Ideas of Euclid, by Theme

[Greek, 330 - 270 BCE, Born in Alexandria. Studied at the Academy in Athens. Died in Alexandria.]

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2. Reason / E. Argument / 6. Conclusive Proof
Proof reveals the interdependence of truths, as well as showing their certainty [Frege]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
If you pick an arbitrary triangle, things proved of it are true of all triangles [Lemmon]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Resnik]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
An assumption that there is a largest prime leads to a contradiction [Brown,JR]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
A unit is that according to which each existing thing is said to be one
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Postulate 2 says a line can be extended continuously [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Bernays]
Euclid's parallel postulate defines unique non-intersecting parallel lines [Friend]
Euclid needs a principle of continuity, saying some lines must intersect [Shapiro]
Modern geometries only accept various parts of the Euclid propositions [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
Euclid's common notions or axioms are what we must have if we are to learn anything at all [Roochnik]