These blogs are offered to show the way in which conceptual, reasoning-led maths can generate interesting, sometimes pivotal, results. In an age of awesomely microscopic, awesomely fast, massive, digital electronics, the manipulation of chosen symbolic configurations will —of course— be mainly handed-over to computers. We can no longer expect to distinguish ourselves by performing chosen sophisticated manipulations. Automated maths on computers will, of course, always be able to out-do the human manipulator.

But where does this maths come from?  Why do some manipulations come to the fore? There is a process of thought which has to begin as putative reasoning —in the heads of people who are acutely curious about the presence of what look like formal coincidences, i.e. mathematicians.  Pure maths arises mainly from intense curiosity about striking patterns in maths… patterns which are obviously crying out for explanations.

And the explanatory solutions to such problems which have been discovered offer a marvellous paradigm… a type of lucid, logical reasoning which has, historically, done much to keep the flame of scientific curiosity alive.  This is a first-class reason for supporting so-called ‘pure maths’ —and for including it in the curriculum to be studied by future scientists.  It is a very strong, unquestionable  “cultural” reason for teaching explanatory pure maths to those who can appreciate it.

The best examples are probably Euclid’s proofs that 2 cannot be a fraction, and that there are an infinity of prime numbers. Both of these results achieve a magic conclusion from simple, easily understood, reasoning.

Unfortunately, though, this unique source of magic has —in recent times— been abused and grossly over-played.

(1) Newton’s calculus and classic laws of gravity attracted much admiration from intelligent observers in Europe.  David Hume experienced this when he took a temporary active role as Chargee d’Affaires in the British embassy in Paris.  It seems to have been particularly envied by the ruling classes in France and Germany.  These dominating hierarchies seem to have decided —consciously or unconsciously— to create their own alternative magic… by consciously promoting exploratory reasoning which broke away from previous norms… moving towards imaginary numbers, non-numerical algebras and non-Euclidean geometries.

This was the dawn of ‘Modern Mathematics’ (or doubly-pure        mathematics), and a result like de’Moivre’s Theorem certainly managed to add previously unguessed new formulae in trigonometry.  This undoubtedly increased the power of maths.  It also, though, “professionalised” maths, because the new concepts were “way-out” and abstruse compared with the simple ideas used by Euclid.  It meant that there was a loss of wide understandability and lucidity… which had been much appreciated since Antiquity.

(2) In the 20^th century a large swathe of doubly pure maths aimed at finding spectacularly elegant results took over. But it could no longer hope to stimulate the kind of self-evident curiosity reflected in the lucid proofs of Euclid… because it could only be understood by a tiny cadre of professional maths experts.  This led in the end to the debacle of ‘New Maths for Schools’ in 1972 and the crash of the good public reputation of higher maths.

The mathematicians of the ancient world (principally the later Roman mathematicians) finally realised that they had come to a brick wall. They had inherited three awkward challenges which did not seem to be soluble.  They were the problems:

A.   How to trisect an angle using only compasses and straightedge.

B.   How to square the circle, i.e. how to construct a square equal in area to a given circle, using compasses and straightedge.

C.  How to duplicate a given cube, using only compasses and a straightedge.

These problems became, in effect, “no-go” areas, and they must have played a significant part in the grievous loss of belief, and the evaporation of an originally oceanic authority. It led to the collapse of the Roman Empire.

In the early 19^th century there was a minimal, valid line of justification for doing doubly-pure maths, the fact that it led, very occasionally, to new kinds of computing power. It also —more admirably—  eventually led to cogent explanations of why A, B and C above were insoluble problems, a very worthwhile result. The downside was, though, that it obscured the principal human reason for doing maths —that it was urgently needed to throw priceless light onto postulated public plans and ambitious future projects.

It also amounted to a huge red herring… helping to obscure the fact that a second 100% abstract logos might exist, and was going to be urgently needed to explain the universe.

CHRISTOPHER ORMELL around January 1^st 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.