Timeline for What did the Intuitionists want to do with applied mathematics?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2018 at 3:31 | vote | accept | bimargulies | ||
| Aug 7, 2018 at 17:33 | comment | added | Noam Zeilberger | @AndrejBauer I am not a Brouwer scholar, but the way I read the passage is that he clearly places value in the ordinary objects of the mathematical world ("ordinary arithmetic or geometry"), and is merely skeptical about certain purported ways of studying them via specific formal systems. I see nothing to suggest that he would want to exclude these classical subjects from the scope of intuitionistic mathematics, and quite the contrary otherwise he wouldn't be so worried about "improper" foundations. | |
| Aug 7, 2018 at 16:42 | comment | added | Mike Shulman | This is a really interesting question; I would also like to know what led Beeson to draw that conclusion about Brouwer's beliefs. Also, it's worth noting that even if it's true, the contrast between Brouwer and Bishop is not quite that extreme: Bishop also believed that constructively one would have to give up various aspects of mathematical practice, such as general topology, where subsequent developments (e.g. locale theory) have proven him wrong. | |
| Aug 7, 2018 at 16:19 | comment | added | Andrej Bauer | @NoamZeilberger: the quoted passages mostly show Brouwer's skepticism of formalism, I think, and not so much what he thought the scope of intuitionistic mathematics (as understood by him) would be. | |
| Aug 7, 2018 at 13:46 | comment | added | Nik Weaver | +1 for the last paragraph. | |
| Aug 7, 2018 at 9:41 | comment | added | Noam Zeilberger | (By the way, it seems to me that this view of Brouwer's is much more in line with the approach of the early pioneers in calculus, who from what I understand did not see their work as proceeding from logical axioms, but rather as based on physical intuitions and backed by empirical evidence, cf. Amir Alexander's "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World".) | |
| Aug 7, 2018 at 9:31 | comment | added | Noam Zeilberger | "So the idea that by means of such linguistic buildings we can obtain any knowledge of mathematics apart from that which can be constructed directly on the basis of intuition, is mistaken. And more so is the idea that in this way we can lay the foundations of mathematics, in other words that we can ensure the reliability of the mathematical theorems." (Brouwer, 1975, pp. 132–133, original emphasis) | |
| Aug 7, 2018 at 9:30 | comment | added | Noam Zeilberger | "And if one succeeds in the construction of linguistic buildings, sequences of sentences proceeding according to the logical laws, thereby departing from linguistic images which could accompany basic mathematical truths in actual mathematical buildings, and if it turns out that those linguistic buildings can never produce the linguistic form of a contradiction, then all the same they belong to mathematics only in their quality of a linguistic building, and have nothing to do with mathematics outside of that building, e.g. with ordinary arithmetic or geometry. [...] | |
| Aug 7, 2018 at 9:30 | comment | added | Noam Zeilberger | Here is passage from Brouwer's thesis that van Atten and Sundholm quote in the introduction: [...] | |
| Aug 7, 2018 at 9:28 | comment | added | Noam Zeilberger | there is Brouwer's essay on "Unreliability of the logical principles", as translated (and with an introduction) by van Atten and Sundholm (arxiv.org/pdf/1511.01113.pdf). From reading that it seems to me that Brouwer was not expecting mathematics to crumble, just that he was skeptical of the reliability of (some of) the logical principles assumed as foundations. | |
| Aug 7, 2018 at 9:22 | comment | added | Andrej Bauer | I asked about sources that would tell us something about Brouwer's opinion on the matter on the constructivenews mailing list, where the experts lurk. I'll report here if anything useful comes up. | |
| Aug 7, 2018 at 8:59 | comment | added | Andrej Bauer | @FrankWaaldijk: I did not write this. I quoted Michael Beeson's book. We should ask him where he got the idea from. Where would we look (as in "references") to find out what Brouwer expected? Hilbert was explicit about expecting mathematics to crumble when you take non-constructive methods from it. | |
| Aug 7, 2018 at 8:58 | comment | added | Andrej Bauer | @DavidRoberts: Ha! I'll blame that one on the spell-checker. | |
| Aug 7, 2018 at 8:58 | history | edited | Andrej Bauer | CC BY-SA 4.0 |
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| Aug 7, 2018 at 8:53 | comment | added | Franka Waaldijk | Andrej, you write: 'Namely both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis).' This is not true, Brouwer did not think this at all. He worked on measure theory himself, and was proficient in real and complex analysis, enough to have constructivized convincing samples of these. | |
| Aug 7, 2018 at 8:31 | comment | added | David Roberts♦ | the constrictive foundations to be insufficient <-- intentional? | |
| Aug 7, 2018 at 8:28 | history | answered | Andrej Bauer | CC BY-SA 4.0 |