How We Learn Mathematical Language

Bibliographic Data

ID10695345
AuthorsVann Mcgee (corresponding author)
Year1997
Volume106
Issue1
Pages35
Publication date1997-01-01
Peer ReviewedYes
Open AccessNo
TypeARTICLE
VenueThe Philosophical Review (JOURNAL)
Journal identifiersISSN: 0031-8108 • E-ISSN: 1558-1470
PublisherJSTOR (PUBLISHER)
DOI10.2307/2998341
OpenAlexW2093568339
LanguageEN
Citations received32

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning (so that 'e' stands for the elementhood relation and the quantifiers range over all the sets there are), each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is En2 or whether there are measurable cardinals, whether or not those facts are knowable by us. There are numerous objections to realism, some of them quite penetrating. Some objections come from philosophers, who frequently find talk about abstract entities either unintelligible or incredible. Other objections come from mathematicians, many of whom are affably willing to go along with realist talk about natural numbers or real numbers, but who find that when we get to set theory their credulity has been exhausted, so that they think of set theory the way we think of group theory, as a formal calculus whose models we examine, without any supposition that any of the models are preferred. Here I do not intend to defend realism but rather to presume it-though this be a very large presumption-and to discuss a problem that arises internally within the realist conception of mathematics. The internal problem is this: the realist conception supposes that the meaning of mathematical terms is fixed with sufficient preci

Contemporary philosophy · Epistemology · Linguistics · Computer Science · History and Theory of Mathematics · Philosophy · Philosophy and History of Science · Philosophy and Theoretical Science

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