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.

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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*

notedscholar- for real?

Before yesterday I had never heard of notedscholar but suddenly he’s commenting everywhere I look. First at Gower’s blog, then Pharyngula, then here. His blog Science Defeated is either the most subtle and brilliant satire of maths and science psychoceramics or it … isn’t.

It’s pretty eclectic stuff. One minute he’s supporting deutsche Physik. The next he’s quoting Noam Chomsky (whose worst put down of certain criticisms of “white male science” was that the term reminded him of “deutsche Physik”) as an infallible authority on physics. Then we have suggesting that the mathematicians of Asian civilizations smuggled phony negative numbers into mathematics because they had a “vested interest in the concept” but also attacking the idea that “colour of light” and “colour of a reflective object” are different uses of “colour” as colonialism.

My favourite is this on infinity.

According to math (and also its feisty sidekick, the English language), the number before infinity would be known as the “penultimate” in the series of all numbers. So in my opinion, the last number in the number line is the penultimate.

The author has clearly read at least a little about an awful lot and appears to have developed a fractally wrong understanding of it all. If we then add his rather self-assured assertion that “While the views expressed here are not always representative of academia at large, the views are nevertheless correct” it’s the most fascinatingly odd blog I’ve seen in ages.

So please, comment. Tell me if you think it’s real.

Pretty Möbius transform video is pretty

This is my favourite maths video ever.

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).

Marcus du Sautoy FTW

The University of Oxford have appointed pure mathematician Marcus du Sautoy to the Simonyi Chair for Public Understanding of Science. This makes him Richard Dawkins’ successor, although I doubt Graun Unltd will be giving him his own category just yet. As well as being a very distinguished researcher, MdS is pure win as populizer. I loved Music of the Primes and his his Faraday Christmas lectures for kids were pitch perfect. Here he is with his Recreativo Hackney team mates and the infinity of primes:

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.