BR TO EARLE RAYMOND HEDRICK, [c.22 APR. 1940]
BRACERS 57265. ALS. Harry Ruja from UCLA Archives
Proofread by K. Blackwell

Dear Dr. Hedrick

Here are some possible answers. Will you make rejoinders to bring out further points? 

1. “Axiom” has two uses in mathematics, one old, the other modern. When Euclid spoke of an “axiom”, he meant a general principle which could be seen to be true although it could not be proved; for instance, “two straight lives cannot enclose a space.” Until non-Euclidean geometry arose, no one doubted that this was always true. 

After non-Euclidean geometry, it came to be thought a matter for observation to find out if two straight lines ever enclose a space; this is a question for the physicist. The pure geometer now says: I can make various hypotheses and deduce their consequences, getting various geometries of which only one can be true of physical space. I shall call these hypotheses “axioms”, though I no longer assert that they are true; whether they are true or false is a matter of indifference to the mathematician, who is only concerned to discover what they imply. 

2. I think however, though some would not agree with me, that in logic we still need axioms in the old sense. The consequences we deduce from hypotheses must really, and not only hypothetically, follow from them; otherwise our reasoning is futile. I should say, therefore, that what is necessary for deductive reasoning must still be secured by axioms that are true, and not mere hypotheses. These purely logical axioms are, on my view, the only axioms (in the old sense) that mathematics needs. 

3. Infinity. A number is finite when it can be got by adding units, infinite when it can’t. 

There are infinite numbers, for instance the number of finite numbers. The number of numbers from 0 to n is n+1, therefore n is not the number of finite numbers unless, possibly, if n=n+1. This happens if n is infinite, but not otherwise. 

There is no difficulty about the arithmetic of infinite numbers; indeed in many ways it is simpler than ordinary arithmetic.

Yours
Bertrand Russell.