Paradoxes of Infinity

— Ansar Fayyazuddin

How a Dangerous Mathematical Theory Shaped the Modern World
By Amir Alexander
Scientific American/Farrar, Straus and Giroux,
Paperback edition, 2015, 368 pages, $16.

MATHEMATICS CONCEIVED AS a closed logical system, based on a set of axioms, may suggest that its history would be an account of the inevitable logical development of these axioms. The pace of this history would then be a function of the abilities of the actors engaged in carrying out these logical steps.

Yet mathematics is decidedly not like that. It is not a closed logical system, and its history is like the history of any discipline — far from the sequential unraveling of truths immanent in its logical structure, it is a narrative of contingent goals and programs, varying material resources, demands of governments and society, the serendipitous coming together of communities of mathematicians, and the psychology and personal circumstances of individual mathematicians.

The abilities of individual mathematicians play a role, but these abilities are fashioned and nurtured (or not) in definite historical and social circumstances as well as subcultures, households and families.

The view of mathematics as a closed axiomatic system was pursued to its utmost limits by Bertrand Russell and Alfred North Whitehead in the early part of the 20th century culminating in their Principia Mathematica, appearing in the 1910s. The project remained unfinished, not only due to the enormity of its own ambitions but irremediably with Kurt Gödel’s theorems from the 1930s showing the incompleteness of all logical systems.

Gödel’s theorems proved that in any axiomatic system (sufficient for arithmetic), there are statements that can neither be proven nor disproven within that system. Thus all axiomatic systems are incomplete in the sense that the truth of all meaningful statements cannot be established.

If mathematics is not an unfolding of truths hidden in its logical structure, what propels it and can its progress be brought to a halt or accelerated? There can be no general answer to this question, but the study of particular developments in historical context can help us understand how and why certain ideas emerge at definite points in history at particular places.

Examination of the historical record, and nuanced attempts to tease out the historical and social forces at play in the development of mathematics, are worthwhile.

A Bitter Debate

Infinitesimal: How a dangerous mathematical theory shaped the modern world, by UCLA professor Amir Alexander, is a history in two parts of certain mathematical ideas that emerged in 17th century Europe with a focus on Italy and England.

This is an idiosyncratic book driven by its two-part thesis. The first part is devoted to showing that the Jesuit brotherhood hindered the development of the precursor of the calculus and ultimately thwarted the development of mathematics in Italy until, I presume, the Risorgimento (Italy’s 19th century unification).

The second part concerns England in the course and aftermath of the English revolution and focuses on debates between Thomas Hobbes and John Wallis. The triumph of Wallis and the mathematics of infinitesimals (“infinitely small” quantities in intuitive calculus) is argued to be central to the flourishing of mathematics and democracy in England.

The thesis of the book goes beyond developments in mathematics and makes a much larger claim:

“From north to south, from England to Italy, the fight over the infinitely small raged across western Europe. The lines in the struggle were clearly drawn. On the one side were the advocates of intellectual freedom, scientific progress, and political reform: on the other, the champions of authority, universal and unchanging knowledge, and fixed political hierarchy. The results of the fight were not everywhere the same, but the stakes were always just as high: the face of the modern world, then coming into being. The statement that “the mathematical continuum is composed of distinct indivisibles” is innocent enough to us, but three and a half centuries ago it had the power to shake the foundations of the early modern world. And so it did: the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms on a scale unknown in human history.” (14)

These grand statements undermine its more reasonable claims. The chapters on otherwise quite fascinating stories are marred by an insistence on claims that are not and ultimately cannot be fully supported. Nevertheless, Infinitesimal is well written and presents compelling accounts that are interesting even if they fail in the end to cohere to prove the central thesis.

The Difficulty of Infinitesimals

The problem of infinitesimals is tied up with its inverse — the problem of infinity. At first blush, infinity is a simple enough concept and is just shorthand for something endless. One quickly discovers that infinity comes in many varieties, and what is perfectly reasonable in finite quantities leads to apparent paradoxes when extended to infinity.

Two famous examples should suffice to give a flavor of these paradoxes. The first is the well-known paradox of Zeno that has Achilles chasing a hare that is moving at half the speed of Achilles. As Achilles crosses the midway point of their initial separation, the hare has moved on to a new point a shorter distance away than their original separation.

In the infinite progression of Achilles crossing the midpoint of their separation, we reach the apparent conclusion that Achilles will never reach the hare. The reason is that we have an infinite sum of periods of time that are not zero. Needless to say, this apparent paradox is based on faulty reasoning — in actuality the infinite sum results in a finite answer.

A second paradox was discovered by Galileo, which he accepted in remarkable stride. Galileo realized that the size of the set of numbers that are squares of natural numbers is the same as that of the set of natural numbers. The reason is quite simply that all numbers that are squares can be put in correspondence with a unique natural number through writing the square number as n2, where n is a natural number.

Intuitively one might expect the number of numbers that are perfect squares to be far fewer than the number of natural numbers (indeed, adjacent such perfect squares occur further and further apart as they increase in size). In this case, there is no resolution to the “paradox,” which lies in our intuition that fails us for infinite sets.

The paradoxes continue. If one asks whether a circle of radius 4cm has a larger number of points than a circle of 2cm? The answer, surprisingly, is “no.” It is easy to see that each point on the small circle has a corresponding point on the larger circle and vice versa by drawing the two circles as co-centric (i.e. sharing the same center). Radial lines from the center intersect the two circles at unique points establishing a one-to-one correspondence between the points on the two circles.

In the above examples, the infinities are all of the same type and the view that all infinities are the same may seem plausible. However, this is not the case.

As shown much later in the 19th century by Cantor, the infinity of the number of real numbers is much “greater” than the number of integers (that is, the real numbers can’t be “counted” in one-to-one correspondence with the integers). In fact, the number of real numbers in any finite interval of real numbers is “greater” than the number of whole numbers (integers), although both are infinite.

The paradoxes of infinity and infinitesimals cast a long shadow on what was considered to be legitimate mathematics. Rigorous mathematics had no place for infinities because the infinite posed too many antinomies.

Mathematics in the 17th Century

The Renaissance engagement with the ancient world brought with it certain sensibilities. These sensibilities included the celebration of mental over physical labor. In Plato’s dialogue Phaedo, for instance, Socrates argues against attempting to avert his imminent execution because death will set his spirit free from the petty material needs of his body.

Ancient mathematics reflected this sensibility. Geometry, confined only to sublime constructions possible with a straightedge and compass alone, held a place of honor. Many problems that defied the greatest practitioners of this form of geometry are trivial to solve in practice once freed from the arcane constraints of the ruler and compass. Mathematics of this cast became the preserve of the leisure class.

But the Renaissance and its aftermath also saw the burgeoning of engineering and the need and development of machines (clocks, for instance). Trade across the seas increased, motivating the invention of instruments that would aid navigation (telescopes, for example). In a few individuals, the practical engineering aspects and rarefied academic mathematics merged. Nevertheless, the tension between the pure and the applied remained.

The 17th century inherited the preoccupations of the Renaissance but it was coupled with the ascendancy of heretical forms of Christianity. In the Roman Catholic centers, the Counter-Reformation tried to protect Catholicism against heresies that it saw lurking everywhere.

The cautionary tales of Bruno and Galileo showed how the provinces of the Church and science were not separate and that the Church could pronounce philosophical, mathematical and scientific ideas to be heretical with dire consequences for the “heretic.”

Central to the Italian narrative of Infinites­imal are the trio: Galileo, Bona­ventura Cavali­eri and Evangelista Torricelli. The latter two were considered to be Galileo’s protégés. Cavalieri and Torricelli’s interests in infinitesimals were inspired by the master’s use of them. When not in his presence, the protégés corresponded closely with him.

Galileo actively encouraged Cavalieri to pursue his book on infinitesimals and Torricelli spent time with Galileo during his exile. Galileo’s interest in infinitesimals grew out of his own preoccupations with the physics of projectiles and motion in general.

Galileo’s contributions, although in many ways preliminary, were in other ways more advanced than what Cavalieri accomplished later. In particular, he understood that infinitesimals could be of different orders – some could be neglected in relation to others. In contrast, Cavalieri’s use of infinitesimals was not an approximation scheme that became more exact in a limiting procedure, rather his methods were exact throughout although they required leaps of logic that he didn’t always justify.

Cavalieri (1598-1647) was a Catholic priest and a mathematician. His enduring contributions include a series of theorems on the areas and volumes of geometric objects.

These theorems were based on a great innovation — the method of indivisibles. Cavalieri conceived of shapes as made up of infinitesimal indivisibles: points, lines, or two-dimensional cross sections. The length, area and volume, respectively, of the objects constituted by these indivisibles were then the “sum” of the corresponding quantity for the indivisibles. This is integral calculus in a nutshell.

One of the many theorems “proved” by Cavalieri is easy to state and, in some ways, quite intuitive. It states that any two objects that have the same cross-sectional area at each height have the same volume.

An intuitive picture of this is as follows. The volume of an upright stack of playing cards is given by multiplying the lengths of the sides of this rectangular cylinder. But it is also simply the sum of the volumes of the individual cards. The cards can be shifted relative to each other to change the three-dimensional shape of the stack of cards. The volume clearly does not change and neither does the height (which is just the sum of the thicknesses of the cards).

The theorem is a generalization of this basic idea. Torricelli also made important contributions using the methods that Cavalieri first discovered.

Jesuits vs. Infinitesimals

The first part of Infinitesimal, besides telling a compelling story of various developments in mathematics in Italy during the late 16th and early 17th centuries, is concerned with how the Jesuit order opposed the method of indivisibles, declared it to be heretical, and managed to thwart the development of mathematics in Italy for a considerable period of time.

The Jesuit opposition to using infinitesimal methods in mathematics is certainly curious and poses a legitimate historical problem that should be addressed. However, the particular resolution proposed by Alexander is not convincing.

First, the book tries to locate in infinitesimal methods itself the seeds of modernity and political ideas that were inherently opposed to the conservative Jesuits. Second, the Jesuits are treated as if their doctrines were not mainstream Catholicism but somehow represented a different more reactionary force. The Jesuits are singled out as the main force blocking the development of infinitesimal mathematics.

In my view the peculiar opposition of the Jesuits to infinitesimals has both a general and a particular aspect to it. Generally, the Catholic Church accepted as dogma the Scholastic Aristotelian obsession with rigor and its apotheosis in Euclidean geometry. This circumscribed legitimate mathematics to be a strictly rigorous and academic discipline. Approximations and practical applications necessarily sullied this otherwise pristine subject.

However, this view was not particular to the Jesuits. One can see elements of this even in Cavalieri’s approach to infinitesimals, which retained a strictly geometric, if not entirely rigorous, view.

The particular circumstances have to do with Galileo, who in his Dialogue Concerning the Two Chief World Systems, ridiculed Scholastic Aristotelian dogma and its insistence on constructing purely philosophical systems of thought that were not subjected to serious comparison with the physical world.

Galileo’s implicit endorsement of the Copernican heliocentric view in this book was viewed as an affront to the Jesuits who were behind the ban on propagating Copernican ideas.

Cardinal Robert Bellarmine, a Jesuit, had advised Galileo in 1616 to refrain from treating Copernican heliocentrism as corresponding to reality rather than a convenient fiction. It was this ban that Galileo defied and for which he was subsequently persecuted and sent into exile. Furthermore, the thinly disguised and bungling antihero of the Dialogue was the reigning Pope who belonged to the Jesuit order.

The fact that the mathematics of infinitesimals originated in Galileo’s thought was thus not an irrelevant detail. Cavalieri and Torricelli recognized their debt to the master. Thus, infinitesimal methods were associated with Galileo and his school, which after Galileo’s censure and exile, carried the indelible mark of this condemnation.

The Italian trio (Galileo, Cavalieri and Torricelli) was engaged not only in developments in purely mathematical subjects and trying to understand the physical world abstractly, but were also engaged in concrete applications of these ideas.

All three were makers of instruments and machines. Galileo not only designed and manufactured telescopes for sale but also made other instruments commercially. Cavalieri is known to have designed a hydraulic pump and produced tables of logarithms for practical applications. Torricelli, like his mentors, was a maker of optical and hydraulic instruments as well as, extremely importantly, the first barometer.

The reason I find this of significance is that experimenters and the makers of machines necessarily have a more practical and less rigorous approach since everything they measure and construct is perforce only an approximation to the idealized system that may have inspired them.

Thus I see many of the mathematical innovations of the Italians in this group as arising, if not directly, then inspired by their forays into the applied and experimental sciences. These characteristics of this group of Italian mathematicians are not given attention in Alexander’s book.

English Debates

Alexander’s description of the state of England in the mid to late 17th Century from the “short parliament” to the Restoration is deft and well written.

From this expansive view of the English revolution and the many radical movements it inspired, Alexander settles to a decidedly narrow focus — the debates between two intellectual giants: Thomas Hobbes and John Wallis. Hobbes is best known as the author of Leviathan, a philosophical tract justifying monarchical or oligarchical rule and the obligations of the citizenry to follow the laws set by these rulers.

Hobbes forms an interesting counterpoint to Wallis, although it is not their philosophical and political differences that distinguish them as Alexander seemingly implies. As far as I can tell, Wallis was a mainstream Protestant who was ordained and retained his sympathy for royalist ideas, becoming Charles II’s chaplain after the Restoration.(1)

Their contrast seems to lie instead in Hobbes being an atheist (or at most a believer in a non-interventionist god), a philosophical materialist and a believer in rigorous reasoning both in philosophy and in mathematics, with geometry representing the highest form of reason. Wallis by contrast was a moderate Presbyterian and, as discussed below, not shy about using methods that were not rigorous in mathematics.

Wallis became familiar with infinitesimals from Cavalieri’s book on the subject. Wallis took these ideas and extended them in ways that were at times baffling.

Cavalieri had shown how to compute the area under the curve y=xn, when n is a positive integer, through careful geometric reasoning. Wallis took these results, culled them from their geometric origins and treated them as algebraic relations. He then extrapolated Cavalieri’s carefully computed results to cases when n is negative, a fraction or even irrational (i.e. not a fraction).

In these extrapolations, or “interpolations,” as he called them, of Cavalieri’s results, Wallis did not rigorously prove the new claims but rather showed that they were sensible and worked out their often fascinating consequences. Among his results was his formula for π (pi) given as an infinite product:

π/2 = (4/3)(16/15)(36/35)(64/63)….

Wallis’ book appeared in 1656 in the midst of an incredible period of intellectual discovery about which Hobbes wrote: “If in time and in place there were degrees of high and low, I verily believe the highest of time would be that which passed betwixt 1640 and 1660.”(2)

The tensions between Hobbes and the Royal Society are the subject of much debate. The Wallis-Hobbes dispute was about a series of issues that became more and more tangled as time passed. The issues ranged from the particular, such as refuting Hobbes’ incorrect claims of solving ancient problems in geometry like “squaring the circle”(3), to more general ideas about rigorous geometry versus “experimental” mathematics and science.

However, one cannot discount the increasingly personal nature of these attacks that set the contenders on an irreconcilable course.

Hierarchy vs. Open Society

Christopher Hill, the preeminent historian of the English Revolution, writes:

“In a hierarchical society each man had his place, and was punished if he left it. How was a society of equal and competing atoms to be kept at peace? The problem was first clearly stated by Hobbes, who had been Bacon’s amanuensis, and saw himself as the Euclid of a new science of politics. “Geometry is demonstrable, for the lines and figures from which we reason are drawn and described are drawn by ourselves; and civil philosophy is demonstrable because we make the commonwealth ourselves.” God made the universe but man made the state: so politics was taken away from the theologians and became a matter of empirical investigation and rational discussion.”(4)

Thus Hobbes was not unsympathetic to observational science and was a proponent of a secular politics, however reactionary his own politics may have been. His criticisms of Wallis and other members of the Royal Society were often couched in terms of attacks on ecclesiastical influences and defending rigorous thought, which need not be reactionary, in my opinion.

In placing emphasis on the intrinsic aspects of infinitesimals and the political ideology immanent in the mathematics, Alexander fails to bring the burgeoning of mathematics in the 17th Century into a larger focus. This is surprising, since the book spends a lot of time discussing the historical and political circumstances of the discoveries.

The intellectual ferment was a Europe-wide phenomenon that had its roots in the growing practical needs of society as mercantile trade and industrial production started to take a greater hold of the economic systems of early modern Europe. In addition, the lack of rigor of the new mathematics had practical origins rather than deep political and philosophical ones in the religious beliefs of the protagonists. (It would only be in the 19th century that the methods of calculus would be put on a truly rigorous theoretical foundation.)

This is clear in the Italian context where both the leading proponents of infinitesimal methods and their opponents were Catholics. Even in the English context, things are not so straightforward since Hobbes’ secular and anti-ecclesiastical views were outside the mainstream of English society even if they defended a Royalist outcome — while Wallis, whose politics and theology lay well within a moderate Parliamentarianism, welcomed the Restoration.

I think we can all agree that the ability to freely pursue mathematics, science and the humanities is inherently good for society, and worth fighting for — especially at this moment. The period described by Alexander exemplified the clash between different interests and where intellectual differences took on an urgency that seems quaint from the perspective of our jaded age.

Although I have outlined some problems I have with Alexander’s historical analysis, I have to end with recommending the book for its many strengths that include its readability that is rare for a book on the history of science and mathematics.

Alexander does a good job of explaining the mathematics of infinitesimals and the conundrums associated with them. The central characters are drawn well and the historical narrative is told with verve. The history of the Jesuit society and the history of the English revolution are self-contained and very well done.


  1. Carl B. Boyer and Uta C. Merzbach, A History of Mathematics, second edition, John Wiley & Sons, Inc. (1989), 380.
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  2. Quoted in C. Hill, The Century of Revolution, (Routledge, 2002), 161.
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  3. Constructing a square with a straightedge and compass with the same area as a circle, ultimately proven to be impossible.
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  4. C. Hill, 180.
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September-October 2017, ATC 190

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