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Single Idea 9546

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

Full Idea

Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions.

Gist of Idea

Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects

Source

report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1

Book Ref

Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.9

A Reaction

Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries.

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]