Gödel’s theorems on “In Our Time” by the BBC

Hat tip to Yemon Choi (via email) for this very interesting discussion of Gödel’s theorems on Radio 4. It features the aforementioned Marcus du Sautoy, John Barrow of Cambridge and Philip Welch of Bristol. It is pitched nicely with interest whether you are a professional mathematician or if you have no training in maths at all.

I was particularly glad to hear someone at least draw some attention to the philosophical trickiness of the usual popularization of the theorems which talks about “true statements that cannot be proved in the system”. This always erks me a bit, because it isn’t entirely clear that a statement in a formal system has a meaning that survives being taken out of the system (and if it does that is a pretty subtle thing for this level of discussion). Thus just hearing the warning (from Welch?) about the theorems and their proofs being basically syntactic rather than semantic was nice. (It was du Sautoy giving the usual “true statement” version with but I will forgive him since he does so much good work and most mathematicians seem to be happy with the truthiness).

Incidentally, I suspect my feelings of awkwardness towards the claim we can talk about “truth” outside of “proof in a given system” may be related to the sort of maths I work in. My “grand-supervisor” Garth Dales discusses here how those who work in abstract analysis (and in algebra) tend to view their work as essentially formal (although using “realistic pictures” to help us).

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In praise of proving that zero equals one.

A chap called Jim Wiseman at some university in Georgia (the American one) has collected together some quotations from linear algebra class and some of them are rather lulsome.

I particularly like

See if you can use this proof to show the square root of three is irrational. Then try the square root of four. If it works, you did something wrong.

There is great wisdom hidden in that statement and it works at a research level as much as with homework problems. It’s very often you think you have proven something (give or take some trifling details to fill in) but then you realise that the same argument would prove something known to be false.

Hopefully you can then pick out why it fails, note what is different between the two cases and mend the argument. Normally this doesn’t happen; normally you just end up banging your head on the desk and crying. All the same, spotting that you have apparently proven a contradiction is a fairly good way of moving forward with a problem and not staying stuck on an idea that can’t work.

This was put nicely by Tony O’Farrell, in the following form:

I consider the day wasted in which I have not proven that zero equals one.*

To see how this is the same you need to know that (in “ordinary” logic) if you were to prove one false statement true (i.e. prove a contradiction) you can infer the truth of ANY statement. Hence we can pick “0=1” as a canonical false statement and refer to all apparent proofs of false statements as “proofs that zero equals one”. I guess mathematicians’ humour can be kind of odd (see Dolphin’s law but note that all the really funny Gauss facts are mine).

Of course this is a rule that doesn’t only work in mathematics. If the an apparently sound argument in any setting can prove nonsense by substituting different terms into it (assuming the specific choice weren’t important to the logical form of the argument) then the argument is flawed. So if your proof that God exists also show the existence of a perfect island, it isn’t really a proof of anything, even if it’s hard to pick out where it fails. There are plenty of day-to-day examples as well; if you refuse to go to one to retailer because you know of bad ethical practices but go to their competition who behave in the same fashion, you’re probably not acting sensibly. (I confess, I’ve found myself doing this).

In short. I think it’s a good rule of thumb to keep with you: “does my thinking here also show that zero equals one”. It can help you to be wrong slightly less often

* Thanks to my PhD supervisor Joel Feinstein for recounting this to me; it’s helped defeat despair many times.