First, I'll briefly introduce myself, seeing as I just joined Physicsforums.com. I am a 2nd-year student at the University of Toronto in the physics specialist program, also pursuing a math major. I haven't actually been exposed to much physics in a formal setting since since my "real" physics courses only start this fall - up to now I've just done the introductory physics survey course and a bunch of math (currently taking ODEs and multivariable calc) - that is to say, forgive me if my knowledge is lacking in the fundaments (i.e. electrodynamics, classical and quantum mechanics formalism, stat mech, etc.).(adsbygoogle = window.adsbygoogle || []).push({});

On to the actual topic: lately I've gained an interest in the philosophy and underlying logical framework of math. Up until a few months ago, I always thought that the basics were entirely agreed upon by those well-versed in the field of mathematics - that is, the set theoretical bases, theory of proof, and principle axioms of mathematics were developed and studied enough to relegate new study of the bare basics to interesting quirks of little relevance or application (example: the formulation of fuzzy set theory). Then I discovered the constructivist school of thought. In this, some include the idea that for mathematical objects (like, for example, real numbers) must be constructable to be validly used when making inferences based on their existence. For example, the mean value theorem could be considered invalid simply because it postulates the existence of a real number without any reference to a method of constructing it. If one gives credence to this (albeit strict) interpretation of certain mathematical philosophies, then doubt is cast on the very existence of the real number system itself. If R is defined as the closure/completion of Q, using convergence of sequences, then there are (in terms of cardinality) "many more" irrationals than rationals, of which not all are constructable. Thus, in the strict interpretation, the very existence of the "continuum" (Rn) is invalid, and so calculus goes out the window.

I don't necessarily advocate this idea. But taking it into consideration for the sake of the argument, we can consider the implications for physics. Considering just about everything in physics is ultimately posed in terms of differential equations (from the most basic a(t)=v'(t)=x''(t) to general relativity), and that the very notion of differentiability (only definable on complete spaces like Rn) is now called into question, physics becomes at best a very good approximation. Any attempts to use physics, even in the most empirical sense, to explain ultimate mechanisms become questionable.

I could go on. I'm interested to hear others' opinions on the matter.

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# Reliability of physics as an exact description of the universe

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