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Reading Amir Alexander’s “Infinitesimal”

June 28, 2014
Roscoe, N.Y.

For as long as I can remember, I have been skeptical about the existence of infinity. I just don’t see any evidence of infinity in the real world. The Big Bang caused only a finite amount of matter and energy to come into being, and the amount can actually be estimated. The number of atoms in the universe is about 1080, and while that’s certainly quite a lot, it’s still short of infinite. Since space is defined by these particles, there is no infinite space either.

Quantum theory seems to deny the existence of the infinitely small. The properties of elementary particles are not continuous but discrete. There is a limit to the subdividing of space.

In recent years, the word “infinity” has been often used in connection with the theory of the multiverse. But I’m not sure it’s justified in this context either. If each universe splits into two every time a wave function collapses, that’s definitely a whole lot of universes, but it’s still not infinite.

So where did the notion of “infinity” even come from? From mathematics, I think. Just start counting 1 2 3, and the concept of infinity raises its ugly head. For every number you can conceive, just add 1 to get another, even if you exceed the number of particles in the universe.

Because we count on and on without reaching an end, the infinity of the counting numbers is always a potential rather than an actual quantity. As Aristotle said in Book III of the Physics (which remains one of the most clearheaded discussions of the subject) “Infinity turns out to be the opposite of what people say it is. It is not ‘that which has nothing beyond itself’ that is infinite, but ‘that which always has something beyond itself.’” (Robin Waterfield translation)

A much thornier problem is the continuum. This is the infinite range of real numbers between any two numbers, and it seems to inherently exist any time we conceive of a number line. What’s worse, Georg Cantor demonstrated that the infinity of the continuum is qualitatively different from the enumerable infinity of the counting numbers. We can’t even devise a system to list all the real numbers that exist between 0 and 1.

Before we get too worried, however, it helps to remember that the continuum is a purely human invention. It is not the analog of anything in the real world of discrete particles and quantum energy levels. It is a construct of the imagination.

But the continuum is also a convenient approximation to the real world, and hence, an essential tool for doing some forms of math. These include the integral and differential calculus, which have proved themselves to be enormously valuable tools for analyzing real-world processes.

Before Newton and Leibniz invented the calculus, earlier mathematicians developed techniques that treated the continuum as an infinite collection of infinitesimals or indivisibles. This was radically different and unusual math, and there were consequently debates about its legitimacy.

This is the prehistory of calculus that is the focus of Amir Alexander’s fascinating book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (Scientific America / Farrar, Straus and Giroux, 2014).

Amir Alexander is in the History Department at UCLA. He knows that the history of science and mathematics does not occur in vacuum but instead takes place within a context of political, religious, and philosophical history. The marvel of Infinitesimal is that it demonstrates how these mathematical debates paralleled — and were actually a part of — the turbulent religious and political conflicts that characterized the 16th and 17th centuries of Early Modern Europe.

The persuasive thesis of Infinitesimal is that rigid political and religious doctrines often used classical mathematics as a model, and that this inhibited the development of useful mathematical tools involving infinitesimals. On the other side, the increased freedom, diversity, pluralism, and (eventually) tolerance that arose out of the Protestant Reformation helped foster a more empirical approach to mathematics and science that welcomed these new tools.

Infinitesimal is divided into two parts. The first focuses on Rome, and begins with the founding by Ignatius Loyola of the Society of Jesus, known commonly as the Jesuits. The purpose of the Jesuits was to establish an intellectual force to reverse the spread of Protestantism, and as such were entirely dedicated to the dictatorial authority of the Catholic Church. As Ignatius wrote, “What I see as white I will believe to be black if the hierarchical Church thus determines it.” (quoted in “Infinitesimal” on page 40).

The Jesuits also founded schools, and still today, many people swear by (or swear at) a Jesuit education. At first, Jesuit colleges focused on theology, philosophy, grammar, and classical languages. Although we today view Aristotle as an early observationalist scientist who also got stuff wrong, at the time he was treated more as an unquestioned authority on many subjects, whose writings had been incorporated into Catholic theology by St. Thomas Aquinas in the 13th century.

Mathematics was originally not a big part of Jesuit schooling until Christopher Clavius (1538 – 1612) began arguing how the subject could fit into Jesuit ideology. To Clavius, the certainty of mathematics and its use of incontrovertible proofs (particularly in Euclidean geometry) established a type of authority that suggested how the universe was governed by rational order and universal truths. Clavius scored a big triumph for mathematics, astronomy, and the Catholic Church in the 1580s, when he was instrumental in the commission that resulted in the establishment of the Gregorian calendar, setting the rules for Leap Years that are still used today.

Yet, during this same period, mathematicians and astronomers (including Kepler and Galileo) started requiring a mathematics that went beyond the Ancient Greeks, and which could deal with movement and change. They began developing mathematical tools that presaged techniques later formalized in calculus.

Galileo was to become the most annoying bee in the Jesuit's bonnet, but Infinitesimal focuses instead on a rather lesser known mathematician, Bonaventura Cavalieri (1598 – 1647). Through a kind of analysis he called the “method of indivisibles,” Cavalieri treated lines as composed of an infinite number of infinitesimal points, and an area as composed of an infinite number of infinitesimally thin lines, and a volume as composed of an infinite number of infinitesimally thin planes.

But what does it even mean to have an infinite number of something that is infinitely small? Even if the techniques worked and yielded good results, were they legitimate? Not to the Jesuits, who continued to believe that only Euclidean Geometry represented the “embodiment of order” (pg. 119). These new techniques threatened the Church’s authority, and despite their increasing popularity, the Jesuits repeatedly issued blanket condemnations:

A major offensive against Cavalieri was launched by Swiss mathematician (and Jesuit) Paul Guldin, which is the subject of a short excerpt from Infinitesimal published in the April 2014 issue of Scientific American. To Guldin, the use of infinitesimals failed to properly build on the foundations established by Euclid, clearly violated Aristotle’s warnings against infinity, and led to disorder and paradoxes.

Yet, the techniques continued to work, and to Amir Alexander, the conflict between dogmatic authority and this more empirical math becomes a microcosm of the battle for modernity.

As the Jesuits suppressed these new techniques, they effectively plunged Italian mathematics into another Dark Ages. As just one example, consider that today we consider the greatest Italian mathematician of the 18th century to be Giuseppe Luigi Lagrangia. But perhaps you know him better by the name he adopted after he left his homeland — Joseph-Louis Lagrange.

The second half of Infinitesimal shifts from Italy to England, which initially looks unpromising because the country is deep into political and religious conflicts of a 20-year Civil War smack in the middle of the 17th century. With an executed king and many extreme and odd religious sects, it is truly “a world turned upside down.”

Yet, following the Restoration in 1660, England seemed to emerge from the Civil War with a strong desire to avoid such conflicts in the future.

How was England to manage this feat?

One approach was described by Thomas Hobbes (1588 –1679). The anarchy of the Civil War frightened Hobbes so acutely that he could only imagine surviving the future under a form of government he called Leviathan — a strong leader who imposed stability and uniformity with dictatorial powers. Hobbes was opposed to the Catholic Church and religion in general, but he agreed with the Jesuits that the rational logic of Euclidean Geometry served as an excellent model for the rigidity and order of his ideal state.

At the other side was John Wallis (1616 – 1703), who was the nascent Whig to the Tory Hobbes. Wallis favored a more tolerant and pluralistic society, and saw science as a model. He was involved in the formation of the Royal Society of London, where scientific consensus could arise not out of blind obedience to authority, but through public experiments, empirical exploration, and discussion.

John Wallis also studied the writings of Cavalieri, and began developing his own mathematical techniques using infinitesimals. In 1665 he even began using a new symbol to represent infinity:

It is a thing of beauty, is it not? It almost makes one want to believe in infinity.

In his decades-long battle with Hobbes, Wallis comes off as the hero of Infinitesimal. But perhaps it’s best not to look at his actual math, featuring an exceptionally promiscuous use of the infinity symbol. Mathematical formulas with an infinity in a numerator cancelling out an infinity in a denominator is the sort of thing that causes the heads of modern mathematicians to explode.

Up until page 272, Amir Alexander has been quite good at getting us into the minds of 16th and 17th century mathematicians, and letting us see their world from a pre-calculus perspective. But on page 272, he seems to give up, and shows some modern renditions of Wallis’s work using lim — the limit operator.

This is very anachronistic! The mathematics behind limits did not appear until the early 19th century, when they finally established a sound foundation for the calculus, replacing what was essentially some extreme hand-waving. Some mathematicians before this time seemed to have an intuitive concept of the limit, but the actual mathematical derivation wasn’t as trivial as it seems in retrospect.

Amir Alexander’s book stops just short of Newton, Leibniz, and the development of calculus. Yet, Infinitesimal leaves the reader reinterpreting even this oft-told story. We now ponder that the first edition of Isaac Newton’s monumental Philosophiæ Naturalis Principia Mathematica was published just a year before England’s Glorious Revolution; and that Newton himself represented Cambridge in the Whig contingent in the Parliament that determined who should subsequently rule England; and that this was the same year that saw the publication of John Locke’s A Letter Concerning Toleration.

Just a coincidence? I think not, and Amir Alexander’s book has given me a better understanding of the trends that connect the history of politics and religion with science and mathematics.


Excellent review. You got me with the opening paragraph. Would be interested in more on infinity. Would agree that there is no evidence for it (like Aristotle on this actually). Big Bang says it too. Everything has a beginning and an end.

Gary Wilson, Wed, 2 Jul 2014 14:40:35 -0400

There's certainly some discussion of infinity in The Annotated Turing! — Charles

I recently heard in an audiobook on cosmology that we can have an infinite space inside a finite volume. Mathematically speaking , we have indeed a countable infinite set of rational numbers inside [0,1] interval , for example. We don't even need to consider the transcendent numbers or algebraic numbers (the algebraic ones are still countable, but the transcendent ones take us into the continuum (which is uncountable)).

I wonder though how we can have infinite space in the physical reality of our universe. Maybe that can happen even if the atoms are not infinitely divisible, because the space might be able to expand / divide itself endlessly.

Cristian Minoiu, Fri, 4 Jul 2014 20:59:13 -0400

+1 for The Annotated Turing. What a great mix of theoretical discussion of the infinite (or lack of it :-) with concrete examples from Turing's machine. I really have to commend you on that book, Charles. It has made me go back to investigate the math I missed and relearn the math I've forgotten.

Jim Dodd, Mon, 7 Jul 2014 17:16:25 -0400

Thanks! — Charles

That was deep. I have often found myself trying to imagine infinite space, and obviously failing miserably. I still am confused by trying to figure out what is beyond the boundaries of the universe. Empty space? For how far? What's beyond that? Infinity. What other answer is there?

— Dan, Mon, 14 Jul 2014 11:18:04 -0400

Consider the surface of the Earth. Is it infinite? No. Wikipedia says the surface area is about 510 million square kilometers.

But is it bounded? No. Assuming you have the appropriate transport, you can go in any direction on the surface of the Earth and never reach a boundary. To ask "What is beyond the surface of the Earth?" is meaningless.

It could very well be that the Universe is a three-dimensional analogue of the surface of the Earth: Finite but unbounded. — Charles

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