Scene: JEAN D'ALEMBERT, French mathematician, and one of the initiators of the "Encyclopédie ou Dictionnaire Raisonné des Sciences, des Arts, et des Métiers", is sitting at his desk, reading out of one of its twenty-eight large volumes. This work was published during the years 1751-1765 (edited by Denis Diderot et al.), as a kind of manifesto, or Bible, of the Enlightenment. D'Alembert was one of its major contributors, and represents the new breed of French mathematicians and scientists, believing Mechanics to be the domain and proper application of mathematics for the future. It is 1783, the year of his death.
D'ALEMBERT [turning pages ]23
...N-A,...N-E,...Ah! Here it is! "Negative"; let me see, now, what did I write?...Hmmm. [Grunts and grimaces, then reads ]
"... the rules of algebraic operations with negative quantities are generally admitted by everyone, and acknowledged as exact, whatever idea we may have about these quantities..." [laughs ]
Whatever idea we may have about these quantities! When I wrote that entry, I wished to express my impatience with those who were detaining our poor negative numbers for interminable philosophical interrogation, instead of permitting them to get on with the work they are so perfectly fitted to perform!
It has to be admitted that it is not easy to get a firm hold on the idea of negative quantities; but those clever philosophers who had given out that a negative quantity is somehow below nothing, or smaller than nothing, were just befogging the issue. When one considered the exactness and simplicity of algebraic operations on negative quantities, one became convinced that the essential idea of them had to be simple, not something arising out of some convoluted metaphysic.
And, as for the outrageous notion that negative quantities are greater than infinity, and other such paradoxes arising out of loose reasonings with divergent series - we do well to distinguish very carefully between convergent series and divergent series. I explained this fully in another article in the Encyclopédie - under the heading "Series" [gestures towards the stacked volumes ]. I said that when the series approaches some finite quantity more and more, and consequently the terms of the series, or quantities of which it is composed, go on diminishing, one calls it a convergent series, and if one continues to infinity, it will finally become equal to this quantity. Thus 1/2 + 1/4 + 1/8 + ... form a series which always approaches 1, and which will become equal to it finally when the series is continued to infinity.
Series which are not thus convergent should be treated with the utmost caution, if at all. Some people find them useful for approximations. As for me, I avow that all reasonings based on series that are not convergent appear very suspect, even when the results are in accord with truths arrived at in other ways.24
No - we can disperse the metaphysical fog very quickly by the light of Mechanics, anchoring our reasonings firmly to physical quantities. As I wrote at the time, in this article on the meaning of "negative"... [pores over the volume again ] ..humm.. ha! Yes - here it is .. I saw that the negative quantities met in calculations - in Mechanics, as in Geometry - only differed from positive quantities by their situation with respect to some point or line. Indeed, I naturally concluded that these so-called "fictitious" quantities are actually real quantities, but with an opposite orientation to that which one might have supposed and represented by a positive quantity.
I must confess, I myself found it difficult, then, to conceive of an absolutely isolated negative quantity - perhaps I still do! What a business these negative quantities have had trying to persuade us that they should be treated as worthy numbers in their own right! Even irrational numbers have had it easier. A plague on the philosophers and their scruples!
But even practical mathematicians like Girolamo Cardano had uneasy consciences. They called them names like "absurd numbers", "fictitious numbers", "invented numbers". Luca Pacioli and others found it necessary to deal with quadratic equations in six different forms - each with its own rule for solution. What a cumbersome bag of tricks! And no European mathematician, right through the sixteenth and seventeenth centuries, could bring himself to accept negative numbers as roots of equations. Blaise Pascal derided those who thought of taking four away from nothing and getting minus four; even René Descartes called his negative numbers "false roots", but he showed the beginnings of acceptance because they turned out to have a use.
Now - there is the secret: in spite of all these scruples, the negative numbers won for themselves more and more recognition over two centuries, because they proved themselves useful. They worked! Especially in Mechanics, when that great science came to the forefront. And no doubt the development of Analytic Geometry was a help.
It is astounding, in view of these long European reservations, to think that six hundred years ago the Hindu mathematician Bhaskara was writing about positive and negative roots, and getting 50 and minus 5 formally as solutions for the equation x2-45x=250. He did say that the second root should not be used as "people do not approve of negative solutions". But those Hindus certainly recognised the role that negatives could play in representing debts, and as early as the seventh century, Brahmagupta was calculating freely with negative numbers and giving rules for operating with them. In contrast, even the man some call the father of algebra - that great Greek Diophantus, in the third century - does not appear to have recognized that any of his equations might have multiple roots, let alone negative roots! But at last, a thousand years after Brahmagupta, we in Europe finally arrived.
[reads ] "... the algebraic rules of operations with negative numbers are generally admitted by everyone..." I wrote that at a time when the philosophical quibblers were at last beginning to subside under the great evidences of success of the new Analysis. It took a long time for some people to learn that we have entered the Age of Mechanics; and outside Mechanics there is no sense or use of Algebra to speak of, hence quibbles about the nature of negative numbers in themselves, without any reference to any physical quantity, are unprofitable. It is enough that they work consistently in the great house of Mechanics!
Generally, the doubting mathematicians seem to have taken my injunction to heart: "Go on!" I said, "Faith will follow!" I do not speak, however, for the English. They remain, regrettably, in a past age of mathematics. They have reverted to studying their hero Newton in place of Nature. There is, not surprisingly, a rearguard action over there: the opposition to negative numbers is kicking still! Their story continues!