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I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics h

I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics has an axiomatic basis. When certain mathematical theories can be "applied" to real world problems, it simply means the relevant axioms and definitions are a robust approximation of reality. To speak to your example directly: machine learning absolutely has an axiomatic basis. You can conduct legitimate research in implementations and software or hardware optimizations thereof; however, fundamentally every experimental result in machine learning is an application of a variety of theorems in linear algebra, probability theory or calculus.

lapinot 7y ago
Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization. On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?

lapinot 7y ago

Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization. On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?