456 SCIENCE PROGRESS 



The Nobel Prizes 



We have been informed by the Nobel Committee at 

 Stockholm that the following gentlemen have been selected 

 for the Nobel Prizes given during 1920. 



Physics : Charles Edouard Guillaume. 

 Medicine : August Krogh. 

 Literature : Knut Hamsun. 



The V-i : A Protest (Amateur) 



Probably modern mathematics differs from past mathematics chiefly in 

 the stress which is now laid upon " Complex Numbers." When algebra was 

 first invented numbers were conceived to be signless ; then gradually the so- 

 called negative numbers were introduced ; then mathematicians went on to 

 separate rational from irrational numbers ; and now their pupils are obliged 

 to twist their brains by the consideration of complex numbers. Every book 

 one reads nowadays commences with a series of paragraphs on these different 

 kinds of numbers, and the reader is often obliged to generalise the simplest 

 functions in terms of the last mentioned. Is there really any advantage in 

 all this ? And, though I am only an amateur, I should like to maintain that 

 there is no such advantage, and, moreover, that complex numbers do not 

 exist at all — though I am aware that such a statement will expose me to ad- 

 verse or even contemptuous criticism. To begin with, it may of course even 

 be doubted whether there are such things as negative numbers — and this 

 doubt has been frequently expressed by the greatest experts. Negativeness 

 is not a property of number itself but merely an expression of the fact that a 

 number has been subjected to the inverse operation of addition. We write 

 — I merely as a convenience and because no method of expressing operation 

 apart from number is now in general use. If A expresses the operation of 

 addition and A^ the inverse operation of addition, then instead of writing - i 

 we should more correctly write A^ (i). This of course would be very cumber- 

 some and we therefore write - i instead oi A^ (i), but merely for convenience 

 and brevity. I mean that in the idea of — i we possess, not only the idea of 

 a number, but also the idea of an operation acting upon a number — that is, of 

 something more than a number. We therefore have no right to say that 

 negative numbers exist by themselves. So also, all fractions are the results 

 of the inverse operation of multiplication. On the other hand, what are called 

 irrational numbers appear to me to be much more real numbers than negative 

 numbers or fractions, because, the notion of a signless number consisting of 

 the sum of an infinity of signless numbers constantly diminishing in magnitude 

 can be immediately comprehended. 



Now let us consider V — i. If there is no such thing as the number — i, 

 then, a fortiori, V — i is not a number. But more than this, even supposing 

 that we assent to the position that — i is a number, still there actually is no 

 number which when multiplied by itself produces — i. In fact V — i is not 

 a number and cannot possibly exist as a number. What our modern mathema- 

 ticians really do is to pretend that it is a number and then to write enormous 

 volumes on the basis of this pretence, thus complicating the whole of the simple 

 and beautiful science of numbers in general — and not only complicating it but, 

 to my mind, falsifying it. Our fathers called complex numbers unreal num- 

 bers or impossible numbers, and they were right. We give them another name, 

 that is complex numbers, and try to persuade ourselves that they are both 

 possible and real. Why not drop the whole pretence so far as algebra is 

 concerned ? 



But of course I am talking only of algebra. Is there no operation, which 

 when once applied to and then once repeated upon a positive number, will turn 



