Reviewed by Charles R. Card
To develop his thesis, Robertson employs something like a "spiral-down" strategy to move from a post-Renaissance philosophical background, past major historical milestones in the development of mathematics and psychology, onward to the specific areas of psychology and mathematical logic that were the provinces of Jung and Goedel. In the first four chapters of his book, Robertson discusses the challenge to medieval Scholasticism posed by the emerging Renaissance ideal--"all our knowledge has its origins in our perceptions"--that found expression in the art of da Vinci and Michelangelo and the astronomy of Copernicus. He sketches the origin of modern mathematical analysis in the discovery of analytic geometry by Descartes, the development of differential and integral calculus by Newton and Leibniz and its extension by the Bernoullis, Euler, d'Alembert, and others, the development of imaginary numbers by Cardan, Gauss, Wessel, Argand, and others, and the later discovery of non-Euclidean geometries by Bolyai, Lobachevsky, and Riemann. As well, he traces the development of modern philosophical thought through the writings of Locke, Berkeley, Hume, Hartley, Reid and particularly Kant, whose postulation of inherent mental categories and consideration of "das ding an sich" established a philosophical precedent for the later development of the archetype concept.
With this background established, Robertson begins to explore the historical developments that led on one hand to Jung's psychology of the unconscious and on the other hand to Goedel's Incompleteness Theorem. As his narrative progresses, it alternates between developments in psychology and those in mathematics, showing point by point the progression of the development of an archetypal mode of thought in each area until in the last chapter both streams are brought together in a discussion of number as archetype. First, he explores the emergence of psychology as a distinct science in the second half of the 19th Century through the experimental research of Fechner, Helmholtz, and Wundt and the clinical research of Charcot, Janet, and Bernheim. Freud's pioneering work in the foundation of psychoanalysis is then examined, with particular attention paid to his interpretation of dreams and theory of sexuality. Jung's personal background and the vicissitudes of his relationship with Freud are then discussed, and the emergence is highlighted of Jung's original theoretical perspective and associated concepts, in particular the concept of archetype, which followed and, in fact, were made possible by his break with Freud and Freudian psychoanalysis. Robertson then presents something of a primer of Jung's analytical psychology, with individual chapters devoted to Jung's model of the psyche and the archetypes related to the development of the personality, namely the Shadow, the Anima/Animus, and the Self. He then examines Jung's studies of medieval alchemy through which Jung came to realize that many of its arcane practices and symbols prefigured elements of his own discoveries in the psychology of the unconscious. In Mysterium Coniunctionis Jung brought his alchemical studies to bear upon the themes of psychic and psychophysical union, and it is with this complex and profound text that Robertson ends his survey of Jung's work.
As the mathematical portion of his narrative progresses to a discussion of Goedel's Incompleteness Theorem, Robertson summarizes the attempts of Weierstrass and Dedekind to redress the problems that were engendered by the treatment in calculus of infinite and infinitesmal quantities. He then reviews Cantor's theory of transfinite numbers by which that mathematician established methods for distinguishing different types of infinity. As an extension of his theory, Cantor proposed that all infinities inherent in the continuum of real numbers are expressable as the power set of the integers, and Robertson highlights this Continuum Hypothesis because of its relevance to the later discussion of number archetypes. Robertson then outlines two bold initiatives undertaken in late 19th Century mathematics to achieve a unified, complete, reductive framework which as well would resolve the problems and paradoxes that result from the concept of infinity. He first examines Russell's attempt to develop a consistent and complete arithmetic from the axioms of symbolic logic and thereby subsume all of mathematics to logic, an effort that grew out of the work of Peano and Frege and that culminated, with the help of Whitehead, in the Principia Mathematica. He next discusses the program of Hilbert to demonstrate the consistency and completeness of arithmetic using a formal axiomatic approach. Hilbert had identified twenty-three problems that needed to be resolved to achieve this goal, and of these, two in particular would be shown later by Goedel to be false or undecidable. Robertson then quickly sketches Goedel's ingenious approach to the proof of the Incompleteness Theorem. That theorem established that if an axiom system such as the one for arithmetic is consistent, then it is incomplete--i.e., there exist arithmetic statements that are true but which cannot be derived from the axiom system--and as a consequence, it effectively derailed the efforts of Russell and of Hilbert and seemingly put an end to any subsequent hopes for a single, unified, all-encompassing, fully reductive account of reality based upon a finite number of axioms.
After surveying the works of Jung and Goedel and displaying each against a panorama of developments in post-Renaissance thought, Robertson reaches at last the place where he can establish a connection between the work of Jung and of Goedel. He first discusses Jung's view that number itself is archetypal: "'It may be the most primitive element of order in the human mind . . . . We [can] define number psychologically as an archetype of order which has become conscious,'" (Jung, quoted by Robertson, p. 270). He then makes the claim that ". . . Jung felt that number might be the primary archetype of order of the unus mundus itself: ie., the most basic building blocks of either psyche or matter are the integers." (p. 270) At this point Robertson reaches the nub of his argument: if the archetypal hypothesis that has emerged from the work of Jung, Pauli, and von Franz holds that the most basic building blocks of psyche and matter are integers, then the properties of integers must be sufficient to account for the nature of the geometrical continuum. In other words, Cantor's Continuum Hypothesis must indeed be true. Here at last Jung meets Goedel. As Robertson relates, Goedel took up the proof of Cantor's Continuum Hypothesis subsequent to his proof of the Incompleteness Theorem, and he came to suspect that within the axiom system of standard set theory, the Continuum Hypothesis was in fact one such formally undecidable proposition, the possibility of whose existence had been established by the Incompleteness Theorem. Goedel was able to complete half of the proof of the undecidability of the Continuum Hypothesis, and Cohen completed the remainder of the proof in 1963. If the Continuum Hypothesis is formally undecidable within standard set theory, then the possibility of fully describing the properties of the continuum on the basis of infinite sequences of integers is thrown into doubt. Goedel believed that an extension to the axiom system of set theory could be constructed in which the validity of the Continuum Hypothesis could be determined, but Robertson points out that no such extended theory has been discovered in nearly fifty years. As a consequence, the prospects for finding representations of the unus mundus based upon the integer nature of the number archetypes is called into question. Robertson then wonders,
". . . if even Jung would have agreed with himself, had he lived longer. For, after all, isn't the attempt to reduce the world to the archetypes of the small natural numbers much the same as the attempt of Russell and Whitehead to reduce mathematics to logic, the attempt of physicists to reduce the physical world to first atoms, then subatomic particles, then most recently to quarks? Isn't it more likely that the world is richer than we can ever hope to comprehend? "
As he brings his book to a conclusion, Robertson reiterates and expands on this perspective:
"Jung thought that there was a unitary reality--the unus mundus--that underlay both psyche and matter, and speculated that the primary archetypes of this unitary reality were the simple counting numbers. In this case, each number is, itself, a true symbol: undefinable and inexhaustible-- a much less reductionistic stance than the hope that all reality can be reduced to logic. But even so, doesn't it seem unlikely that we will ever find any lowest level to reality? . . . The world is a place of magic and wonder. . . .The archetypal hypothesis is a starting point to explore that wonder, not an end point to circumscribe its possibilities."
Robertson's argument raises important issues for understanding the archetypal hypothesis and its implications. If the number archetypes are to be understood to be completely equivalent to the integers, then Robertson's evocation of Goedel's and Cohen's work on the Continuum Hypothesis is valid without qualification. The archetypal hypothesis would then be seen to be a recycled version of Pythagoreanism, brought to the dust by the undecidability of the Continuum Hypothesis, in much the same way as the original dream of the Pythagoreans foundered with the discovery of the incommensurability of the diagonal of a square with its side. The crucial issue, then, is whether the number archetypes are simply the integers. It is unfortunate that Jung does not express himself clearly on this point, and even von Franz, whose work significantly clarifies and advances Jung's initial thoughts about number archetypes, is not altogether consistent in her discussion of the relationship of number archetypes to the integers. This issue, in fact, is the re-emergence of a ghost that haunted Jung's efforts to have his concept of archetype generally understood and accepted by a wide community of scholars; this recurrent problem is the failure to distinguish between the concept of the archetype-as-such and the representations of the archetype, the images and ideas that are the specific realizations of the archetype, each pointing toward the existence of the abstract archetype-as-such but none being strictly equivalent to it. In the context of number archetypes, this problem emerges as the failure to distinguish between the number archetypes-as-such and a specific representation of the number archetypes such as the integers. Thus the number archetypes are not the integers; the integers are only one specific representation of the number archetypes. As with specific representations of any archetype, the integers are symbols with an unlimited potential for expression, but their properties do not exhaust all of the possibilities for representation implicit in the number archetype-as-such.
The application of the undecidability of the Continuum Hypothesis to one representation of the number archetypes, namely the integers, does not, in fact, lead to a conundrum for the archetypal hypothesis. Rather, this result is in concordance with the above-mentioned fact about representations of archetypes: no representation is a complete representation. Apart from what is expressable by any specific representation of an archetype, there are other aspects of the archetype that are valid but unrepresented. Here, it seems, Jung's empirical findings resonate with Goedel's Incompleteness Theorem.
From this perspective, the answer to Robertson's question, ". . . isn't the attempt to reduce the world to the archetypes of the small natural numbers much the same as the attempt of Russell and Whitehead to reduce mathematics to logic...", is simply, "no". It is an important distinguishing feature of any prospective archetypal science that it entertains no dreams of a Final Theory and has no presumptions of being a Theory of Everything in the sense that contemporary physics presently imagines. An archetypal science would be, by its very nature, a self- reflective science that is not only aware of its own epistemological capacities and limitations but actually incorporates them--eg. archetypes--in its representations of phenomena.
Given these considerations, the question naturally arises. "If integers are simply one representation of the number archetypes, what other representations might there be? Finding such representations would be a prerequisite for any future development of science from an archetypal perspective. Some hints in this direction can be found in the work of von Franz. As briefly mentioned by Robertson (p. 269-270), von Franz identified a quaternio of ordering archetypes, naturally associated with the first four integers but whose qualitative representations are often dynamical in nature. She summarized their characteristics as follows:
Numbers then become typical psychological patterns of motion about which we can make the following statements: One comprises wholeness, two divides, repeats and engenders symmetries, three centers the symmetries and initiates linear succession, four acts as a stabilizer by turning back to the one as well as bringing forth observables by creating boundaries, and so on. (p. 74, Number and Time)
The fourth ordering archetype is particularly intriguing because its various representations seem to be concerned with feedback and self-reference. As Robertson has shown so well, self-reference is a recurrent issue in the developments of set theory and symbolic logic that led to Goedel's Incompleteness Theorem. It is worth noting that it also plays a crucial role in the `chaotic' dynamics of nonlinear systems that has attracted much attention within many areas of science in the past decade.
In this book Robertson has covered a vast intellectual terrain, producing thumbnail sketches of five centuries of developments in philosophy, mathematics, and psychology with an enviable compactness and clarity. In constructing his argument, he has expanded the discussion of the archetypal hypothesis beyond the biological and physical sciences, taking it to the foundations of mathematics--set theory and symbolic logic. His work is an original and valuable contribution to a growing literature concerned with the archetypal hypothesis and its implications.