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Single Idea 10302
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
]
Full Idea
Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both are situated.
Gist of Idea
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists'
Source
report of Euclid (Elements of Geometry [c.290 BCE]) by Paul Bernays - On Platonism in Mathematics p.259
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.259
The
12 ideas
from Euclid
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8623
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Proof reveals the interdependence of truths, as well as showing their certainty
[Euclid, by Frege]
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13907
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If you pick an arbitrary triangle, things proved of it are true of all triangles
[Euclid, by Lemmon]
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6297
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Euclid's geometry is synthetic, but Descartes produced an analytic version of it
[Euclid, by Resnik]
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9603
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An assumption that there is a largest prime leads to a contradiction
[Euclid, by Brown,JR]
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14157
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Modern geometries only accept various parts of the Euclid propositions
[Russell on Euclid]
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8738
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Postulate 2 says a line can be extended continuously
[Euclid, by Shapiro]
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22278
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Euclid relied on obvious properties in diagrams, as well as on his axioms
[Potter on Euclid]
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8673
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Euclid's parallel postulate defines unique non-intersecting parallel lines
[Euclid, by Friend]
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10250
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Euclid needs a principle of continuity, saying some lines must intersect
[Shapiro on Euclid]
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10302
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Euclid says we can 'join' two points, but Hilbert says the straight line 'exists'
[Euclid, by Bernays]
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9894
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A unit is that according to which each existing thing is said to be one
[Euclid]
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1600
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Euclid's common notions or axioms are what we must have if we are to learn anything at all
[Euclid, by Roochnik]
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