The comment thread for my post about good writing has turned into a fascinatingly well-focused discussion on writing about math. A mathematician arrived, rending his garments in despair, and now others–both writers and readers–are responding. I’ve always considered math the toughest subject a science writer can tackle, so I find the conversation especially interesting. Check it out.

I had a quick browse of the comments. It seems to me that the topic of semi-simplicial objects is a secondary level development in algebraization of topology. If I were to explain that sort of program, I would first start with examples of topological spaces like cylinder, Mobious Band, how both are formed from two squares (modulo bending and stetching) by different identifications and then how spaces like projective spaces which cannot be visualized can be seen by glueing a cell to a Mobius Band. Then I would go on with more examples of such spaces obtained by glueing or identification (identification spaces) and how many of them occur natuarally; one of the first examples of Poincare is the product of three dimensional projective space with three dimensional euclidean space in his study of celestial mechanices. After a while such spaces are difficult to distinguish visually and one of the first mechanisms found by Poincare was a set of algebraic invariants called Betti numbers which are sort of the number of various dimensional holes in the topological space. The technique is to decompose the space in to familiar ones called polyhedra and finally simplexes which are higher dimensional versions of triangles. These are prefered to squares and general polyhedra since it is easier write doen boundaries. The more formal association of associating algebraic invariants to topological spaces took several decades with one of the crucial suggestions coming from Emmy Noether ; that the Betti numbers are the invariants some groups associated with the spaces. The result is associating groups with spaces so that topological problems associated with spaces can be converted to problem in the algebraic objects groups. The advantage is that equivalence problems in groups can be done mechanically without the pitfalls of trying to visulaize complicated spaces. However it is not a complete equivalence problem which is in some sense better as one has hopes of getting in to more tractable problems. After this one can go on to explain how semisimplicial objects come in as a part of this algebrization program.
I think that the whole program is heavily influenced by Galois Theory about solving equations by radicals. Galois showed how this can be seen as a problem in extensions of fields, a topic which he created. He showed how the problems in these extensions of fields ( which are infinite) can be related to finite groups and whether a particular, called the Galois group of the equation has a simple structure which he called sovable. In the process, he created two topics, field theory and group theory and how in this case, the problems in fields are completely equivalent to related problems in finite groups ( in the case of topology from spaces to groups and other algebraic objects.)
I have not been keeping track of popular or technical writing in mathematics for a while, but off and on I have seen some fine writing by Julie Rehmeyer, Erica Klarreich in Science News while browing about other topics.

Dear gaddeswarup, the area that I work in uses the combinatorial/categorical theory of simplicial sets as a substrate for building up other structures (in particular, weak higher categories). The extent to which this is involved in the classical applications of simplicial sets is somewhat minimal, but for instance, given a simplicial set that is _not_ a Kan complex, its “fundamental groupoid” (the left adjoint of the nerve functor) is not necessarily a groupoid, but a category. This gives us some of the motivation to rebuild category theory out of the theory of simplicial sets.

I write about math for a living, so naturally this discussion is fascinating to me.

The first question you have to answer, always, is “Why should my reader care?” If you can’t answer that, then stop. Whatever else you say, it’s going to be a failure.

It’s true that answering that question with current pure math research can very challenging, because it’s so very far from removed from ordinary experience. To some extent, I solve that problem by picking my subjects carefully. For example, I’ve written about a computer that plays poker using game theory (http://bit.ly/f9ozKs); how mathematicians restored the only known live recording of Woody Guthrie (http://bit.ly/grammy_in_math); and how baseball players might be able to round the bases faster by following a bizarre path mathematically shown to be faster (http://bit.ly/i4VTQD ). Those are all inviting topics where the math is connected to everyday experience.

But recently, I was asked to write summaries of the work of the Fields Medalists. The Fields Medals are one of the highest honors in math and are often called the mathematicians’ equivalent of the Nobel Prize. I accepted the assignment with some trepidation — after all, this was extremely abstract stuff, chosen not for its potential interest to laypeople but for its inherent mathematical significance. Who the winners were was still a secret when they hired me, so I didn’t even know what the topics were until I said yes. Would I be able to find reasons for my readers to care about these topics?

I got lucky in that some of the topics had applications, which made things much easier. But not all of them — take, for example, the fundamental lemma. I’ll be honest with you: I don’t know what the fundamental lemma is. Heck, very few mathematicians know what the fundamental lemma is! It would take me years of hard work to really know that, and when I was done, I’d have to ask my readers to spend just as long in order to share what I’d learned. Hopeless!

And here’s the truth: Almost no one cares what the fundamental lemma is. Really. Even mathematicians (except for the very few working in that immediate field). It’s a technical tool, a very powerful theoretical one that was extremely hard to build. What the tool does exactly, how it’s built — doesn’t matter.

But of course, mathematicians *do* care about the result even if they don’t know quite what it is, and here’s why: The lack of a proof has been a huge stumbling block to a mind-blowing theory that aims to unify mathematical fields that appear to be only distantly related, and the development of this theory is what’s led to huge breakthroughs like the proof of Fermat’s Last Theorem. And the importance of the fundamental lemma is particularly amazing because the result itself is so boring and seems so much like a small technical problem that it got named a “lemma” (what mathematicians call a small, boring, technical result). Then, when mathematicians banged their head on this thing for decades and couldn’t prove it, the title got elevated to a kind of oxymoron: The Fundamental Lemma, or, translated, “The Really Important Little Boring Thing.” The lack of a proof was such a logjam that many mathematicians responded by simply assuming the thing was true and working out the consequences — building a huge edifice of theory that would come crashing down if it turned out to be false.

So my readers care (I hope) about the fundamental lemma because its amazing that something that appeared easy and boring could turn out to be so hard and so vital. They care because they can get a glimpse of the beauty of the field that it’s a part of. They care because they can enter the emotional world of the mathematicians whose life work had teetered on top of this unproven theorem.

My write-up of this is here, if anyone would like the bigger story: http://bit.ly/i5uDo7

In my student days I used to read popular science writing by scientists like Jame Jeans, A.S. Eddington, George Gamow and E.T. Bell. It is a puzzle to me how non-scientists can pick up interesting topics, somehow get the gist of the arguments and convey in an interesting way to lay men. Thanks to Julie Rehmeyer for explaing it to some extent. There is also a slightly technical piece by her which may be of interest to bloggers on evolution: http://www.sciencenews.org/view/generic/id/8501/title/Math_Trek__A_Grove_of_Evolutionary_Trees

Dear Julie, I think that giving the drama of the discovery without discussing the meaning is seen by mathematicians as fluff journalism. The fundamental lemma has a moral meaning, even if you don’t want to give the exact technical statement. For instance, take a look at this post from Bill Lawvere on the categories mailing list: http://rfcwalters.blogspot.com/2010/10/old-post-why-are-we-concerned-fw.html .

Instead of filling up pages with “mathematics news!”, why not actually include some mathematical content? If you’re just publishing about the drama, you might as well publish about fake breakthroughs. Mathematicians don’t care if the news about the mathematical community is celebrated in newspapers. That time rated the fundamental lemma among the top scientific discoveries of the year cheapens the accomplishment. Mathematicians don’t want recognition from the general public. They want people to understand what they do.

I don’t know if you’ve ever read Hardy’s _A Mathematician’s Apology_, but we would love to see expository articles giving fun and interesting little proofs of things like the infinitude of primes. I’ve shown many people that proof, and they always get a huge kick out of it. “Is that what math is about?” they ask with interest.

Write an article about the two-dimensional crystallographic restriction theorem, which states that wallpaper patterns can only have certain symmetries of 1,2,3,4, and 6. The formal proof may be hard, but the general idea of how it works isn’t. Draw a lattice and show that a fivefold symmetry does not fit on the lattice.

You’d be doing the mathematical community and the general public a huge favor. If they want to read about drama they can read the sports section. Science should teach and enrich, not convince people how impossible it is to think scientifically.

Sorry if I came off a bit harsh, it wasn’t directed specifically at you.

Different articles call for different treatments. If you take a look at the other Fields Medals write-ups, you’ll see that they have significantly more mathematical content, because I was able to find ways of connecting the mathematical content to things people understand and care about. My point is that one way or another, you’ve always got to give people a reason to care.

And drama is certainly a part of science and math! God forbid that we ban it to the sports page.

“Fun and interesting little proofs of things like the infinitude of primes” are only one small bit of what mathematics is really about. I do that from time to time, when there’s something new along those lines. But if that’s all I did, I’d be misrepresenting what math is really about. It’s also about art and stopping genocides and saving lives and how many ways you can tie your shoelaces. It’s a huge, fabulous, rich thing, and the only way non-mathematicians have a chance of getting a glimpse into it is if those of us who know something about it learn to present it invitingly, to stop using jargon, and to connect it to the lives of non-mathematicians. I think there’s always a way of doing it, though the connection may be more or less mathematically precise depending on the subject.

When you teach people about math in ways that are relevant to their lives, you make it seem passé and pedestrian. Have you ever read this: http://www.maa.org/devlin/LockhartsLament.pdf ?

Or how about Hardy’s apology? I refer you to a very nice paragraph from that book (reproduced below), which discusses the value of the theorems of Euclid and Pythagoras, on the infinitude of primes, and on the irrationality of 2^{1/2}:
————————————————–
There is no doubt at all, then, of the ‘seriousness’ of either theorem. It is therefore the better worth remarking that neither theorem has the slightest ‘practical’ importance. In practical
application we are concerned only with comparatively small numbers; only stellar astronomy and atomic physics deal with ‘large’ numbers, and they have very little more practical importance, as yet, than the most abstract pure mathematics. I do
not know what is the highest degree of accuracy ever useful to an engineer-we shall be very generous if we say ten significant figures. Then

3.14159265

(the value of pi to eight places of decimals) is the ratio

314159265/1000000000

of two numbers of ten digits. The number of primes less than 1,000,000,000 is 50,847,478: that is enough for an engineer, and he can be perfectly happy without the rest. So much for Euclid’s theorem; and, as regards Pythagoras’s, it is obvious that irration-
als are uninteresting to an engineer, since he is concerned only with approximations, and all approximations are rational.
—————————————————————–

The fact is, ordinary people can do without mathematics and mathematical proof just fine, but throughout the ages, we have seen that people from every culture and every continent have delighted in the puzzles of mathematics proper. They were not inspired by problems of engineering but simply because math by itself is enjoyable.

You are doing your readers a disservice by making your mathematics articles about saving lives or stopping genocides or how many ways you can tie your shoelaces.

Here are some more passages from Hardy (a book that I very much suggest that you read (at least §20-30).
—————————————————
It will probably be plain by now to what conclusions I am coming; so I will state them at once dogmatically and then elaborate them a little. It is undeniable that a good deal of
elementary mathematics – and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus – has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have the least aesthetic value. The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his
work.
———————————
I can remember Eddington giving a happy example of the unattractiveness of ‘useful’ science. The British Association held a meeting in Leeds, and it was thought that the members might like to hear something of the applications of science to the ‘heavy woollen’ industry. But the lectures and demonstrations arranged for this purpose were rather a fiasco. It appeared that the members (whether citizens of Leeds or not) wanted to be
entertained, and the ‘heavy wool’ is not at all an entertaining subject. So the attendance at these lectures was very disappointing; but those who lectured on the excavations at Knossos, or on relativity, or on the theory or prime numbers, were delighted by the audiences that they drew.
———————
What parts of mathematics are useful?

First, the bulk of school mathematics, arithmetic, elementary algebra, elementary Euclidean geometry, elementary differential and integral calculus. We must except a certain amount of what is taught to ‘specialists’, such as projective geometry. In applied
mathematics, the elements of mechanics (electricity, as taught in schools, must be classified as physics).

Next, a fair proportion of university mathematics is also useful, that part of it which is really a development of school mathematics with a more finished technique, and a certain amount of the more physical subjects such as electricity and hydromechanics. We must also remember that a reserve of knowledge is always an advantage, and that the most practical of mathematicians may be seriously handicapped if his knowledge is the bare minimum which is essential to him; and for this reason we must add a little under every heading. But our general conclusion must be that such mathematics is useful as is wanted by a superior engineer or a moderate physicist; and that is roughly the same thing as to say, such mathematics as has no particular aesthetic merit. Euclidean geometry, for example, is useful in so far as it is dull-we do not want the axiomatics of parallels, or the theory of proportion, or the construction of the regular pentagon.

One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot. But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. “Imaginary” universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
—————————————–

I hope you will at least consider what I’m trying to say.

I’ve read “A Mathematician’s Apology” and both enjoyed it and found it short-sighted.

The article on the number of ways to tie your shoelaces, by the way, is about cutting-edge, pure math. I love writing about pure math — but I still believe that you have to find a way to connect it to things readers start off knowing they care about. Otherwise, they’re not going to keep reading.

I had a quick browse of the comments. It seems to me that the topic of semi-simplicial objects is a secondary level development in algebraization of topology. If I were to explain that sort of program, I would first start with examples of topological spaces like cylinder, Mobious Band, how both are formed from two squares (modulo bending and stetching) by different identifications and then how spaces like projective spaces which cannot be visualized can be seen by glueing a cell to a Mobius Band. Then I would go on with more examples of such spaces obtained by glueing or identification (identification spaces) and how many of them occur natuarally; one of the first examples of Poincare is the product of three dimensional projective space with three dimensional euclidean space in his study of celestial mechanices. After a while such spaces are difficult to distinguish visually and one of the first mechanisms found by Poincare was a set of algebraic invariants called Betti numbers which are sort of the number of various dimensional holes in the topological space. The technique is to decompose the space in to familiar ones called polyhedra and finally simplexes which are higher dimensional versions of triangles. These are prefered to squares and general polyhedra since it is easier write doen boundaries. The more formal association of associating algebraic invariants to topological spaces took several decades with one of the crucial suggestions coming from Emmy Noether ; that the Betti numbers are the invariants some groups associated with the spaces. The result is associating groups with spaces so that topological problems associated with spaces can be converted to problem in the algebraic objects groups. The advantage is that equivalence problems in groups can be done mechanically without the pitfalls of trying to visulaize complicated spaces. However it is not a complete equivalence problem which is in some sense better as one has hopes of getting in to more tractable problems. After this one can go on to explain how semisimplicial objects come in as a part of this algebrization program.

I think that the whole program is heavily influenced by Galois Theory about solving equations by radicals. Galois showed how this can be seen as a problem in extensions of fields, a topic which he created. He showed how the problems in these extensions of fields ( which are infinite) can be related to finite groups and whether a particular, called the Galois group of the equation has a simple structure which he called sovable. In the process, he created two topics, field theory and group theory and how in this case, the problems in fields are completely equivalent to related problems in finite groups ( in the case of topology from spaces to groups and other algebraic objects.)

I have not been keeping track of popular or technical writing in mathematics for a while, but off and on I have seen some fine writing by Julie Rehmeyer, Erica Klarreich in Science News while browing about other topics.

Dear gaddeswarup, the area that I work in uses the combinatorial/categorical theory of simplicial sets as a substrate for building up other structures (in particular, weak higher categories). The extent to which this is involved in the classical applications of simplicial sets is somewhat minimal, but for instance, given a simplicial set that is _not_ a Kan complex, its “fundamental groupoid” (the left adjoint of the nerve functor) is not necessarily a groupoid, but a category. This gives us some of the motivation to rebuild category theory out of the theory of simplicial sets.

I write about math for a living, so naturally this discussion is fascinating to me.

The first question you have to answer, always, is “Why should my reader care?” If you can’t answer that, then stop. Whatever else you say, it’s going to be a failure.

It’s true that answering that question with current pure math research can very challenging, because it’s so very far from removed from ordinary experience. To some extent, I solve that problem by picking my subjects carefully. For example, I’ve written about a computer that plays poker using game theory (http://bit.ly/f9ozKs); how mathematicians restored the only known live recording of Woody Guthrie (http://bit.ly/grammy_in_math); and how baseball players might be able to round the bases faster by following a bizarre path mathematically shown to be faster (http://bit.ly/i4VTQD ). Those are all inviting topics where the math is connected to everyday experience.

But recently, I was asked to write summaries of the work of the Fields Medalists. The Fields Medals are one of the highest honors in math and are often called the mathematicians’ equivalent of the Nobel Prize. I accepted the assignment with some trepidation — after all, this was extremely abstract stuff, chosen not for its potential interest to laypeople but for its inherent mathematical significance. Who the winners were was still a secret when they hired me, so I didn’t even know what the topics were until I said yes. Would I be able to find reasons for my readers to care about these topics?

I got lucky in that some of the topics had applications, which made things much easier. But not all of them — take, for example, the fundamental lemma. I’ll be honest with you: I don’t know what the fundamental lemma is. Heck, very few mathematicians know what the fundamental lemma is! It would take me years of hard work to really know that, and when I was done, I’d have to ask my readers to spend just as long in order to share what I’d learned. Hopeless!

And here’s the truth: Almost no one cares what the fundamental lemma is. Really. Even mathematicians (except for the very few working in that immediate field). It’s a technical tool, a very powerful theoretical one that was extremely hard to build. What the tool does exactly, how it’s built — doesn’t matter.

But of course, mathematicians *do* care about the result even if they don’t know quite what it is, and here’s why: The lack of a proof has been a huge stumbling block to a mind-blowing theory that aims to unify mathematical fields that appear to be only distantly related, and the development of this theory is what’s led to huge breakthroughs like the proof of Fermat’s Last Theorem. And the importance of the fundamental lemma is particularly amazing because the result itself is so boring and seems so much like a small technical problem that it got named a “lemma” (what mathematicians call a small, boring, technical result). Then, when mathematicians banged their head on this thing for decades and couldn’t prove it, the title got elevated to a kind of oxymoron: The Fundamental Lemma, or, translated, “The Really Important Little Boring Thing.” The lack of a proof was such a logjam that many mathematicians responded by simply assuming the thing was true and working out the consequences — building a huge edifice of theory that would come crashing down if it turned out to be false.

So my readers care (I hope) about the fundamental lemma because its amazing that something that appeared easy and boring could turn out to be so hard and so vital. They care because they can get a glimpse of the beauty of the field that it’s a part of. They care because they can enter the emotional world of the mathematicians whose life work had teetered on top of this unproven theorem.

My write-up of this is here, if anyone would like the bigger story: http://bit.ly/i5uDo7

And, should you really want your math fix, here are my write-ups of the other Fields Medalists: bit.ly/lindenstrauss_fields, bit.ly/smirnov_fields, and bit.ly/villani_fields.

remarkably well written

In my student days I used to read popular science writing by scientists like Jame Jeans, A.S. Eddington, George Gamow and E.T. Bell. It is a puzzle to me how non-scientists can pick up interesting topics, somehow get the gist of the arguments and convey in an interesting way to lay men. Thanks to Julie Rehmeyer for explaing it to some extent. There is also a slightly technical piece by her which may be of interest to bloggers on evolution:

http://www.sciencenews.org/view/generic/id/8501/title/Math_Trek__A_Grove_of_Evolutionary_Trees

Dear Julie, I think that giving the drama of the discovery without discussing the meaning is seen by mathematicians as fluff journalism. The fundamental lemma has a moral meaning, even if you don’t want to give the exact technical statement. For instance, take a look at this post from Bill Lawvere on the categories mailing list: http://rfcwalters.blogspot.com/2010/10/old-post-why-are-we-concerned-fw.html .

Instead of filling up pages with “mathematics news!”, why not actually include some mathematical content? If you’re just publishing about the drama, you might as well publish about fake breakthroughs. Mathematicians don’t care if the news about the mathematical community is celebrated in newspapers. That time rated the fundamental lemma among the top scientific discoveries of the year cheapens the accomplishment. Mathematicians don’t want recognition from the general public. They want people to understand what they do.

I don’t know if you’ve ever read Hardy’s _A Mathematician’s Apology_, but we would love to see expository articles giving fun and interesting little proofs of things like the infinitude of primes. I’ve shown many people that proof, and they always get a huge kick out of it. “Is that what math is about?” they ask with interest.

Write an article about the two-dimensional crystallographic restriction theorem, which states that wallpaper patterns can only have certain symmetries of 1,2,3,4, and 6. The formal proof may be hard, but the general idea of how it works isn’t. Draw a lattice and show that a fivefold symmetry does not fit on the lattice.

You’d be doing the mathematical community and the general public a huge favor. If they want to read about drama they can read the sports section. Science should teach and enrich, not convince people how impossible it is to think scientifically.

Sorry if I came off a bit harsh, it wasn’t directed specifically at you.

Cheers,

fpqc.

Different articles call for different treatments. If you take a look at the other Fields Medals write-ups, you’ll see that they have significantly more mathematical content, because I was able to find ways of connecting the mathematical content to things people understand and care about. My point is that one way or another, you’ve always got to give people a reason to care.

And drama is certainly a part of science and math! God forbid that we ban it to the sports page.

“Fun and interesting little proofs of things like the infinitude of primes” are only one small bit of what mathematics is really about. I do that from time to time, when there’s something new along those lines. But if that’s all I did, I’d be misrepresenting what math is really about. It’s also about art and stopping genocides and saving lives and how many ways you can tie your shoelaces. It’s a huge, fabulous, rich thing, and the only way non-mathematicians have a chance of getting a glimpse into it is if those of us who know something about it learn to present it invitingly, to stop using jargon, and to connect it to the lives of non-mathematicians. I think there’s always a way of doing it, though the connection may be more or less mathematically precise depending on the subject.

Dear Julie,

When you teach people about math in ways that are relevant to their lives, you make it seem passé and pedestrian. Have you ever read this: http://www.maa.org/devlin/LockhartsLament.pdf ?

Or how about Hardy’s apology? I refer you to a very nice paragraph from that book (reproduced below), which discusses the value of the theorems of Euclid and Pythagoras, on the infinitude of primes, and on the irrationality of 2^{1/2}:

————————————————–

There is no doubt at all, then, of the ‘seriousness’ of either theorem. It is therefore the better worth remarking that neither theorem has the slightest ‘practical’ importance. In practical

application we are concerned only with comparatively small numbers; only stellar astronomy and atomic physics deal with ‘large’ numbers, and they have very little more practical importance, as yet, than the most abstract pure mathematics. I do

not know what is the highest degree of accuracy ever useful to an engineer-we shall be very generous if we say ten significant figures. Then

3.14159265

(the value of pi to eight places of decimals) is the ratio

314159265/1000000000

of two numbers of ten digits. The number of primes less than 1,000,000,000 is 50,847,478: that is enough for an engineer, and he can be perfectly happy without the rest. So much for Euclid’s theorem; and, as regards Pythagoras’s, it is obvious that irration-

als are uninteresting to an engineer, since he is concerned only with approximations, and all approximations are rational.

—————————————————————–

The fact is, ordinary people can do without mathematics and mathematical proof just fine, but throughout the ages, we have seen that people from every culture and every continent have delighted in the puzzles of mathematics proper. They were not inspired by problems of engineering but simply because math by itself is enjoyable.

You are doing your readers a disservice by making your mathematics articles about saving lives or stopping genocides or how many ways you can tie your shoelaces.

Here are some more passages from Hardy (a book that I very much suggest that you read (at least §20-30).

—————————————————

It will probably be plain by now to what conclusions I am coming; so I will state them at once dogmatically and then elaborate them a little. It is undeniable that a good deal of

elementary mathematics – and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus – has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have the least aesthetic value. The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his

work.

———————————

I can remember Eddington giving a happy example of the unattractiveness of ‘useful’ science. The British Association held a meeting in Leeds, and it was thought that the members might like to hear something of the applications of science to the ‘heavy woollen’ industry. But the lectures and demonstrations arranged for this purpose were rather a fiasco. It appeared that the members (whether citizens of Leeds or not) wanted to be

entertained, and the ‘heavy wool’ is not at all an entertaining subject. So the attendance at these lectures was very disappointing; but those who lectured on the excavations at Knossos, or on relativity, or on the theory or prime numbers, were delighted by the audiences that they drew.

———————

What parts of mathematics are useful?

First, the bulk of school mathematics, arithmetic, elementary algebra, elementary Euclidean geometry, elementary differential and integral calculus. We must except a certain amount of what is taught to ‘specialists’, such as projective geometry. In applied

mathematics, the elements of mechanics (electricity, as taught in schools, must be classified as physics).

Next, a fair proportion of university mathematics is also useful, that part of it which is really a development of school mathematics with a more finished technique, and a certain amount of the more physical subjects such as electricity and hydromechanics. We must also remember that a reserve of knowledge is always an advantage, and that the most practical of mathematicians may be seriously handicapped if his knowledge is the bare minimum which is essential to him; and for this reason we must add a little under every heading. But our general conclusion must be that such mathematics is useful as is wanted by a superior engineer or a moderate physicist; and that is roughly the same thing as to say, such mathematics as has no particular aesthetic merit. Euclidean geometry, for example, is useful in so far as it is dull-we do not want the axiomatics of parallels, or the theory of proportion, or the construction of the regular pentagon.

One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot. But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. “Imaginary” universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

—————————————–

I hope you will at least consider what I’m trying to say.

Cheers,

fpqc

I’ve read “A Mathematician’s Apology” and both enjoyed it and found it short-sighted.

The article on the number of ways to tie your shoelaces, by the way, is about cutting-edge, pure math. I love writing about pure math — but I still believe that you have to find a way to connect it to things readers start off knowing they care about. Otherwise, they’re not going to keep reading.