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Single Idea 9553

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry ]

Full Idea

With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert).

Gist of Idea

Analytic geometry gave space a mathematical structure, which could then have axioms

Source

Charles Chihara (A Structural Account of Mathematics [2004], 02.3)

Book Ref

Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.43

A Reaction

The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this?

The 22 ideas from Charles Chihara

8758 We could talk of open sentences, instead of sets [Chihara, by Shapiro]
10265 Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro]
8759 We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro]
10264 Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro]
18151 Could we replace sets by the open sentences that define them? [Chihara, by Bostock]
10192 We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
9547 Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara]
9549 The set theorist cannot tell us what 'membership' is [Chihara]
9550 We only know relational facts about the empty set, but nothing intrinsic [Chihara]
9551 What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara]
9552 Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara]
9553 Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara]
9559 If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara]
9561 The mathematics of relations is entirely covered by ordered pairs [Chihara]
9562 In simple type theory there is a hierarchy of null sets [Chihara]
9563 A pack of wolves doesn't cease when one member dies [Chihara]
9566 No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara]
9568 I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara]
9571 ZFU refers to the physical world, when it talks of 'urelements' [Chihara]
9573 The null set is a structural position which has no other position in membership relation [Chihara]
9572 Realists about sets say there exists a null set in the real world, with no members [Chihara]
9574 'Gunk' is an individual possessing no parts that are atoms [Chihara]