Types of Realism in Mathematics

5 06 2005

The issues noted in my “Eight Views of Mathematics” have of course been noted by other people before (after all, I was talking about the programs of Field, Steel/Woodin/Martin et al, and Dummett). However, it seems that at least the second one of these questions has been somewhat overlooked in past. Some people (I don’t recall who off the top of my head) refer to the first position as “anti-realism in ontology” and the third as “anti-realism in truth value”, but they don’t seem to have a name for the middle one. Similarly, Mark Colyvan (on p. 2 of his The Indispensability of Mathematics) talks about “metaphysical realism” and “semantic realism”. I suppose I’m tempted to call the third a sort of question of “epistemological realism”, or “realism in evidence” or something of the sort.

The first question is about what objects mentioned in mathematical statements exist, the second is about what mathematical statements we can know, and the third is about what mathematical statements are true or false. Dummett seems to want to link the first and third question. But it seems that most people try to assimilate the middle question to one of the other two. I suppose that’s where it falls to me to defend my position that we can know (or believe or accept) mathematical statements much beyond ZFC without being committed to mathematical objects, or to the claim that all mathematical statements are either true or false. (Though I’m probably agnostic about that last claim, rather than opposed.)

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4 responses

10 06 2005
logicnazi

Grr…why can I not make line breaks. Could you either allow HTML codes or turn on the convert line breaks option in the formating stuff?

Stewart Shapiro discusses most of these issues in his book philosophy of mathematics. The cool thing is you can read the entire text online through UCB proxy server.

http://www.oxfordscholarship.com/oso/private/content/philosophy/0195139305/p021.html

This isn’t the front page but where I am in the book. You can get to the start easily enough though.

11 06 2005
Kenny

Now I can return that book to the library and just read it online! I can’t remember if Shapiro is where I got the terms “realism in truth-value” and “realism in ontology”, or if that was somewhere else.

As for the line breaks, it looks like they worked out fine on this post. So I think that means the “convert line breaks” option is set where it should be, right?

5 07 2005
Peter McB.

Ken —

Reading this post, it occurs to me that the debate between Frege and Hilbert over the status of non-Euclidean geometry is relevant to your third question. Frege took the very strong line that one, and precisely one, of Euclidean and non-Euclidean geometry was true; he opted for Euclidean, and even claimed that belief in non-E geometry was akin to astrology.

Hilbert, establishing a position now held by most mathematicians, said that the question of truth was not relevant. The axioms of a geometry, provided they were consistent, effectively defined a class of objects (that class of objects which satisfy the axioms) and so different axioms would potentially define different classes.

There is a nice account of these different views and the debate in a paper by Alberto Coffa, in a book edited by Robert Colodny on the philosophy of physics. See bibtex below.

@INBOOK{coffa:geom86,
AUTHOR = “Alberto Coffa”,
editor = “Robert G. Colodny”,
TITLE = “From Quarks to Quasars: Philosophical Problems of Modern Physics”,
CHAPTER = “From geometry to tolerance: sources of conventionalism in nineteenth-century geometry”,
pages = “3–70”,
PUBLISHER = “University of Pittsburgh Press”,
YEAR = “1986”,
address = “Pittsburgh, PA, USA”}

22 07 2005
Kenny Easwaran

Thanks for the reference! Sorry about taking so long to approve the comment – your comment came in the middle of a huge comment spam attack that took down the site, and I only just now found it. I’ll have to think more about what sort of realism (if any) these positions correspond to. It sounds like Frege supported a strong realism in ontology (the physical world, in fact!) and truth-value, while Hilbert suggests an anti-realism of a certain sort in both types. As for epistemology, I’d have to read more on their particular views.

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