## LMS150

The London Mathematical Society is 150 years old. This year there will be various events, competitions, fellowships, etc. to commemorate the anniversary.

Yesterday, on the 150th anniversary of the inauguration of the Society, there was an event in Goldsmiths’ Hall in the City of London, consisting of several lectures by mathematicians and users of mathematics, hosted by Maggie Philbin, presenter of TV shows on science and maths. The event was streamed live; if you missed it, I expect it will all be available from the website at some point.

The Goldsmiths’ Hall is a grand building, full of wide staircases, rooms with coats of arms, gold cherubs, and busts of kings. They have a list of Prime Wardens of the Goldsmiths’ Company dating back to 1327, in the time of Edward III. You can take a virtual tour of the building here.

I want to say a bit about the event, and then discuss one piece of literature we were given: a facsimile of the printed version of the first President’s Address to the newly constituted Society.

### The event

Since the event was streamed live, and since attendance was mostly restricted to a ballot of LMS members (apart from a few guests and pupils from a school in Coventry), there was a bit of a mismatch between the live audience and people watching remotely. Maggie Philbin encouraged us to tweet about it using the hashtag #lms125; I suspect that many in the audience had little idea what she was on about, and in any case it goes strongly against the grain to fiddle with electronic equipment during a lecture, so I suppose this was addressed to the remote viewers.

I will say a bit about the three talks I considered the best.

#### Nigel Hitchin

Nigel talked about how you go about doing mathematics: use of analogy, eureka moments, and so forth. This could have been a bit fatuous, but it wasn’t, because he kept it tied to real examples: complex numbers and quaternions, Pythagoras, higher dimensions, and so forth.

I will describe one example which might be familiar to many. Descartes unified algebra and (Euclidean) geometry by his technique of setting up coordinates for systems of points in the plane. This gives us a quick route to a motivating example for the connection between algebraic geometry and number theory.

A Pythagorean triple consists of three whole numbers (a,b,c) satisfying a2+b2 = c2; that is, numbers which are the sides of a right-angled triangle, according to Pythagoras’ famous theorem. The best-known example is (3,4,5), which was probably used since ancient times by surveyors and builders to construct a right angle. Can we describe all such triples? Yes, this was done in antiquity, but here is a more modern approach.

Dividing the equation by c2 and putting x = a/c, y = b/c, we get x2+y2 = 1. This is the equation (in Cartesian coordinates) of a circle with centre at the origin and radius 1. Our job is to find the solutions of this equation in positive rational numbers (fractions). If we can do this, we can get the Pythagorean triples by multiplying up by a multiple of the denominators.

The trick is to forget “positive” momentarily. There is one obvious solution, namely (0,1). If (x,y) is any other solution, then the line joining these two solutions will meet the X-axis in a point, say (t,0), as in the figure.

Elementary coordinate geometry gives the equation of the line to be y+x/t = 1. The points where the line and circle meet can be found by solving this equation and the equation of the circle as two simultaneous equations in x and y: for example, express y in terms of x using the linear equation and substitute into the equation for the circle. Now here comes the trick. In general, this will give us a quadratic equation, whose solution will require a square root. But since we know that x = 0 is a solution, the quadratic will factorise, and we can find the other solution without extracting any square root: we find x = 2t/(t2+1) and y = (t2−1)/(t2+1). Plugging in any positive rational value of t gives a positive rational point (x,y) on the curve, and all such points are obtained.

I have misrepresented Nigel’s talk in order to explain this argument in detail. But I think that, even if it is not part of many mathematicians’ toolbox, it is part of our culture. It really shows that, from the point of view of algebraic geometry, the line and the circle are equivalent: both are “rational” curves.

I recently re-read Lee Smolin’s book The Trouble with Physics. In it, he claims that sometimes the real world forces physicists to give up the most beautiful (i.e. most symmetric) solutions; for example, Kepler had to replace the “perfect” circles of Ptolemy by ellipses. Of course, to a projective geometer, all conic sections (including circles) are “the same”; and to an algebraic geometer, they are all the same as a line!

#### Rob Pieké

Rob Pieké works for visual effects company MPC. He explained how to simulate things like smoke by solving the Navier–Stokes equations computationally. Well, what he really did was to explain the data structures required (recording temperature, density and velocity), and how they are updated frame-by-frame. Each step in the updating corresponds to a term in the Navier–Stokes equation, and so at the end of it we understood exactly where these equations come from, as well as how they are used in practice. At the end of the lecture, I almost felt that I could program a simulation of billowing smoke myself! I have never had a clearer lecture on fluid dynamics!

#### Rob Calderbank

In my (maybe biased) opinion, Rob’s lecture was the best of all. He was billed as explaining how some aspect of mobile phones works (to be precise, the space-time coding which means that two antennae are much better than one). In fact he explained how mathematics works.

After many introductory examples, he arrived at Hamilton’s construction of quaternions, which followed on well from Nigel’s talk. (I suspect that more of this was pre-planned than meets the eye.) He told us how Hamilton had become obsessed with quaternions; one of his discoveries was their representation by 2×2 matrices over the complex numbers.

The way that space-time coding works is as follows. The signal to be transmitted is encoded as a pair (x,y) of complex numbers. Time is divided into short segments, two of which are used for the transmission of one pair of numbers. In the first time slot, the two antennae send x and y; in the second, they send (−y*,x*). This is precisely the matrix representation of the quaternion x+yj.

### Augustus De Morgan’s address

De Morgan spoke 150 years ago about his hopes for the new London Mathematical Society. He remarked that it would survive if the interests and enthusiasms of the founders satisfied the “disposition of the members”. He pointed to the fact that

… the English mathematical world of the present day takes its tone principally from the Cambridge examinations; there is no doubt of that, and there is no use in denying it. The Cambridge examination is nothing but a hard trial of what we must call problems—since they call them so—between the Senior Wrangler that is to be of this present January, and the Senior Wrangler of some three or four years ago.

But, he claimed, there is more to mathematics than this; he gave as an example the fact that Wallis had written a book on the Centre of Gravity, which (following the development of integration) “merged at once … in the integral calculus … and mathematicians now do with a few strokes of the pen what it formerly took at least three or four pages of Geometry to effect.”

He presented to his audience three or four examples of mathematical topics which lie outside the Cambridge Tripos view of the subject.

• Mathematical logic, in his view the “Laws of Thought” as contrasted with the “Matter of Thought”, space and time. He gives as an example the fact that Euclid, that stickler for proofs complete in every detail, does not give an argument to establish the proposition “If there is but one A and but one B, if the A be the B, then the B is the A”. De Morgan provides the missing proof.
• History of mathematics: he is convinced that some knowledge of the history of the subject brings benefit to the practising mathematician: “If he be to have his own researches guided in the way which will best lead him to success, he must have seen the curious ways in which the lower proposition has constantly been evolved from the higher”. He gives several comments from his own researchers, including the fact that in the century of Tartaglia and Pacioli, multiplication and division (for fractions) were written as = and × respectively. The reason is clear from an example: $\frac{2}{3}=\frac{4}{5}$ is $\frac{8}{15}$, while $\frac{2}{3}\times\frac{4}{5}$ is $\frac{10}{12}$. (Look at the positions of the numbers which have to be multiplied!) He traces the use of the modern symbols + and − to a German book of 1489.
• Mathematical language. I am not quite clear what the argument is here: he says “If we do not attend to extension of language, we are shut in and confined by it”, and gives the example that “of” and “but” are positive and negative forms of the same thing: “all of men”, or “all men” for short, compared with “all but men” (the complement of the class of all men).
• Improving elementary mathematics. He gives a principle which many teachers of mathematics will agree with: better to have a clear argument in words than a long chain of computations!

A couple of smaller observations:

• The word “mathematics” is almost always plural for De Morgan: see the next quote.
• He admitted applications into the purview of the LMS: “Our great aim is the cultivation of pure Mathematics and their most immediate applications”. After a period of following Hardy’s dictum that real mathematics must be completely useless, the LMS has strongly reverted to De Morgan’s viewpoint, as many of the lectures showed.

## About Peter Cameron

I count all the things that need to be counted.
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