Maths for Renewing Reason – 57

Maths for Renewing Reason – 56
03/01/2025

These blogs are offered to show the way in which conceptual, reasoning-led maths can generate interesting, sometimes important, results. In an age of awesomely microscopic, awesomely fast digital electronics, the manipulation of chosen symbolic configurations will —of course— be mainly performed by computers. We can no longer expect to distinguish ourselves merely by personal performance feats involving chosen new sophisticated manipulative options. Automated maths on computers will, of course, always be able to upstage the human manipulator.

So we urgently need to re-formulate the central personal challenge posed by maths.  The notion that <<applying ultra-sophisticated manipulations to abstract configurations is the heart of the subject>> will no longer do.

It won’t do, because Ulam’s Dilemma tells us that the abstruse subject “higher maths” is already vastly over-loaded and effectively indecipherable. There has already been far too much effort devoted to creating novel sophisticated new manipulations and configurations: the “novelty” of this is wearing thin. This open-ended creativity has resulted in an oceanic amount of supposedly “significant” new results, which, however, have long since passed the point where they could be widely understood, let alone properly appreciated. That anyone could ever get an <<overview of what it all means>> is out of the question. No single person, however talented, can, or ever could, understand what it all means.  There are far too many such results, and the situation is getting worse and worse by the year, because the mystique of the higher subject has now gone into decline. This diminishes the attraction of a career in higher maths for the brightest students, because the former cachet of becoming a “high priest” of a much venerated abstract quasi-religion has effectively disappeared.

Conclusion: fewer and fewer people can still make sense of occasional patches of the total field of known results.

Up to the 1960s there was still a partial justification for creativity in higher maths —as a potential source of unexpected simplifications, thus prompting new calculability.  This was, according to G. H. Hardy, the main justification for doing higher maths. Pure Maths, he claimed, was “more useful than applied maths” because it was pure maths which provided the wherewithal to obtain actual answers to the questions posed by the applied mathematicians. But a long, hard, searching look at the usable “new calculability” which this creativity produced,  never produced very convincing results. There were occasional discoveries like logarithms, difference methods and de Moivre’s Theorem, which led to new kinds of calculability, and some tricks for solving differential equations, but this harvest was a disappointingly small reward for a great deal of exploratory effort.

After 1960 the credibility of this justification for research in higher maths virtually disappeared… because the new, reliable solid-state computers increased the calculability available a thousandfold.  The tiny scraps of calculability previously credited as the main reason for research in higher maths looked pathetic by comparison.

So the central tenet of G.H.Hardy’s charming book A Mathematician’s Apology (1940) had been demolished, in effect, soon after at Bletchley Park.

It appeared that the only credible modern reason for pursuing difficult problems in higher maths was the aesthetic quality of new configurations which occasionally came to the fore.

This might seem retrospectively to justify the historic switch of the goal of higher maths which was, effectively, legitimised by the maths establishment around 1900. They tacitly changed the official purpose of higher maths at that time. Henceforth it would be a supreme aesthetic intellectual artform.

But this earlier (at the time controversial) switch had the unintended, unnoticed, effect of removing much of the perceived rigour needed in higher maths.  Aesthetics is about appearance.  But if it is merely the appearance of surprisingly elegant results which counts, there is inevitably going to be little urgency about , or appetite for, insisting on rigour.

This may have been the reason why so little rigour was applied to Cantor’s transfinite theory at the turn of the century. Rigour told everyone who considered the issue that there was only a countable totality of potential definitions of mathematic objects. So Cantor’s transfinite sets could only be fully populated by counting (legitimising) “indefinable, fairylike” mathematical objects.

It should have been a logical shock of the greatest magnitude that Cantor’s theory rested on an acceptance of such “indefinable” mathematic objects.

But it wasn’t.  A few leading mathematicians, such as Poincare, Borel, Weyl… saw that this was unacceptable, and that its effect was to demolish the credibility of Cantor’s exotic fantasy. But the Official Line didn’t change:  and all those thousands who subsequently followed the Official Line, effectively swallowed the notion that “indefinable mathematic objects” could be accepted as bona fide.

They had forgotten the primary truth that maths’ credibility-with-the-intelligent-public must rest squarely on its observance of the principle that all mathematic objects are well-defined. If the maths fraternity can quite happily swallow indefinable objects, what other kinds of nonsense might they have up their sleeve?

So where do we go from here?

The ultra-pure gurus of 1900 were distinctly averse to regarding higher mathematics as justified mainly by its role in solving problems in mathematical physics. But they were also, incidentally, right in their realisation that higher maths was not a sort of empirical science, as argued by John Stuart Mill.  Their serious lapse of judgment was that they had failed to realise that Charles Peirce had hit the nail on the head when he announced (probably in the 1880s) that <<mathematics was the science of hypothesis>>. This should have been a major turning point for higher maths.

It was missed. It was a major insight which slipped-by, virtually unnoticed.

Peirce was the first heavyweight thinker to see that mathematics’ main job was not to worship conceptual elegance (appreciatable only by a tiny elite), but to tease-out the predictable implications of well-grounded hypotheses, which were about the real world…hypotheses about possible explanations and possible useful innovations.  (There were also, of course, in-house hypotheses in pure mathematics itself… like Fermat’s, Goldbach’s and the Reimann Conjecture.) These mathematically-intriguing “hypotheses” were guesses about possible explanations of what looked like strikingly unusual formal patterns.

In science the hypotheses were about potential explanations of puzzlingly patterned phenomena.

The main body of hypotheses, though, were ambitious practical building and organisational projects. This was the job maths had been doing with great effect and popular support since 2,500 BCE, when the Pharaoh Kuhfu used it to create the immense “wave of belief” needed to sustain the Egyptian population’s confidence that the Great Pyramid of Giza could be built in 20 years. It is easy to forget that Kuhfu’s project would have been dismissed as <<far too ambitious to be worth considering>> by most of the regimes of the Ancient World. It required an oceanic belief, a rock-solid commitment…  showing that such a thing could be brought to fruition using bare-hands methods in a relatively poor agrarian society.

There were a few arithmeticians and geometers who spent time exploring new operations with numbers and new combinations of figures. They occasionally found ways to streamline common calculations, and Euclid gave splendid examples of elegant proofs which were discovered, such as that the square root of 2 can never be an exact fraction.  This, though, was really only a sideline. Plato sought to make it the central reason for studying maths… for maths’ sake. It was the heart of this kind of in-house activity. This seemed ideal to the small dilettante body of rich, amateur “pure mathematician” explorers. So it became the dominant, deeply entrenched approach —the supreme justification— for “pure mathematics”.

It was, in effect, a quasi-religious quest which comfortably survived for more than 2,000 years.

In the 20th century, though, it was suddenly cruelly upstaged by the arrival of digital computers. (And after they had made it look artificial and unnecessary, the IT sector virtually airbrushed-out any mention of the role of maths in their computers.) Worse still, higher maths drove itself to self-destruction by a self-inflicted over-production (Ulam’s Dilemma) of unwanted formal explorations. This became an immense cloud of hyper-complex, over-elaborated notions, far beyond what any single mind could ever understand. It was a disastrous own-goal, which could only happen because the leading gurus of maths were not looking at the big picture, and not listening to the main body of intelligent people.

We need to give-up this precious, myopic in-house mentality, and revert to the solid, commonsense concept of maths as the chief pathfinder which creates the priceless confidence needed to develop projects, theories and innovations of all kinds.  Maths should be seen as the most important and most interesting subject on the school curriculum, because its central purpose is… to provide the confidence-building illumination needed to bring humanity’s vital future projects to fruition.

CHRISTOPHER ORMELL around February 1st 2025.  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.