We need to renew our commitment to reason, which means, in effect, rigour, much stamina and open-mindedness in grappling with deep problem issues. Neglect of reason leads in the end to the pathology of personal hedonism, which is blind and inward-looking. The world which sleep-walked into Covid-19 in January 2020 was a world in which hedonism was the dominant attitude.
There is a feeling, though, that we need some visible success in this quest at the present time, because we have lived through nearly four years of demoralising Trumpist, ‘post truth’ leadership in the democratic, liberal world.
Reason didn’t go into decline by accident. It came about as a result of the scientific gurus giving up trying to believe in their own freewill (=authenticity) in 1919, and the mathematic gurus giving up trying to understand their own subject by accepting ZF theory in the mid 1920s.
This abdication of reason led to the most horrible, brutal century in human history, the 20th century.
We need a dramatic reversal of this extreme defeatism about reason.
Well two dramatic successes have been gained during the lockdown. One is the author’s elementary proof of Fermat’s lost theorem. No mistake has been found in the reasoning. The main section of the argument has been around since 2016 and no mistake has been found, even though a reward of £600 was offered by the author. All that can be asked of mathematic reasoning which constitutes a proof is that it should be mistake-free. It was sent to the Mathematical Gazette and the Editor and two referees could not find a mistake. The presumption must now be that it is a definitive proof. The onus, on anyone who disagrees, is to find a mistake.
The second is the author’s new monograph A First Handbook of Actimatics, another fruit made possible by the extra time available in lockdown. It means that an era which started in about 530 BCE —during which mathematics was believed to be the only 100% abstract way to describe the universe— has come to an end. There is now a second credible, rigorous, 100% abstract way to describe the universe. The new abstract language is based on tallies like the first.
Kurt Godel showed in 1931 that the whole of mathematics could be arithmeticised —i.e. expressed in terms of numbers. But numbers are a generalised way of speaking about bundles of tallies. For example, when we say that there are twelve months in the year we are tacitly comparing the months with the bundle ////////////, because the word ‘twelve’ is associated with this particular bundle. So, in effect, the entire body of mathematics is a language crafted out of tally bundles.
The trouble with this is that they are inert and lumpen. Mathematicians try to make a virtue out of it by describing mathematical truths as ‘timeless’, but this means that they are only good for describing timeless things. ///+///+///+ /// = ////+////+//// is a timeless truth, but it doesn’t begin to capture anything like the extraordinary restless activity of the natural world —meteors, waves, rain, storms, novae, seasons, volcanoes, fish, insects, animals… Then there is the ‘willful, restless and creative’ activity of human beings. This is still further away from anything like the timeless truths which can be constructed using tally bundles.
So mathematics reduces in the end to a rather static, wooden, timeless collection of nouns, associated in the end with immensely complex tally bundles. When used as a language, it has to rely on human inputs, for example a determination to go on and on adding tallies… indefinitely. This is where we signal <<and so on, ad inf>> and which gives us the concept of infinity.
Descartes and Newton discovered 300 years ago that they could liven-up mathematics to a remarkable extent by imagining that a number t –representing time in a descriptive formula– was increasing. But this was maths + imagination, and the imagination part depended on human beings. You got one picture of the situation when t was small, but another when t was large. If you iumagined t increasing you could have a mental video of what was going to happen. A mental activity was being imposed onto the maths, and it allowed anything out there with a regular pattern of activity (predictable activity) to be described.
When Newton showed that this could accurately account for the motion of the Moon, the orbital motion of the planets, the trajectories of meteors, the regularity of tides… it was a moment of amazing revelation, a moment when mathematics seemed to be able to encompass the entire natural world. We call this feeling of amazing revelation ‘The Enlightenment’. So this tally-bundling practice called ‘mathematics’ seemed at this juncture to be capable of throwing light onto everything. Yes, but… Rosa Peter, the Hungarian mathematician, famously said that to be a mathematician you have to have <<iron in your soul>>. She, like everyone else, forgot that it was mathematics + imagination which had pulled off these amazing triumphs, not just wooden mathematics. There was also a grim side-effect in the 18th century when Blake’s ‘Satanic Mills’ treated their employees like machines, and when slave galleys transported thousands of young Africans across the Atlantic to work like robots in the sugar, chocolate, coffee fields and cotton plantations. The new predictive powers of mathematics opened the way for the industrial revolution in Britain, and the new wealth it produced meant that the middle classes wanted their sugar, their coffee and their fancy cotton garments. It was hard-nosed fixers who supplied the need, using slaves in America and a slavish worker under-class in Britain. So this so-called ‘Enlightenment’ was not all ‘sweetness and light’, but rather ‘brutality and light’ for the unfortunate slaves and underclass.
Similar remarks could be applied to the Roman Empire. It, too, was founded on the managerial and military discipline which stemmed from earlier (Euclidean) mathematics. It, too, flourished with a universally brutal mathematics-based ‘Enlightenment’.
We mentioned earlier that the 18th century ‘Enlightenment’ came to an end in the 1920s. It was openly discarded by the scientists who stoically accepted that they had to wear the rigid straitjacket of spacetime —which reduced them to automatons— and by mathematicians who gave up even trying to understand set theory.
Whyever did they choose to abandon reason so casually? To be fair, they did not “choose” this at all: they seemed to be forced to take this reluctant stand by the need to recognise that Einstein’s equations worked, and by the failure of the best brains in the world to find any kind of credible explanation for Russell’s Paradox.
Now a way has been found to conceptualise the physical world with relativity, but without postulating that the future is already there. It is called ‘Actimatics’ and it starts afresh with tallies, this time long strings of utterly unpredictable “random dancing” tallies.
A way has also been found to explain Russell’s Paradox as a dynamic contradiction, in which statements with contrary meanings self-stultify serially instead of in parallel as in ordinary ‘contradiction’.
So the possibility of reason is back. But this doesn’t mean that reasoning is going to resume instantly on the scale which it could boast when it was formerly in the ascendant. It will take a long time to rebuild reason as the prime cognitive basis of policy and thinking in society. We have had more than a hundred years of so-called ‘cognition’ with reason in visible decline. This inevitably sapped the vitality of education, and after the nadir of the 1970s, it has led to the post-modern era when the only mantra which could count on the support of the high-profile public intellectuals was <<Anything Goes!>>. If this was correct, Reason, even in its previously weakened form, vanished. <<Unreason Rules, OK!>> became the supposed standard.
It won’t be possible to rebuild reason quickly via education, which has, incidentally, virtually disappeared during the post modern era. However a much-needed radical reform in maths education would help to rebuild reason. It would convey to children from an early age that maths with much imagination can be a source of illumination in the real world. The stumbling block facing a quick return of reason is that indecision on all issues involving culture and values has spread widely throughout society. It is a kind of fog, which obscures previously honoured landmarks and makes it harder to convey to children a ‘big picture’ view about almost anything. This fog and lack-of-cutting-edge is the worst possible way to conduct education. (Private schools usually manage to retain more cutting-edge than state schools, but even here the indecision of the post-modern society has taken its toll.)
The most surprising area where indecision has become endemic is in professional mathematics. The author has found that leading figures in mathematics are commonly unable or unwilling to reason decisively, even using the maths they learnt at school aged 17 and 18. This has happened during a period when when rigour in maths has been tacitly equated with mere mechanical checking process, and has consequently lost its previous charm and authority. Mathematics used to be the Heartland of Truth, and it can only be an item of the highest priority to try to re-create this form of culture in mathematics.
There is good news, and a way forward can be seen.
CHRISTOPHER ORMELL AUGUST 2020