...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Sunday, March 1, 2015

Richard Elwes.... Writing Maths For the Masses

 Math-Frolic Interview #28
"...it's time to slay some ghosts and kill some prejudices. Let's admit that any field of human endeavor worth investigating eventually gets to a level of technicality that becomes challenging, and it's fair to say that many people hit that ceiling earlier in mathematics before they do in other, perhaps more verbal, subjects. But before that level is reached, there is a whole accessible world of mathematics that can astonish with its diversity, captivate with its mystery, and enthrall with its beauty."

-- Richard Elwes from "Mathematics Without the Boring Bits" (American title)

Long time readers here know that one of my favorite math popularizers is Richard Elwes, a Brit who isn't well-enough known here in the States.
I've read four of Richard's books (below), and loved them all!. The first is an encyclopedic reference source of math terms/topics (I really know of no other book quite like it); the second is a super fleshing-out of 100+ key math concepts; the third is a wonderful introduction to a few dozen of math's most interesting (and non-boring), topics; and the fourth,"Chaotic Fishponds..." was among my favorite 2-3 math-reads of all last year:

1)  Mathematics 1001
2)  Math In 100 Key Breakthroughs
3)  Mathematics Without the Boring Bits
4)  Chaotic Fishponds and Mirror Universes
(note: the above are all U.S. titles, that may differ elsewhere) 
All Richard's books through Amazon are here:
1)  For any who aren't familiar with you, tell us a little about your current position, and about your background/path to becoming a professional mathematician?

My path started off along the usual route for mathematicians: maths, maths, more maths, followed by yet more maths. I enjoyed the subject at school, and so went on to study it at university (Oxford). I enjoyed that too, so decided to do a PhD (in Leeds), and then went on to hold a postdoctoral position (in Freiburg in Germany). Then my career turned several corners. Firstly, I met a wonderful woman, and didn’t really fancy living on the other side of the world from her, which basically meant turning down my next postdoc opportunity. After a bit of dithering, I left academia to train as a high-school teacher. But just as I got qualified, I was offered the opportunity to write a book (Maths 1001). So I did that, and then went on to work as a full-time writer for a few years. Now I’m back inside the academic fold, but I’m still trying to find my niche there – a position which lets me balance the three things I love: teaching, researching, and writing.

2)  One of your main research interests is "Model Theory." Can you explain a bit about what that is, and what 'real world' applications it has?

If we’re going to use the old-fashioned terminology of ‘pure’ and ‘applied’, then model theory is about as pure as it gets! For model theorists, “applications” usually mean applications in number theory, group theory or some other area of (apparently utterly pure) maths. And maybe years down the line, those have applications in theoretical physics or computer science… or maybe they don’t. Who knows!

Having said which, there are some topics within model theory which have turned out to have genuine applications to real world, for example to neural networks. All of which only goes to show how flawed and fuzzy the old pure/applied distinction is.

Anyway, what is it? It’s an area of logic. Basically it works like this: you think of a mathematical structure, let’s pick the system of real numbers, but it could be anything. Then you write down all the rules they obey, in some highly formal logical language. And then you ask “what other systems can I find that obeys these same rules?” and then lo and behold, you’re confronted with a whole load of interesting structures no-one had ever thought of before! These are sometimes called “non-standard models” (of the real numbers, or whatever it is). What is more, studying these new structures can tell you a surprising amount about the original structure – which is probably the one you really care about.

Initially, model theory took place at a really high level of abstraction: while number-theorists study the natural numbers, and geometers study the real numbers, model-theorists worked with very general “structures”, which just means an object obeying some system of logical axioms. That stuff still happens, but more recently people have been working on the model theory of… various things: the model theory of groups, the model theory of graphs, the model theory of derivative maps,….

3)  Your writing is some of the best, clearest math explication I've seen... Have you always had a knack for good clear writing, or is that something you developed over time... or is a skilled editor very much involved?

Thank you Shecky! I really appreciate the kind things you’ve said about my work over the years. I’ll answer the second question first: in my books, what you’re reading is pure Elwes. An editor might make suggestions about the broad structure, but they won’t usually tinker with the text, except to correct typos, etc.. (In some magazines it’s a totally different story – the first time around I was shocked that the editing process essentially amounts to wholesale rewriting. Of course you can get used to that too, so long as the editor’s someone you feel you can work with.)

As to whether I’ve always had a knack for it… maybe, but it’s something I didn’t really cotton on to until a few years ago. I’d done the occasional bit of (non-scientific) writing during my PhD days, just bits and pieces for my own amusement – but it was enough to convince myself that it was something I was reasonably good at, and enjoyed. Then in 2006 I entered the Plus Magazine New Writer’s competition – that was the first time I made a serious effort to write about maths for a general audience. I loved the process of taking something which looked horrifyingly complex, and drawing out the beautiful, simple ideas which underlie all those technicalities.

As an ambitious young researcher, you can miss the wood for the trees. When you’re presented with a theorem, your inclination is to dive straight into the proof, and start grappling with the toughest ideas in there. But when you’re addressing a general audience, you have to step back, pause, and ask “What is the point of this theorem? Where did it come from? Why should we care?” You’re forced to adopt a different perspective on the subject, and that’s refreshing. It’s also healthy to learn something about the social side of the subject – the people who made the breakthroughs. We usually think of mathematics as such a dry area, but there’s actually a lot of human interest in there. For instance, I’ve just read this great article by Alec Wilkinson about Tom Zhang’s proof of the bounded prime problem. When I first heard about that, like many people I was excited about the mathematics. But it turns out it’s also a wonderful human story: 

Anyway, I won that competition, which was a huge encouragement for me to write more. You can read my article here: http://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups

4)  Your books are often hard to find in the U.S. or appear here much later than in the UK. (and then often under a different title) -- I love your last book, "Chaotic Fishponds and Mirror Universes," but have never seen it in a bookstore here; instead I got a used copy online. Can you explain a bit of the whole publishing process involved, and why the books don't have better US distribution (also, why the occasional title changes)? And do you ever visit the states to lecture, study, or for book tours or conferences?

The answer to your first question is: not really, I’m afraid! The publishing side of things is mostly a mystery to me too. I am not involved in most of these decisions. All I can tell you is that publishers do a lot of wheeler-dealing with each other on different editions. So for instance when Maths 1001 was published in the UK by Quercus (who commissioned it), it also came out in the US via a different publisher, Firefly, under the title “Mathematics 1001”. I gather Chaotic Fishponds doesn’t currently have a proper US distribution. That’s disappointing. But I hope it may change – one thing I do know is that these things sometimes take time. (For instance, I’ve just heard that there is to be a Japanese edition of Maths 1001 -- five years after it first came out. I have Japanese family, so am extremely excited about this!)

The title changes are also largely beyond my control (in truth I wouldn’t have picked all of them myself!). One of my books “How to Build a Brain” has been sold under at least three titles, excluding foreign language editions.
[In the US the book is called, "Mathematics: Without the Boring Bits" and it is a GREAT introduction for general readers to a few dozen fun mathematical topics.]

On the US: I once went to a maths conference in Tucson, Arizona while I was a PhD student, otherwise I’ve only ever traveled to the US to see friends on holiday (and I’ve not done that for over 10 years). I’d love to come back if the opportunity arose, and if I could give a talk or two, or sign a few books, that would be fantastic. No immediate plans though.

5)  Do you fall clearly into one or the other of the math Platonist vs. non-Platonist categories? And why? (...or do you prefer a different philosophical handle?)

I would say that I was typical of most working mathematicians, firstly in that I don’t often think very deeply about this stuff, and secondly in that I am a Platonist for practical purposes, even if not necessarily by conviction. When I’m studying mathematical objects, it certainly feels as if you’re working with things which really exist, somehow or somewhere.

It’s funny how the type of maths you’re currently thinking about affects your thoughts though. I’ve written several times about the deep logical foundations of mathematics, and one’s really forced into grappling with these questions there. Can one really say that enormous mathematical objects called “large cardinals” truly exist? That seems a big call to make, because these things are completely unlike anything within our direct experience. All the same, certain large cardinals’ existence turns out to be logically equivalent to relatively simple statements about the natural numbers. So, if you held my feet to the fire, I’d be inclined to say “yes”… which I suppose must make me a Platonist of some sort.

6)  What are some of your own favorite math books to read for enjoyment?  Putting aside math, what other book-reads do you enjoy for pleasure or learning?

Two of my all-time favourite maths-related books are The Princeton Companion to Mathematics (edited by Tim Gowers) and Gödel, Escher, Bach by Douglas Hofstadter. Let me also throw in the works of the game-theorist Thomas Schelling (Micromotives and Macrobehaviour and Strategies of Conflict). I recently reread Longitude by Dava Sobel, which is a delightful (and short) book, which strongly makes the point that mathematical and scientific questions are not just interesting puzzles, but can be matters of life, death, and political power. (And that’s still true: see the recent fuss about the NSA and cryptography.

In broader science, and as he has been in many people’s thoughts recently, I would also like to mention the marvelous writer and neuroscientist and Oliver Sacks. I love several of his books, including the most famous: The Man Who Mistook His Wife For A Hat. I wrote a short review of another brilliant work, Musicophilia, on my website.

The human mind seems a perpetually good topic for books: I also enjoyed Thinking Fast and Slow by Daniel Kahneman, and the writing of evolutionary psychologist Stephen Pinker.

More generally I’m a fan of science fiction, particularly “hard science fiction” of which Greg Egan is the master. I also enjoy Philip K. Dick, Arthur C. Clarke, Iain M. Banks, etc.., and am rediscovering Ursula Le Guin – I adored the Earthsea cycle about 25 years ago, and have recently started exploring her other works.

In terms of guilty pleasures, I enjoy ghost stories and so-called weird fiction, MR James and Charles Dickens from UK and the US’s Edgar Allen Poe and HP Lovecraft are the classics of the genre. I’m currently reading the Castle of Otranto by Horace Walpole, which is supposedly the first gothic horror story. It’s not exactly a masterpiece, but it’s interesting to see where several of these gothic tropes – haunted castles and dark and stormy nights – started.

I’m also exploring Japanese literature. Shusako Endo is maybe my favourite Japanese novelist so far. The horror writer Edogawa Ranpo (quite a funny pseudonym – try saying it out loud) also ticks several boxes!

[Quite an array of book choices! Also, always interesting to me how many, diverse people include Hofstadter's "Gödel, Escher, Bach" on such a list -- his very first book (1979), written in his mid-30's, winning several awards, including the Pulitzer, and still classic to many of us.]

7)  Your "Mathematics 1001" volume is quite a tersely-rendered (and wonderful) encyclopedic overview of mathematics. How long did it take you to write it? And was it difficult to know when to stop or when you had included everything you wanted to cover?

Thank you. The writing took about 6 months – and a very intense 6 months it was too -- followed by a lengthy editing process. In terms of content, there was an immovable limit, since I knew I had to include exactly 1001 things, and I knew how many words I was permitted! Because it was intended to be encyclopaedic there was an awful lot of stuff which I simply had to put in: you can’t imagine e.g. Fermat’s last theorem (FLT) being absent. But it was a good opportunity to fill in the gaps between the well-known highlights, and include material which doesn’t get aired so often. So, for instance, I was able to set FLT within a much broader context. Everyone knows that FLT is a long-standing and difficult puzzle, which a brilliant mathematician (Andrew Wiles) took a long time to solve. But it is just one of a whole class of problems called Diophantine equations, which have connections no numerous other topics, from logic (via Hilbert’s 10th problem, which takes us to Turing machines) to algebraic geometry (via Diophantine geometry, which we can trace back to the study of Pythagorean triples) to complex analysis (via Modular forms, and elliptic curves), and so on. So it’s not just a free-standing puzzle, but one feature within a much broader landscape. These are the sort of connections I wanted to draw out. It was my first book, and probably still the one I’m proudest of.

8)  Are you currently working on a new book, and if so can you tell us about it?

I can tell you… that I am not currently working on a book. Currently my focus is on teaching and research – traditional academic stuff. But I promise I will write more, in future! I think I’d like to write something with a narrower focus – all of my books so far have been quite wide ranging with one chapter devoted to one topic and the next about something completely different. I’ve got a few ideas, but no firm plans yet.

9)  Lastly, since I really want my readers to be familiar with you, I'll just offer an open-ended chance to say anything additional you wish about yourself or your books that you would want them (as math enthusiasts) to know:

Since Chaotic Fishponds is not easily available in the US, perhaps I can say something about it here, and if there’s enough interest, maybe it might eventually pick up a proper distribution. It’s about how maths is used “in the real world”, meaning, for instance, within the modern technology of mobile phones, internet-search, space exploration, weather forecasting, robotics, computer graphics,… The impetus behind the book was that everyone knows that maths is important in all these areas, but if you then ask them how it is used, or what sort of maths is involved, most people have no idea. So, my book is an attempt to answer that question.


Thanks for the in-depth answers Richard, and I do hope more folks on this side of the pond become familiar with your work. I think Richard's books can interest math enthusiasts at various levels, and are especially good for fostering the interest of young math up-and-comers.
(Seriously, some American publisher/distributor should be all over these books; moreso than seems to be the case!)

Richard's main website is: https://www.richardelwes.co.uk/blog 
And he can also be found at:
and recently even on Youtube:

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