Ideas from 'Elements of Geometry' by Euclid [290 BCE], by Theme Structure
[found in 'Euclid's Elements of Geometry (Gk/Eng)' by Euclid (ed/tr Fitzpatrick,R) [Lulu 2007,9780615179841]].
green numbers give full details 
back to texts

expand these ideas
2. Reason / E. Argument / 6. Conclusive Proof
8623

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
13907

If you pick an arbitrary triangle, things proved of it are true of all triangles [Lemmon]

6. Mathematics / A. Nature of Mathematics / 2. Geometry
6297

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
9603

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
9894

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
8738

Postulate 2 says a line can be extended continuously [Shapiro]

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
10302

Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Bernays]

22278

Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter]

8673

Euclid's parallel postulate defines unique nonintersecting parallel lines [Friend]

10250

Euclid needs a principle of continuity, saying some lines must intersect [Shapiro]

14157

Modern geometries only accept various parts of the Euclid propositions [Russell]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
1600

Euclid's common notions or axioms are what we must have if we are to learn anything at all [Roochnik]
