1899 | Foundations of Geometry |
p.17 | 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid |
p.42 | 13472 | Hilbert aimed to eliminate number from geometry |
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 | works |
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 |
p.156 | 10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions |
6.7 | p.154 | 8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) |
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 |
1918 | Axiomatic Thought |
[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 |
1925 | On the Infinite |
p.184 | p.66 | 9636 | My theory aims at the certitude of mathematical methods |
p.184 | p.184 | 12455 | The idea of an infinite totality is an illusion |
p.184 | p.184 | 12456 | I aim to establish certainty for mathematical methods |
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 | 12458 | Kant taught that mathematics is independent of logic, and cannot be grounded in it |
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 |