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Single Idea 17967
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
]
Full Idea
It is necessary to study the essence of mathematical proof itself if one wishes to answer such questions as the one about decidability in a finite number of operations.
Gist of Idea
To decide some questions, we must study the essence of mathematical proof itself
Source
David Hilbert (Axiomatic Thought [1918], [53])
Book Ref
'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1115
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]
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