Proofs Without Refutations
I made a passing remark during a conversation recently and it has shone an interesting light on the acceptance of mathematical proofs. The remark was, I thought, totally innocuous and incontrovertible: that 4.99999… (recurring) was just another way of writing 5. You may wonder about the sort of conversations I have but I assure you that this did occur in a natural way.
My statement caused not only a local explosion of disbelief but, at a party a few days later with various genera of engineers (mechanical, electrical, civil, software) present, my original disbelievers recruited a whole band more: “do you know what Chris is saying? He thinks that 4.999…. is the same as five.” This caused much ribaldry and hilarity until I offered to prove it. With a large audience, and with neither preparation nor safety net, I gave the following well-known “proof”:
(1) Let x = 4.9999999…..
(2) Then 10x = 49.999999…..
(3) i.e., 10x = 45 + 4.999999….
(4) i.e., 10x = 45 + x
(5) i.e., 9x = 45
(6) Therefore x = 5
Quod, as they say, erat demonstrandum. End of discussion? Of course not.
I could reflect on the legitimacy of this “proof” but I’m more interested here in the reaction of my audience. Since this argument clearly proved something that was wrong (in their eyes), they started searching for the step which was in error. Having failed to find it, they then hit upon an ingenious idea. If a proof is valid then the steps should work backwards (6) to (1) as well as forwards (1) to (6). Although I had probably implicitly assumed something like this, it wasn’t a concept that I had ever explicitly codified. Anyway, they applied the technique and found the error: having started at step (6) (x = 5), you can get back to step (5) easily enough and thence to (4). But you can’t get back from step (4) to (3). This convinced them further that the error in the forward argument was between step (3) and step (4). They couldn’t identify the error but it was clearly there.
Now this “proof” is certainly not one that I would man the barricades to defend when the great unwashed rise up to hang mathematicians from the lamp posts but the incident raises two interesting points:
1. people can accept a proof without accepting its consequences. Hardy speaks about this in his Mathematician’s Apology and I can add nothing new but there is another paragraph (section 13) which also addresses this issue: “…. Here [Cantor’s theorem on non-enumerability] there is just the opposite difficulty. The proof is easy enough, when once the language has been mastered, but considerable explanation is necessary before the meaning of the theorem becomes clear”. Perhaps I should have worked less hard on the proof and harder on the meaning of the theorem.
2. a proof is valid only if it can be reversed. There are clearly irreversible steps in some forms of proof (e.g., induction and some forms of reductio ad absurdum) but intuitively I feel as though there should be some basis for this belief. Effectively it is saying that all steps are one-to-one so that we don’t encounter any one-to-many steps in the reverse direction. I’ve just been browsing through Aigner and Ziegler’s Proofs from THE BOOK to see whether I can find a proof other than by induction or reductio that doesn’t work backwards and I’m fascinated by the fact that almost all of the proofs therein are of one of those types.
So, to conclude the story, I said “why can you accept that 0.33333…. is exactly one-third but not that 4.9999…. is exactly 5?” I’m sure my reader is ahead of me: 0.33333… is apparently not exactly one-third but is, so I’m told, pretty close to it.