Timeline for Proofs that require fundamentally new ways of thinking
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Dec 11, 2010 at 23:26 | history | edited | Timothy Chow | CC BY-SA 2.5 |
Added a paragraph
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| Dec 11, 2010 at 17:26 | comment | added | Terry Tao | (and more relevantly, computer programmers need not be doing any mathematics in order to create that software.) | |
| Dec 11, 2010 at 17:19 | comment | added | Terry Tao | Well, the mathematics only comes in at the metalevel rather than at the field itself. Another example would be software engineering; this is a discipline that benefits tremendously from the presence of formal computer languages, which can then be studied mathematically from a computer science standpoint, but the software itself need not have any mathematical content. | |
| Dec 11, 2010 at 17:11 | comment | added | Michael Hardy | But if you axiomatize some portion of some other science, doesn't that axiomatization constitute mathematics? So it seems almost tautologous to say that only mathematics "benefits" from formalization. | |
| Dec 11, 2010 at 16:59 | comment | added | Terry Tao | Conversely, this type of introspection and formalisation is much less effective outside of mathematics (Weinberg has called this the "unreasonable ineffectiveness of philosophy".) Attempts to axiomatise science, the humanities, etc., for instance, usually end up collapsing under the weight of their own artificiality (with some key exceptions in physics, notably relativity and quantum mechanics). The fact that mathematics is almost the sole discipline that actually benefits from formalisation is indeed an interesting insight in my opinion. | |
| Dec 10, 2010 at 22:39 | comment | added | Michael Hardy | I think some people deny the existence of such "levels" of introspection. | |
| Dec 10, 2010 at 14:29 | comment | added | gowers | This is an example that has bothered me in the past, and I have to admit that I don't have a good answer to it. The ability to introspect seems to be very important to mathematicians, and it's far from clear how a computer would do it. One could perhaps imagine a separate part of the program that looks at what the main part does, but it too would need to introspect. Perhaps this infinite regress is necessary for Godelian reasons but perhaps in practice mathematicians just use a bounded number of levels of navel contemplation. | |
| Dec 10, 2010 at 13:58 | comment | added | ndkrempel | There's something amusing about the idea of a computer coming up with the idea that computability can be formalized. | |
| Dec 10, 2010 at 5:05 | comment | added | Michael Hardy | Whether the formalization of computability by Turing and various others in the '30s is the "right" one is a philosophical question, which maybe nobody knows how to think about. | |
| Dec 10, 2010 at 2:52 | history | answered | Timothy Chow | CC BY-SA 2.5 |