The difference between maths and sociology; plus notes on √2, Möbius transformations, quaternions, and complex numbers

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On 20 May 2016 I went to a lecture in Cambridge by Dana Scott on “Why Mathematical Proof?”

Scott is an American mathematician, specialising in mathematical logic. He was a student with two of the great figures from the heroic period of mathematical logic in the mid-20th century, Alonzo Church and Alfred Tarski. (Anyone remember the Banach-Tarski theorem from our lessons on proof at the start of Year 12 Further Maths?)

In a way Scott’s lecture was disappointing, since it wasn’t an attempt to expound a new idea or set of ideas, but designed as a miscellany of interesting titbits.

However, I enjoyed it. And I was very struck by the contrast between its manner and the manner of a sociology lecture I attended the day before, 19 May, at the London School of Economics.

The two events had many similarities. Both were public lectures by elderly but celebrated professors from the USA who were visiting Britain. Both were to a degree “ceremonial” events – Scott deliberately ran a sort of scrapbook session, and the lecturer at LSE, Moishe Postone, really just summarised the argument of his famous book from 1993, Time, Labour, and Social Domination – and both were organised by posh and somewhat pompous institutions.

But when I arrived at the maths building in Cambridge, the cafe area, which is also the entrance hall, was abuzz with groups of mathematicians sitting round cafe tables, scribbling, arguing, explaining to each other.

The crowd in the lecture theatre was maybe twice as big as at LSE, and there too, it was a hubbub of chat until the lecture started. One of the Cambridge lecturers, dressed informally (no tie), got up on the stage; told a few jokes about Louis Mordell (it was the Mordell memorial lecture: see here for the “Mordell equation”) and about Scott; and asked if anyone in the audience knew how the AV equipment worked. Then he handed over to Scott.

Scott asked questions of the audience during his lecture, got answers, got heckled… People laughed.

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Moishe Postone is a radical left-winger, a Marxist who is unafraid to dispute with other Marxists, a sharp, critical-minded person, and not personally pompous at all. (I met him before the lecture to do an interview). Yet the LSE event was ten times more stuffed-shirt than the Cambridge one.

Before it, people waited in a reverent hush outside the lecture hall. When we were let in, about half a dozen people wearing “event steward” badges scurried round, fussing about the AV equipment, directing certain people to certain seats. Then the Director of LSE arrived, looking more like a businessman than an academic (he’s paid nearly half a million pounds a year).

He introduced Postone in pompous terms – “distinguished professor”, etc. Postone gave his lecture to reverent silence. All the questions after he finished, apart from mine, were along the lines of asking him to explain further, rather than directly criticising what he said.

Sociology is supposed to be the leftish, critical-minded field of study. Back in the 1960s some conservative lecturers opposed even having a sociology course at Cambridge with the slogan “Sociology today, Cohn-Bendit tomorrow”. (Dany Cohn-Bendit was one of the leaders of the student revolt in France in 1968). And maths is supposed to be staid and strait-laced.

But it’s not like that. Mathematicians get things wrong every day. They know they get them wrong: it can be proved. When they make discoveries, those come only after days or weeks or years of getting things wrong. Regularly they start on solving problems with this situation:

In other subjects, people speak “with authority” because they are famous or they hold prestigious positions. In maths, what you say has authority only when you can prove it. A solid proof by the humblest always beats the “voice of authority” from the most famous. The subject is by its very nature more democratic, more a field for critical thinking, than other subjects as they are studied today.

Enough of that. What did Scott say?

He told us about generalisations of the Pythagoras theorem where, instead of studying the squares on the sides of the right-angled triangle, we study, for example, semicircles on those sides.

He discussed proofs without words – proofs which convince us just from a diagram, without words. (Tim Gowers talked about those when he came to CoLA.)

He described the geometrical proofs that √2 is irrational.

He told us about a proof by David Hilbert that Möbius transformations map circles to circles. Rather than the proof using two-dimensional geometry which I give on this website, Hilbert’s proof uses three dimensions and works by showing that stereographic projection maps circles on a sphere into circles in a plane. See this video for an explanation:

He talked about how Möbius transformations are useful for the study of hyperbolic geometry, and how they are illustrated in the art of M C Escher.

He told us that quaternions can be conceptualised as quotients of three-dimensional vectors, and complex numbers as quotients of two-dimensional vectors. (To get from one 2-vector to another, you have to “multiply” the first vector by a rotation and an enlargement. But that is exactly what complex numbers are: operations of rotation-and-enlargement. See http://matheducators.stackexchange.com/questions/1883/are-there-disadvantages-to-teaching-complex-numbers-as-purely-geometrical-object).