Can a formalist obtain results in, say, real analysis that a Platonist cannot or vice versa?
12
-
1No, because both use Classical Logic.– Mauro ALLEGRANZACommented Nov 16 at 11:26
-
1It 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.– ConifoldCommented Nov 16 at 12:05
-
@Conifold So, is it just a matter of personal preference in one's narrative style?– Dan ChristensenCommented Nov 16 at 18:19
-
4Platonism 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.– BumbleCommented Nov 16 at 23:12
-
1@mudskipper The latter. The public part of mathematics. That which is published or otherwise presented to some audience.– Dan ChristensenCommented Nov 17 at 20:36
|
Show 7 more comments
1 Answer
To pick up on user Bumble's comment on infinitesimals, I would point out that there would indeed be practical differences between a Formalist and a Platonist, as determined by their respective philosophical positions.
Consider, for example, the mathematical entity R (the real number line). A Platonist believes that he has a detailed grasp of the latter that accounts for the type of elements R may contain and its further properties. Similar remarks apply to "the" natural numbers N.
A Formalist considers an infinitary entity like R necessarily reference-less (either in the physical realm or any putative Platonic realm), and refrains from asserting that he possesses a detailed grasp of R, which to him is only a theory (since a Formalist would reject the notion of a standard model a.k.a. Intended Interpretation of an entity like R).
How do such views play out in practice? Consider, for example, the axiomatic theories of nonstandard analysis (NSA), as developed by Karel Hrbacek and Edward Nelson (independently) in the 1970s, about a decade after Robinson's original work on NSA. In the axiomatic approaches, infinitesimals are found within R itself (rather than in an extension, as in the model-theoretic approaches to NSA), and nonstandard numbers in N itself (rather than an extension); see this introduction.
A Formalist would likely take this in stride. After all, she never claimed to possess a detailed grasp of each and every element of R (or of N); that there turn out to be elements she has not noticed before is an interesting idea.
A Platonist is as likely as not to shrink away in horror: "What, the categorical and unique R that I knew and loved for so many years, all of a sudden contains things I never dreamed of?" It is clear that it will be harder for him to work in such a framework or to accept its legitimacy.
-
A bit of a tangent, but: What you mentioned about Hrbacek and Nelson axiomatization which has infinitesimals “hiding” (so to speak) within R sounds fascinating. Wondering if you know of resource(s) (book, paper, etc.) that you would recommend as a good introduction to this topic.– NikSCommented Nov 18 at 4:09
-
-
Indeed a Platonist is as likely as not to shrink away in horror, since Hrbacek & Nelson works in internal set theory (IST) with different axioms from ZFC and IST axioms already internalize many nonstandard or ilimit entities at the proof-theoretic level, so there's no surprise at all infinitesimals will hide in the Platonically intended standard R... Commented Nov 18 at 9:20
-
@DoubleKnot, just to be sure we agree about the facts. (1) Nelson's IST incorporates all the axioms of ZFC (and adds three axioms to govern the interaction of the new predicate with the axioms of ZFC); (2) IST is conservative over ZFC; this means not merely that it is consistent if and only if ZFC is consistent, but that IST (in theory) proves exactly the same statements as ZFC (so long as the statements don't involve the new predicate). Are you with me so far? Commented Nov 18 at 9:37
-
Indeed IST conservatively extends ZFC results that don't involve the st predicate. It essentially wraps ZFC to handle infinitesimals and other nonstandard elements without altering the standard mathematical universe structure as understood by ZFC. With the added 3 axioms, idealization and transfer principle stipulate and internalize infinitesimals and infinities into R and N for every IST's possible model which is nonstandard via the choice of different ultrafilters same as NSA, and standardization standardizes them, yet ZFC's nonstandard models implicitly follow transfer and idealization... Commented Nov 19 at 6:05