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Single Idea 16901

[filed under theme 6. Mathematics / A. Nature of Mathematics / 2. Geometry ]

Full Idea

Geometrical concepts appear to depend in some way on a spatial ability. Although one can translate geometrical propositions into algebraic ones and produce equivalent models, the meaning of the propositions seems to me to be thereby lost.

Gist of Idea

The equivalent algebra model of geometry loses some essential spatial meaning

Source

Tyler Burge (Frege on Apriority (with ps) [2000], 4)

Book Ref

Burge,Tyler: 'Truth, Thought, Reason (on Frege)' [OUP 2001], p.384

A Reaction

I think this is a widely held view nowadays. Giaquinto has a book on it. A successful model of something can't replace it. Set theory can't replace arithmetic.

The 14 ideas from Tyler Burge

9382 Subjects may be unaware of their epistemic 'entitlements', unlike their 'justifications' [Burge]
16892 Is apriority predicated mainly of truths and proofs, or of human cognition? [Burge]
16901 The equivalent algebra model of geometry loses some essential spatial meaning [Burge]
16902 Peano arithmetic requires grasping 0 as a primitive number [Burge]
9159 You can't simply convert geometry into algebra, as some spatial content is lost [Burge]
17622 We come to believe mathematical propositions via their grounding in the structure [Burge]
13479 Given that thinking aims at truth, logic gives universal rules for how to do it [Burge]
3115 Are meaning and expressed concept the same thing? [Burge, by Segal]
14349 If there are no finks or antidotes at the fundamental level, the laws can't be ceteris paribus [Burge, by Corry]
8126 Anti-individualism says the environment is involved in the individuation of some mental states [Burge]
8127 Broad concepts suggest an extension of the mind into the environment (less computer-like) [Burge]
8129 Anti-individualism may be incompatible with some sorts of self-knowledge [Burge]
8131 Some qualities of experience, like blurred vision, have no function at all [Burge]
8132 We now have a much more sophisticated understanding of logical form in language [Burge]