That there is a crisis in school mathematics is hardly news. It has been around for so long that it has long since become part of the given status quo. The last significant attempt at introducing a “new brush” into the teaching of maths in schools was the so-called ‘Practical Maths’ of the mid 1980s.  It was, probably, a by-product of —or, more precisely, a bowdlerised response to— the palpable classroom interest raised by the author’s Mathematics Applicable project of the 1970s. By the end of the decade this project was making waves and visibly importing vitality into maths teaching.  The “waves” could be seen in the fact that about 500 schools had contacted the project in the UK and Australia had seen even higher sales of the project’s ten-book series than the UK.  This much was fairly obvious. What was (much) less obvious was how this effect had been conjured-up. It came about because Mathematics Applicable was based, from the beginning, on an exciting  new philosophical basis: the adoption of a Peircean Interpretation of mathematics. 

Charles Peirce was for many years up to his death in 1912 and afterwards up till the mid 1930s a much under-appreciated US philosopher. Then it began to be realised in America and elsewhere what an original, perceptive talent he was:  America’s greatest 19^th century philosopher no less.

His revolutionary new “take” on the meaning of mathematics, though, still lay in obscurity …until 1956 when the four-volume World of Mathematics (Simon Schuster, New York) was published.

It contained an essay by Charles Peirce which declared roundly that <<Mathematics was the science of hypothesis>>.  This was in complete contradiction to the de facto interpretation of higher maths which had been in place since the end of the 19^th century —the Official Story that mathematics had been re-tasked and had become a special intellectual artform. (This “re-tasking” was no minor matter. It meant that mathematics had ceased to be regarded as any kind of ‘science’. The most brilliant researchers could now enjoy a great widening of their opportunities —for showing-off their skills, creativity and sophistication— because all sorts of bizarre, even wacky, ideas in hyper-abstract configurations could become the legitimate targets for their research.)

Unfortunately few of those who read Peirce’s essay in the World of Mathematics seemed to realise just how revolutionary this new insight was. They treated it as a truism, because mathematical truths are, of course, predicated of the truth of the axioms from which they are deduced. So all mathematical truths have the implicit form <<if p, then q>> where ‘p’ stands for the axioms involved.

That this was a superficial level of comprehension hardly needs to be said. This wasn’t the central point Peirce was making.  He had rumbled something which the priesthood of higher maths had overlooked for more than two millennia, actually since the time of Pythagoras in the 6^th century BCE: namely, that mathematics had a human purpose, that it was not just an elegant quasi-religious stamping ground for brilliant, introspective logicians who preferred to engage in careers operating under the precise rules of mathematics rather than face the messiness and sometimes squalor of ordinary human activity. 

Yes, Ludwig Wittgenstein had seen clearly in the 1920s that the meaning of any language is a direct product of its use. 

But the higher mathematicians, who were the superstars of academia at the time, refuted this thesis, because they pointed out that their research into the logical implications of hyper-abstract reality was much more important than any merely mundane “use”.  This was strictly “useless knowledge” held in very high regard.

Unfortunately —for the higher mathematicians of the 20^th century and their families— this assumption, that hyper-abstract reality carries high public regard, has now all-but disappeared. Computers have stolen mathematics’ thunder, and spending time on abstract implication games is nowadays regarded as little more than a form of mental embroidery.

So what Charles Peirce had seen in a brilliant Eureka Moment in the 1880s fully vindicates Wittgenstein.

Yes, mathematics does have a “really significant use” —namely, that it is our way of teasing out the full implications of promising ideas (hypotheses). It is also a home truth —whether we like it or not— that new ideas in technology, commerce and development provide the crucial adrenalin which keeps modern economies turning. New ideas in politics and social thinking are also a perennial subject of interest.  So mathematics is near the heart of the positivity and inquiry which keeps us all on our toes. 

Mathematics Applicable found a way to harness this source of interest. This was the secret of its success.    

CHRISTOPHER ORMELL 6^th July 2023

If you would like an online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com asking for this.