How much mathematics is there?

And further, by these, my son, be admonished: of making many books there is no end; and much study is a weariness of the flesh. (Ecclesiastes 12:12; King James Bible.)

I’ve memorised all the digits of pi. Now I just have to remember what order they come in. (Trad.)

If you’ve done or are doing a maths degree, at some point you may have had to disillusion a doting parent or an over-optimistic younger relative who’s asked you something like “so, do you know everything about maths yet?” Clearly the answer to that question must be “no”: every mathematician in the world can point to somebody else who knows something about maths that he or she doesn’t. It suggests, though, a different question: “what would it mean to know all of mathematics?” — or, if you prefer, “how much mathematics is there to know?” In one sense this is also a silly question, for reasons I’ll discuss below, but trying to find a way to answer it is surprisingly interesting.

Is mathematics infinite?

The easy and boring answer is simple: there’s an infinite amount of maths. The reasoning goes something like this. Arithmetic is part of maths, so at the very least every simple equation we can write down is part of maths. To take just a few of these: 3\times 1 = 3; 4\times 2 = 8; 5\times 3 = 15… Since there is clearly an infinite number of such equations, maths is infinite. QED.

If you’re a mathematician, this answer isn’t very satisfying, because this infinite family of equations can of course be boiled down to a single expression: (n+1)(n-1) = n^2-1\forall \ n \in \mathbb{N}. By moving up one level of abstraction, we’ve compressed an “infinite” quantity of information into a single expression. We can see, then, that any answer we get to the question “how much maths is there?” will depend on the level of abstraction we’re allowed — for each increase in abstraction we will reduce the number of simple “facts” we have to list, but at the expense of hiding away a lot of complexity in our increasingly abstract notation.

One approach to this kind of compression is the axiomatic approach, which aims to express large swathes of mathematics as the logical consequences of the smallest possible set of definitions and assumptions. For instance, arithmetic and number theory can — in a sense — be compressed down to a set of axioms, such as Peano’s axioms, that define what’s meant by a natural number; still wider tracts of mathematics can be derived from, say, the ZFC axioms of set theory. As it happens, there are some problems with this compression process — but even without worrying about these, it’s clear that to say “maths is…” and then list the ZFC axioms wouldn’t be terribly satisfying, because all the really interesting results (like Fermat’s Last Theorem, or 2 + 2 = 4) can only be found by doing a lot of work to “unpack” these axioms.

It seems that we’ll have to compromise, and to try to measure how much maths there is in terms of the mathematical results that people think are worth working out and recording in detail. (The philosophically-minded may note that I’m not taking sides on whether mathematics is “discovered” or “created”: you can replace “how much maths is there?” with “how much maths is known?” if you prefer!) These published results might range from, say, the elementary derivatives you’ll find on a first-year calculus formula sheet, up to the highly abstruse theorems published in mathematical research journals. In these terms, how much maths is out there?

I can think of two approaches to the question, which I’ll call the cumulative and the geographical approaches.

The cumulative approach

The idea of the cumulative approach is simply to count the mathematical theorems (or papers, or books) that have been published.

A famous estimate was made by the Polish mathematician and physicist Stanisław Ulam in the early 1970s, and is recorded in his book Adventures of a Mathematician. Based on a rough count of mathematical journals, the typical number of papers per journal and the typical number of theorems per paper, Ulam estimated that around 100 000 to 200 000 mathematical theorems were being published every year. About a decade later, Philip Davis and Reuben Hersh, in their book The Mathematical Experience, estimated that the accumulated body of mathematical knowledge represented about 100 000 volumes of books: for comparison, they estimated that a specialist researcher might know the equivalent of between 60 and 70 volumes of mathematics…

These numbers will, of course, have increased since the early 1980s, both as mathematical knowledge has accumulated further and as the world population of mathematicians increases. A quick search for “Mathematics” in the catalogue of the British Library tells me that it holds 51 106 books, 5839 theses and 3081 journals on the subject, which is not too far from Davis and Hersh’s estimate of 30 years ago. A vast amount of maths, however, never finds its way into books (or at least hasn’t yet done so), and to quantify it we must look at the output of the research journals.

One measure is to count the preprints (draft papers) uploaded to the arXiv server, which many researchers use to disseminate their work before it’s officially published. Under the category “Mathematics” there were 24 176 such papers in 2012; 21 287 in 2011; 18 765 in 2010; and 16 319 in 2009. This suggests a rapid rate of increase, but it may simply reflect the increasing popularity of the arXiv. A slightly more comprehensive approach is to look at databases. The ISI Web of Knowledge database contains only articles published in peer-reviewed journals, and it records 56 433 mathematics papers in 2012; 56 526 in 2011; 58 567 in 2010; and 62 014 in 2009. (No, I’ve no idea why the number seems to be going down!) Alternatively, the American Mathematical Society’s MathSciNet database, which has a slightly wider coverage and also includes books and conference proceedings, lists 81 142 items in 2012; 99 588 in 2011; 98 281 in 2010; and 98 945 in 2009. It seems that somewhere between 50 000 and 100 000 maths papers are being published every year — and according to MathSciNet, that’s about three times as many as in the early 1970s when Ulam made his estimate. Assuming that the number of theorems per paper hasn’t changed significantly, we could perhaps estimate that between 300 000 and 600 000 theorems are now published every year… Oh, and as a final set of staggering numbers, Web of Knowledge lists a total of 1 342 406 mathematics papers since 1900, while MathSciNet lists 2 888 464 entries for the same period, and the British Library holds a total of 1 333 240 mathematics articles.

So who on earth is producing all this material? An output of 100 research papers would be a decent haul for a mathematican’s working lifetime, and many people publish only one or two before leaving academia, so we might expect there to be tens or even hundreds of thousands of mathematicians out there. There was an interesting discussion of this recently on MathOverflow, with estimated numbers worldwide ranging from 80 000 to 300–400 000. (That’s in contrast to the US Bureau of Labor figure of 3100 mathematicians employed across the whole USA… but they seem to be taking quite a restricted definition of “mathematician”.) Lonely mathematicians can take some encouragement from this — there are a lot of us around!

The geographical approach

The other sense in which we might ask “how much mathematics is there to know?” is “how far does mathematics extend?” In particular, how far does it extend beyond what we’ve learned by the time we finish our first (or second, or third…) degree in mathematics? Are there whole regions of mathematics nobody’s told us about?

So, what might we mean by a “region of mathematics”? Fortunately, some hard thought has already gone into answering this question. The definitive attempt to categorise mathematics is probably the AMS’s Mathematics Subject Classification scheme. This is a hierarchical system: at the top level there are 64 disciplines, described in rather general terms (e.g. “mathematical logic and foundations”, “nonassociative rings and algebras” or “ordinary differential equations”). At the next level, each of these disciplines divides into second-order categories: under “ordinary differential equations”, for example, we have twelve categories ranging from “General theory” to “Dynamic equations on time scales or measure chains”. Each of these, finally, divides further into third-order categories: for example, “General theory” divides into eighteen categories ranging from “Explicit solutions and reductions” to “Differential inclusions”, plus the ever-useful “None of the above”. I haven’t counted the total number of third-level categories, but the full list covers roughly 46 double-column pages and I counted 63 entries in a single typical-looking column, so I’d estimate there are 5000–6000 such categories.

While this hierarchical system is useful, it doesn’t really capture how these areas fit together, or their relative size (whatever either of these terms means). A nice attempt to visualise these is Prof. Dave Rusin’s MathMap, which is based on an  older version of the MSC scheme: each top-level discipline is shown as a circle proportional in size to the number of papers published in that discipline over the last 20 years, and the disciplines are arranged on the page to show how closely related they are. (Prof. Rusin has also provided a lower-resolution but interactive version of the map, a gentle introduction to the MSC, and a guided tour of mathematics.)

An alternative perspective on both the spread of mathematics and its interconnections is can be found in any of the various mathematical encyclopedias that exist. Probably the definitive one at present is the MathWorld website, which contains 13 150 entries organised under eleven main headings (from Algebra to Topology, alphabetically). Because these entries are extensively hyperlinked, a site like this gives some indication of how intricately woven together the disciplines of mathematics can be — they don’t just split off successively from their parent branches, but they share ideas and sometimes even merge with one another. (For many mathematicians, the most exciting research results are precisely those that demonstrate a “deep” connection between areas of mathematics that most of us had thought were separate branches. For example, the deep aspect of Andrew Wiles’s proof of Fermat’s Last Theorem was not that it had solved this specific problem, but that it had done so by proving the Taniyama–Shimura conjecture relating ideas from topology (MSC category 54) and number theory (MSC category 11).) A good way to explore MathWorld is to set yourself the task of finding the shortest route from one randomly-chosen entry to another…

So where does this leave us?

“How much maths is there” is in many ways a silly question, but in trying to answer it we have to ask more sensible ones: about the nature of mathematics; about the rate at which mathematics is developing; and about the relations between different branches of mathematics.

However we ask the question, it seems, the brutal fact is that none of us is ever going to “know” more than a small part of mathematics in any meaningful sense, and even if we could somehow cram the whole of existing mathematics into our brains at once, they’d be sure to burst under the slew of half a million new theorems arriving every year. The doting parents of new maths graduates are going to have to cope with a whole lot of disillusionment, it seems. (The doting parents of more experienced mathematicians have probably got used to it by now.)

That’s the bad news. The good news is that if you enjoy learning maths at least as much as knowing it, then you’re never going to run out of new maths to learn. The other good news is that maths isn’t simply a list of results: it’s a vast multidimensional landscape of intricately-connected ideas, and though we may only get to know a few small areas well, we can at least learn how the other parts fit together, and maybe gaze in admiration at them from a safe distance now and again… Maths may or may not be of infinite extent, but the relationship between it and the human brain guarantees that there will always be as least as much mathematics as we can possibly want!

(DP)

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6 Responses to How much mathematics is there?

  1. Vincent Keenan says:

    A good way to explore MathWorld is to set yourself the task of finding the shortest route from one randomly-chosen entry to another…
    Perhaps a good exam question for the MM409 class…

  2. Richard says:

    “is in many ways a silly question, but in trying to answer it we have to ask more sensible ones…”

    This is basically the job description for many a mathematics gradute working in fields that require interaction with more normal human beings….

    • strathmaths says:

      With a cynical attitude like that, anyone would think you worked in government statistics or something. (I’m not saying you’re not correct, mind…)

  3. Richard says:

    🙂 I actually didn’t think I was being cynical.

    What does a mathematical education offer and how is it useful to people? Genuinely I think that the ability to look at a complex question and articulate simpler ones, that one can then examine to make progress towards the original one is a significant and rare skill.

    Is it the central strategy in any kind of mathematical proof? (or indeed in the development of any CS algorithm) – and is it therefore a bit more central to our worldview than in the population at large?

    This doesn’t engage with the point of your original post. I might do that in a minute.

  4. Richard says:

    So back to the topic – I prefer the second approach as it seems to address a more interesting aspect of the question. Ulam’s approach isn’t so much quantifying how much mathematics there is as how much mathematics human beings have done to date. The geographical approach I like better because I think its a more interesting question to ask – how much of the globe is there, rather than how much of it have we explored already (sometimes several times).

    If you count papers – or theorems in papers – or pages in books there are a couple of obvious problems. One is that there will be many cases of theorem rediscovery, even ignoring out and out plagiarism. More interestingly I think, there will also be a lot of cases where things are expressed in subtly different languages but are actually just expressions of the same idea. The question arises of how you count these cases in your enumeration of amount of mathematics.

    Do you want to consider and therefore count statements about Diophantine equations and their equivalent restatement in the land of the elliptic curve as being the same – and how do you count the discover of the equivalence? (I read this so I could pretend to know what I’m talking about http://www.math.uci.edu/~krubin/lectures/psbreakfast.pdf ).

    So x^n+y^n=z^n is a fact mathematically equivalent to a statement that the elliptic curve y^2=x(x-a^n)(y+b^n) isn’t modular – and we (human mathematicians) learn after much effort that the first bit doesn’t ever hold for any integers x,y,z,n if n>2 only via the observation of the equivalence and then the proof that all elliptic curves are modular. How do you count these various facts ? The equivalence between the two is what gets you to the proof as you can do the second one but not the first. Is the first a separate bit of ground to be enumerated? Perhaps each theorem is the top of a mountain and each proof is a path up the mountain – it’s only when you get to the top from one side and meet a traveler that started on the bottom of the other that you realise you’re both in the same place.

    To pick another idea from elliptic curves there is the fact that you can reduce each elliptic curve to its Weierstrass form by performing rational substitutions on the co-ordinate space. Basically you are stretching and scaling the axis to get the description (in an equation) to take a certain canonical form. The way I think of it is that an individual curve can arise in a lot of contexts and may be described in these contexts by various different equations depending on what angle and scale you look at it from – its only by reducing to the Weierstrass form that you realise exactly which curve you are looking at. Now there is an invariant you can calculate (en.wikipedia.org/wiki/J-invariant) which is unique to the curve that you are looking at and that function has certain symmetries – it’s invariant under certain group actions on its argument (in much the same way that a periodic function is invariant under translation). The nice thing is that you can look at the space where the function is described and “divide out” the symmetry to get a fundamental region in the plane (see the wikipedia article) – similarly you can divide out the “translation by an integer multiple of the period” symmetry from the real numbers and understand that actually when thinking about a periodic function you only really need to consider the unit circle.

    Now – why am I blethering half remembered stuff about elliptic curves?

    I think between a lot of the theorems published (or theorems that exist) there is going to be some kind of fundamental symmetry – one to one in cases of plagiarism or rediscovery – more complicated in cases where a theorem is a restatement of an equivalent result in some other branch of mathematics. If you accept this – the task of asking how much mathematics is there is the same as counting all of it then fully understanding the properties of the symmetry group and then identifying the quotient space. You then get something akin to the fundamental region in the upper half plane http://upload.wikimedia.org/wikipedia/commons/a/ad/ModularGroup-FundamentalDomain-01.png and that’s the thing that you want to measure (and add on the size of the symmetry group if you think the equivalence results are also part of the thing you are measuring.)

  5. Hiro says:

    If I should someday comprehend all of what you have just remarked, I shall be satisfied with my grasp of mathematics.

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