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has had an attraction for reflective minds, viz., what 

 was the origin of mathematical ideas ? Are they to be 

 regarded as independent of, or dependent upon, ex- 

 perience? The question has been answered sometimes 

 in one way and sometimes in another. But the absence 

 of any satisfactory conclusion may after all be under- 

 stood as implying 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 cer- 

 tainly 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 multipli- 

 cation table, fractions, and proportion ; how we were 

 afterwards amused to find that common things con- 

 formed 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 apparent diflference in the mode of first apprehend- 

 ing them and in their ultimate cogency arises from the 

 difference of the ideas themselves, from the preponde- 

 rance 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 



