### New Directions in the Philosophy of Mathematics

#### Posted by David Corfield

To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one of these workshops on the Philosophy of Mathematics.

So, on Saturday 4 October, we began with Mary Leng, a philosopher at Liverpool, talking about whether the creation of mathematical theories, e.g., Hamilton’s quaternions, gives us any more reason to think mathematical entities exist than does the discovery of new consequences within existing theories. She concluded that it does not – both concern the drawing of consequences from suppositions, e.g., “Were there to be a 3 or 4-dimensional number system sharing specified properties with the complex numbers, then…”.

George Joseph, author of The Crest of the Peacock, then told us about sophisticated work in 16th century Kerala concerning expansions of trigonometric functions, and better proved equivalents to the proto-calculus of Wallis. He then moved on to China where in search of the equally tempered scale, the 16th century Chinese mathematician Zhu calculated the value of $2^{1/12}$ to an extraordinary number of decimal places. Discussion centred around the question of why we continue to ignore non-Western roots of modern mathematics.

Marcus Giaquinto, a philosopher from University College London, talked about visual intuition and proof. He explained how a great deal of care is needed in assuring ourselves of the validity of a geometric demonstration. This follows up on the work of Ken Manders (Pittsburgh) to understand when Euclidean diagrammatic reasoning is valid.

Angus MacIntyre, a model theorist at Queen Mary College London, told us about the limited interest for mainstream mathematics of incompleteness results. He argued that the kind of Diophantine equations appearing in incompleteness results are not of the kind number theorists deal with, and he explained to us how he is trying to show that Wiles’ proof of Fermat’s Last Theorem can be written in first-order Peano arithmetic.

I finished off the talks by speaking about my paper Lautman and the Reality of Mathematics. This argues that philosophy should study mathematics less as (potentially) concerning abstract entities, but rather as concerning the development of certain ideas. While Lautman pointed us to some excellent examples of this phenomenon, the Galoisian idea and the idea of duality, it seems less clear to me that they are situated somehow superior to mathematics as he believed.

We ended with a brief general discussion, which included a deliberately provocative comment from the philosopher John Kennedy that the majority of mathematicians don’t think hard enough about basic concepts of identity and relation, and that this marked an impoverishment of mathematics since the days of Hilbert.

It occurred to me just before I began my talk that I should have said more about Michael Harris’s comments, discussed at my old blog:

Perhaps we might find richer pickings in answering a challenge posed by Michael Harris in “Why Mathematics?” You Might Ask that philosophers “…have a duty, it seems to me, to account for terms like “idea” and “intuition” – and “conceptual” for that matter – used by human mathematicians (at least) to express their value judgments.” (p. 17). Take the term idea:

Nothing in the life of mathematics has more of the attributes of materiality than (lowercase) ideas. They have “features” (Gowers), they can be “tried out” (Singer), they can be “passed from hand to hand” (Corfield), they sometimes “originate in the real world” (Atiyah) or are promoted from the status of calculations by becoming “an integral part of the theory” (Godement). (p. 14)

Harris makes the very useful point that my own use of the term is liable to a certain slippage:

Corfield uses the same word to designate what I am calling “ideas” (“the ideas in Hopf’s 1942 paper”, p. 200) as well as “Ideas” (“the idea of groups”, p. 212) and something halfway between the two (the “idea” of decomposing representations into their irreducible components for a variety of purposes, p. 206). Elsewhere the word crops up in connection with what mathematicians often call “philosophy,” as in the “Langlands philosophy” (“Kronecker’s ideas” about divisibility, p. 202), and in many completely unrelated conections as well. Corfield proposes to resolve what he sees as an anomaly in Lakatos’ “methodology of scientific research programmes” as applied to mathematics by

a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas… (p. 181)

He sees “a kind of creative vagueness to the central idea” in each of the four examples he offers to represent this shift of perspective; but on my count the ideas he chooses include two “philosophies,” one “Idea,” and one which is neither of these. (p.16)

Point taken. I’ll see what I can do.

## Re: New Directions in the Philosophy of Mathematics

The talk of Marcus Giaquinto got me thinking a little about Euclid’s first proposition. This says that the you can construct an equilateral triangle ‘on’ any line segment, by which I mean you can draw an equilateral triangle which has that line segment as one of its sides. The proof is apparently notorious because it requires you to deduce something from a diagram.

Marcus Giaquinto was arguing that you can get round the philosophical problems associated with arguing from the diagram, but as far as I could see there was still a mathematical problem.

The proof of the proposition goes as follows (and I deliberately won’t give you a picture to go with it). For each of the two endpoints of the interval, construct the circle with that point as its centre and with the the interval as its radius. This gives you two circles. These intersect at two points. Pick one of the intersection points and take that to be the third vertex of your equilateral triangle. QED

There is at least one dodgy step in the above proof. The one I’m thinking of is the assertion “These intersect at two points.” Where does that come from? Well you draw a picture and see that it’s true. Hmmm. I thought Euclid was all about deducing things from axioms. So let’s step back a little.

We could try to do the same construction in spherical geometry rather than Euclidean geometry. This is the geometry of the surface of the Earth, so “straight line” means part of a great circle, that is part of a biggest possible circle such as the equator or any circle passing through both the north and south poles. This kind of geometry satisfies Euclid’s axioms

exceptthe so-called parallel postulate.However, Euclid’s first proposition

failsfor spherical geometry. If you take your interval to be greater than a third of the circumference of the sphere then there is no equilateral triangle on that interval. If you construct the two circles on that interval then theydo not intersect.I’ve tried to illustrate this with a little animation on YouTube.

So we could play at being Euclid and assert that just because we can draw an interval on the ground (think of doing it in the sand on the beach with a stick), then construct the two circles as above and observe that they intersect, we can therefore conclude that it will be true for any interval we draw on the ground. But that’s not true. If we draw a line from the South Pole to Sheffield then it won’t work.

This seems to mean that any proof of the first proposition must require the parallel postulate. Euclid certainly doesn’t mention it.

So Euclid’s first proof is decidedly iffy. Does anyone know any way around this? I mean I guess this is a very well known problem (for instance, this webpage mentions that the proof doesn’t work for other geometries), so there must be better proofs.