MAIN LINES OF MATHEMATICS — COOPER 325 



axioms, every step being a statement implied by the axioms and the 

 statements previously proved. 



This procedure is not quite exclusive to mathematics. It can be 

 followed equally well in other disciplines, and is sometimes found in 

 works on mathematical physics, in particular in dynamics, though a 

 strict axiomatic procedure in which no assumptions are introduced 

 apart from the axioms which are stated initially is rare outside mathe- 

 matics. There are, however, two crucial differences between an 

 axiomatic system in mathematics and one, however strictly carried 

 through, in physics. The first is that in mathematical physics some 

 at least of the undefined terms are intended to be names of specific 

 things or relations in the physical world and the second is that the 

 validity of the system depends on the correspondence between the 

 deductions made within the systems — the theorems of the system — 

 and the observed behavior of these physical things. The correspond- 

 ence may be pretty remote and abstract, in the most developed 

 theories, but it must in some sense be there. In a mathematical 

 theory, on the other hand, the undefined terms need not be the names 

 of specific things, and no observational evidence can, therefore, affect 

 the validity of the theory. 



The classical example of a mathematical system is Euclidean geom- 

 etry. In its traditional form the words used in the system — point, 

 line, distance, and so on — were held to refer to things which exist in 

 the real world or at any rate are approximately copied by real things ; 

 and the axioms of the geometry are supposed to be true statements 

 about these real things. They were also supposed to be self-evidently 

 true, with the exception of the axiom of parallels; it was hoped for 

 centuries that it could be shown that to deny the axiom of parallels 

 would lead one to a contradiction with the other axioms, but eventually 

 geometries were constructed in which the axiom of parallels was 

 denied — for example, Lobachevskian geometry, in which the existence 

 of an infinite number of lines through a given point parallel to any 

 given line is asserted. What is more, these systems were proved to 

 be free from contradiction. 



Here is one way in which we can prove the consistency, that is, the 

 freedom from contradiction, of Lobachevskian geometry. Draw a 

 circle T in the ordinary Euclidean plane. Now we make a miniature 

 dictionaiy, interpreting words which occur in Lobachevskian geom- 

 etry by assigning to them meanings in terms of figures inside r ; for 

 clearness, the words occurring in Lobachevskian geometry will be 

 enclosed in inverted commas to distinguish them from the same words 

 in their normal Euclidean sense. (See fig. 1.) 



"Point" and "line" are to mean, respectively, pomt inside and part 

 of a line inside r. We say that two "lines" are "parallel" if they 



