Conventionalism in geometry

Either:

How do we decide between alternative geometries, according to Poincaré?

or:

1. According to Kant, geometrical truths are known via a peculiar capacity of the human mind (intuition);
2. Everything that is dependent on the capacities of the human mind is empirical'
3. Therefore, geometry is empirical, according to Kant.

What, if anything, is wrong with this argument? Compare the attitudes that Kant and Poincaré could take towards it.

Your answer may include discussion of the following questions:

• Is there geometrical truth?
• What is the rôle of intuition in geometry? How is it related to the notion of pure intuition in Kant?
• What is the basis of rigour in geometry?

Readings

(Torretti, 1978, 320-40 (esp. 328-35));

(Poincaré, 1978a, chs. 3, 5);

(Poincaré, 1978c, chs. III, IV(§§1-6 only));

(Poincaré, 1963, ch. III, §§1,4-6);

(Kant, 1997a, A19/B33-A30/B45)

Further readings

(Poincaré, 1978a, ch. 4);

(Poincaré, 1963, ch. II);

Stump (1991)