How do we know when we’ve proved something?
For most people, it’s hard to think of something more black and white than mathematics, a domain that, with its celebrated distinction between “right answers” and “wrong answers” is often invoked as nothing less than the acme, in some cases even the very definition itself, of objective tests of knowledge.
And yet, it is not always quite as simple as that.
You might be relieved to learn that this piece will not try to convince you that claims such as 2+2=4 are merely experientially-determined subjective opinions, but rather explore something somewhat more nuanced: the very notion of mathematical proof itself. How do we know, in short, when something has been mathematically proven?
If you talk to a mathematician, more often than not this question won’t seem any more profound than asking her what 2+2 is, as she will unthinkingly rattle off a list of the various long-established techniques for mathematical proof that everyone agrees on: deduction, induction, contradiction, and so forth.
But what about if I do something different from all of that? What if, as the philosopher of science James Robert Brown suggests, I rely upon a picture instead as proof of my claim?
Well, it’s one thing to discuss these things in the abstract, but quite another to grapple directly with a concrete example, and the one that Professor Brown highlights in the associated video clip Proof by Picture on Ideas Roadshow’s IBDP Portal will, I’m quite certain, make you agree with him: a careful examination of the diagram he provides will convince you, just as much as any other “established” mathematical technique, that the theorem he states at the outset is successfully proven.
And suddenly, our answer to the question of “how do we know we’ve proven something?” is much less obvious. Because now we’re aware – at least sometimes and in some cases – that there might be additional ways, much less easy to objectively quantify and assess, that can somehow provide us with that very same sense of certainty that our established mathematical toolkit of proofs does. And suddenly, instead of being in possession of a clear, objective decision procedure for mathematical certainty, our gut feelings have begun to play a curiously significant role in our convictions of what is true.
For additional examples of how TOK overlaps with mathematics, see Ideas Roadshow’s TOK Connections Guide for Mathematics which you can find in the Teacher Resources section and the Student TOK section of Ideas Roadshow’s IBDP Portal.
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