28 report — 1878. 



what was the origin of mathematical ideas ? Are they to be regarded as 

 independent of, or dependent upon, experience ? The question has been 

 answered sometimes in one way and sometimes in another. But the 

 absence of any satisfactory conclusion may after all be understood as im- 

 plying that no answer is possible in the sense in which the question is put ; 

 or rather that there is no question at all in the matter, except as to the 

 history of actual facts. And, even if we distinguish, as we certainly 

 should, between the origin of ideas in the individual and their origin in a 

 nation or mankind, we should still come to the same conclusion. If we 

 take the case of the individual, all we can do is to give an account of our 

 own experience ; how we played with marbles and apples ; how we learnt 

 the multiplication table, fractions, and proportion : how Ave were after- 

 wards amused to find that common things conformed to the rules of 

 number ; and later still how we came to see that the same laws applied to 

 music and to mechanism, to astronomy, to .chemistry, and to many other 

 subjects. And then, on trying to analyse our own mental processes, we 

 find that mathematical ideas have been imbibed in precisely the same way 

 as all other ideas, viz., by learning, by experience, and by reflexion. The ap- 

 parent difference in the mode of first apprehending them and in their ulti- 

 mate cogency arises from the difference of the ideas themselves, from the pre- 

 ponderance of quantitative over qualitative considerations in mathematics, 

 from the notions of absolute equality and identity which they imply. 



If we turn to the other question, How did the world at large acquire 

 and improve its idea of number and of figures ? How can we span the 

 interval between the savage who counted only by the help of outward 

 objects, to whom 15 was " half the hands and both the feet," and Newton 

 or Laplace ? The answer is the history of mathematics and its successive 

 developments, arithmetic, geometry, algebra, &c. The first and greatest 

 step in all this was the transition from number in the concrete to number 

 in the abstract. This was the beginning not onlv of mathematics but of 

 all abstract thought. The reason and mode of it was the same as in the 

 individual. There was the same general influx of evidence, the same 

 unsought-for experimental proof, the same recognition of general laws 

 running through all manner of purposes and relations of life. No wonder 

 then if, under such circumstances, mathematics, like some other subjects 

 and perhaps with better excuse, came after a time to be clothed with 

 mysticism ; nor that, even in modern times, they should have been placed 

 upon an a priori basis, as in the philosophy of Kant. Number was no 

 soon found to be a principal common to many branches of knowledge that 

 it was readily assumed to be the key to all. It gave distinctness of 

 expression, if not clearness of thought, to ideas which were floating in the 

 untutored mind, and even suggested to it new conceptions. In " the one," 

 "the all," "the many in one," (terms of purely arithmetic origin,) it gave 

 the earliest utterance to men's first crude notions about God and the 

 world. In " the equal," " the solid," " the straight," and " the crooked," 



