The overall theme of these articles on the Renewal of Mathematics is that the subject has been grossly over-mystified and its most important manifestation —mathematical modelling— has been grossly under-valued.

Pythagoras —the first significant polymathematician*— founded  a Brotherhood of colleagues which has, in a sense, survived for more than 2,500 years, though it morphed into a joint Brother-Sisterhood during the 20^th century.  The result of the imbalances (too much mystique, too little modelling) has been that a sequence of disastrous decisions have been made by the Brother-Sisterhood starting in the early 20^th century. 

‘Disastrous’ is a strong word, but it is not too strong to denote the damage done by:

1 Re-launching higher maths as ‘Intellectual Artform’.  This inexorably led to Ulam’s Dilemma which has produced so much impenetrable fog that it has reduced the immense body of 20^th century higher maths to absurdity. Whatever is the point of creating new definitions which add new fog to an already fog-bound situation? No academic subject can afford to get a reputation for being in the business of generating fog on a massive scale.

2 Trying to insert an axiom into mathematics (ZF set theory) which contradicted logical commonsense. It became a Party Line, which was ruthlessly enforced, and thereby effectively ended the era of rational mathematics.

3 Backing New Maths for Schools in the 1960s, which unexpectedly lit a fuse which led to an explosion of progressivism —which destroyed the credibility of mainstream thinking in education.  Chaos followed. Eventually the politicians stepped in and made the situation worse, by favouring so-called Cognitive Science, a crude, substandard system still in place today, which should not have been allowed to go anywhere near the schools.

It is, incidentally, the effect of item 3 —to spawn Post-Modern nonsense— which has led to a partial collapse of confidence in Western Civilisation.

The climate of opinion about mathematics and its mystic aspect has also subtly changed.  There is nothing sillier than invoking an aura of mystic implications which is no longer there.  The God-fearing populations of the past have gone. The idea that mathematics was the language used by God to create the universe no longer makes any sense.  And this is before the discovery of Actimatics, which has, in any case, seized the role of the best abstract modelling discipline available.

What is needed now is a quiet switch to the Peircean Interpretation of maths, the adoption of some processes which this implies, and new criteria for research in higher maths.  This is crucial, because we have never previously had a realistic, rounded account of meaning in mathematics. It makes clear what the final “use” of mathematics is, thus in effect completing Wittgenstein’s project of demystifying the meaning of words.

There is still a place for higher pure maths because there are striking patterns to be found in advanced configurations which cry out for explanation. This role of mathematical research to explain striking observed patterns in mathematics itself was the original prompt for science. The power of simple logical arguments like that which establishes the irrationality of 2 is intellectually inspiring. The ancients saw that if such light could be achieved within mathematics, perhaps something similar could be achieved in the cosmos. But it is only pure maths patterns which have universal recognition and common intelligibility, whose explanations are going to add to this general inspiration.  Busywork adds to the overall fogginess of higher maths. 

Mathematics has hundreds of lucid, understandable, strikingly unexpected theorems, which satisfy this condition (universal recognition + common intelligibility) and they are, in effect, the Heartland of Truth. They may be correctly described as <<absolutely true>> because they can be checked 2, 20 or 200 times to ensure that there are no mistakes.  

The outcome of all this recent progress in understanding maths need not be a cause for gloom. There can be a great future for post-platonic mathematics.  

*A ‘polymathematician’ may be defined as a polymath who is also a mathematician. It is striking that most of the great names of the past were polymathematicians (Pythagoras, Archimedes, Aristarchus, Descartes, Newton, Leibniz, Gauss). But since the dawn of ‘modern mathematics’ around 1830 few of the big names have been polymathematicians. Since then we have been fortunate in the UK to have polymathematicians such as Charles Hutton, Charles Babbage, George Boole and Alan Turing.  We are fortunate, today too, to have three active polymathematicians, Roger Penrose, Ian Stewart and  Marcus de Sautoy.

CHRISTOPHER ORMELL December 1^st 2021.