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 which is devoted to 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 delightful 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.

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.