This weeks finds in General Mathematics

Today I took the stupid O’clock train from Braga to Lisbon, actually managed to tick off some of to so list and went to ArXiv to see if there were any new papers I should be aware of. After “Functional Analysis”, I clicked “GM”. This may have been unwise.

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Induction on buffalo

First- and second- year undergrads often think that proof by induction* is extremely hard, even if they can actually do it without thinking. More than once I’ve had conversations with a student along these lines:
MJH- So what’s the problem?
Undergrad – I can see that this result is true but I don’t know how to prove it.
MJH – OK, why is true?
UG – Well it’s trivial if n=1, and [something simple] means that shows we have when n=2 and [the same simple thing] means we can get n=3 and keep repeating up to any number.
MJH – Exactly. That’s why it’s true.
UG – So how do I write that out?
MJH – Well it’s a basic induction. Write it out like all the examples of induction.
UG – INDUCTION!? I don’t know how to do proof by induction. It’s hard!
MJH- ARGH! You just did!!!one

So anyway, maths undergrads reading this, you can do induction. Well done. Good. Let’s do induction on strings of the word “buffalo”.

The single best title of any article on Wikipedia is Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo
. The notable thing about this string of “buffalo” is that it can be parsed as a sentence. This fact was first noted by the computational linguist William Rapaport of (naturally) the University of Buffalo. We use the facts that:
“buffalo” is a plural noun (1 buffalo, 2 buffalo,…);
“to buffalo” is a (somewhat rare) verb meaning “to bully”;
“Buffalo” is a city in the American state of New York and in English we may use city names as adjectives (e.g. “the London mayor is a laughing stock”).

This lets us parse the sentence as follows (this is taken form Wikipedia).

[Those] (Buffalo buffalo) [whom] (Buffalo buffalo buffalo) buffalo (Buffalo buffalo).

or

THE buffalo FROM Buffalo WHO ARE buffaloed BY buffalo FROM Buffalo ALSO buffalo THE buffalo FROM Buffalo.

. OK, but what does that have to do with proof by induction? Well we claim that, for each natural number n, we can repeat the word “buffalo” n times and then (give or take capitalisation) parse it as a sentence**.

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.

Can a kid who writes “MoM” know why odd plus odd is always even?

Here’s some discussion of (primary) maths education from off the usual mathsblog trail. It’s the this post from John Holbo at Crooked Timber. He offers this excerpt from his daughters homework:

3. What do you get if you add two ODD numbers together? – an even number
4. Do you think this is always true? – yes
5. Why do you think so? My MoM Said So

The question is what she could possibly be expected to have put that would be better. Could a child so young be taught even a vague understanding of a better reason? Some informal “first draft” of what a proof is? Is it worth the effort to try?

On a side note, sorry for the not posting for a while.

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!)

Annoyingly brilliant comment to the last post

I can see absolutely no flaw in the following argument by Daryl McCullough.

My complaint about rephrasing Godel’s result as saying that the Godel sentence G can neither be proved nor disproved is that, to me, the ontological status of “G cannot be proved” is *exactly* the same as the status of “G is true”. To say that G is true is to say that there does not exist a natural number satisfying such and such a property. To say that G cannot be proved is to say that there does not exist a proof satisfying such and such a property. I don’t see how the latter makes any less ontological commitment than the former.

GARGH! Thus (unless someone can furnish me with a reason why it may be wrong) I probably going to have to accept that there isn’t really a good reason not to say “true”. After all there is no alternative rendering that avoids the same potential confusions and we don’t expect popularisations to define things perfectly – just to give the “shape” of the idea, so any confusion between a technical use of “true” and an everyday one is acceptable (they are different but close enough to get some understanding – it’s not like “group”).

It still seems squicky to me but I guess that’s my problem.

The truth about mathematics.

I often read paraphrases of one or other of Gödel’s theorems that talk about true, unprovable statements. I’ve said before that I’m a formalist of sorts. Talk of undecidable statements in a system being true gives me headaches. And I’m an analyst so I work in ZFC. If I say “statement X is true” I’m telling you that there exists a proof of statement X in ZFC. If you ask me if I think the continuum hypothesis is true, I’ll explain to you that it’s known to be undecidable. If you tell me you know it’s undecidable but still want to know if I think it’s true, I’ll look at you as if you asked me what colour integrity is.

Not really pseudomaths at SB

Rebecca Skloot has replied to my comment on her blog in a way that makes me feel like she is, in fact, one of the good guys. She seems to have genuinely believed that the formula seller in question was really trying to make people think about body image and not just taking money to try to get newspaper space for the company that hired him while using his credentials to make people think it was “real” science. I think she is wrong to trust him on this, but as I understand this sort of scam is mostly a British thing so she would reasonably be less sensitive to it.

Also if it is just a one off bit of silliness in a magazine it is quite different to every week in the news pages of the dailys.

Just for kicks here is a classic of the PR-driven equation-for-X genre

EDIT: Actually Skloot’s further comment is even more interesting. She says the researcher in question was very open about the PR-ness and claimed to be taking advantage of the situation to make a point of his own.

Pseudomathematics sympathiser at ScienceBlogs?

EDIT: I maybe went off half-cocked with this. See the following post./EDIT

I’m worried about Culture Dish. In her welcome message newest member of the SciBorg, Rebecca Skloot, links to an article she wrote entitled Tushology, which is about a Manchester Met professor’s mathematical equation for the perfect human arse.

*sigh*

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?