Projects & Research A Radical Approach to Real Analysis Macalester College


Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

>2.6 Emerging Doubts

Problem with Series

Vibrating String

Cauchy's Counter-example

2.6 Emerging Doubts

Calculus derives its name from its use as a tool of calculation. At its most basic level, it is a collection of algebraic techniques that yield exact numerical answers to geometric problems. One does not have to know why it works to use it. But the question of why kept arising, partly because no one could satisfactorily answer it, partly because sometimes these techniques would fail.

Newton and his successors thought in terms of velocities and rates of change and talked of fluxions. For Leibniz and his school, the founding concept was the differential, a small increment that was not zero yet smaller than any positive quantity. Neither of these approaches is entirely satisfactory. George Berkeley (1685–1753) attacked both understandings in his classic treatise The Analyst, published in 1734. His point was that a belief in mechanistic principles of science that could explain everything was self-deception. Not even the calculus was sufficiently well understood that it could be employed without reliance on faith.

“By moments we are not to understand finite particles. These are said not to be moments, but quantities generated from moments, which last are only the nascent principles of finite quantities. It is said that the minutest errors are not to be neglected in mathematics: that the fluxions are celerities, not proportional to the finite increments, though ever so small; but only to the moments or nascent increments, whereof the proportion alone, and not the magnitude, is considered… It seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities. But the velocities of the velocities, the second, third, fourth, and fifth velocities, &c, exceed, if I mistake not, all human understanding. The further the mind analyseth and pursueth these fugitive ideas the more it is lost and bewildered; the objects, at first fleeting and minute, soon vanishing out of sight. Certainly in any sense, a second or third fluxion seems an obscure mystery. The incipient celerity of an incipient celerity, the nascent augment of a nascent augment, i.e., of a thing which hath no magnitude; take it in what light you please, the clear conception of it will, if I mistake not, be found impossible; whether it be so or no I appeal to the trial of every thinking reader. And if a second fluxion be inconceivable, what are we to think of third, fourth, fifth fluxions, and so on without end!

“The foreign mathematicians are supposed by some, even of our own, to proceed in a manner less accurate, perhaps, and geometrical, yet more intelligible. Instead of flowing quantities and their fluxions, they consider the variable finite quantities as increasing or diminishing by the continual addition or subduction of infinitely small quantities. Instead of the velocities wherewith increments are generated, they consider the increments or decrements themselves, which they call differences, and which are supposed to be infinitely small. The difference of a line is an infinitely little line; of a plane an infinitely little plane. They suppose finite quantities to consist of parts infinitely little, which by the angles they make one with another determine the curvity of the line. Now to conceive a quantity, or than any the least finite magnitude is, I confess, above my capacity. But to conceive a part of such infinitely small quantity that shall be still infinitely less than it, and consequently though multiplied infinitely shall never equal the minutest finite quantity is, I suspect, an infinite difficulty to any man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust.”

This is only a small piece of Berkeley's attack, but it illustrates the fundamental weakness of calculus which is hammered upon in the second paragraph: the need to use infinity without ever clearly defining what it means. The abuse of infinity has yielded rich rewards, but it is abuse. Berkeley recognizes this.
No one was prepared to abandon calculus, but the doubts that had been voiced were unsettling. Many mathematicians tried to answer the question of why it was so successful. Berkeley himself suggested that there was a system of compensating errors underlying calculus. Jean Le Rond d'Alembert relied on the notion of limits. In 1784, the Berlin Academy offered a prize for a ``clear and precise theory of what is called the infinite in mathematics.'' They were not entirely satisfied with any of the entrants although the prize was awarded to the Swiss mathematician Simon Antoine Jean L'Huillier (1750–1840) who had adopted d'Alembert's limits.

To the reader who has seen the derivative and integral defined in terms of limits, it may seem that d'Alembert and L'Huillier got it right. This was not so clear to their contemporaries. In his article of 1754 on the différentiel for Diderot's Encyclopédie, d'Alembert speaks of the limit as that number that is approached “as closely as we please” by the slope of the approximating secant line. We still use this phrase to explain limits. It can be found in this book. But it is imprecise. What do we mean by “approach”? Are we implying movement? Does it ever actually arrive? If there is an infinite—read endless—sequence of approximations getting us ever closer to the limit, then how can we get to the end?

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