Two valuable, but neglected, sides of mathematics

Mathematics, insofar as it is an asset to the human race, has two distinct sides.  It serves both as the Heartland of Truth (“pure” mathematics) and as the main Pathfinder for Progress (“applicable” mathematics).

It would be more accurate, though, to say that <<it can serve in these two roles>> because today’s priesthood of mathematicians have allowed the Heartland of Truth to go to seed, and have long since washed their hands of any involvement with the real world, let alone with any “Pathfinding for Progress”. (They have tended to treat anyone who champions applications of mathematics as self-serving and commercially driven, thus ignoring the great contribution mathematics has made to the common good.)

If we define a ‘mathematical blossom’ as a line of reasoning which can be followed by (and whose surprising denouement can be enjoyed by) any mathematically well-educated person, the current crop of state-of-the-art results in pure mathematics lack blossoms. Today’s state-of-the-art results can only be understood by those who have ascended the abstract stratosphere via trails of technical, specialisation. In this way the results are bloomless (except to their own specialists), and from the point of view of the ordinary mathematically educated person, the garden has gone to seed.

Does it matter?  Yes, humankind needs an accessible Heartland of Truth to maintain and envalue the presence of truth as an uniquely important social and personal goal. If such a presence had been around in 2016 it is unlikely that the US electorate would have voted Trump into the White House. 

The inhabitants of the Roman Empire had a distinct Heartland of Truth which was, in effect, Euclid’s Elements. The truths included in this book —e.g. that the square-root of 2 cannot ever be expressed as an improper fraction, and that the prime numbers continue to infinity— are much better, paradigmatic truths than empirical generalities like <<water runs downhill>> and <<night follows day>>.  The latter have forced themselves onto our attention only by signalling the brute consistency of what happens. But it is, for all that, unexplained. The best truths —those which deserve the title ‘Heartland’— are those which can be understood and enjoyed by the reasoning mind.

That the priesthood of mathematicians is not interested in the real world is hardly a closely guarded secret:  their unworldly concern with “pure” mathematics and rejection of “applicable” mathematics says it all. For countless centuries they were on the side of the majority, because this was Christendom, and Belief led the public to value truth as the final, pre-eminent good. Pure mathematics was generally reckoned to be the pinnacle of human knowledge, and it was also generally regarded as the language which God must have used when He created the universe. 

This now sounds over-the-top and self-serving. The priesthood has been left high and dry by the fading of Belief, at least in the advanced societies.

They need to wake up and recognise that it would be possible to establish an accessible core sub-set of today’s gone-to-seed pure mathematics. The sub-set could be listed on a Special Catalogue, each item furnished with lucid proofs and explanations. A campaign would be needed to make its presence felt in public.

The gamechanger, though, is Actimatics, a new 100% abstract, 100% rational language which looks altogether more likely to underpin the physical reality of living organisms, plants, animals and human beings.  This new post-mathematic discipline also makes possible a spectacular explanation of the existence of the universe as a self-defining system, which is in effect controlled by the human mind (plausibly itself an actimatic construct) .

But this doesn’t mean that modelling mathematics can no longer be the main Pathfinder for Progress, albeit as implemented on computers.  It can, and will, still maintain this practically essential role. 

If society itself wakes up to what has been missing from education, the groundwork for a much more pro-active kind of mathematical modelling could also be laid. (There is a great way to harness modelling to maths education, because when students are naturally interested in particular narratives, they can apply their modelling skills to teasing-out the consequences —a sure-fire way to motivate involvement.)

This could be as momentous in its practical effect as the emergence of agriculture in pre-history. Before the dawn of agriculture, homo sapiens was a hunting-gathering species, the gathering part searching for edible plant resources which existed naturally in the wild. Agriculture arose from the realisation that edible resources need not rely on the accidents of nature. They can be controlled and planned to suit human needs. The same logic could underly mathematical modelling. 

Given a school mathematics regime based mainly on developing maths as a modelling tool, there would be a much larger pool of well-prepared modellers around.  They would not be idle. There is a huge field of “interesting modellable possibilities” which are not currently being modelled, because good modellers are in short supply, and the best rewards for their effort are to be found in existing projects.  Here, in effect, modelling the implications of good ideas would be like growing crops: an extensive, accepted, pro-active side of the subject.

CHRISTOPHER ORMELL 1^st February 2022