Mathematical Antirealism
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What is it?
The view that mathematical reality is never unknowable (i.e. that there do not exist mathematical facts/objects that we will never discover). To put it another way: there can be no unknowable mathematical truths. (This is Moore’s interpretation in the Routledge article at least.)
References
MOORE, A.W. (1998, 2004). Antirealism in the philosophy of mathematics. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved February 28, 2008, from http://www.rep.routledge.com/article/Y065
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Variations -Intuitionism: math. objects mind-dependent creations, thus nothing to them beyond proof. -Dummettian: meaning of math. statements created publicly (cf. Wittgenstein on meaning in general). Hence cannot go beyond this social construction of meaning to formulate unprovable math. truth. - Infinitism(?): math. relies on infitinite domains. Cannot “get at” all members of infinite domain at the same time. Thus any statement about all members of a particular infinite domain has to be proved in some way. - A further distinction hinges on whether the anti-realist holds that what can be proved is true or that what has been proved is true. Also, the anti-realist must stipulate whether the “can” is an in principle one or an in practice (which seems to lead to strict finitism) one.
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Problems
It is argued that all the versions above have revisionary implications for the practice of mathematics. - e.g. One cannot assume math. statement T v F. — Math. statement T only in light of accepted proof.
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What do I think? - Moore does bring out a nice point regarding the fact that mathematical objects are really reliant upon the techniques and proof-procedures of math. in order to be understood by us (this does not necessarily imply ~realism though). - I really don’t think Moore is doing justice to some of the anti-realist views here. In particular I think his focus on analysing the Dummettian position merely in terms of this every truth is provable stuff is unsatisfactory. Indeed, it feels like he has chosen one aspect of these conceptualisations of mathematics and then tried to use it to analyse the conceptualisations as a whole. - As it stands the initial statement is not necessarily anti-realist. If one takes a very liberal attitude to provability, then one could still have a wide range of real, mind-independent mathematical objects. (Moore actually raise this as a criticism of certain anti-realist positions at one stage, which seems a little unfair given he has created this fairly artificial position and then tried to cram various pre-existing views into it.)
{[pink good stuff — jason ]}
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I recently came across a book, Platonism and Anti-Platonism in Mathematics by Mark Balaguer [Oxford: OUP] 1998. He contends that there are no good arguments for or against the view that abstract, non-spatio-temporal objects exist and that mathematical theories are descriptions of such objects, because both platonism and anti-platonism are defensible positions.
Is this even vaguely relevant to this discussion?
I am an undergraduate student of Jason’s, and happened across this posting while browsing Xeny Net. Hope you don’t mind my intrusion.
Dick Parker
{[green Hi Dick, ]}
{[green I just had a quick look at a review of this (by Dieterle in Vol. 50 (4) of the British Journal for the Philosophy of Science if you want to check it out). From this, it looks like the position Belauger advocates as anti-Platonist is a version of fictionalism (a kind of error theory that Brian was talking about in the tute today) that I believe I’m right in saying is actually anti-realist (which of course implies anti-Platonism). ]}
{[green It certainly does seem to be a unique position: a sort of argument from dead-heat! I will definitely investigate it further in the future but it’ll have to go on the back-burner at present as it’s not quite directly involved in the direction my thesis is heading. (Another one on the to-read list!) ]}
{[green Thanks for letting me know about it, ]}
{[green Chris ]}
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