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Single Idea 17963

[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation ]

Full Idea

The facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, electrodynamics into a theory of statics, electrodynamics, or the facts of the physics of gases into a theory of gases.

Gist of Idea

The facts of geometry, arithmetic or statics order themselves into theories

Source

David Hilbert (Axiomatic Thought [1918], [03])

Book Ref

'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1108

A Reaction

This is the confident (I would say 'essentialist') view of axioms, which received a bit of a setback with Gödel's Theorems. I certainly agree that the world proposes an order to us - we don't just randomly invent one that suits us.

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]