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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 Counterexample

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 selfdeception. 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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