In this instalment we look at the fundamental question <<Where did mathematics start to diverge from its proper destiny?>>. Part of the answer is the Artform Turn which occurred around 1900, because it has led to a tendency to valorise strange, counter-intuitive concepts in mathematical research. In this way it has finally resulted in a vast accumulation of baffling, foggy, exotic definitions. In this way the menacing, demoralising, insoluble problem of Ulam’s Dilemma has emerged. Young researchers knew that they would be judged by aesthetic appearances, not by any epistemological significance: so their emphasis naturally turned to trying to find mazy, unusual, intriguing, novel concepts. But this is not the whole story, because the Artform Turns was not inevitable, and was, at the time, recognised as a clear break away from  traditional assumptions about the purpose of mathematics.

So the roots of this problem, it has become clear, were put down much earlier than 1900.  Mathematics, as we know it today,  really began in the 17^th century with Descartes’ coordinisation of geometry. Before that there was only Euclidean Geometry and fragments of algebra, e.g. the solution of quadratics, cubics and ways to factorise expressions like aⁿ – bⁿ. 

Mathematics started with a process of abstraction at its heart, because the introduction of tallies meant that a stroke like / could “stand for” a person, a camel, a fish or a jar of wine.  This had a clarifying effect, because sorting out whether a crowd of soldiers could be organised into chariot trios (say) could be done without considering their names, their family connections, their boot sizes, etc. Using numerals enabled organisers to represent the gist of many otherwise diverse, messy situations. The most telling effect of such clarification occurred in relation to organising armies and naval fleets.  The Romans, and many others before them, discovered that a well-organised army was much more effective than a large armed but muddled horde.  In this way mathematics acquired a reputation for being very “important”, although today we would only describe the kind of “mathematics” involved as ‘logistic modelling’.

Once mathematics came to be socially regarded as “important” its practitioners became a well-regarded professional enclave, but one which was not closely accessible or readable, either to the mass population or to its leaders. The enclave had its dry, abstract definitions, objects, procedures, etc. which did not appeal much to the average person, and into which she/he was not inclined to delve.  In effect from earliest times, the activity of professional mathematicians was not monitored by anyone except other professional mathematicians. Their status was essentially secure, but what they did most of the time was “their own thing”.

This lack of external social pressure has its advantages, because determined geniuses like Descartes could worry-away at the essence of difficult questions and finally devise marvellous new ways of using maths to clarify thinking. 

But the outcomes were not always so rewarding. The most pressing geometric questions of the ancient world had been how to trisect the angle (using only a straightedge and compasses), and similarly to duplicate the cube, and square the circle. Probably millions of hours were wasted by talented geometers in antiquity trying to solve these  notorious problems. The result was that attempting to solve high profile, but unpromising problems, acquired a bad name. The paradigm example of this syndrome during the last 350 years has been Fermat’s Lost Theorem often mistakenly called his ‘last’ theorem.  Peter Medawar famously defined research in pure science as an activity which depended on recognising “the art of the possible”. He meant that scientists had to use their judgment in deciding which problems were likely to be capable of being solved.  But mathematicians long ago realised that research in pure mathematics could easily fall into “wasting time on the impossible”.

There is a less hazardous option in professional mathematical research. In every area of the subject, some plainly possible, little-studied, configurations can usually be identified, ones which have not previously been explored. So nowadays only the masochistic tend to take a risk, and spend time on unpromising questions. The majority are attracted towards more routine problems, which involve exploring visible but not particularly significant, formal configurations. (In the case of high profile questions the main difficulty is often discerning which obscure configurations are going to be apposite to the problem.)  Euclidean Geometry used to offer endless opportunities for such busywork, for example, in defining new categories of circles associated with a typical triangle. (E.g. the incentre of a triangle defines three interior triangles. Each of these has an orthocentre. Now take the centroid of the triangle defined by these three orthocentres… What can we learn about its properties? There was a cult of amateur mathematics in Japan in the 19^th century called ‘Temple Geometry’. Large numbers of problems of this busywork kind were posed and successfully solved by ordinary templegoers.)

So research in pure mathematics can target problems across a range which stretches from easily visible busywork to the ‘difficult significant’ category. Since pure mathematicians ceased to treat their subject as a science —or as a handmaiden to science— they have moved further and further away from the kind of work which opens new perspectives across the common scientific landscape. The nub of the issue is the question whether mathematics is better served when it pursues problems of its own devising, or raw problems which originate from the outside world and the elementary, common parts of the subject.  The long-term effect is now clear: mathematics as a whole slowly enriches its public meaning when it takes on problems from the outside, and conversely gradually dilutes its perceived ‘meaningfulness’ in the eyes of society when it concentrates only on “in house” thinking. Unfortunately the elites in France and Germany in the 19th century were determined to bolster this otherwise declining clout by political support and favour from above. Ulam’s Dilemma is a dreadful warning about the long-term consequences of not genuinely growing the meaning of the subject via involvement with the problematic experiences of the whole society.

The ironic effect of the ground-breaking work of Descartes and Newton was that it diminished the need for creativity and reasoning in elementary maths. Descartes opened-up large areas of potential busywork, because coordinate methods combined with trigonometry were much more reliable in geometry than the hit-or-miss methods of Euclid. Solving problems in Euclidean geometry, however, did more to build-up confidence, creativity and stickability than practising standard processes. A compensatory school reform should have been introduced in the 18^th century to ensure that the challenges posed for the brightest youngsters during their formative teen years always involved plenty of exercise in conceptualising and searching for solution-paths —of the kind offered by hard challenges in Euclidean geometry.  But there was little reform, and Euclidean geometry had been largely ossified into learning standard theorems by the late 19^th century.  It was not until 1871 that the Mathematical Association was formed in the UK, to make sure that the school curriculum included valuable problem-solving in geometry.

But its gradual neglect meant that the great forward strides taken by Descartes, Newton and Leibniz in the 17^th century probably harmed the quality of school maths. They tipped the balance in school work in favour of “routine methodology” (learning ‘handle turning’) rather than gradually building-up problem-solving confidence.  This would explain why there was an increasingly “in-house”, “institutionalisation” in maths in the 18^th and 19^th centuries, especially in France and Germany. In the UK, Charles Hutton was an outstanding exponent of the problem-solving approach (he claimed that Newton had been a distant relative), but as the 19^th century progressed, the extent of the recognition accorded to in-house mathematics in France and Germany began to exert a social pressure on UK mathematics. This was, disastrously, a fertile field for the fateful Artform Turn of 1900.

Today maths education has regressed still further mainly as a result of the ‘cognitive science’ dogmas of the managerialists who dominate schooling. Nothing, probably, would do more for the cause of renewing reasoning than a fundamental reform of school mathematics. Everywhere around the world maths is being taught either as an artificial ritual with ornamental overtones, or as a practical chore which bores children.   

CHRISTOPHER ORMELL February 2021