Infinitesimal, by Amir Alexander (2015)

There was a time when the average school child learned before high school the story of the Catholic Church's persecution of
Galileo Galilei
for his advocacy of the Copernican view of the solar system  for saying the earth revolves around the sun rather than viceversa.
I don't know
what is taught today, but I suspect it is far less than the whole story. The subtitle of Infinitesimal hints at much more to come: How a
Dangerous Mathematical Theory Shaped the Modern World.
That subtitle speaks ironically, because theories of infinitesimals and invisibly small numbers shaped the modern world not because they are
dangerous but because they are key to everything in science, math and technology today, including integral calculus and its twin brother,
differential calculus. Indeed, differentials are nothing but indefinably small differences. What made this concept so dangerous  or who?
The Jesuits. As author Alexander relates, before infinitesimals were used to solve the specific problems of orbital mechanics
and the broader
problems of calculus, they were studied in great detail by Galileo and his followers, Bonaventura Cavalieri, who wrote the definitive precalculus
work on the subject; and Evangelista Torricelli. By the time the Jesuits finished destroying their efforts, Italy had lost its preeminent position
in mathematics and ceded it to the Royal Society of England.
The Society of Jesus was less than a century old when it took on Galileo and his adherents, but it had already built an important network of
schools throughout Europe. Jesuit schools were based on Aristotelian principles and the mathematics of Euclidean
(plane) geometry: order and
predictability. Ironically this absolute nature of geometry is what attracted young British philosopher Bertrand Russell, an atheist,
to mathematics
in the nineteenth century. Early European mathematics, going back to Greek times, were based on geometric proofs that started with simple,
obvious facts and deductive reasoning to build more complex structures and concepts. This orderly framework, a seemingly unchallengeable
foundation, was nourished by the Jesuits and considered an essential part of their theology. In school today mathematics emphasizes modern
algebra, which did not exist in either Aristotle's or Galileo's day, so it is easy to overlook the importance of the absolute certainty that seemed
to accompany Euclidean geometry. Absolute certainty has always been highly coveted and claimed by religious orders of all stripes.
These concepts were at the heart of the English Civil War in the seventeenth century, which is usually depicted as
Protestants versus Catholics,
but which was actually a much broader conflict of ideas and power. One philosopher greatly influenced by that conflict was
Thomas Hobbes,
who conceived his book Leviathan as a solution for the anarchy that he grew up fearing and abhorring. Hobbes was so scarred by what he saw
that he believed an absolute police state, a leviathan, was a desirable alternative. He was also a lay mathematician who fancied himself a bit of
a savant. Throughout his life he claimed to have squared the circle, to much ridicule, and as a result his other theories were never welcomed in
mathematical circles. Alexander includes an extended description of Hobbes' works in both arenas. Although this side trip is interesting, its
connection to the main story is peripheral at best; it rounds out the century and fleshes out the book, but it did not need fleshing out.
Alexander's historian roots show. He spends a lot of time talking about the religious, political and military conflicts of the sixteenth and
seventeenth centuries, so much so that you start wondering where the infinitesimal part of the book would begin; then it does. He digs
deeps into some of Aristotle's early geometry, demonstrates how it forms the basis for the theoretical conflicts that follow, and then shows
how later proofs by Cavalieri and Torricelli deviate from a onceorderly approach. I was surprised that he never got around to discussing what
makes infinitesimals important to calculus. For him that's a different story, which is what makes this book a work of history first and mathematics second.
What Alexander does not do, and does not try to do, is convince the reader of the utility of this "dangerous idea" of continuous functions being infinitely
divisible. He seems to take it for granted, but I still see it as controversial in some respects. As a result I can agree with the Jesuits to a point, but only
to a point because their interest was in defending previously captured intellectual territory, while mine is in arriving at the approach that works best.
At the heart of the problem with infinitesimals is the idea that something can be infinitely small. The first problem is that infinity is a
mathematical concept with no provable physical basis. The second problem is that when you have a continuous function, as soon as you begin
dividing it, it ceases being continuous. Integral calculus requires a continuous function. So the early ideas of infinitesimals as being real things  infinitely
small and thus unmeasurable, but still real  I do not accept as viable. And yet these were the early concepts of Galileo and Cavalieri, concepts that
eventually revolutionized modern mathematics. Euclidean geometry is no longer king. After all, Euclid's plane geometry only works on a flat surface,
and the Earth is not flat.
The later stage of infinitesimal theory goes to a place that makes a lot more sense to me: infinitesimals as paradigm rather than as real things. For me
the classic example comes from solving an equation of integral calculus. The solution is equal to the area beneath the curve between two points (which
may be zero, infinity, "negative" infinity, or any real value in between). If the curve is a straight line the calculation is obvious, but if it is nonlinear
the calculation was impossible until solved, independently, by Isaac Newton and Gottfried Wilhelm Leibniz.
The solution comes from dividing the curve into slices  thin rectangles  that together are almost exactly the same size as the area under the curve.
When you make the slice thinner, the error is reduced, and so the approximate calculation comes closer to the correct answer. Today we think of this
as approaching a limit  for instance, the area under the curve is the limit of the sum of the slices as the slice width approaches zero. We consider a
calculus equation solved when we can fit it into one of the innumerable solved forms (for instance, the integral of [x to the power of n] is
[x to the power of n+1], divided by [n+1], plus an unknown constant c). As long as the width of the slice is assigned a finite amount, no matter how small,
there will always be an error. Only by fleeing the slices and going to more powerful concept of limits to we get exact solutions. So using the idea of
infinitely small slices  infinitesimals  works as an idea only. You could say it has limits, pun intended. Once you start calling those slices "real", you
lose your connection to reality.
Infinitesimal does a decent job of covering this ground, but the story is erratic and incomplete, much like the modern maths that the Jesuits feared.
It should be part of your math/science/history library, but if you read it as I did you will give it 3.5 stars.
