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Ideas of David Hilbert, by Text
[German, 1862  1943, Professor of Mathematics at Königsberg, and the Göttingen.]
1899

Foundations of Geometry


p.9

9546

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


p.17

18742

Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew]


p.25

18217

Hilbert's geometry is interesting because it captures Euclid without using real numbers [Field,H]


p.42

13472

Hilbert aimed to eliminate number from geometry [Hart,WD]

1899

Letter to Frege 29.12.1899


p.51

15716

If axioms and their implications have no contradictions, they pass my criterion of truth and existence

1900

On the Concept of Number

p.183

p.129

22293

Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Potter]


p.148

10113

The grounding of mathematics is 'in the beginning was the sign'


p.153

10115

Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman]


p.156

10116

Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [George/Velleman]

6.7

p.154

8717

Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Friend]

1904

On the Foundations of Logic and Arithmetic

p.130

p.130

17697

The existence of an arbitrarily large number refutes the idea that numbers come from experience

p.131

p.131

17698

Logic already contains some arithmetic, so the two must be developed together

[03]

p.1108

17963

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

[05]

p.1108

17965

The whole of Euclidean geometry derives from a basic equation and transformations

[05]

p.1108

17964

Number theory just needs calculation laws and rules for integers

[09]

p.1109

17966

Axioms must reveal their dependence (or not), and must be consistent

[53]

p.1115

17967

To decide some questions, we must study the essence of mathematical proof itself

[56]

p.1115

17968

By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge

p.184

p.66

9636

My theory aims at the certitude of mathematical methods

p.184

p.184

12456

I aim to establish certainty for mathematical methods

p.184

p.184

12455

The idea of an infinite totality is an illusion

p.186

p.186

12457

There is no continuum in reality to realise the infinitely small

p.191

p.65

9633

No one shall drive us out of the paradise the Cantor has created for us

p.192

p.192

12459

The subject matter of mathematics is immediate and clear concrete symbols

p.195

p.195

12460

We extend finite statements with ideal ones, in order to preserve our logic

p.196

p.174

18112

Mathematics divides in two: meaningful finitary statements, and empty idealised statements

p.200

p.200

12461

We believe all mathematical problems are solvable

p.201

p.201

12462

Only the finite can bring certainty to the infinite

1927

The Foundations of Mathematics

p.476

p.285

18844

You would cripple mathematics if you denied Excluded Middle
