Timeline for Formalist or Platonist--does it make any difference in mathematical practice?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Nov 17 at 20:36 | comment | added | Dan Christensen | @mudskipper The latter. The public part of mathematics. That which is published or otherwise presented to some audience. | |
| Nov 17 at 17:52 | comment | added | mudskipper | @DanChristensen - I'm tempted to write an answer, but your question seems ambiguous to me: Is it a question about the process of discovery (=solving unsolved mathematical problems) or a question about the process of proving (=rigorously verifying a discovery and establishing that it is true)? Those are two pretty different perspectives. | |
| Nov 17 at 16:45 | comment | added | Bumble | @DanChristiansen It is correct to say that intuitionistic proofs are all classically valid, but not the other way round. Consequently, some theorems of classical mathematics are not theorems of intuitionistic mathematics, so there is a difference in practice. Intuitionists are not platonists: they regard mathematical truths as products of rational intuition, not as statements about an abstract reality. As mudskipper says, it leads to a rather different understanding of mathematics. | |
| Nov 17 at 16:41 | comment | added | mudskipper | @DanChristensen - The way continuity/continuous functions is seen is also pretty different and not completely compatible. A focus on logic hides these aspects. Besides, logic has no foundational role in the intuitionist's view. | |
| Nov 17 at 16:31 | comment | added | mudskipper | @DanChristensen - Intuitionism is more than that. The intuitionist view of the continuum is also pretty different from the classical one. Cantor's continuum hypothesis has no immediate meaning, for instance. Also, to describe intuitionist logic as "just a subset" fails to do justice to the fact that in intuitionism mathematical theorems may mean something different. | |
| Nov 17 at 15:58 | comment | added | Dan Christensen | @Bumble IIUC intuitionist logic is just a working subset of classical logic. It simply disallows certain axioms of classical logic that, for reasons I don't understand, are found to be somehow suspect. Every proof based on intuitionist logic is also a proof in classical logic, is it not? | |
| Nov 17 at 13:22 | answer | added | Mikhail Katz | timeline score: 5 | |
| Nov 16 at 23:12 | comment | added | Bumble | Platonism and formalism are not the only approaches to the philosophy of mathematics. While different approaches do not result in different practices in most subjects, they can lead to differences. For example, intuitionists have a different standard of what counts as a mathematical proof and they reject non-constructive proofs. ¬(∀x)Fx does not entail (∃x)¬Fx intuitionistically. Some approaches to the philosophy of mathematics might lead to rejecting transfinite set theory or to rejecting infinitessimals. | |
| Nov 16 at 23:04 | comment | added | Conifold | It is neither personal nor a preference, mathematicians can do little to 'prefer' their 'style', and it is not about a narrative. Their dispositions and experience structure their thinking in certain ways, which fall into some common patterns. The narrative comes after the results they are led to are worked out and there they have more room to exercise preferences. | |
| Nov 16 at 22:07 | comment | added | Double Knot | It may be a deeper and more interesting question to swirl your question, that is, does difference in mathematical practice make the practitioner to form different philosophy of mathematics? The practice of whomever is obviously the means at least colloquially speaking, and the mature philosophizing thereof shall be the ends... | |
| Nov 16 at 18:19 | comment | added | Dan Christensen | @Conifold So, is it just a matter of personal preference in one's narrative style? | |
| Nov 16 at 13:02 | review | Close votes | |||
| Nov 22 at 3:06 | |||||
| Nov 16 at 12:05 | comment | added | Conifold | It is not a question of "cannot", that is not what mathematical practice is about. Platonists and formalists have different kinds of intuitions and motivations, so the types of results they tend to pursue may differ. Or they may pursue the same results in different ways. But one can, in principle, obtain results of the other and certainly can follow the other's proofs of them. | |
| Nov 16 at 11:35 | review | Low quality posts | |||
| Nov 16 at 22:16 | |||||
| Nov 16 at 11:26 | comment | added | Mauro ALLEGRANZA | No, because both use Classical Logic. | |
| Nov 16 at 11:14 | history | asked | Dan Christensen | CC BY-SA 4.0 |