In this instalment we continue the line of thinking of No. 9 in relation to the current phase of epidemiological mathematical modelling. 

One small positive by-product of the current tragic pandemic has been that it has shown mathematical modelling in a surprisingly  favourable light. Before the pandemic there was a growing tendency for some jaundiced lay critics to treat mathematical modelling as a bogus, deceptive, substandard activity. This was most likely a consequence of there being a great deal of bogus, substandard mathematical modelling circulating, especially in perverse (secret) online algorithms. (It was also a natural consequence of training maths specialists at school in an epistemological vacuum.) As a result charlatans were tempted to  use the residual authority of the word ‘mathematical’ as a respectable cover for their knavish “modelling” tricks. Cathy O’Neill’s book The Weapons of Math Destruction (2016) focused much attention onto this insidious phenomenon.                                                                                                                                                                                                       

One factor which has probably spawned a lot of disreputable modelling was that we have no agreed common adjective to distinguish a well-based, grounded mathematical model from an ill-based, flaky one. They are both called, and are, ‘mathematical models’.  (They can both be studied validly mathematically to tease-out their implications.) The first, though, is thoroughly grounded on trustworthy empirical knowledge, while the latter is not. It would help if those who have devised well-grounded models always mentioned this fact, and routinely described their models as such.   

Now the models developed by epidemiologists during the pandemic —and featured in Downing Street press conferences— have, of course, been properly grounded on genuine epidemiological data. They have turned out to be a major tool in understanding and anticipating the spread of the virus. 

This has clearly put some much-needed positivity back into the good name of ‘mathematics’, but there is a proviso… namely that this is commonly classified as being “only” mathematics in its so-called ‘applied’, lesser, modelling role.  

But… isn’t there a thinly disguised disdain implied by the term ‘applied mathematics’?  Where does this come from? 

Thinking about the proviso can only prompt the implicit question, which is whether modelling hasn’t now become mathematics’ principal, essential role. In other words, whether we should stop dismissing it as ‘applied mathematics’. We don’t say that we ‘apply’ a car to the road.  Travelling on roads is what cars do —this is their raison d’etre.  So should we change the way we talk about cars, and, following mathematical practice, say that cars are “at their best” under halogen spotlights in a well appointed, smart showroom, where their sheer elegance and elan can make the greatest impression on potential customers?  Should we start saying that driving a car is only an “applied automotive experience”? We might say part of this —with the proviso that cars may be seen in their showrooms at their ‘aesthetic best’, but it still wouldn’t tempt us to say that cars are operating in a secondary, ‘applied’ role when they are used to take us about on roads.

So, shouldn’t we wake up, and recognise that modelling misty, impenetrable, important situations is what mathematics quintessentially does best, i.e. its principal contribution to the welfare of humanity?  It can serve as a headlight, showing the way ahead, both in science and technology.

This question has been strangely out of sight for a long time, probably because today all serious mathematical modelling is, of course, done with the help of computers.  From the beginning of the computer age the computer industry started calling this activity ‘computer modelling’, and a wide public misconception has built up about this.  A computer is essentially just a device which uses the wonderful medium of modern (micro) solid-state electronics to automate logical and mathematical processes. Everything is handled on the machine by means of digital registers which are, in effect, numbers. For example, in word processing every word is represented by a number.  So to describe mathematical modelling performed on computers as ‘computer modelling’ is a considerable semantic put-down —one not unlike the long-time put-down implied by the term ‘applied mathematics’. It was adopted from the 1960s onwards as part of computer salestalk which tried to impress on customers that “computers have nothing to do with mathematics”. This was a brazen lie, but computer salespeople knew that any mention of, or dependence on, ‘mathematics’ would worry the average customer. What they should have said was that <<you don’t need to know anything whatever about maths to use a computer successfully>>.  This would have disarmed the potential anxiety which undoubtedly surrounded “maths” in the mind of the average person.

The gurus of mathematics in 1960 —who were ironically at the time being mistakenly lionised by the public as a result of the supposed reflected glory which derived from computers —went along with this deliberate public disinformation.  They too believed that “computers have nothing whatever to do with real mathematics” because ‘real mathematics’ —for them— was higher mathematics treated as an aesthetic artform.  They definitely didn’t want to have anything to do with computers.

But things have gradually changed during the computer era. A vast amount of spectacular, effective, grounded mathematical modelling, e.g. successfully landing rovers on Mars, has been done.  Once the artificial occlusion of mathematical modelling —as a result of its misdescription has been removed— it becomes clear that mathematical modelling is what a majority of professional mathematicians nowadays do.

And in any case it can no longer be credibly pronounced that “computers have nothing to do with mathematics” because since the arrival of Mathematica 40+ years ago computers have been doing complex classical and “modern” algebraic processes (manipulations), of a kind which would stretch the patience of professional paper-and-pencil mathematicians to the limit. 

In 1960 the gurus of higher mathematics tried hard to snuff out the importance of mathematical modelling. They treated it as just another “applied maths” activity. They strongly resisted any suggestion that it was central to the meaning of ‘mathematics’. (It was just “applied maths”, wasn’t it?) They set roadblocks in the path of modelling, when it looked as if it might revolutionise school mathematics. They fully acquiesced in the myth that computers “have nothing to do with mathematics”.

Sixty years later the situation is entirely different. A huge body of mathematical modelling has been successfully accomplished. As a result, mathematical modelling —largely hidden from view by its common misdescription as “computer modelling”— is now the dominant element.

So has mathematics fundamentally changed during the last sixty years?  

No.  It has been doing this all along. It certainly started in this modelling role  —on a  simple level— when it was first used in a very primitive way as tally bundles— probably as long ago as 30,000BCE. A bundle of tallies told a shepherd at the end of the day whether we had rounded up all his sheep, or whether a few stragglers were still missing. Tally bundles progressed and eventually became numbers, and the advantage achieved by handling the organisation of armies became a source of great social respect.  This accurate logistic modelling provided Kings, Princes, Generals… with a kind of vision which turned out to be decisive when they confronted larger, disorganised armed hordes.  Leading arithmeticians naturally registered and enjoyed the respect they were accorded, and they also naturally assumed it sprang from a deep appreciation of the aesthetics of their activity. But those who commissioned their work were mostly thugs, and weren’t in the least interested in the aesthetic qualities of the maths processes employed. 

When Eudoxus started modelling the movements of the planets with simple geometry (circles), a new kind of much ‘purer’ mathematical modelling emerged. The planets were heavenly bodies, and understanding their movements had no military pay-off.  This kind of geometric modelling was, nevertheless, a source of great fascination. It developed gradually, via Ptolemy’s modelling scheme, to Copernicus, who used the vision it provided correctly to place the Sun at the centre of the planetary system. Again there was no commercial or military pay-off, but intelligent people everywhere were unmistakably drawn to it —as part of trying to understand this baffling universe they found themselves in.

Logistic and geometric modelling kept gaining pluses throughout the BCE. It was, and was seen as, a source of priceless vision in commerce and building. Its clear distinction between ‘right’ and ‘wrong’ evidently began to transfuse into moral codes, once the first cities emerged, and moral codes were necessary to maintain any kind of civic order.  When monotheistic religion arrived, it was criticised for being too abstract. (It was much more abstract than the primitive tribal religions which preceded it.)  The religious authorities of the ancient world had always been happy to co-opt some of the clarity and glamour of mathematical thinking (e.g. talk about infinity) into their narratives.  

Such considerations help to bring out the admirable quality of the contribution mathematics actually made to society from the earliest times.  Using mathematics in the public space was not typically the minor, messy, unworthy, “merely applied” business it has often been routinely portrayed as.  On the contrary, it clarified the muddy waters of intuitive thinking, generated synoptic vision, supported elegant building projects, enabled national defensive military pride, and brought a glimmer of understanding of the outlines of an extraordinary physical universe.

CHRISTOPHER ORMELL March 2021