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Single Idea 15716
[filed under theme 3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
]
Full Idea
If the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist. For me this is the criterion of truth and existence.
Gist of Idea
If axioms and their implications have no contradictions, they pass my criterion of truth and existence
Source
David Hilbert (Letter to Frege 29.12.1899 [1899]), quoted by R Kaplan / E Kaplan - The Art of the Infinite 2 'Mind'
Book Ref
Kaplan,R and Kaplan,E: 'The Art of the Infinite' [Penguin 2004], p.51
A Reaction
If an axiom says something equivalent to 'fairies exist, but they are totally undetectable', this would seem to avoid contradiction with anything, and hence be true. Hilbert's idea sounds crazy to me. He developed full Formalism later.
The
29 ideas
from David Hilbert
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17963
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The facts of geometry, arithmetic or statics order themselves into theories
[Hilbert]
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17964
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Number theory just needs calculation laws and rules for integers
[Hilbert]
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17965
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The whole of Euclidean geometry derives from a basic equation and transformations
[Hilbert]
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17966
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Axioms must reveal their dependence (or not), and must be consistent
[Hilbert]
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17967
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To decide some questions, we must study the essence of mathematical proof itself
[Hilbert]
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17968
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By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge
[Hilbert]
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13472
|
Hilbert aimed to eliminate number from geometry
[Hilbert, by Hart,WD]
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9546
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Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects
[Hilbert, by Chihara]
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18742
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Hilbert's formalisation revealed implicit congruence axioms in Euclid
[Hilbert, by Horsten/Pettigrew]
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18217
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Hilbert's geometry is interesting because it captures Euclid without using real numbers
[Hilbert, by Field,H]
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17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience
[Hilbert]
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17698
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Logic already contains some arithmetic, so the two must be developed together
[Hilbert]
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18844
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You would cripple mathematics if you denied Excluded Middle
[Hilbert]
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22293
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Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency
[Hilbert, by Potter]
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9636
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My theory aims at the certitude of mathematical methods
[Hilbert]
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12456
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I aim to establish certainty for mathematical methods
[Hilbert]
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12455
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The idea of an infinite totality is an illusion
[Hilbert]
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12457
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There is no continuum in reality to realise the infinitely small
[Hilbert]
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9633
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No one shall drive us out of the paradise the Cantor has created for us
[Hilbert]
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12459
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The subject matter of mathematics is immediate and clear concrete symbols
[Hilbert]
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12460
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We extend finite statements with ideal ones, in order to preserve our logic
[Hilbert]
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18112
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Mathematics divides in two: meaningful finitary statements, and empty idealised statements
[Hilbert]
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12461
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We believe all mathematical problems are solvable
[Hilbert]
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12462
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Only the finite can bring certainty to the infinite
[Hilbert]
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15716
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If axioms and their implications have no contradictions, they pass my criterion of truth and existence
[Hilbert]
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10116
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Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions
[Hilbert, by George/Velleman]
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10113
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The grounding of mathematics is 'in the beginning was the sign'
[Hilbert]
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10115
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Hilbert substituted a syntactic for a semantic account of consistency
[Hilbert, by George/Velleman]
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8717
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Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted)
[Hilbert, by Friend]
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