189-338A: History and Philosophy of Mathematics

Course blog

A disclaimer about this course, and about the textbook.

In spite of its title, Math 338 is primarily about mathematics and the evolution of some important mathematical ideas, and only secondarily about history and philosophy. Inevitably, it offers an overly simplified account of a subject that is vastly richer and more complex than any single course or textbook could convey. We can only discern the past through the clouded lense of contemporary prejudices and perspectives, misled and informed in equal measure by our present understanding of how mathematical ideas have resonated--and been amplified, or forgotten-- through the ages. You are invited to take the historical insights imparted during lectures or gleaned from the textbook with a healthy dose of skepticism!

Like any historical text, Journey through genius must be envisaged at best as an unreliable narrator. (Albeit, it is hoped, a companionable one.) The critical reader will note its marked emphasis on the western contributions to mathematics and the almost complete absence of women among the dramatis personae. (The metaphor deployed in the second paragraph of page 24 also reminds us of how views of gender have evolved since 1990 when the book was written.)

For a stimulating discussion of issues germane to a deeper and more nuanced understanding of the history of mathematics, see Minhyong Kim's excellent essay in the Notices of the American Mathematical Society, and this response to it.

I will closely follow the textbook during the lectures, devoting roughly one week of lectures to each chapter. This serves a purpose of pedagogical convenience and narrative consistency. But in the blog I will also propose supplementary readings which you are strongly encouraged to consult for a more complete understanding of the subject. (Please also feel free to suggest such supplementary texts yourselves for eventual inclusion in the blog!).

Week 1 (August 27 and 29). Dunham, Chapter 1. Ancient Egyptian, Babylonian, and Greek mathematics. The Pythagorean theorem. Geometric constructions by ruler and compass. Hippocrates and the quadrature of the lune.

For further discussion about the quadrature of the circle and how its impossibility was eventually established towards theend of the 19th century, read the last pages of epilogue of chapter 1 of Dunham. The problem of squaring the circle had a powerful hold on the imaginations of aspiring mathematicians for many centuries. Abraham Lincoln, who was largely self-taught but could reproduce by heart the proofs of all the propositions in Euclid, and whose intimate knowledge of Euclid was in fact central to his political philosophy, once spent several days in a fuitless attempt to square the circle.

Here is another discussion of the quadrature of the lune by Barry Mazur, one of the greatest mathematicians of the 20th century.

Week 2 (September 3 and 5). Dunham, Chapter 2. The geometry in book I of Euclid. Proofs of the Pythagoras Theorem.

After a brief discussion of the impossibility of squaring the circle (which we will return to later in the semester), this week was devoted to a discussion of Euclid's elements, in particular the first book of Euclid concerned with plane geometry. The books of Euclid introduced the axiomatic method, whose impact on medieval Arab scholars and later on western mathematics was profound and persisted for over two millenia, in realms going even beyond the development of mathematics itself. Euclid's Elements was the best selling book in the western world after the bible, undergoing over 2000 editions, and an integral part of an educated person's world view well into the 20th century. There are echoes of Euclid, for instance, in the stirring opening lines of the American Declaration of Independence, ``We hold these truths to be self evident: that all men are created equal...". Can you think of other examples of the impact of Euclid's axiomatic method on human civilization?

Here is a lovely numberphile video giving yet another proof of Pythagoras' theorem told by Barry Mazur.

Week 3 (September 8-12). Dunham, Chapter 3. The 13 books of Euclid. Further results in geometry. Books 7-9, and the foundations of number theory (arithmetic).

Attempts to prove Euclid's 5th postulate eventually led, under the hands of Gauss, Bolyai, Lobachevski and Riemann in the 19th Century, to the discovery of non-Euclidean geometries. The possibility of alternate geometries represents a paradigm shift in the sense of Thomas Kuhn's The structure of scientific revolutions. But the germs of this revolution were already present in Euclid, and the advent of non-Euclidean geometries can be envisaged as a broadening of the earlier framework rather than a repudiation of Euclid's elements, which can still lay claim to being the most influential mathematics textbook of all time.

This week was devoted to a brief survey of the contents of the other books of Eiclid's elements: book II on geometric algebra, book III on circles, book IV on constructions of plane geometric figures and regular polygons, book V presenting Eudoxus's theory of proportions, and book VI which explores the notion of similarity of plane geometric figures. Books VII to IX contain a remarkable discussion of arithmetic and number theory. Highlights include the notion of prime and perfect numbers, the Euclidean algorithm for calculating the gcd (still one of the most fundamental and widely used results in computational number theory), the fundamental theorem of arithmetic asserting that all integers can be uniquely broken up as a product of primes, and the proof that there are infinitely many prime numbers.

Books X-XIII are concerned with solid geometry, the calculations of volumes and Eudoxus's method of exhaustion, and the five regular Platonic solids.

Last week we had the first of Samy's tutorials. Take a look at his notes about constructible numbers.

Week 4 (September 15-19). Dunham, Chapter 4.

This week was devoted to the immortal Archimedes of Syracuse, whose mathematical achievements c.a. 287-212 BC were unmatched in originality in sophistication during his lifetime, and remained so for almost 2000 years afterwards, a distinction that marks him as a singular figure in the history of ideas.

Among his most important contributions are the comparison between the area and circumference of a circle, as well as a similar results on the volume and surface area of a sphere, revealing that a single universal constant governs all these quantities, the storied number $\pi$ which we celebrate each year on March 14.

Archimedes' theorem that the area and volume of a circumscribing cylinder are exactly $\frac{3}{2}$ as large as the area and volume of a sphere it contains, is one of his signature results, which was engraved on his tombstone at his request.

In addition to revealing the fundamental and ubiquitous nature of $\pi$, Archimedes calculated this constant numerically, with an accuracy of two significant decimal digits of accuracy, a stunning computational tour de force with the notations and tools he had at his disposal, and a marked improvement over the biblical value of 3. We then discussed the calculation of $\pi$ throughout the ages, culminating with modern times where this mysterious number is now known to over 10 thousand billion significant decimal digits.

Further references.
In class, we explored the calculation in Archimedes' The cylinder and the sphere of the area of the sphere, and how this area is related to its volume. This leads to the modern formula ($\frac{4}{3}\pi r^3$) for the volume of a sphere of radius $r$. In 1906, an old goat skin parchment was uncovered containing religious writings written over what appeared to be Archimedes' original calculation of the volume of a sphere, based on mechanical engineering principles that he had discovered himself, notably his law of levers applied to a judiciously chosen balncing of a hemisphere, a cylinder, and a cone. An explanation of Archimedes' brilliant approach is given in this highly recommended youtube video.

In our discussion of the calculation of $\pi$ over the ages, I described the celebrated formula of Leibniz for $\pi$: $$ \pi = 4\times \left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7} + \cdots \right),$$ which we obtained by calculating the Taylor expansion of the inverse tangent function. You may enjoy this delightful alternate proof based on Fermat's Christmas theorem, a striking number theory result in its own right which gives a closed formula for the number of ways in which an integer can be expressed as a sum of two integer squares.

Week 5 (September 22-26). Dunham, Chapter 5.

This week concludes our tour of the mathematics of ancient Greece, with discussions of the Greek mathematicians Eratosthenes, Apollonius, and Heron of Alexandria. Although none of them attained the towering stature of Archimedes, several elegant results are attributed to them: the Sieve of Eratosthenes to calculate prime numbers, and Apollonius's celebrated eight volume treatise on conic sections, were briefly touched upon. But the main result we discussed at length is Heron's formula for the area of a triangle with side lengths $a$, $b$ and $c$, $$ {\rm Area} = \sqrt{s(s-a)(s-b)(s-c)}, \qquad s:= \frac{a+b+c}{2},$$ whose chief virtue is that it involves only the side lengths and not the less readily calculated height of the triangle. This leads to what is arguably one of the more intricate geometric proofs we have seen so far, involving a complicated dance between the triangle's inner (inscribed) circle, one of its outer circles, and two different pairs of similar right angled triangles that were extracted from the resulting figure. This proof of Heron's formula is given in the book of Dunham, and is also explained very nicely in this mathophilia video.

The portion of that proof involving the outer circle and the various similar triangles is without question a bravura performance and might have left you feeling a little dizzy. A somewhat more motivated and symmetrical proof (treating all three sides of the triangle on a more equal footing) involving complex numbers was also proposed, as explained in this other mathophilia video. (Complex numbers will take center stage as the key dramatis personae in next week's lectures, when we turn to the algebraists of the Italian renaissance and their remarkable solution of the cubic and quartic equations.)

If you want to see some further intriguing geometric ramifications of Heron's formula, this mathologer video is also a must watch.

Week 6 (September 29-October 3). Dunham, Chapter 6.

This week marks a big leap, both chronological and thematic, from the geometry of the ancient Greeks to the algebra of the Italian Renaissance in the 16th century. We discussed the solution by Scipione del Ferro, Niccolo Tartaglia, Gerolamo Cardano, and Ludovico Ferrari, of the general cubic and quartic equations by radicals. This discovery is assigned a central role in the history of mathematics because it marked one of the first radical improvements over the mathematics of the ancients, and led in short order to many key innovations: better algebraic notations, the (reluctant at first) introduction of the complex numbers, the deep study of higher degree equations culminating in the development of Galois theory, and the gradual seperation of algebra from geometry that allowed mathematics to attain ever higher degrees of abstraction, for the tremenous benefit of both subdisciplines.

In addtion to Chapter 6 of Dunham's book, the following videos give a lovely account of the material covered this week and its implications:

500 years of not teaching the cubic formula, by Mathologer.

How imaginary numbers were invented, by Veritasium.

Week 7 (October 6-October 10). Dunham, Chapter 7.

This week's lectures were devoted to an unavoidably sketchy and incomplete tour of the explosion of mathematics in the 17th century, with an emphasis on Isaac Newton, the unquestioned genius of that era. We discussed Newton's daring extension of the binomial theorem to negative and fractional exponents, based on what might be viewed as an instance of the murky mathematical principle of permanence of identities, and how he used this along with his newly discovered theory of fluxions -- integral and differential calculus -- to give a remarkably accurate calculation of $\pi$, making decisive strides beyond the approach of Archimedes.

This engaging video by Veritasium describes Newton's approach and is highly recommended.

Passing mention was also made of several influential French mathematicians of that century, notably René Descartes (noted for his Discours de la méthode and its introduction of analytic geometry), Blaise Pascal (whose famous wager applies probabilistic reasoning to pressing theological questions), and, last but not least, Pierre de Fermat, who is arguably the founder of modern number theory and would have deserved a more thorough treatment in the text.

The outdatedness of our textbook (published in 1992), already commented on earlier, is nowhere more glaringly evident than in Dunham's discussion of Fermat's Last Theorem, which concludes with the line ``Should anyone resolve the issue -- and even in the late 20th century interest in the problem runs high -- he or she will surely merit a page in all subsequent histories of mathematics." A year later, Princeton professor Andrew Wiles announced a proof of Fermat's Last Theorem. You can find out more about the dramatic backstory that led to his discovery in the following popular BBC video. (I had the privilege of witnessing the making of mathematical history first hand, being Wiles' postdoc in Princeton at the time.) Some of the flavour of the mathematical ideas that go into Wiles' proof is conveyed in this lecture given in Oslo on the occasion of Wiles receiving the Abel Prize, the mathematics equivalent of the Nobel Prize.

Reading Week (October 13-October 17). Dunham, Chapters 1-7

This reading week you will take some time to study for the midterm exam. The midterm will be given in class, on Friday October 24 from 8:30 to 9:30 AM. It will be comprised of 5 questions, each worth 20 points for a maximum total of 100. This is strictly a closed book exam, no calculators, cheat sheets, or computers are allows.

If you wish to practice with the material and challenge yourselves (at a level of difficulty somewhat greater than for the quizzes and midterm), I encourage you to try to go through some of the problems that are proposed in this addendum to Dunham's book.

I also advise you to take some time to go over the extra material that is proposed in this course blog so far, if you have not already done so.

Week 8 (October 20-October 24). Dunham, Chapter 8

The main mathematical topic we covered in this relatively short week (since we had the midterm on Friday) was the summation of various infinite series by mathematicians of the late 17th century: the infinite sum of the reciprocals of the triangular numbers by Leibniz, and the study of the harmonic series and the proof of its divergence by the Bernoullis. This material, and more, is covered in the following video which Will Gray kindly brought to my attention. Our discussion of infinite series prefigures the work of the great Leonhard Euler in the 18th century, who brought an unaparalleled virtuosity and undisputed genius to the study of infinite series.

By popular request, I am also posting this timely Mathologer Halloween video about Fermat's little theorem, which (you may recall) we proved in week 7 using necklace colorings.

Week 9 (October 27-October 31). Dunham, Chapter 9

This week, we covered Euler's proof of the infinite series identity $$ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = \frac{\pi^2}{6},$$ which takes the infinite series evaluations of the seventeenth centuries to the next level. Euler's proof relies on a completely original and visionary infinite product expansion of $\sin x$ as $$ \sin x = x \left(1-\frac{x^2}{\pi^2}\right) \left(1-\frac{x^2}{4\pi^2}\right) \left(1-\frac{x^2}{9\pi^2}\right) \left(1-\frac{x^2}{16\pi^2}\right) \left(1-\frac{x^2}{25\pi^2}\right) \cdots,$$ and on comparing the coefficient of $x^3$ in this product expansion to the same coefficient in the Taylor series expansion of $\sin x$. Euler did not fully justify his infinite product expansion but it was a decisive breakthrough that spurred the rigorous development of the theory of functions of a complex variable in the 19th century.

No less important for mathematics is Euler's celebrated infinite product identity $$ \zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \ {\rm prime}} (1-p^{-s})^{-1},$$ relating an infinite sum over the integers to an infinite product over the primes, which can be viewed as an analytic formulation of the fundamental theorem of arithmetic asserting that every positive integer can be uniquely expressed as a product of prime powers. It immediately leads to a new proof of the infinitude of primes in a more precise quantitative sense, notably the fact that the subseries $\sum_{p} \frac{1}{p}$ of the harmonic series continues to be divergent (and hence the primes are not as thinly spread out, say, as the set of perfect squares of integers).

This reasoning relied on considering the quantity $\zeta(s)$ as a function of a real variable $s$. Riemann in the 19th Century had the stunning insight that further information about the prime numbers might be obtained by viewing $\zeta(s)$ as a function of a complex variable. In doing this Euler and Riemann essentially jump-started the field of analytc number theory, which remains extremely active today. The Riemann zeta function is one of the most important objects in this central branch of pure mathematics. For more on this fasinating object, you can watch the following video suggested by Lucas Malatesta.

If you want to challenge yourselves with a beautiful, highly original physical solutioh of the Basel problem, this video is highly recommended!

Week 10 (November 3-November 7). Dunham, Chapter 10

This week was devoted to Euler's number theory. As André Weil commented, ``Euler's achievements in number theory consisted in no more nor less than giving a rigorous proof of all of Fermat's claims." We explained his proof of Fermat's little theorem, which is the more traditional proof on sees in a first abstract algebra class. We then explained how Euler factored the Fermat number $$2^{32}+1 = {4294967297} = 641 \cdot 6700417,$$ thereby showing that Fermat was spectacularly wrong in his prediction that $2^{2^n}+1$ is always a prime number. Ironically, the key conceptual tool that Euler used to invalidate Fermat's prediction is Fermat's little theorem!

Among the other number theory results, we discussed on Friday how Fermat (following in the foosteps of the Indian mathematicians Bramagupta in the 8th century and Bhaskara in the 12th century) studied what is now sometimes known as Pell's equation, the equation $x^2-Dy^2=1$ where $D$ is a positive integer and one seeks integer solutions $x$ and $y$ to the equation. We described the general algorithm found by Bhaskara and rediscovered by Fermat, which allowed the latter to show, in particular, that the smallest non-trivial solution to the equation $x^2-61 y^2 =1$ is $$ x = {1766319049}, \qquad y = {226153980}.$$ Another noteworthy example of the same kind, which I didn't mention in class, is the equation $x^2-109 y^2=1$, where the smallest solution is $$x= {158070671986249}, \qquad y = {15140424455100}.$$ We described the remarkable algorithm based on continued fractions that allowed Fermat to find, using only pencil and paper, such huge solutions which clearly lie beyond the scope of a naive search.

Week 11 (November 10-November 14). Dunham, Chapter 11

This week we turn to the mathematics of the 19th century, which was characterised by an explosion of activity, spurred by a growing abstraction and formalisation of the subject. This had a liberating effect by freeing mathematics from the strictures of immediate relevance to the description of the physical world. Important philosophical developments include the introduction of non-Euclidean geometries, the inception of abstract algebraic structures like rings and groups which gave the proper conceptual foundation for increasingly abstract notions of number (for instance, the complex numbers which had been an important if somewhat controversial object in the mathematician's toolkit for several centuries by then, and Hamilton's quaternions), higher dimensional spaces, and so on. This century also ushered in more solid foundations for analysis and the calculus, with a more rigourous notion of limit proposed by Cauchy and Weierstrass, and a better working definition of the real numbers. Such a flurry of activity could only lead to a renewed urgency for a better, more systematic understanding of the foundations of the subject, notably, a better understanding of the philosophically subtle but absolutely crucial notion of infinity whose importance in mathematics can hardly be overstated.

We discussed the great mathematician, logician, and philosopher Georg Cantor, who discovered that there is a tremendously rich (infinite, in fact!) hierarchy of infinities, leading to the so-called infinite cardinals or transfinite numbers. We presented Cantor's celebrated diagonal argument, which led him to his proof of the non-denumerability of the continuum, i.e., the proof that it is impossible to draw up an infinite list of real numbers, as one can for integers, rational numbers, or polynomials with rational coefficients, etc. This gives a sweeping but highly non-constructive proof of the existence of transcendental numbers.

The following video gives a nice account of Cantor's diagonalisation argument. You might also want to watch the somewhat more advanced video describing some of Cantor's ideas (along with related topics, like the equally celebrated and notorious axiom of choice), in order to gain further perspectives on this week's philosophically subtle material.

Week 12 (November 17-November 21). Dunham, Chapter 11

This week we proved Cantor's celebrated theorem that the cardinality of a set is always strictly less than the cardinality of its power set, a statement whose stunningly simple proof belies its mathematical depth and philosophical import. It unleashes a bewildering array of different orders of infinity going well beyond the countable infinity of the integers and the ``cardinality of the continuum" describing the size of the set of real numbers. Mathematicians, logicians, and philosophers are still digesting the implications of the remarkable richness of the concept of infinity, which had entirely escaped the notice of thinkers before the 19th century.

The proof of Cantor's Theorem is perhaps the most perfect embodiment of the eponymous diagonalisation argument: if every element $a\in A$ of a set $A$ is matched with a subset $S_a \subset A$, then the set $$ S^* := \{ a\in A, \mbox{ satisfying } a \notin S_a \}$$ is not in the image of the assignment $a \mapsto S_a$, so this assignment can never be surjective.

The continued relevance of the ideas of Cantor is nicely illustrated in this expository article in Quanta Magazine, describing a surprising connection between Cantorian set theory and computer science and featuring a McGill professor (Anush Tserunyan) among the dramatis personae.

Week 13 (November 24-November 28).

The closing week of the course was devoted to the mathematics of the late 19th and 20th centuries, focussing on the modern evolution of the concept of number (with the advent of the abstract notion of ring), space (with the appearance of non-Euclidean and Riemannian geometry, and its remarkable applicability to the physics of general relativity), and symmetry (formalised and abstracted in the notion of a group). We discussed the mathematician David Hilbert, one of the last mathematicians to lay claim to a general grasp of the mathematics of his time, and the celebrated list of 23 open problems which he presented at the 1900 International Congress of Mathematicians in Paris. This list of problems set the stage for some important mathematical developments in the 20th (and even 21st!) Centuries, a few of which were touched upon.

The following Scientific American article by Richard Courant, one of the Jewish disciples of Hilbert who was forced to leave Gottingen shortly after the Nazis took power, takes stock of the state of mathematics around 1964, raising timeless questions about the nature and mission of mathematics. These questions are not very different from the ones we continue to ponder more than 60 years later and have only become more urgent with the advent of AI, in the midst of growing speculation about its eventual impact on mathematics.

Week 14 (December 1-3).

In the last two days we revisited the harmonic series, Leibniz's series for $\pi$, and Euler's Basel sum (the sum of the reciprocals of the squares of the positive integers) from the point of view of Fourier series, giving essentially 19th century proofs of some 17th and 18th century results discussed in previous weeks.

A colleague who is an expert in logic and foundations has recommended the book Naming Infinity: a True Story of Religious Mysticism and Mathematical Creativity by Loren Graham and Jean-Michel Kantor, which gives a revealing account of how the ideas of Cantor were understood and developped by two different groups of mathematicans: a French rationalist school that gradually came to regard Cantor's unbridled infinities and their attendant paradoxes with a measure of suspicion, and a more mystical Russian school that embraced them more fully, leading to the birth of what is now known as descriptive set theory which (as you learned if you read the Quanta Magazine article recommended in week 12 of this blog) has now found unexpected relevance in the theory of computer networks. A worthwhile reminder of how mathematics is a fundamentally human activity and is deeply influenced by its practitioners' philosophical and even religious perspectives, and highly recommended if you were intrigued by the more philosophical questions surrounding the nature of infinity that were barely touched on towards the end of the course.