Ellis Hillman: Mathematics and the Imagination

The history of mathematics has been tackled with varying degrees of success by literary mathematicians over the last few hundred years. Perhaps the most successful and readable of these histories is Dr. J. Struick’s A Concise History of Mathematics (London, G. Bell & Sons Ltd. 1954). Dr. J. Struick is Professor of Mathematics at Massachussetts Institute of Technology, and in the compass of 299 pages he has succeeded in describing “the main trends in the development of math ematics throughout the ages and of the social and cultural setting in which it took place” (Introduction p. xii). His book begins with the following sentences:

Our first conceptions of number and form date back to times as far removed as the Old Stone Age, the Paleolithicum. Throughout the hundreds or more millenia of this period men lived in caves, under conditions differing little from those of animals, and their main energies were directed towards the elementary process of collecting food wherever they could get it. They made weapons for hunting and fishing, developed a language to communicate with each other, and in the later paleolithic ages, enriched their lives with creative art forms, statuettes and paintings. The paintings in caves of France and Spain (perhaps of 15,000 years ago) may have had some ritual significance, certainly they reveal a remarkable understanding of form. (Ibid p. 1)

The early numerical systems arose in connection with the development of the crafts and of commerce. Numbers were arranged and bundled into larger units, usually by the use of fingers of the hand or of both hands, a natural procedure in trading. This led to numeration first with five, later with ten as a base, completed by addition and sometimes by subtraction, so that twelve was conceived as 10 + 2, or 9 as 10 – 1. Of 307 number-systems of primitive American peoples investigated by a W. C. Eels, 146 were decimal, vigesimal and quinary vigesimal.

From these primitive beginnings, arithmetic, mensuration and geometry developed, or rather the stimulus for each new advance, or rather the stimulus for the application of each new “breakthrough,” coming from some social or national crisis within society.

It is not our intention to review the history of mathematics, and draw out the continual struggle between the scientific, rational approach and the mystical religious approach to numbers, to numerical or algebraic operations – but rather to extrapolate, as it were, into the future, the next advance in mathematical thinking.

Dr. J. Struick shows how the fundamental concepts of number, of numerical operation and of function have evolved over thousands of years. Each new advance was met by a blank wall of incomprehension on the part of the Establishment of the time, quite apart from the general level of ignorance and superstition which eyed all, new ideas or theories with grave suspicion.

Numerical concepts, as “abstractions,” are most likely to have arisen in connection with everyday problems such as evaluating the “work” of a neighbour in terms of so many cattle, or so many wares. With the development of social intercourse between men, with barter and the general exchange of goods, concepts such as negative number could be more easily grasped. Once the exchange of goods became general people would find themselves owing 2 horses to a neighbour, or two stone implements to the friend “across the road,” and owing came to be expressed as a minus sign.

Multiplication and division were later developed as transactions took on more complex forms. Multiplication as a repeated addition, or as a “shorthand” for successive additions – and division as shorthand for successive subtractions – must have grown up alongside the increasing scope and complexity of transactions within communities and between tribes.

Although some mathematical advances or problems – the solution to many a problem is often determined or given in its presentation – have taken place in the pure field, in the realms of abstract thought long before their practical significance was recognised or applied – this is not true of all advances. Many practical mechanical problems presented pure mathematics with difficulties which had to be “settled” or resolved outside the framework of the then existing mathematical systems.

The next revolutionary breakthrough in mathematical thinking found itself confronted with the concept or problem of negative growth. All the generally known and used functions in mathematics, such as log x, sin x, cos x, th x etc., have been reduced to a sort of universal “base function,” E^x. E is one of the universal constants, one of the bricks upon which functional mathematics has rested for some hundreds of years.

It is the exponential growth function e^x which holds the key to the concept of “negative growth.” Hitherto decay has been accepted as the polar opposition, the “negative pole” as it were, of .positive growth. More careful reflection reveals the full relation of “decay” to “growth.” Decay is but retarded growth. All metabolic processes reach a maximum point of activity, and then slowly or otherwise reduce their activity to the point or level from which the activity had started. This reduction of metabolic activity has been regarded as “negative growth” when, in fact, it is but a retardation of growth, just decay.

These processes are a “closed circuit” operation in that the final level of activity, or end-level, is the same as the original zero level. Is it possible then to conceive a growth “behind,” or through the looking glass – a reversed image growth?

Lewis Carrol did not only write Alice’s Adventures in Wonderland and Through the Looking Glass and What Alice Found There; he was also a mathematician. At one level his writing betrays an outlook on the world which corresponds closely to this “bicameral” view of positive and negative or reversed image exponential growth.

In “The Garden of the Live Flowers” from (Through the Looking Glass) the reader is given this description of Alice’s extraordinary experiences:

For some minutes Alice stood without speaking, looking out in all directions over the country – and a most curious country it was. There were a number of tiny little brooks running across from side to side, and the ground between was divided up into squares by a number of hedges, that reached from brook to brook.

“I declare, it’s marked out just like a large chess board!” Alice said at last. “There ought to be some men moving about somewhere – and so there are!” she added in a tone of delight, and her heart began to beat quick with excitement as she went on. “It’s a great huge game of chess that’s being played – all over the world – if this is the world at all, you know. Oh, what fun it is! … How I wish I was one of them! I wouldn’t mind being a Pawn, if only I might join, though of course I should like to be a Queen best!” She glanced rather shyly at the real Queen as she said this, but her companion only smiled, pleasantly, and said: “That’s easily managed. You can be the White Queen’s Pawn, if you like, as Lily’s too young to play, and you’re in the Second Square to begin with: when you get to the Eighth Square you’ll be a Queen.” Just at this moment, somehow or other, they began to run. Alice never could quite make out, in thinking it over afterwards, how it was that they began: all she remembers is that they were running hand in hand, and the Queen went so fast that it was all she could do to keep up with her: and still the Queen kept crying “Faster!”; but Alice felt she could not go faster, though she had no breath left to say so.

The most curious part of the thing was, that the trees and other things around them never changed their places at all: however fast they went, they never seemed to pass anything. “I wonder if all the things move along with us?” thought poor puzzled Alice. And the Queen seemed to guess her thoughts, for she cried: “Faster! Don’t try to talk.”

Mathematics and the Imagination by Ellis Hillman

This description of the Looking Glass Chess Board fits in perfectly with the reversed image growth function, E (x), which is:


and its “real” world equivalent e^x i.e.:

1+x+x^2/2!+x^3/3!+ …

The zero multiplying the divergent series (a well-known asymptotic series) performs the function of an active zero operator which in real world terms would appear as a rapid divetgent movement (Faster! Faster! said the Queen) without in fact shifting from the point of departure (“The most curious part of the thing was that the trees and the other things round them never changed their places at all, however fast they went, they never seemed to pass anything!” [my emphasis]).

In real world terms, an increasingly divergent series for increase of speed, which is apparent only, the movement being, in fact, zero. It is possible to reconstruct the mirror-image, the “trigonometric and hyperbolic functions” using E^(x) as the new base function.

“Meaningless” problems, such as how much interest will accrue to a person investing 5% of his £x capital over a period of -5 years, say, can be solved by using the special mirror version of the Bicameral Series. Physical problems such as the concept of abolute zero, or the absolute limit to the speed of light may well find their solution by regarding these “absolute zeros,” or absolute limits as “mirrors,” behind which “matter reconstitutes itself” as “through a looking glass” taking on properties which bear the same relationship as E^(x) does to extend to e^x.

– – –

DID FREUD fake the evidence on infantile sexuality? Ought his famous couch be displayed in London or Vienna? Should his letters be embargoed? One after another the rows break out and none of them seem to get settled. Contradiction is built into the movement; as Nick Isbister reminds us (TLS 8 Aug 86): “The ‘great revealer’ was self-confessedly ‘a careful concealer.'” It is the pattern of behaviour of the neverending quarrels of the left, confirming that psychoanalysis and political reform/revolution spring from the one ideological root.

from Ideological Commentary 25, January 1987.