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Single Idea 12457
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
]
Full Idea
A homogeneous continuum which admits of the sort of divisibility needed to realise the infinitely small is nowhere to be found in reality.
Gist of Idea
There is no continuum in reality to realise the infinitely small
Source
David Hilbert (On the Infinite [1925], p.186)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.186
A Reaction
He makes this remark as a response to Planck's new quantum theory (the year before the big works of Heisenberg and Schrödinger). Personally I don't see why infinities should depend on the physical world, since they are imaginary.
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
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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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