Timeline for Has philosophy ever clarified mathematics?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 13, 2017 at 8:55 | comment | added | Mikhail Katz | @ಠ_ಠ, an approach to synthetic differential geometry via Robinson's framework can be found here. | |
| Oct 10, 2017 at 17:21 | comment | added | მამუკა ჯიბლაძე | Well, the self-proclaimed greatest ignoramus of all times was Socrates, for that matter, but this does not prevent us from being interested in his opinions on a wide range of topics. | |
| Oct 10, 2017 at 17:10 | comment | added | მამუკა ჯიბლაძე | Would you use the word "synthetic" in context of Robinson's nonstandard analysis? If yes, why? If not, why do you think I did? From my quote, it should be clear that by "synthetic" I meant an axiomatic description of a system with desired properties. Primarily I had in mind systems like synthetic differential geometry. | |
| Oct 10, 2017 at 14:20 | comment | added | მამუკა ჯიბლაძე | Yes, the synthetic approach is cool. Still, another famous quote comes to my mind - "The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil." (Bertrand Russell, Introduction to Mathematical Philosophy (1919)). I would not mind that in fact, but what raises my suspicions is that regarding the most interesting synthesized entities it is still not known whether their existence leads to contradiction or not. And what to do if it is consistent but is not compatible with, say, ZFC. | |
| Oct 10, 2017 at 8:33 | comment | added | Mikhail Katz | @მამუკაჯიბლაძე, the phrase you mentioned is out of date. What is more relevant today is the observation that historians of mathematics are involved in a quest for the ghosts of departed quantifiers. Namely, they seek to interpret the work of Leibniz, Euler, Cauchy, and other pioneers of infinitesimal analysis in such a way as to replace their use of infinitesimals by long-winded quantified paraphrases. Related literature can be consulted here. | |
| Oct 9, 2017 at 3:58 | comment | added | მამუკა ჯიბლაძე | I believe the famous quote from The Analyst: a Discourse addressed to an Infidel Mathematician by George Berkeley (1734) is appropriate here: "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" | |
| Oct 8, 2017 at 22:06 | comment | added | ಠ_ಠ | William Lawvere, another philosophically-minded mathematician, also developed another elegant approach to infinitesimals: synthetic differential geometry. | |
| S Jun 16, 2016 at 14:53 | history | answered | Mikhail Katz | CC BY-SA 3.0 | |
| S Jun 16, 2016 at 14:53 | history | made wiki | Post Made Community Wiki by Mikhail Katz |