EULER [with heavy sarcasm ]
Indeed? We all have these experiences when our pens appear wiser than ourselves, and all the world is made clear and simple in a flash. On the other hand - I will be honest with you - I have had moments when my pen seemed to have run away with me and betrayed me - when my head would ache fiendishly in the battle to come to terms with these negative numbers and infinities. See this: we know by indisputable arguments, as (for example) letting x take value minus one in the well-known series:
|=1-2x+3x2-4x3+ ¼, x=-1
that infinity is the sum of the positive whole numbers.
But if I let x
take value two
in that geometric series we considered previously, I find that minus one is the sum of all the powers of two!12
Now in this series the numbers being summed on the right are greater than those being summed in the first series, and therefore I deduce that the second sum (namely minus one) must be greater than infinity. I am forced, therefore, to conclude that infinity is a kind of limit or threshhold between the domains of positive and negative numbers, and in this respect infinity resembles nothing!13
Quite so! We appear to have come full circle back to my original assertion, and (if my eyes do not deceive me) with much the same arguments as I used myself - extrapolating from quantities growing towards infinity.
Not so, Dr Wallis! You were drawing unwarranted conclusions from most unruly sequences; I must beg to point out that my arguments are more rigorous. I reason from standard series such as everyone uses. It is outrageous and intolerable that such well-known series should lead to contradictions. (Although I do sometimes entertain doubts as to how freely we may assign values to variables14... possibly there exists some peculiar limitation of domain - but this smacks of metaphysics again!).
Come, I will give one more unpalatable example:15
We already have minus one as the sum of the powers of two:
Now, if I let x
, in the well-known series
then I find that minus one is also the sum of another series:
Two distinct series, each with sum minus one! I cannot possibly equate them and I am left to face an insoluble contradiction.
LEIBNIZ [in attitude of gloom and despondency ]
... I was so certain that no harm could come from calculating with these negative numbers ... Perhaps the difficulties all arise with this thorny notion of infinity? Or has some property broken out of its proper domain? Back to metaphysics, again...
Whatever will become of my dream of a universal rational language? I have sought to discover this language - founded upon a logic of the imagination - which will be able to handle everything in the whole domain of human thought that is capable of exact expression. But there can be no hope of ever succeeding in martialling the human reason into safe paths, if even the logical tools and symbols of the mathematicians cannot keep their minds from getting lost!16
[EXIT in deep thought ]
ARNAULD [shaking his head ]
I am a great deal more confused than I was. It began with negatives not behaving in the same fashion as positive numbers, and it ends with their breaking the very laws of logical thought. Perhaps one would be wiser to eschew them altogether!
[EXIT in agitated manner ]
Methinks I will return to solid ground: let us have less of this metaphysical torture and restrict ourselves to the healthy and fruitful physical and geometrical application of these quantities. Only by rooting our speculations in Nature can we become properly schooled in their true uses.
[EXIT counting paces ]
EULER [to audience, rubbing his eyes and holding his head in his hands ]
I really must be getting back to my own century. It has been fascinating and instructive to see how these problems were perceived by my mathematical ancestors. I can take a grain of comfort, I think, in the progress we have made since their day; but it is a sobering fact that we are still beset, forty years later, by the same major difficulties. What manner of being is a negative number in itself? Upon what logical basis can we justify our rules for working with them? How, above all, can we avoid heading into these painful contradictions?17
I tell you - I do not believe we shall entirely circumvent the mathematical problems and escape the metaphysical swamps by Dr Wallis' road; Nature herself is only too ready with her next riddle - challenging our mathematical resources to their limits. That is my experience.
Nor, I believe, can we hide for long from the confrontation with negatives and infinities by simply running away, as Monsieur Arnauld seems to be tempted to do! If their time has come, they will surely find us! But learning to live with them is proving much more difficult than I, for one, ever suspected. We shall need all our mathematical ingenuity and even a dose of Herr Leibniz' metaphysics; I vow to keep my courage up - and to keep a firmer grip of my mischievous pen; but I wish that my head would not ache so!
[EXIT, CURTAIN ]