236 SCIENCE PROGRESS 



probable course of thought as soon as this process has com- 

 menced in any important branch of mathematical knowledge. 



A people once existed, civilised and addicted to learning, 

 who, as it happened, had only thought of number in connection 

 with fishes. They would say two-fishes and three-fishes make 

 five-fishes ; but never once had they thought of two and three 

 and five, and never had it occurred to them that two stones 

 and three stones make five stones. Their philosophers, erudite 

 men steeped in the classical languages and ancient wisdom, 

 had elaborated subtle theories of the a priori nature of fish 

 existence. Also the fish-thinkers, as the mathematicians were 

 termed, were wont to commence their more complete treatises 

 with some consideration of deep-sea dredging. In time other 

 aggregates were considered but at first without reference to 

 fish-arithmetic. Some fish-thinkers elaborated a disconnected 

 theory, rather curious than important as it appeared to them, 

 of deux-stones, trois-stones, quatre-stones, etc. ; and, finally, in 

 connection with fish-arithmetic a bold man denied that a fish 

 was an ultimate thought-unit and introduced the notion of 

 fractions— an idea which he utterly failed to make intelligible, 

 as it was conclusively proved by philosophers that any portion 

 of a fish was not a fish, and that it was illegitimate to apply 

 to it the fish-concept. 



Unfortunately the narrow specialism of a mathematical 

 education left many mathematicians utterly unable to appreciate 

 the crushing force of this philosophic refutation. Also the 

 confusion was increased by the effort of an eminent mathe- 

 matician to explain the idea of fractions in some popular 

 lectures. He asked his hearers to imagine a universe in which 

 it was impossible for any being to see at one and the same time 

 the head and the tail of a fish. For more than a generation 

 afterwards all philosophers and many mathematicians imagined 

 that the theory of fractions was inextricably bound up with 

 the existence of such a fantastic world. In short, no sound 

 thinker— of the sort that carries weight — ever treated fractions 

 seriously ; and in standard philosophic treatises they were 

 habitually dismissed as mathematical fads without any real 

 import for serious thought. 



At this point we may dismiss the fable and return to the 

 central problem. How can mathematical truths be completely 

 disengaged from all adventitious ideas? Twenty years ago, 



