324 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1961 



a sense which does not make it either absurd or trivial. Literally, it 

 is certainly absurd. It is characteristic of a language that it has no 

 concepts of its own, that anything said in one language can be trans- 

 lated into another. A statement in English, when expressed in French, 

 is no longer an English statement ; but a mathematical statement can- 

 not be made into a nonmathematical statement by any translation, 

 while perfectly good mathematical statements — for instance: "the 

 sum of the angles in a triangle is two right angles," "given any prime 

 number, there is a prime number larger than it" — can be expressed, as 

 here, in perfectly good English, and remain the same when expressed 

 in Russian, Greek, or Japanese as far as mathematical content is con- 

 cerned. As far as I know, the origin of the phrase is an aphorism of 

 Willard Gibbs, "Mathematics is the language of the Sciences"; but 

 this is a statement about the sciences, not about mathematics, just as 

 the sentence "English is the language of Shakespeare's plays" is not 

 a description of the English language. What it means is that mathe- 

 matics has a highly developed notation of its own, which is used in the 

 other sciences ; and this is clearly true, unless it is misinterpreted into 

 an identification of mathematics with its notation. 



The underlying meaning of the phrase "mathematics is a lan- 

 guage" is that mathematics has no content of its own. This is ex- 

 pressed with greater precision in positivist and related philosophies 

 by saying that mathematical theories are purely a matter of definition, 

 or consist of a set of tautologies ; or, and this comes to much the same 

 thing, that in a mathematical system the truth or falsity of the state- 

 ments made depends only on the form of the statements, just as the 

 grammatical correctness of an English sentence is determined by the 

 form of the sentence in relation to the rules of syntax; it is this last 

 formulation which links this view with the linguistic theory of 

 mathematics. 



To understand the degree of truth, and the errors, in these theories 

 we must consider the structure of a mathematical theory if it is 

 presented in an ideally exact form. Suppose that we have such a 

 theory — say the theory of the integers, or Euclidean geometry, to give 

 examples — set out with the greatest possible precision and rigor. We 

 should find, to begin with, that the theory had a set of words or con- 

 cepts in terms of which all the other expressions of the theory are 

 defined : for since any concept can be defined only in terms of other 

 concepts, and the process must start somewhere, there must be some 

 terms which are not defined ; these we call the undefined terms of the 

 theory. Next we find a set of initial assumptions, or axioms of the 

 theory, which take the form of a set of statements about the undefined 

 terms. The theory itself then follows in the form of a chain of 

 logical arguments proceeding by deduction step by step from the 



