Monday, October 05, 2015

Diamonds are forever, and so are mathematical truths?

Over the next few weeks, Leo Corry, historian of mathematics and  author of A Brief History of Numbers, helps us understand how the history of mathematics can be harnessed to develop modern-day applications today. In this first post, he explores how the differences between perceptions of truth in maths and history affect the study of those subjects.

 Diamond by nafets. CC0 Public domain via Pixabay.

Try googling ‘mathematical gem.’ I just got 465,000 results. Quite a lot. Indeed, the metaphor of mathematical ideas as precious little gems is an old one, and it is well known to anyone with a zest for mathematics. A diamond is a little, fully transparent structure, all of whose parts can be observed with awe from any angle. It is breathtaking in its beauty, yet at the same time powerful and virtually indestructible. This description applies equally well to so many pieces of mathematical knowledge: proofs, formulae, and algorithms.

Leonhard Euler, for instance, was the greatest mathematician of the eighteenth century and we associate his name nowadays with many beautiful mathematical gems. Think of the so-called Euler Formula: V – E + F = 2. This concise expression embodies a surprising property of any convex polyhedron, of which V represents the numbers of its vertices, E of its edges, and F of its faces. But probably the most famous gem associated with his name is the so-called ‘Euler identity': eip + 1 = 0.

Beyond the mathematical importance of this identity it is remarkable how often it is known and praised, above all, for its beauty: “the most beautiful equation of maths”, we read in various places. A most impressive diamond!

But we can compare mathematical ideas to diamonds not only in terms of beauty. Diamonds are also, as you surely remember from the James Bond film, forever. And so are proved mathematical results. Indeed, the theme song of the Bond film defines very aptly, I think, the way in which mathematicians relate to those ideas with which they become involved and invest their best efforts for long periods of time:

Diamonds are forever,
Hold one up and then caress it,
Touch it, stroke it and undress it,
I can see every part,
Nothing hides in the heart to hurt me.
— Shirley Bassey, Diamonds Are Forever

Of course, before reaching the point where mathematical ideas become diamonds, likely to remain forever, there is a period of groping in the dark. This period may sometimes be long and the dark may be deep, before light is finally turned on and the diamond becomes transparent. You can then touch it, stroke it and undress it, and you will truly understand the necessary interconnection between all of its parts.

In a recent TED video, the Spanish mathematician Eduardo Sáenz de Cabezón tells his audience that “if you want to tell someone that you will love her forever you can give her a diamond. But if you want to tell her that you’ll love her forever and ever, give her a theorem!” (Unfortunately, in spite of the accompanying English subtitles, his most successful jokes are lost in translation from Spanish.)

And so, it is the eternal character of mathematical truths and the unanimity of mathematicians about them that sets mathematics apart from almost all other endeavors of human knowledge. This unique character of mathematics as a system of knowledge may be stressed even more sharply by comparison to another discipline, like history for example. At its core, mathematical knowledge deals with certain, necessary, and universal truths. True mathematical statements do not depend on contextual considerations, either in time or in geographical location. Generally speaking, established mathematical statements are considered to be beyond dispute or interpretation.