Tricki is up (kinda sorta)

I was starting to think it was some kind of vapourware, but the Tricki is up to view (via Tim Gowers ). You can’t edit yet, and it’s small. The articles there at the moment are well written (when they are complete).

I hope the project works. The sort of information that it is designed to contain is not easily available anywhere else. It’s the nature of maths that you can give a completely convincing explanation of why something is true (a proof) without giving away very much at all about how you came to know it was true (the endless headdesking before the proof). AFAIK there is no other field of study where this is true to anything like the same extent. For better or worse, the normal style of a maths journal paper positively encourages hiding the process that lead you to the result. Thus, the only way you can usually find out tools for proof is by having somebody show you, and it’s obviously rather hit-and-miss as to whether you will ever explain your problem to the person with the right trick up their sleeve. Somewhere to pool this sort of information and a decent way of navigating it will be very useful indeed.

I also think it has a fairly good chance of working well. I can see possible problems; it would be hard (and probably counter-productive) to set down very firm rules about how articles should be( such as exist at Wikipedia) s conflict resolution will rely on people being reasonable. I think this won’t be too bad though because (IME and compared to academics in other disciplines) mathematicians have, typically, as a group, fairly good habits with respect collaboration.

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Apparently being a mathematician is really, really great

Well maybe it’s just being a mathematician in the United States that’s great. Not just great, in fact, but the best job in America according to a study by a job search website.

Well I like my job, but I was a bit surprised by this. Then I saw that the methodology is ridiculously arbitrary . Physical exercise is intrinsically bad? I knew there was a reason nobody wanted to be a professional footballer. Apparently meeting the public is also very bad. Yeah, people suck don’t they?

Basically, according to this survey mathematician is the best job available but only because there isn’t full time employment available as the subject in an experiment on sensory deprivation (a job that would actually share many of the downsides of mathematics but would lack the cycles of manic optimism and crushing disappointment).

Oh and $94,160? That’s Zimbabwean dollars, right?

Hat tip to Edge of the American West (who are historians and philosophers amongst other things; pwned!)

Clear but inelegant writing of maths – should I care?

Happy New Year, folks.

I’m (still) editing a paper that is mostly stuff from my PhD and I saw the following phrase, which I had forgotten writing:

… equivalence classes with respect to equivalence.

UGH! That’s not nice, is it? The equivalence relation is established (although not quite unanimously so) under the name “equivalence” in this context. Also there is another equivalence relation that I will be using on the same class of objects (so can’t refer to “the equivalence class” without ambiguity).

The question is, should I care? It’s perfectly clear. People don’t read maths papers for the joy of the prose (although very occasionally it is a nice extra). Does this sort of thing matter?

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).

It’s [almost] time (to rock a rhyme)³

It’s tricki!.

If Tim Gowers’s project for a wiki-like compendium of proof techniques catches on (and the search features work well), it could be a really big deal. These sorts of tricks are traditionally not recorded anywhere. Their use is buried deep in the folds of proofs where it’s difficult to see the scope of the idea. To learn the use of a particular technique you have to have the good luck to work with someone who knows the right tool and shows you it.

If used well, Tricki could speed up problem solving considerably. I don’t want to sound too excitable, but this is like the mathmos’ LHC.

btw. I lost my passport while I was in England and got stuck there (bad) and a bunch of other stuff is going on (good) so I doubt I’ll write the post I mentioned last time for a while. The moment has kind of passed.

EDIT: I could have sworn “It’s Tricky” went “It’s time to rock around”. Apparently it’s “It’s tricky to rock a rhyme” which makes more sense. I’m only half correcting the title though.

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.