PHYSIOLOGICAL PARADOXES. 



argument must be wrong. The question is 

 sound as the argument seems in every detail, 

 what is the detail in which it begins to go 

 wrong, and how ? 



The mathematician, driven, in spite of his 

 learning, to recognise the real nature of the 

 difficulty, resorts to an artifice in the form of 

 a mathematical convention, which he attempts 

 to palm off upon us as the real solution. In- 

 finitely small things, he says, having no real 

 size at all, are equal to one another, so that 

 when the distance between Achilles and the 

 tortoise has become infinitely small, it is equal 

 to the distance between him and a point in 

 front of the tortoise, and he can traverse the 

 latter distance in the same time as the former. 



But mathematical conventions can only be 

 allowed to pass while we remember that they 

 are conventions, brief statements summing up 

 the facts with an error so small that for prac- 

 tical purposes it does not matter. When at- 

 tempts are made to use them as strict state- 

 ments of exact truth, and arguments are 

 founded upon them, they lead us astray. 



When we speak of infinitely small spaces, 

 if we mean spaces which, though very small, 

 are nevertheless real and actual, it is obviously 

 not true to say that they are all equal. A real 

 space can be divided into two, and by an in- 

 evitable axiom, it is greater than either of its 

 parts. If, however, we mean spaces too small 



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