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Full Idea
Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge.
Gist of Idea
Rationalism tries to apply mathematical methodology to all of knowledge
Source
Stewart Shapiro (Thinking About Mathematics [2000], 1.1)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.3
A Reaction
Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach.
Related Idea
Idea 22 Trained minds never expect more precision than is possible [Aristotle]
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |