Timeline for In what ways is physical intuition about mathematical objects non-rigorous?
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| Jun 13, 2010 at 9:21 | comment | added | jeremy | Similarly, when I took mechanics, it disturbed me a little that we'd use things like delta-function "impulses." The argument given for why that's okay was "in real-life the impulse isn't really infinite" but to me that seemed to make the problem even worse! Of course, the real "physical" reason (and the one a good professor would've given!) is in terms of thinking about what it means in terms of Newton's laws being satisfied, position being continuous, it being okay if e.g. velocity isn't, etc. That's not good enough for an ODE class, but if your careful it leads to the real proofs. | |
| Jun 13, 2010 at 9:15 | comment | added | jeremy | Absolutely; what I should have said was: "it does not matter to physicists." As in, personally, it's not something they explicitly think about. Clearly, in terms of the actual structure, everything matters, whether they know it or not! But the average physicist is, really, only barely aware of why the delta function is not a function, let alone what would tell them what kind of functions their framework tells them they can use. For example, in quantum mechanics, knowing only "conservation of probability" and "things are normalized" is typically enough to settle these kinds of issues. | |
| Jun 13, 2010 at 5:03 | comment | added | Tim Perutz | Jeremy, whilst physicists typically do not care to declare a class of functions, I contest your assertion that the class rarely matters. In wave mechanics, the probabilistic interpretation of overlap integrals dictates that wavefunctions should live in a subspace of $L^2$ (this is not guaranteed by the Schroedinger equation alone), with its standard inner product. If I write down a different inner product, you will have to figure out an isometry to the standard $L^2$ product before you can do physically meaningful calculations. | |
| Jun 13, 2010 at 3:59 | comment | added | jeremy | The specific classes of functions we want rarely matter, aside from "functions which somehow satisfy this equation" where somehow may include distributionaly, or in some cases "not at all." Though from a theory-building point of view, they're more critical; general relativity is built off of diffeomorphisms after all... | |
| Jun 13, 2010 at 3:54 | comment | added | Theo Johnson-Freyd | I recently attended a talk by Graeme Segal in which he pointed out that you should never ask physicists to decide what type of functions they want. | |
| Jun 12, 2010 at 22:05 | history | edited | Dan Piponi | CC BY-SA 2.5 |
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| Jun 12, 2010 at 18:17 | history | edited | Dan Piponi | CC BY-SA 2.5 |
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| Jun 12, 2010 at 18:05 | history | answered | Dan Piponi | CC BY-SA 2.5 |