This is the twelth essay offered in the cause of this series: increasing the use of reasoning in maths, and also increasing the perceived relevance of thinking in maths to social progress. The original idea was to assist in raising our faith in reasoning to the level of 20-20 vision. 

This also marks twelve months of presence in the public domain of the author’s 2020 proof of Fermat’s Conjecture by elementary reasoning of the kind Fermat himself probably found in the 17^th century. No disproof has been found so far.

The response of those, especially professional mathematicians, who have noticed the proof and realised that they could, if they wished, check it … is not exemplary. Here is an elementary path of reasoning which, if valid, can only serve as a fascinating way to inspire young scholars. The immense surprise is… that it works!  (It took the author 29 years to find a quite strange way through this labyrinth.)

This proof only uses 17^th century maths ideas, i.e. things we were taught at school aged 17-18.  This is really easy maths. There is no excuse for pushing it aside and begging the conclusion that it harbours a supposed simple mistake.

But there has been an excess of conclusion begging. 

The predominant response seems to have been to assume —with little hesitation— that the narrative <<must, of course, contain a mistake somewhere>> even though two sets of referees at the Mathematical Gazette… and those who have worked the proof online, have been unable to find a mistake.  There is an unspoken assumption here: that <<a mistake must exist>>.  Actually it is not a recommended part of mathematical method to assume, without good reason, that something, x, exists. On the contrary: good mathematicians are, of course, existence sceptics. It should be pointed out, that the lame assumption here… that the proof contains a mistake… is gradually losing its credibility, as no such mistake has emerged.

Those who have dismissed the new proof as nonsense are advised to think again and sit down and actually work through the argument. They will find that it is a surprisingly neat, coherent narrative which finally leads to strikingly simple results. They show clearly —even before formal impossibility appears— that it would require a mathematical miracle to the nth degree for any set of natural numbers, a, b, c to satisfy Fermat’s equation.   (For example, for a solution to exist when n=113 a number —which has no special reason to have the unlikely factor 113— would have to have it as a factor to the power 112.)

But there is also a wider point: that the virtual abandonment of interest in reconstructing Fermat’s elementary lost proof, is an implicit gesture against the central point and magic of pure mathematics —which is surely to set an example of explanation, elucidation and enlightenment. The wonderful examples of antiquity (Pythagoras’ proof that the square root of 2 cannot be expressed as a ratio, Euclid’s proof that there is an infinity of prime numbers)  show that simple mathematical reasoning can deliver strikingly unexpected, powerful results. And if strict logical reasoning can achieve this much in the man-made world of mathematics… what might it be able to deliver —via the same kind of explanatory approach— to throw light on how the universe works!

This was the very origin of modern science, the germination insight which led eventually to today’s world.

It worked.  The entire history of scientific inquiry illustrates how mathematics has, again and again, made previously deeply puzzling things plain and clear.  It is therefore very sad that the superstars of modern mathematics have, in effect, abandoned this cause.  Instead of standing firm and “worrying” elementary problems until solutions have emerged, they have often chosen the escape route of looking at a more generalised, bland version of the same problem.

<<If you can’t solve a simple problem, generalise it, and switch to trying to solve that instead!>> is the virtual motif of modern mathematics since the time of Gauss.  But there is a downside.  The effect of this generalisation-letout is that, when a solution is eventually found, it is far too sophisticated and abstruse for the result it enables to be really appreciated by the mass of ordinary mathematically-literate bystanders.  

The dismal result has been that mathematics, pursued in this way, has gradually lost its former proud reputation as the Common Heartland of self-evident, lucid truth.  Instead it has become the obscure, impenetrable, privileged language of an increasingly isolated, sidelined elite.  Letting this transformation happen in today’s essentially democratic age has been a major blunder.   It has led inexorably to the marginalisation of the subject, and the current abysmal crisis in school mathematics. 

 In the 1920s the philosopher Martin Heidegger was so disgusted with the fudging involved in Zermelo-Fraenkel set theory (which made it appear that the problem of Russell’s contradiction had been solved when it hadn’t) that he announced <<The End of the Enlightenment Project>>.

Do we need to spell out the implications of this <<end of enlightenment>> warning? It led to the most brutal, destructive century the human race has ever endured. 

Long live feet-on-the-ground pure mathematics!

By comparison, obscure, invisible pure mathematics in the logical stratosphere at 60,000 ft has little special appeal to reflective members of the human race, still less a cooling and calming effect on the whole.

CHRISTOPHER ORMELL May 1^st 2021.