Thought experiments putting us in the shoes of our prehistoric ancestors are the best way to get a secure, grounded view of how maths began. If we do this the “essential nature” of maths becomes plain. It is a kind of kit of written symbols which can be used to extract the essentials of changing situations and thereby predict the outline of what is going to happen in the future —-assuming that deep-seated physical behaviours and invariants remain actively in place.  

We know that realistic cave paintings of animals have been found dated from as far back as 25,000 years or more. This tells us that our prehistoric ancestors probably stumbled onto the idea of tallybundles like Illlllllllll… marks scratched on rock, mud, sand or papyrus to represent a collection of things. Why would they do this? Well, a tallybundle of a valuable collection of amphorae in a cellar is useful, because it tells the owner of the cellar whether any of her amphorae have been stolen —-since she checked last time.
     Each tally represents an amphora, so this is the simplest possible exercise in abstraction. Later the abacus was invented, which allowed each bead to represent a single amphora on the lowest rail, ten amphorae on the second rail, a hundred amphorae on the third rail, etc. This was quite a large step forward in maths.  (It meant that merchants in over-crowded places like Babylon —with massive stores of wine— could organise their reserves of wine efficiently.) The abacus could also be used to divide a collection of things into three equal parts, say the inheritance of three children after their remaining parent had died. (This could be done by trial-and-error symbol manipulation in the first instance… afterwards replaced by time-saving algorithms. Finding suitable efficient algorithms became a stamping ground for the arithmetic experts. Schools were set up to train these abacus specialists.)
    So far we have mentioned using tallybundles to check collections against theft, and to establish fair, equitable shares in inheritance…but the major area of use for tallybundling was preparing resources for the future, like the supplies of food needed by an army, or for an ordinary family to keep them alive during the bleak winter months.  It was also a way of organising the construction materials and working drawings needed for building projects. We know that a great deal of elementary maths was used in building the pyramids.
    Some of it was geometry, which no doubt began with the perception that the horizon at sea was a horizontal line, and a thread on which a pebble hung was a vertical line.  Planning buildings prior to their actual building must have been developed in pre-history, because the vastly ambitious building schemes of the ancient Egyptians tell us that the architects who showed such confidence in their methodology must have been standing on the shoulders of predecessors, perhaps going back several hundred years. We are also aware that the ancient Egyptians already knew that a 3, 4, 5 triangle had a right angle at the join of the two shorter sides. This was no doubt used as a handy gadget to check right angles during building.
    The classical Greeks were the first to discover that reasoning was a possible exploratory tool which could be used in simple arithmetic and in geometry.  Reasoning amounted to a kind of ‘over-view’ of what was happening in the busy rooms full of abacus specialists and draughts-people. It soon turned out to be an amazingly lucid, elegant way to learn new spatial truths, like that the angles subtended by a given chord of a circle were all equal, and that a triangle enclosed in a semi-circle contained a right angle.  

    But this insight also had social repercussions. It palpably added up to a superior form of knowledge. The ruling families of the time unsurprisingly wanted to put their stamp of approval on it and its language. (They were bent on some of the shine falling back on them.) When Euclid put together his Elements —the most famous textbook in history— it created a concentrated, elegant source of instrumental knowledge and promoted the associated disciplined thinking.  It became the essential Bible of the engineer-architects who kept the Roman show on the road… which they did more or less successfully for a thousand years.
    Eugene Wigner famously queried why essentially pure, conceptual mathematics turned out to have such unexpectedly potent “applications” in physics. If he had taken the trouble to re-think the problems of the ancient world, he would have realised that he had got the puzzle back-to-front. Maths began as a simple instrumental subject. The “uses” weren’t “applications” of an already special, protected, rarely opened, kit: they were the chief point in having the kit in the first place. Maths had grown as an instrumental subject for thousands of years before Classical Greece. If anything is unexpected about this, it is the apparent take-over and valorisation of this ‘superior’ language by kings, princes, rulers, etc. throughout history.

    The ‘kings, princes, rulers, etc.” obviously favoured it because maths was the ‘X factor’ which made all the difference in their military campaigns. Equipping an army was a logistic nightmare. It was essential to have the right food supplies, weapons, chariots, horses, elephants, etc.  A lot of logistical arithmetic was needed to get this right. 

    After 1700, the new methods pioneered by Descartes and Newton (coordinate geometry and calculus) transformed mathematics unexpectedly into a wonderful tool for designing machines and industrial systems.  This happened first in the UK, and it seems to have provoked a lot of envy in the capitals of continental Europe. So the mathematicians of continental Europe tried to outshine Newton’s mechanics by developing a new kind of ultra-abstract, ultra-pure mathematics.

    This was the beginning of ‘modern mathematics’ a sustained attempt to explore mathematical truths which were more mysterious, ineffable, abstract, sublime, etc. than the previous set. It grew and grew during the 19^th century. and reached its peak with Georg Cantor’s concept of transfinite immensities, which, later turned out, disappointingly, to be only composed of shadowy items which were indefinable and un-nameable. There were only at most a countable totality of bona fide names available to attach to real numbers. Without definitions, or even names of any kind, these extra so-called ‘real numbers’ provoked much scepticism.  

    Some very bad luck followed: in June 1901 Russell discovered that there was a logically necessary contradiction in simple set theory.  It was a problem which was never solved. It was a situation the maths hierarchy couldn’t stomach. In the end Zermelo-Fraenkel (ZF) set theory axioms were introduced in a dodgy attempt to limit the damage. 

    Unfortunately this deception was noticed outside the confines of mathematics. 

     It has turned out to be the undoing of Western Civilisation. It had previously been an in-depth kind of rigorous reasoning which prospered for two millennia. But there was nothing ‘fearless’ about ZF set theory. It was visibly deceptive and motivated entirely by fear.  Far-sighted savants like Heidegger shook their heads: “was this the best the gurus of higher mathematics could do?”. 

    Other blunders followed. Like: the attempted revolution in school maths (1960-1980) which would have changed school maths away from numbers… into the study of sets. It was a forlorn hope, which eventually, predictably, collapsed… leaving educational, political and intellectual mayhem in its wake.

    Like: letting the computer industry tell everyone that “computers have nothing to do with mathematics!”, probably the largest, most naively swallowed, most preposterous, lie in human history. 

    So the former holy grail of ‘modern mathematics’ has turned out to be a nightmare.

    Is this really the case?  Yes, the word ‘mathematics’ has almost dropped out of ordinary conversation. It is equally absent from the words of the opinion-leaders, pundits, the commentariat, the critics, reviewers, futurists, etc. By following the false gods of mystique and aesthetics, the gurus of higher maths have sowed the seeds of society-wide conceptual muddle, irrationality and theory-phobia. We have been left with an imperious AI establishment —with an inflated opinion of its own sagacity— casually running automated mathematics of a billion different shapes and sizes, but without the essential fearless rigour and sense of responsibility which that activity requires.

    Today’s top priority is to restore the rude health of school mathematics by presenting it to mathsphobic children in ordinary, interesting, meaningful language. The P E R Group launched a Manifesto in October which sets out the full case for this reform.     

CHRISTOPHER ORMELL 1^st February 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.