Nobel prize for mathematics

There's no Nobel Prize for Mathematics

This is a common statement. I've certainly used it myself. Recently it occurred to me to be annoyed with this.

Nobel Prizes are awarded in physics, chemistry, medicine, literature, peace and economics, but not mathematics.

On the other hand, mathematics is widely applicable and I think I could convince you it is certainly used in physics (career), chemistry (career), biology (career), medicine (career) and economics (career). (Links to the excellent Plus Magazine and Maths Careers.) The case for literature and peace might be a bit harder to sell. But even without these two we still have a majority.

So perhaps from now on I will try to remember to say:

Most of the Nobel Prizes are for Mathematics¹

[1. there is a fallacy here: for example, saying that some mathematics can be applied to economics does not mean that all economics involves mathematics. But, shh!]

Christmas presents

What did I get for Christmas (mathematically, at least)? My big present was an Acme Klein Bottle, whose website claims it to be one of "the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions". This is a 3D representation of a 4D Klein bottle; a pale shadow, of course, although the cheerful and entertaining information leaflet that came with the bottle claims this an advantage: "You can actually hold an Acme Klein Bottle in your hand. Those highfalutin' 4-dimensional ones can only be held in your mind". Here is a photo of mine:

I also got a set of physical puzzles which are nice to have a supply of. I find I am sometimes laying puzzles on a table for students to play with and solid, physical puzzles, while perhaps not the most mathematically interesting, are certainly an attractive draw. People can't resist picking up and playing with wooden blocks, it seems!

I got a copy of The Great Mathematicians by Raymond Flood and Robin Wilson, a past and present President of BSHM, which claims to present "mathematics with a human face, celebrating the achievements of the great mathematicians in their historical context". You can watch a lecture given by Raymond and Robin at the launch of the book at Gresham College.

As a bit of Christmas day craft, inspired by the escapades of the Manchester Maths Jam, we made dodecahedron star lanterns. Unlit, these are like this:

And lit they come alive like this:

Merry Christmas everyone! What mathematical presents did you receive and what mathematical activities did you do this year?

Pictorial proofs

I received this message from Alan Stevens, Nottingham Maths Jam attendee. I am putting it here so readers of this blog and the other Maths Jams might consider the topic as well.

Although I won't be able to make the next MathsJam at Nottingham I've thought of a theme you might like to consider. I don't know if you have themes, but, if you do, how about "Pictorial proofs and derivations"?

Probably the most obvious pictorial proof is of Pythagoras's theorem (in fact there are probably several such). Do your mathsjammers know of any more?

I thought of this while viewing James Tanton's YouTube channel, where he has a very pictorial way of looking at maths, including a very nice pictorially based derivation of the geometric series 1/3 + 1/3^2 +1/3^3 + ... = 1/2.

If you haven't seen someone in a t-shirt displaying a pictorial proof of Pythagoras, you haven't been going to the right sort of conferences! The James Tanton video reminds me of a pictorial demonstration of summing 1/2 + 1/4 + 1/8 + ... which I used in a lecture, after Zeno's paradox of Achilles and the tortoise, when I was trying to get across the idea of an infinite series summing to a finite amount. Shading half a square, then a quarter, then an eighth, and so on it looks like you will eventually shade the whole square and nothing more, a useful illustration that the series converges to one. In fact, I repeated this in the micro-teaching session of my Postgraduate Certificate in Higher Education course and a screenshot of the slide after I drew on it using the interactive whiteboard is below. I'm not sure if this constitutes a proof, though.

Why I like some bad maths stories

My two most recent posts here have been about a story reporting a coincidence as more exceptional that it is and 'bad maths' reported in the media. Both are examples of mathematical stories being reported in a way that is not desirable. Somehow, though, I like the whist story and dislike the PR equations. I have been thinking about why this might be the case.

The PR-driven, media-friendly but meaningless equations from the first article are annoying because they present an incorrect view of mathematics and how mathematics can be applied to the real world. Applications of mathematics are everywhere and compelling, yet the equations in these sorts of equations seem to present little more than vague algebra. The commissioned research with seemingly trivial aims I find more difficult because, as commenters on that article pointed out, it is really difficult to decide what is trivial. Still, reporting that a biscuit company has commissioned research into biscuit dunking is either meaningless PR or else a matter of internal interest, and certainly nothing like what I expect mathematicians do for a living.

Coming back to our Warwickshire whist drive: what do I like about this story? It too presents incorrect information about mathematics and the real world, claiming that the event, four perfect hands of cards dealt, is so unlikely that it is only likely to happen once in human history (and it happened in this village hall!).

I think the difference is that the mathematics used, combinatorics and probability, appear to be correctly applied. The odds quoted, 2,235,197,406,895,366,368,301,559,999 to 1, are widely reported and I see no reason to doubt them.

The problem, then, is one of modelling assumptions. Applying a piece of mathematics to the real world involves describing the scenario, or a simplified version of it, in mathematics, solving that mathematical model and translating the solution back to the real world scenario. In this case, the description of the scenario in mathematics assumes that the cards are randomly distributed in the pack. This modelling assumption, rather than the mathematics, is where the error lies.

The result is still a bad maths news story, presenting a mathematical story as something other than what it is, but while the PR formulae are of little consequence, this incorrect application of a correct combinatorial analysis is something we can learn from.

'Bad maths' news stories

On the Math/Maths Podcast, we frequently cover 'bad maths' stories. A recent example was the bobbing apples story we spoke about in episode 71: Halloween Fruit Special. This proposed a "mind-bogglingly complicated equation", provided by a supermarket, for finding the perfect bobbing apple:

D = 3 x (2 + T^2) x M / (10 x T), where D is diameter, T is typical texture of an apple, and M is average mouth size.

Along with another formula from a rival supermarket (I reproduce this as reported; I imagine the second equals is supposed to be a plus):

B = (BU + S) x (C = BI), where B is bobability, BU is buoyancy, S is size, C is colour and BI is biteability.

A headline like "Mathematician finds formula for the perfect bobbing apple" is a tell-tale sign of bad maths in the news. Or is it? Actually there are several types of story that all appear under this same style of headline.

A couple of years ago Simon Singh launched a mini-campaign against 'phoney formulae' being reported in the news. Simon describes being asked by a PR company for an equation to say that the perfect shopping day coincided with the launch of a shopping exhibition, saying to the PR company,

I would engineer the equation so that the graph peaked on the day you require. There would be no real science behind the equation, but it would look sensible and convincing.

and getting the response,

Your ideas and formulas are perfect and exactly what we are looking for and it would be great to confirm you working with us.

As well as calling them "absurd PR equations", Simon also gave a more serious warning, saying this,

demeans mathematics and science by giving the impression that academics waste their time on frivolous topics and are willing to come up with the appropriate answer if someone is prepared to pay them enough money.

A 2004 BBC article, "Formula for the perfect formula", claims the origin of this media fascination with formulae was a piece of work by Len Fisher on how to dunk biscuits, sponsored by a biscuit company. What is described in that article is, I think, something subtly different from what Simon is railing against.

Len seems to be taking corporate sponsorship to do real experiments (the original article describes a "two-month investigation"), just that they are experiments with fairly trivial goals. The article describes a scientist who is aware that newspapers just want "to make a story look scientific" but motivated by a desire to communicate science to the public through any means available. In the case of the biscuit dunking, Len is quoted describing the physical processes and saying: "As with most things in physics, we can write equations which govern this".

I doubt the biscuit manufacturer cared in advance how long a biscuit should be dunked for (though they are presumably interested in getting their brand in the news) and Len claims to be doing real science to answer the question. I have also heard people describe being approached by PR companies with ready-made meaningless formulae to which the approachee is asked to put their name, on behalf of some brand. This seems similar to but slightly different from Simon’s case, where he is apparently asked to manufacture a realistic-looking piece of pseudo-mathematics to back up a pre-determined conclusion, or Len’s, where a brand name is associated with some trivial piece of research.

Simon also speaks about the danger of confusing real research with these PR equations. In Math/Maths 52: World's Smallest Klein Bottle we covered side-by-side the formula for the perfect cup of tea and that for the perfect golf putt. The former appears to be just PR for a milk company but the latter, although you could imagine a headline like "Mathematicians discover the formula for the perfect putt" being dumb PR for a golf equipment company, appears to be based on genuine research. Using mathematics to improve the performance of sports people and manufacturing of sports equipment is big business. The author, a physics professor at Yale, filed a patent in 2005 through the university for a golf training aid, marketed through a spin-off company. This was described by the Dean of the Engineering Faculty at Yale as "a great example of the joy in practical application of basic science and engineering". Over in the UK, I expect this would be an example of 'knowledge-transfer' and much sought-after evidence of the positive impact university research has on wider society.

There is another interesting case, when a real piece of research is explained using a metaphor which then gets reported as its real purpose. This may be what happened in a 2010 story in the Guardian, which is about a paper on Recursive Binary Sequences of Differences published in Complex Systems that we spoke about on Math/Maths 7. Attempting to explain potential applications to blending problems, the researcher seems to have given the trivial example of pouring coffee. The work is reported under the headline "How to pour the perfect cup of coffee" with explanation: "Years of research have resulted in the definitive way to pour the best second cup of coffee". I don't know what happened here but it looks to me like some number theory with potential applications to chemistry is being presented in a way that might lead you to believe it is one of Simon's "absurd PR equations".

So what have we learned? Rail against bad maths in the media when you see it, but be careful to check you've found what you think you've found and be aware there are levels of severity.

Four perfect hands: An event never seen before (right?)

A couple of weeks ago there appeared several reports of an astonishing coincidence. Reports in the Daily Express, The Sun and the Daily Mail tell of a game of whist at the village hall at Kineton, Warwickshire. In whist, one deals 52 cards equally between four players. During this particular game, all four players were dealt one entire suit each.

All three reports refer to an analysis by Dr Alexander Mijatovic of Warwick University. It is always difficult to know how much of what is reported is faithful, but the fullest account of his words was given in The Sun:

The chances of this happening are so humongous that it is almost impossible.
The event can only be compared to natural occurrences.
It would be the same as a person having a tiny drop of water and then finding that same drop of water in the Pacific Ocean.
I would question whether the cards were shuffled the correct number of times but if they were, and the people involved are sure they were, then it is probably safe to say this is the first time this hand has ever been dealt in the history of the game.

It is this last sentence, in particular, that caught my eye: "it is probably safe to say this is the first time this hand has ever been dealt in the history of the game". I took a quick look on Google News, which indexes old newspapers. I obeyed the following rules: I ignored results when only one perfect hand was dealt (hardly remarkable at all!); I didn't pick a second result from the same decade (although there were plenty, particularly in the 1920s and 30s) and I didn't spend very long at all on this. Here are a set of articles I found:

• The Evening Independent - Nov 27, 1929: Four Perfect Bridge Hands Held On The Same Deal In London Game
• The Montreal Gazette - Mar 15, 1935: Perfect Bridge Hands: Four of Them at One Table Break Up Party
• Toledo Blade - Jul 17, 1949: Perfect Bridge Hand Is Dealt At Party
• The Vancouver Sun - Mar 13, 1954: 4 Perfect Bridge Hands Dealt Simultaneously
• The Press-Courier - Apr 5, 1963: Four perfect bridge hands reported second time in less than a week
• The Milwaukee Journal - Jan 25, 1978: A Magnificent 4 in Bridge World
• BBC News - January 27, 1998 Card trick defies the odds

I was particularly taken with an account by Catherine Ford in The Calgary Herald of 29th November 1983, which contains,

Every bridge player fantasizes about the perfect hand - being dealt the 13 cards of one suit - and the perfect game, in which each of the four players receives all 13 cards of one suit. The odds of this happening are 2,235,197,406,895,366,368,301,559,999 to 1, which explains why a plain brown envelope, sealed in 1946, is among my mother's prized possessions. It contains the cards which dealt one perfect suit to each player.

William Hartson, commenting in The Independent on one such incident in 1998, said:

There are about six billion people in the world. If they all played one hand of cards every five minutes, 12 hours a day, such a coincidence would happen about once every ten trillion years. On the other hand, there are a good few practical jokers around who would love to sneak a doctored pack of cards to four unsuspecting players to create the perfect whist hands when dealt. I know which possibility my money is on.

It is tempting to suggest that someone made these stories up, or stacked the deck as a joke. However, it turns out these assumptions aren't needed to explain what is happening.

Essentially, Dr Mijatovic was right to question whether the cards were shuffled correctly (so I wonder if this was actually the main thrust of what he said). Basically, whist is a game in which the objective is to stack the deck. A card is played and the other players must follow suit if they can, meaning the cards at the end of the game are particularly well ordered into suits. If the shuffling does not completely randomise the deck (and it often doesn't) then the probability of a perfect game occurring is increased greatly. There is a good summary of this on the MAA website at Ivars Peterson's MathTrek.

Samuel Hansen pointed out on the Math/Maths Podcast when we spoke about this that this is still very unlikely and may even be worthy of note in a local newspaper, so we should let people have their fun. He's right, of course - I am mostly just amused by the claims of just how unlikely this is and the way an event that happens every few years is set up as unlikely to happen during the lifetime of the human race.

Shifting decline of mathematical preparedness?

Last year I wrote On the Decline of Mathematical Studies, and ever was it so, which looked at several examples of people complaining that the new generation of mathematics students were not as well prepared as the current one, with quotes from the late 20th C, mid 20th C. and even from the early 19th C. I wondered whether the problem was one of perception, or whether mathematics teaching could really be in constant (or, as Tony Mann pointed out, cyclical) decline.

I have just read 'Mathematics at the Transition to University: A Multi-Stage Problem?', an essay by Michael Grove (of the National HE STEM Programme, which supports my project) which offers an interesting view on this question. Though the complaint, that students are not prepared for university courses, sounds the same, Michael suggests the root cause and manner in which this problem manifests itself has changed. He backs up his argument with findings from several recent reports. His essay is worth a read if you are interested in this issue.

Having identified a possible root cause for the current situation, Michael also makes recommendations for what can be done to address this and points to relevant work the Programme is doing.