Timeline for Usage of set theory in undergraduate studies
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 9, 2013 at 1:09 | comment | added | Monroe Eskew | If you want a more precise notion of "ordinary math," I suggest you look at Simpson's book on reverse math, and also consult your school's undergraduate course listing. | |
| Jan 9, 2013 at 1:07 | comment | added | Monroe Eskew | What I am saying is not 100% precise, but I think the only way one can fail to catch my meaning is if one is blinded by ideology. How was it that Lebesgue and Borel were able to do such wonderful rigorous mathematics if they did not believe in set theory? Did you guys really learn math by first learning set theory? Of course our standards of rigor evolve over time as we notice more issues we must take care of. But the remarkable thing is that once it is in place, we all agree about it. One example to ponder: Euclid's proof of the infinity of primes was perfectly rigorous and correct. | |
| Jan 8, 2013 at 15:19 | history | made wiki | Post Made Community Wiki by François G. Dorais | ||
| Jan 8, 2013 at 10:58 | comment | added | Marco Caminati | @mbsq: I'm sorry, but I just can't figure out what you mean by eternal and absolute math, so it's difficult to reply. I am interested to your point that you can pursue rigor without sticking to no "particular theory": how would you do that? To me, it sounds like saying that you can write a poem without using any particular alphabet. Anyway, reading my comments now, they sound a bit too polemical; I apologize for that. | |
| Jan 8, 2013 at 7:06 | comment | added | Asaf Karagila♦ | What? Where did I equate set theory and rigor. I only used your argument of "it's new, and it's not real mathematics anyway, so we should start with something which is real mathematics" and compared that to the fact that rigor is a relatively new invention to mathematics. Besides that, what is "real mathematics" anyway? Is it what Euclid did? Newton? Gauss? Is it actually physics or chemistry? Or does real mathematics is what mathematicians do? | |
| Jan 8, 2013 at 4:27 | comment | added | Monroe Eskew | @Marco: I go further and say that particular theories like ZFC are NOT necessary for rigor, or creativity. I do not deny that they are "exciting," but I say they are somewhat removed from the absolute math for which they serve as foundation. Now, to address the "subjectivity," if you know anything about relative consistency results, or even just some history of math, it is clear that basic math is much more eternal than current foundations, which may actually die. Finally, I DO NOT DISMISS THE IMPORTANCE OF FOUNDATIONAL STUDIES. You totally misunderstand me. | |
| Jan 8, 2013 at 4:21 | comment | added | Monroe Eskew | @Asaf, Weierstrass was not a set theorist, and his pathological functions had no inspiration in set theory, in fact Cantor was his student. Rigor in analysis is its own thing, and owes little to set theory. To equate set theory with rigor itself is nothing more than ideology. Find your local complex analysis expert and ask when was the last time they used anything more than a small fragment of ZC, or even went beyond what can be formalized in 2nd order arithmetic. I am just saying that the current foundations may collapse, but more basic things will live on. | |
| Jan 7, 2013 at 21:19 | comment | added | Marco Caminati | @mbsq-part two: The fact that multiple, particular choices to encode foundations exist should not mislead anyone into thinking that the results obtained by studying them (not only by applying them) are less "eternal" than the results you arbitrarily classify as such: your whole post is ultimately pivoted on such a subjective (and scarcely elaborated on) distinction. Finally, I find ironic that you deployed the fruits of reverse mathematics, arguably a foundational discipline, to dismiss the importance of foundational studies. | |
| Jan 7, 2013 at 21:19 | comment | added | Marco Caminati | @mbsq-part one: It seems to me that you have a merely utilitarian view of mathematical logic and foundational disciplines: as if they are "only" needed to attain consistency and rigor (which, by the way, is, in my opinion, as vital as creativity to mathematics: I dare to say that mathematics is a beautiful interplay between rigor and mental pure creation). What makes them even more exciting, to me at least, is that they are a beautiful subject of mathematical investigation themselves, giving them a special "double face" status. | |
| Jan 7, 2013 at 20:11 | comment | added | Asaf Karagila♦ | Oh you are reading too much into my analogy. Let me try once more (with feeling), rigor is a new invention in mathematics. It actually dates roughly as set theory. Should we start by being non-rigorous? May we assume that all functions are continuous and infinitely differentiable almost everywhere like the good people of the late 18-th and early 19-th people assumed? Mathematics moves forward, and while we cannot ditch the past completely, there is so much to teach an undergrad that starting with "real mathematics that will survive the collapse of the current foundation" seems strange to me. | |
| Jan 7, 2013 at 17:23 | comment | added | Monroe Eskew | No Asaf, this is not a good analogy. First these are political/moral issues, not science, and I absolutely disagree that freedom may fail to withstand the test of time, but we can debate that elsewhere. Technologies change, but these are empirically demonstrable things unlike set theory, and new technologies build on old. Furthermore I am not saying you can't teach foundations, only that you should do that after they know a good deal of "real" mathematics. If a new revolution in mathematics comes, people should be equipped to deal with it by knowing the basic truths that must be preserved. | |
| Jan 7, 2013 at 17:17 | comment | added | Asaf Karagila♦ | Equally speaking, the relative freedom of speech and from slavery, as well the wonders of the technological era, all those are quite new and will certainly fail to withstand the test of time. Should we all go live about our lives in a cabin somewhere in the woods? :-) | |
| Jan 7, 2013 at 17:11 | history | answered | Monroe Eskew | CC BY-SA 3.0 |