axiom. Where you consider your idea to fall on this spectrum determines how precisely you need to state it. Your choice of calling it an "axiom" led me to believe that you are in fact considering it as something much mores imilar to the latter. In that case, expressions like "maps to statements", which, I agree, are perfectly acceptable in physics, require some work to be intelligible to mathematicians. Your expression reminded me that set theory is not the only way you can try to incorporate your idea into the foundations of mathematics. Although I know at this time still very little about category theory, from what I do know I have the impression that "maps to statements" might be more easily accommodated there than in set theory.

As a final note, part of my interest in this probably reflects the fact that I am myself struggling with incorporating a general philosophical principle as a rigorous axiom into the foundations of mathematics.

Best,

Armin

Armin Nikkhah Shirazi

Dear Chris,

I noticed via Researchgate that recently it appears you took an interest in some of my articles.

Thank you for your interest, and I would like to note that my earlier papers, (pre-2015) particularly pertaining to the foundations of quantum theory, do not yet reflect an appreciation that (in my current view) the challenges in this field have their origin in the foundations of mathematics, particularly in the fact that mathematics, in its current standard form, does not have sufficient expressive power to describe in a formal language certain kinds of distinctions that are important in quantum theory.

I only began to acquire such an appreciation toward the end of 2013, and have accordingly shifted much of my focus to this area to learn more, having in the process found that the same distinctions have much broader applicability than I originally thought.

My essay in this contest did already outline many of the ideas that I am referring to, but since I wrote it, my aim has changed somewhat: I am trying to use as little additional logical machinery as possible in order to achieve essentially the same outcome. The reason for that is simple: The less you need to use, the more the new ideas that are introduced will be palatable to a wider audience, thereby increasing the chances of their general acceptance. Also, the less likely it is that the new ideas introduce new problems in some unanticipated way in other areas of mathematics.

Should you have any questions about any of the works, feel free to contact me.

Happy holidays,

Armin

Christopher Duston

Dear Odessa,

Thanks for showing an interest in this - yes, I agree, the next step in this project would be to find a formal system that naturally includes urelements of the sort needed. It's interesting, my main thought when writing the essay was how to carefully define the nature of truth in an experimental context, whereas most people's response has been like yours, worried more about the formal system.

In any case, it is a bit outside of my area of expertise but when I return to this project I feel that there are two major pieces:

1) Determine which variant of set theory is appropriate and carefully formulate it.

2) Use language within the formal system to carefully define the axiom of measurement. This would likely be a recursive sort of definition, since the truth in experiments only comes from the truth in earlier experiments. The choice of set theory would have to include such axioms.

After that, maybe try to fit a few physical theories (GR, QFT, etc) into this framework to illustrate how it can be used in practice to formalize these systems. Beginning with axiomatic field theory might be natural, since that philosophy is rather built-in already.

Again, thanks for you interest and your comments.

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