MAIN LINES OF MATHEMATICS — COOPER 329 



is easily avoided. This important construction provides us with a 

 way of giving a model for any part of mathematics in terms of con- 

 cepts derived solely from set theory. 



This process was called by Frege and Russell tlie reduction of math- 

 ematics to logic : but this is rather misleading, for what they meant by 

 logic was set theoiy, which is vastly different from traditional logic. 

 "Wliat the construction gives us is a proof of the consistency of mathe- 

 matics depending only on the assumption that set theory is consistent : 

 and if set theory were identical with logic in the traditional meaning 

 of the word, its consistency would not be open to reasonable doubt. 

 Unfortunately, hardly had the new theory of sets been established 

 before serious antinomies were discovered arising from the arguments 

 used in it; and even apart from these antinomies some mathemati- 

 cians considered certain arguments of set theory to be suspect. These 

 difficulties have not yet been overcome. Certainly, axiom systems for 

 set theory have been constructed which avoid all the known antino- 

 m.ies and do not appear to contain any new ones; but we have no 

 proof that these axiom systems are self-consistent, and there is strong 

 reason to believe that we can never have such a proof. 



Let us now assess the degree of truth in the theories that mathe- 

 matics consists of tautologies or that it is solely a matter of definition. 

 These theories are frequently coupled, though they are in fact dis- 

 tinct. They will be found, for example, in Ayer's Language^ Truth 

 and Log'tc^ where they are supported by the picturesque, if scarcely 

 verifiable, statement that "A being whose intellect was infinitely power- 

 ful would take no interest in logic or mathematics. For he would be 

 able to see at a glance everything that his definitions implied, and, 

 accordingly, could never learn anything from logical inference which 

 he was not fully conscious of already." This couples the definition 

 theory of mathematics with the theory that, within a given set of 

 definitions — and axioms are regarded here as definitions — the truth 

 or falsehood of any statement can be decided by a purely mechanical 

 process — say, that a computer could be built which, on having fed to it 

 the axioms of mathematics, could settle the truth of any theorem, given 

 enough time; for a computer with indefinite time is the nearest we can 

 get in this world to Ayer's being with infinite intelligence; and, if his 

 philosophy needs divine help to save it, we are not better off in the next 

 world, if theology is correct in holding that divine omniscience ex- 

 tends to the physical world as well as to mathematics. 



The tautology theory of mathematics is essentially a result of gener- 

 alization from two special cases. It is based, first, on the obvious fact 

 that it is true of elementary aritlunetic; the truth of an arithmetical 

 calculation, however complicated, can be checked by a purely mechan- 

 ical decision process. Second, it is based on the fact that the same is 



