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Ideas for 'works', 'Abstract Entities' and 'Thinking About Mathematics'

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7 ideas

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

8744	Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
	Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1)
	A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism.

6. Mathematics / C. Sources of Mathematics / 7. Formalism

8749	Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
	Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names?
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1)
	A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up.

8750	Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
	Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2)
	A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare.

8752	Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
	Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2)
	A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

8753	Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
	Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
	A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism

8731	Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
	Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
	A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

8730	'Impredicative' definitions refer to the thing being described [Shapiro]
	Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
	A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise.