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Geometrical Psychology: Mathematical Models of Consciousness by the 19th-Century Psychologist Benjamin Betts

“Imagination receives the stream of Consciousness, and holds apart and compares the different experiences.”

Geometrical Psychology: Mathematical Models of Consciousness by the 19th-Century Psychologist Benjamin Betts

“What makes a mathematician is not technical skill or encyclopedic knowledge,” mathematician Paul Lockhard observed, “but insatiable curiosity and a desire for simple beauty.” But what if this mathematical curiosity and desire for beauty were applied to questions that have perplexed scientists and philosophers for millennia — questions about consciousness, what it is, how it works, and how it shapes our lives? That is precisely what the New Zealand psychologist Benjamin Betts set out to do in the 19th century with his unusual diagrams of consciousness, collected in the 1887 volume Geometrical Psychology, or, The Science of Representation (free ebook) — a predecessor to Julian Hibbard’s geometric diagrams of love.

Educated in England and trained in architecture and decorative art, Betts spent time in India and the Far East immersed in the Eastern philosophical traditions before settling in Auckland to work as a “trigonometrical computer” for the Civil Service department of the local government. He became intensely interested in developing a “science of representation” that would trace the successive stages of the evolution of human consciousness. After completing his first set of diagrams, he sent them to the great Victorian art critic John Ruskin. Ruskin dismissed them. Betts persisted. He next shared the diagrams with his sister, who was close with Mary Everest Boole — the widow of the great English mathematician George Boole. Fascinated more by the mathematical than the metaphysical aspect of the diagrams, she entered into lengthy correspondence with Betts, cited his diagrams in her 1884 reference book on symbolic methods of study, and showed them to her circle of scientific friends, including the president of the Royal Society, as well as to a number of artists.

In his letters to Boole, Betts describes the ultimate objective of his project — not a scientific objective but a spiritual one, aiming to serve as a clarifying force for the human spirit:

When you are thoroughly able to understand these diagrams and the truths they inculcate, when you look at any forms of humanity it will not be at their outward appearance, neither at their hapless struggles after vanities, but at their unhappy Ideal, which is giving them such trouble, and which they would almost fain be rid of that they might eat and sleep undisturbed.


It is not true that there is an Earth in space with individuals wandering about on it ; it is not true mathematically, but each carries its own world with it, and if there is any ground of relation between my world and your world or other worlds, that ground exists in you or me and not in the world, except only as it is a part of each independently.

Animated by his conviction that “Love is the Substance of all things” and Wisdom is Love’s counterpart in the evolution of human consciousness, Betts drew on his time in the East to position his diagrams as an integration of Eastern and Western thought necessary for the advancement of wisdom and love in the service of attaining higher consciousness:

There appears to me to be a fundamental antithesis between Eastern and Western Thought. This would only be carrying out the necessary conditions of all existence, without which Existence would lapse in Being Western Thought has sprung from the Hebrew “I am,” crude and arbitrary at its first promulgation, but subdued, and humanised, and spiritualised in its latest announcement, so that now this Western idea is taking root as a demand for harmony, and is breaking out on all sides as emotional activity, and is even getting quite unanswerable in its demand, but let but the Lily show herself and you will find a wonderful change come over modern history. A sudden breeze springing up, our ship shall again obey her helm and spring forward toward her horizon.

Eventually, Betts enlisted the help of a woman by the name of Louisa S. Cook to edit what became the published volume. Primitive and metaphysically clouded as they may be, his diagrams endure as a visionary early attempt to map human consciousness at the improbable intersection of mathematics and moral philosophy, long before the birth of neuroscience and even before the dawn of modern psychology as we know it.

Cook writes in the introduction:

Mr. Betts has spent more than twenty years in studying the evolution of Man. He contemplates Man, not from the physical, but from the metaphysical point of view ; thus the evolution of Man is for him the evolution of human consciousness. He attempts to represent the successive stages of this evolution by means of symbolical mathematical forms. These forms represent the course of development of human consciousness from the animal basis, the pure sense-consciousness, to the spiritual or divine consciousness; both which extremes are not man — the one underlying, the other transcending the limits of human evolution.

Mr. Betts felt that consciousness is the only fact that we can study directly, since all other objects of knowledge must be perceived through consciousness.

Mathematical form, he considers, is the first reflection and most pure image of our subjective activity. Then follows number, having a close relation to linear conception. Hence mathematical form with number supplies the fittest symbols for what Mr. Betts calls “The Science of Representation,” the orderly representation by a system of symbolisation of the spiritual evolution of life, plane after place. “Number,” Philo said, “is the mediator between the corporeal and the incorporeal.”

Cook notes the form of Betts’s forms:

The symbolic forms which Mr. Betts has evolved through his system of Representation resemble, when developed in two dimensions, conventionalised but very scientifically and beautifully conventionalised leaf-outlines. When in more than two dimensions they approximate to the forms of flowers and crystals.

These mathematical curves might serve as a truer and more scientific basis of classification for Botany than de Candolle’s system or any other yet employed, many so-called amorphous developments of the Flora being readily reducible to law according to this method. For instance, the simple corollas, the horn-shaped corollas, and the bi-axial corollas would supply three main classes of flower forms, each of which might be divided into various distinct sub-classes.

The fact that he has accidentally portrayed plant-forms when he was studying human evolution is an assurance to Mr. Betts of the fitness of the symbols he has developed, as it affords presumptive evidence that the laws he is studying intuitively admit of universal application.

She considers Betts’s model of the imagination:

After the repeated recurrence of any sensation, though slightly varying in form, the individual develops the consciousness of its identity, and he begins to form an image or idea, both of the subjective sensation and of the accompanying objective perception, which he can retain in his mind though the sense affection of which it is the counterpart is transitory. Mr. Betts calls this power of ideation Imagination, using it in the literal sense of the word. As a prism receives a beam of light and deflects the rays, holding them apart so that the colours of the spectrum are separated and distinguished, so Imagination receives the stream of Consciousness, and holds apart and compares the different experiences.

Comparison is represented in the diagrams by the angle-; Consciousness from one-dimensional becomes two-dimensional, the line is expanded to a surface.


And since every idea is dual — e.g., the positive idea of light brings with it the negative complementary idea of darkness — of a colour, its complementary colour — therefore the positive representative line on the right hand of the diagram is duplicated by a counterpart line on the left. The sensation of the present moment is not yet reflected as an idea, nor distinguished by comparison. In the diagram it is the apex of the form. When more than two senses occupy Consciousness the lines representing them are arranged radially round the centre. Although the distinction must then be represented by a smaller angle, it does not follow that it is less in amount, as the form itself of Consciousness has become enlarged. At the same time it is quite possible that when the number of modes of manifestation is very limited the sensations are more vivid, and consequently the distinctions more marked, than when more modes of consciousness are differentiated.

Whether abstracting something as complex as consciousness in such concrete physical forms is a reasonable proposition remains a question for the metaphysicians — but the forms themselves are, unequivocally, pure visual mesmerism.

Betts’s diagrams are now in the public domain and the book is available for free in multiple digital formats from Open Library.

Published November 20, 2012




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