These blogs are intended to show the way in which reasoning-led maths can generate striking, even sometimes major, results. In an age of awesomely microscopic, awesomely fast, digital electronics, authentic, traditional maths cannot compete with the latest hardware —if the contest is simply about the efficiency of complex manipulative process.  But we are not handle-turners. The essence of the challenge of maths is not to find technically difficult manipulations.  Maths has made some of its most spectacular forward leaps   —like Euclid’s demonstration that the set of prime numbers is infinite—  by marshalling the full power of rational human insight… or if you prefer, the careful use of transparent logical reasoning.

This instalment is based on an article in The Spectator (September 2024) page 15 by David Whitehouse. His theme is that there are extraordinarily difficult problems in maths connected with prime numbers. He highlights the still unproven Riemann Hypothesis as being probably the most difficult. He quotes David Hilbert as saying that <<If he woke up after sleeping for 500 years, his first question would be whether the Riemann Hypothesis had been solved>>!

But this charming thought experiment only serves to draw our attention to the absence of priority surrounding challenges in pure, non-applicative mathematics. If Hilbert did wake up after another 400 or so years, and if he did find that the Riemann Hypothesis had been solved, there would be little benefit to celebrate for humankind in general… merely we would learn that the individual who pulled the trick had been awarded the Fields Medal.

This, I’m afraid, is a conjectured mis-use of the best human penetrative, problem-solving capacity we have.  This kind of research has been reduced since the collapse of credibility in 1972, but some remnants of it are still being promoted. We are in crisis:  passing through a phase in which the world is erupting in violence, unrest and confusion… on such a wide scale, and of such severity, that it could easily result in human extinction. In these exceptionally frought circumstances there is nothing admirable about devoting thousands of Ogdens of attention to solving the Riemann Conjecture… while our civilisation is burning down.

Whitehouse introduces his thesis with the claim that the prime numbers are <<the building blocks of all numbers, as atoms are of the physical universe>>. He goes on to a (verbfree) mention <<The scaffolding of a mathematical universe that reaches to infinity —because mathematics is the only thing which can be truly infinite>>.

I’m afraid this is a misguided, rather over-dramatised, account of mathematics.  Mathematics arose originally from several millennia of tally-bundling, which was used by our prehistoric ancestors to keep tabs on flocks of sheep, fish catches, amphorae stores, etc.  The greatest achievement maths brought about in Antiquity was to create the confidence (in what was a poor agrarian society) to launch and sustain a project which would take more than 20 years to complete … building the Great Pyramid at Giza… a massive tetrahedral mathematical statement… a building which was the tallest structure in the world for more than 3,000 years . Mathematics can do this.  It is a fantastic pathfinding, illuminative activity, which can motivate and materialise projects which look absolutely impossible.  It is not a science in the ordinary sense of ‘science’, and it is philosophically naïve to treat it as the science of an imagined, mythical abstract parallel universe.  Speaking about mathematics as if it handled objects was a device conceived in Antiquity to give its much appreciated experts some status. (By introducing ‘The Number Three’ they reified the adjective ‘3’ and added gravitas to those who studied it.) Numbers became nominal ‘objects’.  They belong to a category of humanly imposed ‘honorary existents’, like marriages, treaties and degrees:  they are not anything to do with metaphysical reality.   The cleverest mathematicians have tended to fall into this ‘Abstract Reality’ fallacy ever since. Lewis Carroll guyed it when the Red Queen observed that Alice must have exceptional eyesight, because she was able to see nobody coming down the road!

The prime numbers can be fancifully credited as being like “atoms”, but this absurdly hypes their actual origin as “those numbers which cannot be portrayed as rectangles of dots”. They are not unique.  There is another infinite family of numbers “which cannot be portrayed as triangles” and another which “which cannot be portrayed as regular pentagons of dots”… and so on ad inf.

There are hexagonal arrays of dots which have three short sides with n dots and three alternate long sides which each have m dots.  So we could, if we were minded-to, define m@n via “hexagonal multiplication” (@ being the symbol for it) as the total number of dots in such an array.  This opens up yet another possible source of “hexagonal prime numbers”.

And so on…  There is an infinite set of quasi-prime numbers of this type.

Mathematics itself can be hyped —as Whitehouse does— as <<the only thing that can be truly infinite>> but here too, we are in danger of over-dramatising it.  ‘Infinity’ is simply a term used in mathematics to indicate a process which can continue indefinitely… as far as you want. To treat it as a ‘thing’ is as misconceived as expecting the two Bishops of Chess to be able to take prayers in Canterbury Cathedral. Infinity has a well-understood meaning within the artificial sub-language of mathematics, much as the two Bishops have a well-understood meaning in the artificial sub-language of Chess.

Higher mathematics is currently in the doghouse with the general public, because it looks too much like a self-serving, arbitrary, artificial ego-trip. (Incidentally there have been famous voices in the past who wryly articulated similar self-doubts:  people like Pascal, Russell, Wittgenstein, Simone Weil.) This is not a healthy situation.

It is sad that, in the 20^th century, the gurus of higher mathematics completely misread the nature of maths and what it means.  As a result they ended-up making eight serious blunders which have been furiously hushed-up to prevent the subject’s reputation from looking broken.

It is essential that we drop many of the absurdities which have attached themselves to the subject, much like the barnacles on the underside of an old boat. Maths will stay in the doldrums and become extinct —unless we can manage to relaunch it in a much more sensible, interesting way.

Mathematics is, as Charles Peirce pronounced in the 1890s, the science of hypotheses.  It is the discipline we need when we are minded to explore interesting untried projects, inventions, possibilities.

Whether the modern world needs mathematics is no longer a live issue:  today almost every routine on which we rely is being controlled, and new arrangements are being projected, using an automated version of maths.  What mathematics actually does very well (whether automated or not) is to illuminate the germs of the good ideas we need to achieve a happy future.  This means that we desperately need a thriving mathematical culture in schools —to produce a body of lively professionals who are able fully to understand what it can, and cannot, do.

CHRISTOPHER ORMELL October 1^st 2024.  If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email requesting this to per4group@gmail.com  Also comments on the reasoning in this or earlier blogs in this series can be submitted by email to the same address.  This includes any counter-argument submitted as a bid for the prize offered in blog 49.