Dear Dr. Moldoveanu,
Thank you for your kind words. Sorry for the delay in getting back to you. I have a ready answer, but I realized that you have also entered this essay in this contest and thought it best to read your essay in detail. I found it very interesting at several levels, but I need to think about it before commenting. However, it is immediately obvious to me that we have very different views on mathematics and physics.
My formal training is in pure mathematics with the emphasis on foundational matters, mostly mathematical logic. After my schooling I continued to study this topic and eventually became somewhat disillusioned with it. This motivated me to switch my interest to physics. As a "Johnny come lately" to physics I still have a great deal to learn, so I must beg your indulgence.
In my study of physics I noticed that I was thinking like a mathematician rather than a physicist. These modes of thinking are quite different and I have struggled to make my patterns of thought more physical than mathematical. I have attempted to look at physics not in a mathematical way, but in a physical way. Shedding my baggage as it were.
My views on mathematics, if not radical, can be described as controversial. As for physics I am, at least in part, not as well informed as I would like. With my clumsy apology in place allow me to address your concerns.
You say that "Mathematics is infinite and supposing physics is math, new content can appear in physics." I do not believe that "physics is math". There are very pronounced differences in content, patterns of thought and method. It is these differences that draw me away from mathematics toward physics.
While mathematics considers infinite objects (e.g., the natural numbers or the real line) I do not feel that it is essentially infinite. One may consider theorems that require an infinitely long statement or theorems that require infinitely long proofs, but this is hardly mainstream. While proofs can be rather long (e.g., the four color theorem) and programs very involved (e.g., the classification of finite groups) they remain finite as all human efforts must.
When I speak of content of a subject I mean to discuss what the subject is about. Classical logic, in my view, is about correct modes of reasoning. This is its content. The classical logician doesn't care what about what you argue (your content), only the the way you go about it. If your reasoning is incorrect, your argument is dismissed out of hand. If your reasoning is correct, then only your basic premise (essential content) may be argued. The maintenance of validity with no regard to content is where logic gains its strength. Indeed, the word validity comes to us from the Latin word for strong.
The mathematician is concerned with the abstract study of things like quantity (as in arithmetic) or form (as in geometry.) The properties of these things and their interrelations, expressed as theorems, is the content of mathematics. In my thinking I find it convenient to regard the axiomatically stated premises of mathematics as expressing the "essential content". This is where the content of mathematics is brought into contact with logic. Unlike Kant, I believe that mathematics has content beyond logic.
The content of physics is, to my eye, vastly different from that of logic and mathematics. It seems clear that physics is about (its content) physical reality. The scope of this content is so large that I feel that mathematic and logic, as they now stand, may not be up to the task. That is why I mentioned modal and paraconsistent logic and mathematics developed in these frameworks. These things lead me to believe that modern (as opposed to classical) logic and mathematics will extend or at least mitigate the limits of physics in the future.
I suspect that my opinions and beliefs will change as I learn more about physics, but for now I hope I have spoken to your concerns.
Sincerely,
Jim De Spears
