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.
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| 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.
Read more...
Source: OUPblog (blog)