I was very sorry to hear of the death last week of the historian Lisa Jardine. Although it wasn't her main focus, she made a big contribution to our understanding of early modern mathematics and especially of key figures like Robert Hooke and Christopher Wren. Her books are wonderful - readable, full of insights, and giving a vivid picture of intellectual life in the seventeenth century.

I was lucky enough to hear her talk, less than a year ago, at the BSHM Christmas meeting last December when she gave an inspiring talk about women in twentieth century mathematics -in particular Hertha Ayrton, Mary Cartwright and Emmy Noether.

Jardine;s scholarship was important, but so was her encouragement of others. I believe she was an exceptional research supervisor, and her writing certainly inspired many, myself included. I experienced her kindness several times, and enjoyed a few conversations with her in coffee breaks at conference. Twice I consulted her by email, and although she can have had no idea who I was, she replied quickly, enthusiastically and helpfully. (On the first occasion I was seeking clarification of a view attributed to her in someone else's book, and on the second I was hoping to persuade her to talk about the novelist Robert Musil at a conference I was organising - she agreed in principle but sadly the dates didn't work out.)

Her contribution to the history of science, direct and indirect, is immense. She is a great loss to the history of mathematics.

## Sunday, 1 November 2015

## Sunday, 25 October 2015

### Outreach: Simon Singh's comments

The

Now, not only did I not hear Singh's talk, so that I am relying on the THE report, but Singh is one of my heroes for his contributions to mathematics and its public reputation. For one thing, his documentary film on Andrew Wiles's proof of Fermat's last theorem conveyed to the general public the emotion and the joy of doing mathematics, and what the profession is about, in a way which was almost unprecedented. His subsequent writings - the book on Fermat's Last Theorem, the book on codes (and the wonderful CD-rom he made with Nicholas Mee), his recent book on the maths of

But on this I disagree with Singh's comments (at least as they are reported). He says that in his view the best science outreach is "largely dirt cheap". Well, there is certainly a lot of excellent public engagement work on mathematics that is done on the cheap (such as the Royal Institution Masterclasses and the British Science Festival, the power of both of which I have seen at first hand), and a large number of people doing it more or less in their spare time for no reward other than the joy of communicating mathematics. But it shouldn't be like that, and it isn't only like that,

Singh is critical of the funding of a ballet about relativity. "People hate physics, they hate ballet, all you've done is allowed people to hate things more efficiently." Well, I believe that science and mathematics are so important that they should feature prominently in art. There should be ballets, novels, operas about mathematics. Happily, thanks to people like Marcus du Sautoy, Scarlett Thomas, Dorothy Ker and a great many others, there are. Not everyone hates ballet: a dance piece about science is potentially reaching a valuable audience. When Singh says of the "Faces of Mathematics" project of portraits of mathematicians that "I don't quite understand how this is really going to have an impact", he is surely not using his imagination. We need to show the world the diverse nature of mathematicians to encourage the aspirations of the potential mathematicians of the future,

Of course Singh is right to suggest that the value for money of any outreach project should be compared with the cost of a teacher. And we certainly need more teachers, and to pay them better. But imaginative (and expensive) public engagement projects are also important. They won't all succeed, Singh's TV programme about Wiles was of inestimable value in showing what mathematics is about. Dance and photography projects promoting public engagement with mathematics and science have similar potential. Even when they fail, they are not wasteful.

*Times Higher*has reported a conference talk by the writer Simon Singh about public engagement in science under the heading "Simon Singh criticises wasteful science outreach". This has even led to the ultimate distinction of a mention in Laurie Taylor's Poppletonian, the comic column which is consistently the highlight of my week.Now, not only did I not hear Singh's talk, so that I am relying on the THE report, but Singh is one of my heroes for his contributions to mathematics and its public reputation. For one thing, his documentary film on Andrew Wiles's proof of Fermat's last theorem conveyed to the general public the emotion and the joy of doing mathematics, and what the profession is about, in a way which was almost unprecedented. His subsequent writings - the book on Fermat's Last Theorem, the book on codes (and the wonderful CD-rom he made with Nicholas Mee), his recent book on the maths of

*The Simpsons*and others - have continued to inform the public and inspire young mathematicians. And his creation of the undergraduate ambassador scheme, which puts maths undergraduates into schools and colleges, is a hugely important contribution to maths education in the UK, directly benefiting school and university students and motivating many outstanding graduates to become maths teachers. So his views on public engagement certainly deserve to be taken seriously.But on this I disagree with Singh's comments (at least as they are reported). He says that in his view the best science outreach is "largely dirt cheap". Well, there is certainly a lot of excellent public engagement work on mathematics that is done on the cheap (such as the Royal Institution Masterclasses and the British Science Festival, the power of both of which I have seen at first hand), and a large number of people doing it more or less in their spare time for no reward other than the joy of communicating mathematics. But it shouldn't be like that, and it isn't only like that,

Singh is critical of the funding of a ballet about relativity. "People hate physics, they hate ballet, all you've done is allowed people to hate things more efficiently." Well, I believe that science and mathematics are so important that they should feature prominently in art. There should be ballets, novels, operas about mathematics. Happily, thanks to people like Marcus du Sautoy, Scarlett Thomas, Dorothy Ker and a great many others, there are. Not everyone hates ballet: a dance piece about science is potentially reaching a valuable audience. When Singh says of the "Faces of Mathematics" project of portraits of mathematicians that "I don't quite understand how this is really going to have an impact", he is surely not using his imagination. We need to show the world the diverse nature of mathematicians to encourage the aspirations of the potential mathematicians of the future,

Of course Singh is right to suggest that the value for money of any outreach project should be compared with the cost of a teacher. And we certainly need more teachers, and to pay them better. But imaginative (and expensive) public engagement projects are also important. They won't all succeed, Singh's TV programme about Wiles was of inestimable value in showing what mathematics is about. Dance and photography projects promoting public engagement with mathematics and science have similar potential. Even when they fail, they are not wasteful.

## Thursday, 8 October 2015

### The Mpemba Paradox

As a mathematician I love mathematical paradoxes because they are disturbing and thought-provoking. For example, Parrondo's Paradox tells us something counter-intuitive about probabilistic games; Simpson's paradox reminds us that we have to think carefully about statistics; and Curry's paradox is just mind-bending.

Paradoxes in the sciences are important because they make us think about our theories and where they don't quite match reality, driving new scientific ideas. My favourites include Olbers' Paradox (why is the sky dark at night?) and the EPR Paradox which shows us just how surprising the world is.

So I was delighted to come across, in an article by Oliver Southwick in the excellent magazine

Like all the best paradoxes, this is amusing but tells us something surprising about our world.

Paradoxes in the sciences are important because they make us think about our theories and where they don't quite match reality, driving new scientific ideas. My favourites include Olbers' Paradox (why is the sky dark at night?) and the EPR Paradox which shows us just how surprising the world is.

So I was delighted to come across, in an article by Oliver Southwick in the excellent magazine

*Chalkdust*, a paradox that was new to me, the Mpemba Paradox. "If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?" (The background story is wonderful - read the article!) Not only is there no agreement on the answer, but it gives insights into the mathematical equations involved, and mathematical modelling may help us understand the effect.Like all the best paradoxes, this is amusing but tells us something surprising about our world.

## Sunday, 24 May 2015

### Greenwich's mathematician in residence

As I write, the mathematician Katie Steckles is in residence at the Stephen Lawrence Gallery of the University of Greenwich, where she is demonstrating some fun maths to the public and getting people to take part in various mathematical activities. It's a fascinating show, covering many different kinds of mathematics, from tessellation to random walks, fractals to Benford's Law, and some graph theory.

For me, what has emerged is the visual connections. My photo above shows to the left images of blackboards relating to some of the maths research at Greenwich: even without a full explanation of the context the blackboards viewers are intrigued by the look of the mathematics. The photo below shows a wall of "doodles" by visitors, showing that all such doodles (formed by closed curves) can be coloured in only two colours so that regions with a common boundary are different colours.

Katie has curated a marvellous collection full of visual and mathematical interest. Hopefully the show will persuade people that there is much more to maths than sums!

## Sunday, 18 January 2015

### Logical paradoxes

I'm talking about logical paradoxes (

It's nice to prove things: suppose I want to prove something which is slightly doubtful (like "Arsenal will beat Manchester City this afternoon" - a very unlikely proposition). Here's a proof from Martin Gardner. Consider these two statements:

A: Both these statements are false

B: Arsenal will beat Manchester City this afternoon.

Clearly A cannot be true, since if it were it would contradict itself. So A is false, and if B were also false, then A would be true, So B must be true.

One proof isn't always enough, So here's another - this one is Curry's Paradox. Consider the statement:

If this statement is true, then Arsenal will beat Manchester City this afternoon.

Is this statement true? It's of the form "If A, then B", and we test that by seeing what happens when A is true. So assume that the first part of the statement above is true - which means that the whole statement is true, because that is what that clause asserts. And if that whole statement is true, and the first part is true, then the second part is true. So we have established the truth of the statement above, And if it is true, then Arsenal will win.

So I've proved in two different ways that Arsenal will win, despite almost all the pundits and 76% of the BBC poll thinking the opposite.

ADDED AT 6pm: Arsenal did win. Which proves the power of mathematical logic.

*This lecture will surprise you: when logic is illogical*) at Gresham College on Monday 19th January, which is tomorrow as I write this on Sunday afternoon. I've been fascinated by these for years, thanks to writers like Martin Gardner, Raymond Smullyan, Douglas Hofstadter.It's nice to prove things: suppose I want to prove something which is slightly doubtful (like "Arsenal will beat Manchester City this afternoon" - a very unlikely proposition). Here's a proof from Martin Gardner. Consider these two statements:

A: Both these statements are false

B: Arsenal will beat Manchester City this afternoon.

Clearly A cannot be true, since if it were it would contradict itself. So A is false, and if B were also false, then A would be true, So B must be true.

One proof isn't always enough, So here's another - this one is Curry's Paradox. Consider the statement:

If this statement is true, then Arsenal will beat Manchester City this afternoon.

Is this statement true? It's of the form "If A, then B", and we test that by seeing what happens when A is true. So assume that the first part of the statement above is true - which means that the whole statement is true, because that is what that clause asserts. And if that whole statement is true, and the first part is true, then the second part is true. So we have established the truth of the statement above, And if it is true, then Arsenal will win.

So I've proved in two different ways that Arsenal will win, despite almost all the pundits and 76% of the BBC poll thinking the opposite.

ADDED AT 6pm: Arsenal did win. Which proves the power of mathematical logic.

## Wednesday, 31 December 2014

### Why I can't have the perfect cup of coffee

Apologies to any regular readers for the paucity of recent posts, due to pressure of work. A suitable New Year Resolution would be to post more often next year.

This final post of 2014 comes from my presentation at MathsJam and was inspired by a puzzle in (I think)

Like most mathematicians coffee is important to me. I like my breakfast coffee to be strong and black, and (since one wants as much as possible of a good thing) I want my cup to be full.

So how much coffee does my cup hold? Well, it is basically cylindrical, so you might assume the volume of coffee it can contain is pi(r^2)h. But just as the surface of the ocean is not flat but part of the surface of a sphere, so is the surface of my coffee. And the volume extra bit I get over the top of the cup depends on the curvature of te sphere - the more curved, the more coffee I get.

What is the curvature? It depends how near to the centre of the earth we are. The closer to the centre of the earth, the sharper the curvature and the more coffee in the cup. (If my cup were infinitely far from the centre of the earth then the surface of my coffee would be flat.)

So the higher my cup is, the less it can contain. Which means that if I fill my coffee cup to the brim, as soon as I lift it to drink from it, it will spill.

Physicists may tell me that I am ignoring effects like surface tension or changes in density with altitude. But I'm a pure mathematician and such incidental matters don't interest me. What is annoying is that in an idealised mathematical universe I can't drink from a full cup of coffee without spilling some. Which is one more way in which the world doesn't work as it should.

This final post of 2014 comes from my presentation at MathsJam and was inspired by a puzzle in (I think)

*Which Way Did the Bicycle Go*by Konhauser, Velleman and Wagon.Like most mathematicians coffee is important to me. I like my breakfast coffee to be strong and black, and (since one wants as much as possible of a good thing) I want my cup to be full.

So how much coffee does my cup hold? Well, it is basically cylindrical, so you might assume the volume of coffee it can contain is pi(r^2)h. But just as the surface of the ocean is not flat but part of the surface of a sphere, so is the surface of my coffee. And the volume extra bit I get over the top of the cup depends on the curvature of te sphere - the more curved, the more coffee I get.

What is the curvature? It depends how near to the centre of the earth we are. The closer to the centre of the earth, the sharper the curvature and the more coffee in the cup. (If my cup were infinitely far from the centre of the earth then the surface of my coffee would be flat.)

So the higher my cup is, the less it can contain. Which means that if I fill my coffee cup to the brim, as soon as I lift it to drink from it, it will spill.

Physicists may tell me that I am ignoring effects like surface tension or changes in density with altitude. But I'm a pure mathematician and such incidental matters don't interest me. What is annoying is that in an idealised mathematical universe I can't drink from a full cup of coffee without spilling some. Which is one more way in which the world doesn't work as it should.

## Monday, 25 August 2014

### Some nice puzzles

Here are some rather nice wooden puzzles found by Noel-Ann on recent trips. I'm not commenting on solutions here!

Noel-Ann found this one in Texas. The object is to get the two marbles into the holes simultaneously, like this:

When one slides the lid open one finds that it is packed full with more wooden blocks. How can one add the red block and close the lid?

### Two Marbles

Noel-Ann found this one in Texas. The object is to get the two marbles into the holes simultaneously, like this:

### A Secret Box

The other puzzles were brought back from a market in the Dordogne. This, amazingly, is a secret box which can be opened by performing a series of operations.### A packing problem

This one starts off as a rather nice box with a red slab on top.When one slides the lid open one finds that it is packed full with more wooden blocks. How can one add the red block and close the lid?

### Another Packing Problem

This one has four pieces each composed of two overlapping slabs. Can one fit them into the tray?
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