PLENO Y DIPUTACIÓN PERMANENTE
INTERPELACIONES URGENTES:
Mathematicians sometimes express opinions about non-mathematical topics to the intense annoyance of those who feel that they should stick to their own business. In 1734, a very annoyed Bishop Berkeley decided to attack mathematicians on their home ground and wrote a short, but extremely clever, pamphlet entitled
THE ANALYST
A Discourse addressed to an Infidel Mathematician. Wherein it is examined
whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived,
or more evidently deduced, than Religious Mysteries and Points of Faith. In it, he argues that the calculus as then conceived was such a tissue of unfounded assumptions as to remove every shred of authority from its prac- titioners. If the reader substitutes our ‘derivative’ for words like ‘fluxions’, ‘differences’ and ‘infinitesimals’ she will get the flavour of his attack
And yet in the calculus differentialis, which Method serves to all the same Intents and Ends with that of Fluxions, our modern Analysts are not content to consider only the Differences of finite Quantities: they also consider the
Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum. That is, they consider Quantities infinitely less than the least discernible Quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding Infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of Infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth etc. Fluxions, so there are Differences, first, second, third fourth, etc. in an infinite Progression towards nothing, which you still approach and never arrive at . . .
It must indeed be acknowledged, the modern Mathematicians do not consider these Points as Mysteries, but as clearly conceived and mastered by their comprehensive Minds. They scruple not to say, that by the help of these new Analytics they can penetrate into Infinity itself: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites. But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species. Nothing is easier than to devise Expressions or Notations for Fluxions and Infinitesimals of the first, second, third, fourth, and subsequent Orders, proceeding in the same regular form without end or limit ˙x, ¨x,...x ,....x etc. [that is to say, our f0(t), f00(t), f000(t), f0000(t), etc.] . . . . These Expressions indeed are clear and distinct, and the Mind finds no difficulty in conceiving them to be continued beyond any assignable Bounds. But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves, which are supposed to be expressed or marked thereby, we shall discover much Emptiness, Darkness, and Confusion.
On the same theme, he says ‘the Velocities of the Velocities, the second, third, fourth, and fifth Velocities, etc. exceed, if I mistake not, all Humane Understanding. . . . He who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity.’ The sting of Berkeley’s attack has been removed by the development of rigorous analysis. However, it is worth taking a little while to think about Berkeley’s remarks about higher derivatives.
From the mathematical point of view there is no problem. Starting from a well behaved function f we can form its derivative f0 which is just another function. If everything is well behaved, we can differentiate the new function f0to obtain f00and then differentiate that to obtain f000and so on. (The result of differentiating n times is called the nth derivative and written f(n).)
From the point of view of intuition, things are less clear. I think that riding in cars, trains and aeroplanes has given me a clear experience of velocity and some experience of acceleration. Thus, I believe (possibly falsely) that I have a certain intuitive idea of a rate of change and some (rather less clear) intuitive idea of the rate of change of a rate of change. However, the notion of the rate of change of a rate of change of a rate of change conveys rather little to me.1
To see that this is not entirely a consequence of my mental limitations, it is instructive to consider the problem of physical measurement. If I ask an 1 ‘In the fall of 1972, President Nixon announced that the rate of increase of inflation was
decreasing. This was the first time a sitting president had used the third derivative to advance his case for re-election.’ Hugo Rossi in the Notices of the AMS, Vol. 43, No. 10, Oct. 1996.
engineer to measure a quantity (distance travelled, say), then she will often be able to do this easily and cheaply. If I ask for the rate of change of the quantity (velocity, say), then the measurement will considerably harder to perform and cost considerably more. If I go further and ask for the rate of change of the rate of change (acceleration, say), the task becomes very much harder and very much more expensive.
The next exercise may help the reader see part of the problem.
Exercise 6.1.1. Let
f(x) D sin x C 1020sin 1012x. Graph f , f0and f00using appropriate scales.
Ever since the work of Galileo on falling bodies, we have known that, hard though second derivatives may be to measure and understand, they form part of the language of nature. Fortunately (either because that is the way nature is, or because that is the way we understand nature) many of our physical theories do not require higher derivatives than these. We may answer Berkeley by saying that we can differentiate as many times as we like, but that we do not expect to have much intuition as to the meaning of derivatives of high order.