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Diamonds are forever, and so are mathematical truths?

Over the next few weeks, Leo Corry, 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.

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’:

e i π + 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.

The discipline of history, on the contrary, deals with the particular, the contingent, and the idiosyncratic. It deals with events that happened in a particular location at a particular point in time, and events that happened in a certain way but could have happened otherwise. Historical statements are always partial, debatable, and open to interpretation. Arguments put forward by historians keep changing with time. ‘Thinking historically’ and ‘thinking mathematically’, then, are clearly two different things.

For historians of mathematics, the comparison between mathematics and history, as two different ways of thought and as two different kinds of academic disciplines, is an important issue. Historians in general do not see just providing an account of “one damn thing after the other” (as the phrase often attributed to Arnold Toynbee goes) as the aim of their intellectual pursuit.  Historians of mathematics, in turn, do not see the aims of their pursuits as just providing a chronology of discoveries. What does it mean, then, to think historically about the ways in which people have been ‘thinking mathematically’ throughout history, and about the processes of change that have affected these ways of thinking? If mathematics deals with universal truths, how can we speak about mathematics from a historical perspective (other than to establish the chronology of certain discoveries)? What is it that changes through time in a discipline whose truths are, apparently, eternal?

“Who was the first to discover the formula for the quadratic equation?” That’s not really the kind of question that historians of mathematics try to be involved with. In fact, rather than “Who was the first to discover X?” we may find it more interesting to investigate a question such as “Who was the last person to discover X?” This latter question involves the understanding that in spite of the eternal character of mathematical results, there is still a lot to be said about the way in which mathematical ideas develop and are understood throughout history. It suggests that something that was mathematically proved at some point was not considered to be so at a later time, that mathematicians are not always aware or do not always care about the existence of a proof that later becomes interesting and relevant. It also suggests that it makes historical sense to try and understand the circumstances of this change in mathematical values. The question also implies that only at a certain point in time a mathematical proof became so fundamentally convincing that it impressed upon that result a stamp of eternal validity. Or, at least, temporarily so.

Featured image: Isfahan Lotfollah mosque ceiling symmetric by Phillip Maiwald. CC BY-SA 3.0 via Wikimedia Commons.