Let’s take stock of some unfortunate glitches which have occurred in the record of Maths and which have been chewed over with some reluctance in this series.  Having surveyed the seriousness of the glitches, the next stage is to consider what can be done to put the record straight.  Maths can be a powerful force for renewing reason, but if we are honest, it has not been such a force for a long time.  So much so, indeed, that a little-mentioned retreat from reasoning has occurred within the subject. We take the glitches in the order of their theoretical importance.

GLITCH 1  This was the extraordinary failure of the leading mathematicians of the mid 19^th century to recognise that what they took to be ‘Set Theory’ was really ‘Applied Set Theory’.  [For details see Instalment 4 of this blog.] It was compounded by the extraordinary failure of late 19^th century giants like Dedekind, Frege, Cantor, Peano, Whitehead, Russell… to recognise their predecessors’ mistake. It was compounded still further by the project these leading figures backed and promoted, namely, the attempt to reduce the total body of mathematics to what they called ‘set theory’ but which was really Applied Set Theory.

GLITCH 2 This was Cantor’s misinterpretation of the Diagonal Argument. He argued correctly that the real numbers could not be counted, but this was not because the ‘set of real numbers’ was larger than the set of rational numbers. It could never be ‘larger than the set of rational numbers’ because the entire universe of humanly-operable mathematics is bounded by the need for proper definition, and the total field of all possible definitions of mathematical objects is countable. Another way to describe the mistake is that there is no such thing as ‘the set of real numbers’ because the totality of real numbers is incompletable: it never settles. 

GLITCH 3 David Hilbert declared at the Maths Congress in Vienna in 1900 that nothing would induce him to give up the paradise which Cantor had opened for us. He was saying in effect that you can believe what you want to believe in mathematics: that emotion can trump reason. At around this time leading figures started describing mathematics as an ‘intellectual artform’. 

GLITCH 4 In Principia Mathematica Part I (1910) Russell declared that formal logic must be stratified into layers of reference.  This was an abandonment of the self-evidence on which we need to be able to rely at all times in mathematics. It was totally unobvious.  It would require an oppressive Thought Police to enforce it.

GLITCH 5 In the 1920s Zermelo-Fraenkel set theory adopted an axiom which forbad a set from being a member of itself. It was introduced because no one could find a proper explanation of Russell’s Paradox. Here again emotion was being used to trump reason.  It, and other versions of restrictive set theory, were subsequently tacitly treated as valid mathematics.

GLITCH 6 In the early 1960s a group of virtually all the world’s most eminent mathematicians threw their weight behind a revolution in school maths called ‘New Math for Schools’. It turned out to be a folly, one which caused immense havoc, far beyond the boundaries of mathematics. The humiliation was so great that, for a while, academic mathematicians were excluded from the curriculum committees which determined the maths syllabi used in schools. It also provoked the post modern pandemonium, because the average person began to lose faith in intellectuals of all kinds, not merely higher mathematicians.  

How did these six specific glitches come to happen?

Probably we need to go back to 1830. Around this time there was a profound revolution in thinking in maths, the emergence of non-traditional (complex) numbers, non-classical algebra and non-Euclidean geometry.  These amazing breakthroughs became the hallmarks of ‘modern mathematics’. They were probably prompted by the generally revolutionary mood of the time, stemming from the French Revolution.  They soon became foci for a kind of defiance, which was in direct conflict with the intensely conservative, practical, feet-on-the-ground character of earlier mathematics. In the UK an ageist gulf opened up between the old guard exemplified by Charles Hutton, and the new guard exemplified by Charles Babbage.

So, after 1830, there were two schools of thought in mathematics, those who clung desperately to the previous status quo, and those who favoured the new, strange, abstract concepts. Gradually, as the 19^th century progressed, the second school gained the ascendancy. The result was that a kind of defiance of self-evidence and practicality became respectable, took over the high moral ground, and eventually emerged as the new status quo… hence acquiring the mantra of conservatism which is the norm in mathematics. This ‘tradition of defiance’ has remained in place to the present day. After Cantor’s exotic ideas were embraced around 1900, it went further and morphed into a kind of triumphalism, which positively flaunted its adherence to anti-practical and anti-commonsense logical ideas.

This translated into an immense diversification in research mathematics, because young researchers were encouraged to chase the most unusual, unexpected, wacky lines of reasoning.

Unfortunately a ‘tradition of defiance’ is a dangerous ethos to  cultivate, and it was probably only a matter of time before it would lead to a clash with mainstream attitudes and inevitable public disgrace.

In retrospect it seems that the powerful coordinate methods introduced by Descartes in the 17^th century quietly turned mathematics into a much more technical business than it had previously been. When Euclid reigned supreme, you had to think creatively. But after Descartes, imagination gradually stopped being the leading, make-or-break role in maths research. By the 19^th century and the eruption of the challenging ideas of 1830, academic mathematics had become much more inward-looking than it had been before Descartes. Its characteristic ideas and aspirations had also become much less accessible to the ordinary intelligent lay person. This may explain why it seems that no one ever questioned whether set theory, as devised by George Boole, was really ‘pure mathematics’. They just assumed that it was pure mathematics. The leading figures were no doubt thrilled to find that a new, non-traditional form of algebra had been discovered. They unconsciously identified it with the abstract algebra of Galois. It never occurred to them to ask whether it was really ‘applied’ maths:  that was something they knew little about, leaving it to fringe lesser-talents in physics and engineering.

It seems that later, around 1900, the sheer depth of the conceptual crisis in physics became known. Conceptual changes of some profoundly strange ilk appeared to be needed if mathematical modelling was to resolve the outlines of the new reality. The average academic mathematician had little appetite for this unsettling prospect.  So a kind of divorce happened between higher mathematics and mathematical physics.  The main body of higher mathematicians chose to swing into ‘artform mode’. 

In retrospect this turn towards aesthetics and away from science has gradually led to a massive problem which finally casts doubt on the integrity of the subject: Ulam’s Dilemma. 

Ulam’s dilemma is the existence of millions of highly technical, abstruse theorems which have been judged to be valid by referees and published in professional journals. They count as the “20^th century legacy of modern maths”, but which no single person can possibly hope to understand more than possibly 0.1% of the whole. The problem gets worse as time passes, because esoteric ideas which seemed important at a particular moment lose much of their appeal once they have been superseded by even more esoteric later notions. An immense degree of diversification has been tolerated which makes it impossible for anyone to oversee the subject.  It is not just an over-production of millions of theorems… whose fate will be to gather dust on the shelves of university libraries.  The most serious outcome is a massive loss of clarity and focus, because no one can now legitimately claim to know where the heart of the subject is, or even what it is really trying to do.

Advanced mathematics has gone out of control: and it is impossible for anyone to gain the confidence which stems from mastery of its main themes —there are just too many and of so many kinds.

So what does the future have in store? Caleb Gattegno asked the a depressive question in the 1960s at a Public Meeting in Islington Town Hall <<Will there be mathematicians in the future?>>.

The artform turn has been a disaster.

The subject needs to rediscover its essence.

Fortunately Actimatics has come along and Actimatics will require a great deal of hard mathematical problem-solving if it is to flourish.  

The fiasco of Ulam’s Dilemma will need to be cleaned-up, by launching a 20-year project to over-view, analyse and classify all the stuff which emerged from the artform turn.  The aim will be to create a software taxonomy which makes some kind of “sense” at last of all this wacky theorising.