Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.
Currently flicking through Seymour Papert’s brilliant ‘Mindstorms: Children, Computers and Powerful Ideas’. I’ve read it before but the whole book is popping with insights and I should probably do this many more times. Plus maybe this time it’ll actually teach me how to juggle too?
In the epilogue on mathematical thinking, he discusses his experiences getting people “with little mathematical knowledge” (enough to rearrange an equation, though) to work towards a proof that the square root of 2 is irrational. In the process, he gives two standard proofs, the second of which I’d never seen before:
Of these I shall contrast two which differ along a dimension one might call ‘gestalt versus atomistic’ or ‘aha-single-flash-insight versus step-by-step reasoning’.
They both start with the usual proof by contradiction: let sqrt(2)= p/q, a fraction expressed in its lowest terms. This is then rearranged to get
p^2 = 2 q^2,
and then you can go down the route of “well p must be even, so p=2r, so q^2=2r^2 WHY IS q EVEN TOO WHEN WE CANCELLED THE TWOS AT THE START?” Which is pretty compelling stuff.
But I really like the second proof, what he calls the ‘flash’ version:
Think of p as a product of its prime factors, e.g. 6=23. Then p^2 will have an even number of each prime factor, e.g. 36=2233. But then our equation p^2=2q^2 is saying that an even set of prime factors equals another even set multiplied by a 2 on its own, which makes no sense at all. Done.
Papert makes the point that if you have the right idea (decomposition into prime factors) ‘pre-loaded’ into your head, the equation is directly seen as absurd just by looking at it. In fact it now surprises me that it didn’t look wrong before! The first, more algorithmic proof gets you there, but without the insight flash.
Anyway, this is a lot of set-up just to say that reading this made me understand more clearly that that dopamine hit of insight is a large part of what I’m attracted to maths for. For me, much of the purpose of studying maths is to pre-load a pile of the necessary structure into my head to enable these insights to occur. Differential geometry is great for this, and this is probably why I like learning it so much! In the process I’m willing to put up with a load of chains of logic (which I am annoyingly poor at) but I’m always really hoping that they will be the means to an aha-single-flash insight.
OK time to go practice juggling (= ‘throwing all the balls on the floor’) again.