330 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1961 



true for elementary logic, the propositional calculus. Any of the 

 theorems of this calculus, which is to be foimd, for example, at the 

 beginning of Principia Mathematical can be checked by a purely 

 mechanical process. It was this which led Wittgenstein to put for- 

 ward the tautology theory of mathematics ; for he accepted Russell's 

 claim to have reduced mathematics to logic, and equated logic with 

 the elementary propositional calculus. However, it has been proved 

 that no mechanical decision process is possible even for the whole of 

 mathematical logic, let alone for all of mathematics. The tautology 

 theory should, therefore, be regarded as one wliich applies to some 

 relatively trivial parts of mathematics, but not to the more interesting 

 parts and certainly not to the whole subject. 



The definition theory has more truth to it ; indeed, it has a validity 

 for any section of mathematics. It is quite correct to say that the 

 undefined terms in any mathematical system have the properties as- 

 sumed in the axioms as a matter of definition, and that any theorems 

 of the system are true as a result of these definitions. A statement 

 about points, lines, and so on which is valid if these things occur in a 

 model for Euclidean geometry may be false for a model of a non- 

 Euclidean geometry. 



However, it would be just as true to say that the formulae of chem- 

 istry are a matter of definition ; for if we assigned the names of the 

 elements in a different way we would get quite different chemical 

 formulae. Nevertheless, a statement that chemistry is a matter of 

 definition would ignore the fact that however the names are jumbled, 

 for a chemical system to be valid there must be some way of assigning 

 elements to the names which makes the formulae correct ; to make the 

 analogy clear, there must be some assigmnent of elements to names of 

 elements which makes the actual behavior of matter a model for the 

 chemical system. In the same way, the assertion that mathematics is 

 purely a matter of definition ignores the problem of the validity of 

 mathematical systems. For particular systems, this validity can be 

 established within mathematics, by constructing models for a given 

 system in terms of another axiom system — say geometry in terms of 

 the real numbers ; but eventually we must come to a primary system of 

 axioms, and if we are to have any sort of guarantee of validity for this 

 it must be found outside mathematics. 



What sort of guarantee can we have ? This is a very difficult ques- 

 tion, and it would be wrong to suggest that mathematics must be tied 

 down to, or that it does imply, any one answer. Certainly there can 

 be no absolute guarantee of the consistency of mathematics : any science 

 is liable to error, and the progress of mathematics in the future may 

 reveal unsuspected inconsistencies, as it has done in the past. However, 

 this does not allow us to dismiss the whole problem of consistency; 



