Tarleton — The Relation of Mathematics to Physical Science. 163 



that the highest aim of scientific investigation is the mathematical expression 

 of the fundamental laws of nature, and the mathematical explanation of the 

 dependence of its observed facts on these laws. 



The truth of what I have said will not, I think, be questioned by those 

 who realize the true nature of Mathematics ; but on this subject considerable 

 misapprehension seems to prevail, not only among the unlearned, but even 

 among men who have attained the most exalted position in the scientific 

 world. 



The opinion that mathematical processes are merely exercises of pure 

 reasoning seems to have come down from the Middle Ages, and harmonizes 

 with the theory that formal Logic is an instrument for making new 

 discoveries. This theory is still occasionally put forward by men who have 

 a considerable knowledge of Mathematics and Physics. If it were true, 

 physicists might well regard Mathematics as of comparatively little value. 

 Pure reasoning taken alone can give only consistency and clearness of 

 thought, but can never lead to the discovery of a new truth ; and if 

 Mathematics were nothing but a process of pure reasoning, its claim to be 

 regarded as Science might be fairly disputed. It is, however, easily seen that 

 in the case of pure Geometry the theory I have mentioned is quite erroneous. 

 No amount of pure reasoning could deduce the first theorem in Euclid's 

 Elements from the Axioms, without supposing the superposition of one 

 triangle on another. This process is not pure reasoning, but is rather of 

 the nature of an experiment. Every geometrical proof which requires a 

 construction is a process of the same kind. 



In the case of Algebra, which starts as the Science of number, and 

 becomes, as it progresses, the Science of quantity in the most general sense, 

 the true nature of the process is not so easily seen. 



Every algebraical expression may be regarded as being of the nature of a 

 series ; and algebraical theorems have to do, in general, with the properties of 

 series and their relations. It is impossible to think at all without passing- 

 through a series of states of consciousness ; and thus the more elementary 

 properties of series are constantly before us. Certain general laws are thus 

 discovered ; and these, together with results obtained by immediate inspection 

 in any particular case, enable us to arrive at new results. In the inspection 

 nothing is immediately before us but the algebraical symbols themselves. 

 They, however, as separate objects of attention, are as well suited as any 

 other objects to enable us to intuite the property of series-arrangement which 

 we require. The processes of Algebra are thus fundamentally, though not 

 perhaps to the same extent as those of Geometry, of the nature of 

 experiiucuLs. 



