Combining Philosophers

All the ideas for Euclid, Thrasymachus and Ryan Wasserman

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18 ideas

2. Reason / E. Argument / 6. Conclusive Proof
Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by 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 [Euclid, by Lemmon]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by 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 [Euclid, by 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 [Euclid]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
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 [Euclid, by Roochnik]
9. Objects / C. Structure of Objects / 6. Constitution of an Object
Constitution is identity (being in the same place), or it isn't (having different possibilities) [Wasserman]
Constitution is not identity, because it is an asymmetric dependence relation [Wasserman]
There are three main objections to seeing constitution as different from identity [Wasserman]
9. Objects / C. Structure of Objects / 8. Parts of Objects / a. Parts of objects
The weight of a wall is not the weight of its parts, since that would involve double-counting [Wasserman]
9. Objects / F. Identity among Objects / 3. Relative Identity
Relative identity may reject transitivity, but that suggests that it isn't about 'identity' [Wasserman]
28. God / C. Attitudes to God / 3. Deism
Clearly the gods ignore human affairs, or they would have given us justice [Thrasymachus]