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Single Idea 18742

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

Full Idea

In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised.

Gist of Idea

Hilbert's formalisation revealed implicit congruence axioms in Euclid

Source

report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2

Book Ref

'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.17

A Reaction

The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy.

The 29 ideas from David Hilbert

17963 The facts of geometry, arithmetic or statics order themselves into theories [Hilbert]
17964 Number theory just needs calculation laws and rules for integers [Hilbert]
17965 The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
17966 Axioms must reveal their dependence (or not), and must be consistent [Hilbert]
17967 To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
17968 By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge [Hilbert]
13472 Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD]
9546 Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara]
18742 Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
18217 Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H]
17697 The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
17698 Logic already contains some arithmetic, so the two must be developed together [Hilbert]
18844 You would cripple mathematics if you denied Excluded Middle [Hilbert]
22293 Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter]
9636 My theory aims at the certitude of mathematical methods [Hilbert]
12456 I aim to establish certainty for mathematical methods [Hilbert]
12455 The idea of an infinite totality is an illusion [Hilbert]
12457 There is no continuum in reality to realise the infinitely small [Hilbert]
9633 No one shall drive us out of the paradise the Cantor has created for us [Hilbert]
12459 The subject matter of mathematics is immediate and clear concrete symbols [Hilbert]
12460 We extend finite statements with ideal ones, in order to preserve our logic [Hilbert]
18112 Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert]
12461 We believe all mathematical problems are solvable [Hilbert]
12462 Only the finite can bring certainty to the infinite [Hilbert]
15716 If axioms and their implications have no contradictions, they pass my criterion of truth and existence [Hilbert]
10116 Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman]
10113 The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
10115 Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
8717 Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]