Timeline for Top-down mathematics, or "Where it all begins"
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2021 at 14:32 | answer | added | rimu | timeline score: 0 | |
| Mar 22, 2021 at 3:10 | history | protected | Yemon Choi | ||
| Mar 22, 2021 at 1:02 | comment | added | xuq01 | It doesn't have to start from somewhere in mathematics! As some intuitionists/constructivists believe, it all roots in the innate notion of computation and reasoning. If you tend to believe in Kant, it's the sort of a priori knowledge that is independent of any experience. You don't have to study math to acquire the conception of quantity, or of (simple) reasoning, right? And this is where it starts, according to some... | |
| Mar 21, 2021 at 22:08 | comment | added | davidbak | It's turtles all the way down ... | |
| Mar 20, 2021 at 9:11 | comment | added | TrayMan | Having dabbled a bit with ELF, it is basically a technical approach to defining various different kinds of logical systems in another system. On a quick glance, GLF seems very similar. They are both founded in the informal logic that the papers are written in. You could found both of them in yet another formal framework, but it just becomes "turtles all the way down". While those systems are interesting, the question you are after seems more philosophical than anything to do with practical mathematics. | |
| Mar 20, 2021 at 8:31 | answer | added | Andrej Bauer | timeline score: 20 | |
| Mar 20, 2021 at 2:43 | comment | added | Adam Chalcraft | I would definitely describe starting with the axioms - whatever you choose them to be - as "bottom-up" rather than "top-down". A "top-down" approach is more like "Suppose you're building a bridge ..." introducing concepts only as they're needed. This is what the terms mean in programming, anyway. | |
| Mar 20, 2021 at 0:54 | comment | added | Erik Walsberg | I also suspect that if the Bourbaki guys were around today, they wouldn't be Bourbaki. Bourbaki came out of a particular time, I don't think anyone would try to take their approach today. | |
| Mar 20, 2021 at 0:52 | comment | added | Erik Walsberg | I don't think that there is a single beginning. Math, or what we now call math, requires a foundation, but there are several possible foundations, and to a large extent mathematics is independent of the foundations. Now, whatever your choice of foundation, you have to be able to do certain basic things, like induction and counting arguments. So maybe those basic ideas are the beginning and foundations are a framework built around those ideas. | |
| Mar 19, 2021 at 22:52 | history | became hot network question | |||
| Mar 19, 2021 at 17:01 | comment | added | Dave L Renfro | Irina's comment has been what my thoughts over this have evolved to in the past 3 decades (I used to spend a great deal of time worrying over this in the second half of the 1970s and throughout the 1980s), and with that said, jwodder's seemingly too simplistic answer to Where to begin with foundations of mathematics is what I've finally decided is, for me, probably best for how to leave things. For an elaboration of the "Philosophy (optional)" step, this answer might be of interest. | |
| Mar 19, 2021 at 16:04 | history | edited | gmvh | CC BY-SA 4.0 |
edited body; edited tags
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| Mar 19, 2021 at 15:56 | comment | added | Irina | While I understand the impulse to sort out foundational issues before you start learning "real" mathematics, I think it is misguided. All the programs you'll find for setting up the foundations came in response to issues that arose in mathematical practice, and it is impossible to understand them without having the mathematical maturity/experience to appreciate those issues. Almost all mathematicians work in basically a naive version of ZFC (of the sort that you can read about in Halmos's "Naive Set Theory"), and I think that trying to do more than that at the beginning just won't work. | |
| Mar 19, 2021 at 15:40 | answer | added | Timothy Chow | timeline score: 40 | |
| Mar 19, 2021 at 8:23 | comment | added | mlk | A thing to keep in mind is that the logical hierarchy of topics is rarely the same as the order they should be studied in. Furthermore you are allowed to obtain basic knowledge in one topic, move on to something else and then come back at a later time. This is true even if you take Bourbaki as a guide (which I wouldn't recommend). Their books weren't written in order. They started with only a short summary of set theory without proofs, then got all the way to integration, before finally writing the proper volume on sets 15 years after the initial summary. | |
| Mar 19, 2021 at 7:46 | comment | added | Pace Nielsen | My personal, current opinion: Formalization of mathematics begins with finite strings of symbols. Mathematics itself is a process of the mind, and the mind is not a finished product, so there is no beginning. | |
| Mar 19, 2021 at 7:38 | comment | added | user44143 | A logical system is a way to designate some strings of symbols as valid proofs...but discussion in that generality is probably not illuminating. | |
| Mar 19, 2021 at 7:09 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Mar 19, 2021 at 6:35 | review | Close votes | |||
| Mar 24, 2021 at 3:05 | |||||
| Mar 19, 2021 at 6:17 | comment | added | Monroe Eskew | This is an interesting line of thought, but in my opinion not appropriate for this forum. I think the answer will be a personal viewpoint arising from years of reading and deep thinking. Keep it up, and have fun! | |
| Mar 19, 2021 at 6:11 | review | First posts | |||
| Mar 19, 2021 at 7:00 | |||||
| Mar 19, 2021 at 6:09 | history | asked | steve | CC BY-SA 4.0 |