The last instalment of this blog (May 2021) aired two brilliant new insights about the possibilities of modelling which have risen to the surface during the pandemic. To say these developments are “hopeful”, is a considerable understatement. (They are the Peircean Interpretation of Mathematics and Actimatics.) When taken in, and properly focused-on, they indicate that the Western Project of trying to understand the World via rigorous abstract modelling has come back to life. Some exciting signs of genuine movement have appeared. This is remarkable. These are wholly unexpected developments —ones which imply a surprisingly rosy future. But they also enter a dried-up, despondent world of ultra-low or non-existent expectations… about the possibility of progress of this kind. It is a world which long since gave up any hope whatever of really understanding mathematics, or physical reality, or why we are here,or how to live appropriately in today’s conflicted, bumpy, contradictory melee.
So against this deadening backdrop there is no easy way to introduce such out-of-the-ordinary new promising developments. They are a quantum jump away from today’s downbeat preoccupations.
What is required in these circumstances is probably a Channel of Ordinary Thoughts which embodies a vein of —critical, qualified— openness to better things. Like the prisoners in Plato’s Cave, who have now been unexpectedly able to escape, a re- or pre- acclimatisation phase may be needed —if they are to accustom their eyes tothis new painfully dazzling light, after enduring for so long the stoic state of peering into darkness.
We know the experience of homo sapiens was, from the earliest times —for many thousands of years— that there was a body of personal wisdom in circulation which provided essential guidelines and facilitated friction-minimised living. It took the early form of rustic, tribal traditions and myths (primitive religions) and these opinions gradually developed until around the year 0 CE (+ some years) when monotheism burst upon the scene. This was a turning point. The new idea was that a single Infinite Supermind or Deity must have created the universe. The ancients knew that powerful minds could create amazing places like the Pyramids, Rome, Persepolis, Alexandria... So a mind could be envisaged behind the creation of the universe, but the scale of this was open-ended, so it must be an infinite mind. (It was difficult to think of any alternative explanation. It also looked like the only possible way toaccount for the sustained enforcement of a suite of Natural Laws, like that an unsupported stone falls downwards.)
But during the last five BCE centuries Greek thinkers had developed mathematics into a cool, elegant, rational logos, and it began to seem that there was another factor deeply involved in these Natural Laws —mathematics. Some of these Natural Laws could be explained by means of mathematics —with the help of “self-evident principles” (sic) which said, for example, that because the circle was the perfect curve, it was the only shape suitable for the trajectories of the “heavenly” (i.e.incorruptible) bodies like the Sun, Moon and the planets. This pointed to the inescapable conclusion: that it must be a combination of God + Mathematics which had been responsible for human beings and the natural cosmos.
So mathematics came into human history as a profoundly authoritative (but technical) logos at a time when primitive religions were dominantoverall. It (maths) then began to exert a benign influence on the thinking of the most reflective people, because it hugely increased the facility of bargaining and dealing —and hence wealth— in the Ancient World. (It also dramatically improved military effectiveness and the design of elegant, iconic buildings.) Eventually its influence seems to have led to monotheism, because primitive religion was a rag-bag of traditions, narratives, images, rites, superstitions… which could hardly stand the comparison with the powerful, clear, unified thinking which had resulted from the general use of maths.
This historic switch to monotheism meant that the destiny of religion and mathematics became intertwined. Mathematics, it was clear, was the “perfect language God must have used when he created the universe”.
But the switch to monotheism was never going to be easy. When St Paul preached in the ampitheatre at Ephesus he was trying to persuade his audience of local Romans that they should give up their extended family of minor Gods and Goddesses, in favour of a single, rather awesome, but abstract Father–and–Son Deity. It was a difficult message to sell.
To these locals St Paul’s core message sounded very like atheism —and it would also mean an abrupt end to the lucrative trade in effigies of their local Goddess, Artemis. (So it provoked audience unrest, which eventually turned to riot, and subsequently the seizing and imprisonment of St.Paul.)
However the implicit comparison between the clarity of mathematics and the muddle of religion was telling. It eventually led to the Christianisation of the Roman Empire. Once this had happened, the alliance between mathematics and religion became a permanent fact.
We can see today with the advantages of hindsight, that this long–time link between religion and mathematics, has been to the detriment of mathematics. Its main effect has been to polarise the subject. On the one hand there was the huge, lower, mundane, utilitarian branch (arithmetic and applied maths), and on the other, a much smaller, semi-mystic, elegant, “pure”, timeless branch (pure mathematics). As a result of this stratification almost all the many millions of Ogdens of intellectual effort which have been devoted to trying to understand the subject, have been lavished onto pure mathematics. (This was true in the past and it remains true today.) Applied maths has been virtually ignored —as a messier, lower, insignificant activity, suited to lesser talents and to those mainly intent on making money.
As a result pure mathematics inevitably became a highly-regarded performance subject. Taking a career in it was like becoming a concert pianist. The performance became the main focus, not the product.
The “truths” which emerge from the pursuit of this performance subject are, of course, noted, but they don’t have major significance in telling us about the nature of the universe. It suffices that they are examples of a timelessly true logos, frought with difficulty, sometimesthe fruits of logical genius. This turns a career in pure mathematics into something like a secular high priesthood.
In the 17th century Descartes and Newton were splendid role models —mathematicians of the highest order who were interested in applying their knowledge to understand the world, as well as developing its(maths’) internal felicity.
But by the end of the 19th century most of the best mathematicians were devoting their time to developing the subject’s internal operations. A good illustration of the change this has brought about is transfinite theory, the notion that there are higher degrees of infinity. This was the brainchild of Georg Cantor a late 19th century mathematician who was obsessed with religion. He discovered an early sign of the incompletability of mathematics, namely that there can be no final, canonical list of real numbers. But he interpreted this as a sign that there must be more than a normal infinity of them. It is now generally agreed that there is at most a normal infinity of properly-defined real numbers. So Cantor’s transfinite sets must be composed mainly of indefinable real numbers. This strains the credibility of maths to the limit, because everyone knows that they have never seen, and never will see, a bona fide example of an “indefinable real number”. To believe in them is like believing in ghosts.
So the result has been to turn the mathematics-religious link into a link between a slightly mystic, elite, cult subject and religion, not a link between mathematics-as-a-way-to-understand-the-world and religion. Ithas had the unforeseen effect that the subset of “pure mathematics”involved has become gradually more abstract, more refined, more obscure, and more arcane: thus incidentally losing most of the awe and public believability which prompted the link in the first place.
In recent times things have worsened.
An unfortunate concurrence of two effects has been disastrous for pure mathematics. First, the computer revolution came along, then there was the decline of religion caused by the four Whammies (atomic energy, the computer, space exploration and DNA) —events of biblical significance nowhere mentioned in the Bible.
The computer revolution has amplified the power of mathematical modelling —the use of abstract skeletons of situations to form working analogies of proposed situations— a thousandfold. This upstages the pure branch, which begins to look relatively insignificant, obscure and artificial. The assumption —so confidently made in earlier times— that “only higher pure maths counts” has become visibly absurd. Higher pure maths is now only a fraction of the whole, as well as beinglumbered with unsolved contradictions and absurdities like transfinite theory.
The two great proofs of pure mathematics discovered in antiquity were that the square root of 2 can never be expressed exactly as a fraction, and that there is an infinity of prime numbers. These brilliant proofs can be understood by anyone who has minimal mathematic literacy. That such simple proofs have the capability of proving things with infinite implications, is a major triumph of the Classical world… indeed a major landmark in the history of human thought. But they are now buried underneath a vast mountain of difficult, inaccessible, unmemorable (for the ordinary person) technical results. Pure mathematics has lost, by over-staying its welcome, its former inspiring, spectacular image.
Recent decisions haven’t helped.
Unfortunately the gurus of higher maths abandoned their applied colleagues in the 1960s when they agreed with the computer industry that their machines <<had nothing to do with maths>>. The result, though, has been quite different from that expected. A golden age of mathematical modelling followed, but it was not called ‘mathematics’. It was all put down to the magic of computer power. (This has grossly and misleadingly enlarged the public’s faith in the power of computers, anddone almost nothing to make maths seem relevant in schools.)
Now the pandemic has revealed the true situation: that mathematical modelling is a powerful way to illuminate the future.
There are some downsides to today’s situation. (1) The public have lost their earlier sense of the special, transparent, even-handed nature of mathematical authority, (2) mathematics, the clearest, most accessible Heartland of Truth, has been devalued, (3) school mathematics is in a mess, (4) remaining traces of the official authority associated with mathematics are being used cynically to give a bogus credibility tooppressive algorithms online.
Does it matter? Yes! The 2,500 year old Western Project which has led to today’s sophisticated world, is founded on the wide use of mathematics to improve and understand the nature of things. But pure mathematics has been hell-bent on becoming more specialised, more useless and more introspective since the early 19th century. It has only succeeded in digging itself into an impossibly deep hole.
Today, at last, the centrality of modelling with mathematics —as the main source of the public’s “sense of meaning” about maths— hasbecome an established fact. And reflection on how mathematical modelling works —by providing lean, abstract analogies— has finally led to the conceptualisation of a new, quite different, kind of random, dynamic, abstract modelling, Actimatics.
For more than two thousand years it was accepted as the most certain,certain truth that pure mathematics was the only possible ‘Ultimate Representation of Reality’ (URR). This was utterly beyond question.
Now it turns out that it isn’t the URR at all. The URR has switched from maths to actimaths.
The way out of the fly-bottle (Wittgenstein’s metaphor for philosophy) is becoming plain.
CHRISTOPHER ORMELL 1st June 2021