NUMERICAL LAWS AND MATHEMATICS 147 



from previous knowledge, we are always assuming some 

 experimental fact which is not clearly involved in the 

 original statements. What we usually assume is that 

 some law is true in circumstances rather more general 

 than those we have considered hitherto. Of course the 

 assumption may be quite legitimate, for the great value 

 of laws is that they are applicable to circumstances more 

 general than those of the experiments on which they are 

 based ; but we can never be perfectly sure that it is 

 legitimate until we try. Calculation, then, when it 

 appears to add anything to our knowledge, is always 

 slightly precarious ; like theory, it suggests strongly 

 that some law may be true, rather than proves definitely 

 than some law must be true. 



So far we have spoken of calculation as if it were merely 

 deduction ; we have not referred to the fact that calcula- 

 tion always involves a special type of deduction, namely 

 mathematical deduction. For there are, of course, forms 

 of deduction which are not mathematical. All argument 

 is based, or should be based, upon the logical processes 

 which are called deduction ; and most of us are prepared 

 to argue, however slight our mathematical attainments. 

 I do not propose to discuss here generally what are the 

 distinctive characteristics of mathematical argument ; 

 for an exposition of that matter the reader should turn 

 to works in which mathematicians expound their own 

 study. 1 I want only to consider why it is that this kind of 

 deduction has such a special significance for science. 

 And, stated briefly, the reason is this. The assumption, 

 mentioned in the last paragraph, which is introduced in 

 the process of deduction, is usually suggested by the form 

 of the deduction and by the ideas naturally associated 

 with it. (Thus, in the example we took, the assumption is 

 suggested by the proposition quoted about proportionality 



1 E.g. "An Introduction to Mathematics," by Prof. Whitehead, in 

 the Home University Library. 



