more on this theme
|
more from this text
Single Idea 17965
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
]
Full Idea
The linearity of the equation of the plane and of the orthogonal transformation of point-coordinates is completely adequate to produce the whole broad science of spatial Euclidean geometry purely by means of analysis.
Gist of Idea
The whole of Euclidean geometry derives from a basic equation and transformations
Source
David Hilbert (Axiomatic Thought [1918], [05])
Book Ref
'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1108
A Reaction
This remark comes from the man who succeeded in producing modern axioms for geometry (in 1897), so he knows what he is talking about. We should not be wholly pessimistic about Hilbert's ambitious projects. He had to dig deeper than this idea...
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]
|