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The Decline and Fall of Mathematics

February 21, 2012

“What a piece of work is mathematics! How noble in reason! How infinite in application! in form, how concise and elegant! In solving enigmas how admirable! in understanding how like a god! The beauty of the world! The paragon of the sciences!  And yet at the same time what is this  quintessence of argumentativeness? How dreadful in its applications! How cold as a cadaver to the fingers! How tortuous in its  reasonings! How pettifogging in its distinctions! How ridiculous in its assumptions! Mathematics delights and yet delights not me; no, nor logic neither.”
                                                Hamlet, Act II, Sc. 2  (slightly adapted) ¹

One is sometimes tempted to consider mathematics as mankind’s greatest achievement : certainly without it our society’s unparalleled mastery of the natural world  would be inconceivable and, apart from that, once you have penetrated the formidable barriers of mathematics it offers much pleasure as well. However, everything has its day and mathematics is surely now in its declining phase, undone largely by its very success : it is an overvalued currency and, once enough people realise this, it will come tumbling down fast enough, indeed this is already happening.
The Greeks, Pythagoras and Plato at any rate and their followers, saw mathematics as the gateway to eternal truth, the first by way of  Numbers and the second by way of geometry. Supposedly, Pythagoras once passed a blacksmith’s shop and, hearing the various sounds coming from the interior, had the sudden intuition that sound was based on relationships between whole numbers and that the simpler relations like the octave, the fifth and the third were superior to more complicated and messy ratios. He, or one of his pupils experimenting with plucked strings, hit upon one of the very first scientific law : that the pitch of a musical string varies inversely with its length. Mathematics, in this case Number Theory, was thus from the very beginning intimately connected with scientific discovery.
By Plato’s time geometry and not arithmetic had become the leading branch of mathematics and Plato himself was responsible for making the first radical separation between mathematics and everyday physical reality. Geometry was the study of perfect forms whereas the natural world was, at best, only a poor imitation of the eternal. This was by no means such a foolish theory as it may sound since, as certain sophists pointed out, the theorems of geometry did not apply exactly to the physical world : a tangent to a circle was bound to touch the circumference at more than one point and, for that matter, a drawn straight line was not perfectly straight. And yet the theorems of geometry were ‘obviously’ true —  they could be proved ! The only way to resolve this dilemma was to posit a transcendent world, more beautiful and more real than this one, and it was this world to which the propositioons of geometry applied.  There lines really were perfectly straight, spheres perfectly round and a tangent to a circle touched it at a single point.
Plato’s conception of ‘eternal forms’ was given new life and became much more plausible when, at the Renaissance, the early scientists associated it with Christian theology. God was the supreme architect and mathematician and the truths of mathematics were literally ‘thoughts in the mind of God’ for people like Kepler and Galileo. “The truth which mathematical demonstrations give us [is] the same which the Divine Wisdom knoweth” as Galileo put it.  The mystery of why mathematics, a creation of the mind, could be successfully applied to the study of natural phenomena was now resolved : God had devised the natural laws that Nature was obliged to follow and these laws were, at bottom, strictly mathematical. There was thus no great separation between pure and applied mathematics which is why all the leading classical scientists were also leading mathematicians and vice-versa, Galileo, Huyghens, Newton and Leibnitz.
Thus to the nineteenth and twentieth centuries when mathematics expanded prodigiously but increasingly severed its connection with physical reality. Today it is fashionable, at any rate amongst pure mathematicians and they rule the roost, to view mathematics as a free creation of the human mind that neither has nor needs to have any connection with the actual world (we think) we live in. “The majority of writers on the subject seem to agree that most mathematicians, when doing mathematicvs, are convicned that they are dealing with an objective reality, but then of challenged to give a philosophical account of this rreality, find it easier to pretend that they do not believe in nit after all….  The typical mathematician is both a Platonist and a formalist — a secret Platonist.with a formalist mask that he puts on when the occasion calls for it”  (Davis and Hersh, The Mathematical Experience).            This is essentially having your cake and eating it, and it is incredible that mathematicians have been allowed to get away with such a sleight of hand. The Formalist approach, which reduces the whole of mathematics to the intellectual equivalent of embroidery, at once sabotages the teaching of mathematics in primary and secondary schools and is responsible more than anything else for the repugnance that mathematics inspires in a lot of people (ioncluding myself when I was at school). It is a great pity that no government would dare to impose on professional mathematicians the obligation to occasionally spend some time teaching eleven-year olds, as the author Lancelot Hogben perjhaps flippantly once suggested. I at one time had a pupil of about eleven who came to me after one session in great indignation, accusing me of having “said something that was not true”. I asked what this was. “You said that the sum of all the angles in a triangle is always 180 degrees, but I measured one and found that it wasn’t”. I asked her what result she got and she said, “One hundred amd seventy-nine and a half”.  I  was rather taken aback by this and mumbled something about the inaccuracy of measurements, finally getting her to agree that her triangle had ‘nearly’ 180 degrees. This girl, with such a strong empirical bent, has doubtless been completely put off studying mathematics and, for that matter, physics since the latter subject is now little but abstruse mathematics. Gauss was so bothered by the selfsame problem, i.e. “Is Euclidian geometry actually true?”, that he used surveying data to plot a giant triangle formed by three peaks in Hanover and test the well-known theorem. Fortunately for him, he found the result correct allowing for slight experimental error and breathed again.
This anecdote puts in relief the staggering change in attitudes to mathematics in little more than two centuries : on the one hand we have the greatest mathematician of his time and a man who toyed with the notion of non-Euclidian geometries bothered by a problem of ‘reality’, and on the other an era like the present where only fractious eleven year old children or fringe figures like myself worry about the ‘truth’ of mathematics.
Contemporary pure mathematicians do not simply have an indifference to ‘reality’ but a positive distaste for it : Mandelbrot, a brilliant thinker if ever there was one, scandalized orthodoxy by actually applying recondite mathematics to the ‘real world’ and, God forbid, even producing ‘pretty pictures’. But there is something more serious here : if mathematics is a ‘free creation of the human mind’, its successful applications in science and technology are utterly mysterious.
In reality (sic), present-day mathematics is a vast Art-Deco hotel with suites of empty rooms leading in opposite directions and some of them taking you straight into the icy waters of a non-existent lake : it is a confused medley of the realistic, the formal, the decorative and the useful. It would be nice if we could separate out those portions which are, or could be, applicable to the real world and those which are not. This is unfortunately not possible, but what we can do is to establish a sort of ‘reality’ grade for different branches of mathematics running from 0 to 1. Whole number theory is, if not 1, at least very close indeed to 1, Euclidian geometry somewhat less than 1, Banach Spaces a good deal less still with Cantor’s Theory of the Transfinite scoring very close to zero. In the past the major technical changes often came  directly or indirectly from mathematics but the Information Technology Revolution has been largely spawned by people outside or on the fringe of official mathematics — I was even told by a computer programmer that his firm actually regards trained mathematicians with some suspicion rather than respect. Similarly, Paulson, the head of the FED, os supposed to have said off the rcord that he would “rather meet with a trader than a mathematician”. The reason for all this is not far to seek : an excessive concern with logic and the formal aspects of mathematics makes one unfit for the hutly-burly of the real world wehere decisions have to be taken rapidly on the basis of inadequate and inaccurate data.
The trouble with mathematics is that it is irretrievably linked to a world-view which was in its time a very advanced and productive one but which scarcely anyone (except mathematicians) believes in today. For Galileo and Newton and Kepler, the world had to obey mathematical formulae and principles laid down once and for all by God. This viewpoint presupposed that these laws did not and could not change and that they applied everywhere and at all eras. However, the current culturalm drift is quite other. The primacy of  biology over physics which is going to be of vast importance in this century, leads one, on the contrary, to believe that this ‘Platonic’, ‘top-down’ view is fatally flawed. There would seem indeed to be certain constraints and broad principles built into the universe but these constraints are not necessarily mathematical in the normal sense, nor are they necessarily unchanging. Nature’s way is to experiment, not to obey, and ‘progress’ comes about, not by ‘obeying God’s laws’ more closely but by ingeniously circumventing them whenever possible. Instead of deductions from eternal truths, we have trial and error, a sublimely messy but brilliantly efficient procedure. Mathematical formulae and equations are, by their very nature, fixed and unyielding whereas we require more fluent techniques to mirror a fluctuating and fleeting reality. These techniques are currently becoming more and more available in the form of increasingly lifelike computer modelling, ‘genetic programming’, ‘evolutionary invention’ and the like. The entire conception of Nature and man ‘obeying’ pre-existing laws is a paradigm that we are moving away from fast : the new paradigm is that of vast amorphous entities, Life, the Universe, Nature, ourselves, that ceaselessly try out all sorts of possibilities until they are stopped dead in  their tracks or, in some cases, manage to break through the very constraints that seemed to be everlasting.  “Notre imagination ne concoit clairement que l’immobilite” (“Our imagination can only get a clear picture of what is motionless”). Or perhaps, “Our minds are so made that we can only conceive clearly what is dead”.  This might be a motto on the gravestone of mathematics.

Notes :

¹ The original reads : “What a piece of work is a man! How noble in reason! How infinite in faculty! in form, how moving, how express and admirable! In action how like an angel! In apprehension how like a god! The beauty of the world! The paragon of the animals! And yet to me what is this  quintessence of dust? Man delights  not me; no, nor woman neither.”   from Hamlet, Act II, Sc. 2

In my poem, The Initiates, I have imagined a group of latter-day Greek aesthetes gathered in the house of one of their number:

“We met always by night: a household slave brought in
A tray of sand, giving each visitor a cane,
With joy we gathered round; the latest theorem
Imported from North Africa was scrutinised,
The argument abridged, occasional points of style
Touched up…Then silence fell, a sense of ultimate peace
Came over us; these lines and circles that we traced
Were clearly images of a superior world,
Indifferent to man, exempt from frailties,
War, death, disease, could not affect them and their truth
Did not depend on trial or experiment,
Each step self-evident, demonstrable and sure.”

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