458 MATHEMATICS 



are telling the truth when they say that the prisoner was in St. Louis 

 at the moment the crime was committed in Chicago, and if it is 

 true that a person cannot be in two places at the same time, it follows 

 that the prisoner was not in Chicago when the crime was committed." 

 This, according to Peirce, is a bit of mathematics; while the further 

 reasoning by which the jury would decide whether or not to believe 

 the witnesses, and the reasoning (if they thought any necessary) 

 by which they would satisfy themselves that a person cannot be 

 in two places at once, would be inductive reasoning, which can give 

 merely a high degree of probability to the conclusion, but never 

 certainty. This mathematical element may be, as the example 

 just given shows, so slight as not to be worth noticing from a prac- 

 tical point of view. This is almost always the case in the transac- 

 tions of daily life and in the observational sciences. If, however, we 

 turn to such subjects as chemistry and mineralogy, we find the 

 mathematical element of considerable importance, though still 

 subordinate. In physics and astronomy its importance is much 

 greater. Finally in geometry, to mention only one other science, the 

 mathematical element predominates to such an extent that this 

 science has been commonly rated a branch of pure mathematics, 

 whereas, according to Peirce, it is as much a branch of applied 

 mathematics as is, for instance, mathematical physics. 



It is clear from what has just been said that, from Peirce's point 

 of view, mathematics does not necessarily concern itself with quanti- 

 tative relations, and that any subject becomes capable of mathe- 

 matical treatment as soon as it has secured data from which import- 

 ant consequences can be drawn by exact reasoning. Thus, for 

 example, even though psychologists be right when they assure us 

 that sensations and the other objects with which they have to deal 

 cannot be measured, we need still not necessarily despair of one day 

 seeing a mathematical psychology, just as we already have a math- 

 ematical logic. 



I have said enough, I think, to show what relation Peirce's con- 

 ception of mathematics has to the applications. Let us then turn 

 to the definition itself and examine it a little more closely. You 

 have doubtless already noticed that the phrase, " the science which 

 draws necessary conclusions, " contains a word which is very much 

 in need of elucidation. What is a necessary conclusion? Some of 

 you will perhaps think that the conception here involved is one 

 about which, in a concrete case at least, there can be no practical 

 diversity of opinion among men with well-trained minds; and in 

 fact when I spoke a few minutes ago about the reasoning of the 

 jurymen when listening to the lawyer trying to prove an alibi, I 

 assumed tacitly that this is so. If this really were the case, no further 

 discussion would be necessary, for it is not my purpose to enter into 



