Chapter 4 - The Logic of Symbol

Cherish that which is within you, and shut off that which is without; for much knowledge is a curse.

Chuang Tzu.

Many spiritual traditions have a concept of a second birth, a spiritual birth, which can only occur after we have reached a low point, a point at which all conscious resources are exhausted. At that point of despair, we have to humble ourselves, admit our inability to advance further on our own, and instead turn to a higher power for an answer. Another way of seeing this, which seems very different but is, in fact, the same, is that we come to a point in our life's journey where there is nothing ahead but an abyss, an abyss so wide and deep that it is impossible to see what lies beyond or below. Despite our fear, we have to step out into that abyss, not knowing whether we will fall to our death.

I can remember when I first came to that point in my life. I have discussed it several times before in my books, but it's worth repeating. I was in my mid-thirties and had reached a point of despair. I no longer knew who I was or how I had come to the point I was at. My values seemed useless to deal with the grey fog of depression that increasingly enveloped me. At that spiritual nadir, resources began to appear from within. I began to consult the I Ching, asking endless questions in a dialogue with what I was eventually to discover was my unconscious. Before I began this process I had dismissed divinatory tools, like the I Ching or tarot, as silly games for foolish people. But now I only knew that I needed answers from someplace deeper than my conscious mind.

Soon after that, I also began meditating, as I have already discussed. Meditation brought back body memories of states I had previously experienced, not only in my drug days but also unremembered experiences from childhood. Throughout my childhood, it wasn't uncommon for me to have unusual visual experiences, usually involving light. Sometimes while staring ahead, I would find my visual field stretching forth, expanding, so that it seemed to extend outward for a very great distance. This would sometimes extend to infinity, where it would culminate in a brilliant white light with which I would find myself suffused. Afterwards, my vision would return to normal, as if nothing had occurred. Often this experience would be coupled with a sense of extreme well-being, a sense of the perfection of reality, which would continue for hours, sometimes throughout the entire day. I never discussed this experience. Perhaps I thought everyone experienced this, or perhaps more simply, some part of me knew this was mine alone. While my experiences during meditation were not, on the surface, similar, something in me knew they came from a similar place.

The unconscious was waking in me, but neither the I Ching nor meditation was sufficient in itself to lift my grey fog. I finally decided I needed some form of psychotherapy to deal with the depression which increasingly enveloped me. I found this need intensely humiliating. As with my low opinion of divinatory tools (and anything else even vaguely "mystical" or "occult"), I had always contemptuously dismissed psychotherapy as unscientific nonsense that appealed only to weak people. In my arrogant view, when strong people like me had problems in their lives, we examined the situation rationally and came up with solutions. If those solutions required changes in our personality, then we would own up to our faults and make the necessary changes in our lives. I had never understood why everyone couldn't deal with life in a similarly rational manner. After all, I could--or so I thought. Acknowledging that I needed help from a therapist was a major defeat for my ego. I was to find that acceptance of defeat was a good first step.

Raymond Lull's Ars Magna

[Raymond] Lull introduces movement into memory. The figures of his Art, on which its concepts are set out in the letter notation, are not static but revolving. One of the figures consists of concentric circles, marked with the letter notations standing for the concepts, and when these wheels revolve, combinations of the concepts are obtained. In another revolving figure, triangles within a circle pick up related concepts. These are simple devices, but revolutionary in their attempt to represent movement in the psyche.⁽¹⁾

Like so many ideas now considered purely scientific, symbolic logic began in mysticism. Like his predecessors, St. Paul and St. Francis, Raymond Lull [1235-1315] was born of wealthy parents and led a sybaritic life until converted by visions. Apocryphal stories about Raymond Lull abound. One concerned the manner of his religious conversion. While a nobleman at the court of James I, King of Aragon, by chance Lull glimpsed the bosom of a married noblewoman, when her neckerchief was blown aside by the wind. He fell madly in love. He wrote her love letters, which she ignored, until he finally accompanied them with poems addressed to the beauty of her breasts. In order to persuade him that a man of knowledge should love only God, she agreed to show him the breasts he admired so much. When she opened her blouse and revealed her breasts, he saw that they were covered with a large cancer. His shock at the sight supposedly converted him to a religious life. He began having visions which convinced him of his divine destiny to convert the Moslems to Christianity, and then to die himself as a martyr.⁽²⁾

Raymond Lull was a strange combination of mystic and scientist, as such perhaps a precursor of those of us who find ourselves occupying that same uneasy middle-ground in these late days of the twentieth-century. Living in the thirteenth century, Lull was far too early for the Renaissance attitudes toward the world which would evolve into science. He still came from the Medieval perspective of scholasticism. Like his ten year older contemporary, St. Thomas Aquinas, Lull believed that "the divine law required man to see God by the rational methods of philosophy.⁽³⁾ Yet he was also far ahead of his time, almost twentieth-century in some ways, and managed to develop an abstract method of extending knowledge through a primitive precursor of the computer.

His book Ars Magna (i.e., the Great Art) presented "ideographs representing the primitive concepts which Lull planned to combine in order to express all other ideas and solve all problems of science, religion and philosophy."⁽⁴⁾ Therein he presented "a mechanical method of exhaustively stating the possible relations of a topic."⁽⁵⁾

Imagine a dart board with an inner circle, a middle circle and an outer circle. Each circle is divided into 9 sections. Then assume that the outer circle is fixed and each of the other circles can rotate freely around the center. By rotating the two moveable circles, any section of the outer circle can be paired up with any section of the middle circle and any section of the inner circle; hence there are 729 possibilities (i.e., 9 X 9 X 9). On one circle, he would put ideographs representing relevant subjects, on another ideographs for relevant predicates; in the last circle ideographs for various relevant questions--Whether? What? Whence? Why? How Large? Of What Kind? When? Where? How?⁽⁶⁾

Obviously, you could make up particular examples of this device for virtually any subject you were interested in. Lull was most interested in using the Ars Magna to construct examples which he felt would demonstrate the infallibility of Christian doctrine, and thus prove the superiority of Christianity over Islam. We might think of it as a sort of missionary computer--something which would unfortunately probably be welcomed today by those religious who proselytize either door-to-door or over the airwaves. Lull lectured on this method widely throughout Europe.⁽⁷⁾ Unfortunately for Lull, the Ars Magna proved less infallible at demonstrating the superiority of Christianity than he would have hoped. In trying to convert the Moslems, he was twice imprisoned and banished, then finally stoned so badly that he died soon afterwards on board a ship sailing home to Palma, Mallorca, where he was buried. At least, he achieved his goal of dying a martyr, if not of converting the Moslems.

At this stage of creation, understandably the presentation was more idiosyncratic than generally useful. But consider what a powerful concept it was: reducing thought itself to a number of general principles, each of which could be combined mechanically in order to demonstrate true conclusions. In mathematics, a similar symbolization process is seen in the concept of a variable; i.e., a symbol which represents not any particular number, but any number whatsoever. Since the concept of a mathematical variable was at best only vaguely understood in Lull's time, his intuitive attempt to create symbolic logic was all the more impressive. And, of course, there were as yet no computers, religious or otherwise!

Beginning Jungian Analysis

You cannot speak of ocean to a well frog; its sphere is limited. You cannot speak of ice to a summer insect; its capacity is restricted by time. You cannot speak of Tao to a pedagogue; his scope is confined by teachings. But now that you have emerged from your narrow sphere and have seen the great ocean, you know your own insignificance, and I can speak to you of great principles.

Chuang Tzu.

I called a psychiatric referral service which referred me to a psychiatrist, formerly a Freudian, who was in the process of becoming a Jungian analyst. I knew none of that at the time; he was simply the therapist to whom I was referred. Why I would have received that particular referral is on the surface hard to understand. But when I was later myself a therapist, I found that patients often seem to end up with the type of therapist they need, without any conscious method of selection on their part. Those who need highly cognitive therapy find highly cognitive therapists; those who need body work find therapists who do body work. And, though I didn't know it at the time, I needed someone who respected the unconscious. So I found a Jungian. What really pushed the boundaries of probability is that he was also an amateur magician! That gave us an additional bond from the start.

The first session was normal enough. I told him my problems, embarrassing myself in the process by becoming teary as I confessed my inability to deal with my situation. Again this was a good start at becoming more vulnerable, regardless of the fact that I regarded it as weakness. He listened closely for most of the session, then gave me a Myers-Briggs personality test which he would evaluate before I returned the following week. The Myers-Briggs is a standard personality test based on a simplification of Jung's theory of psychological types. Jung felt that rather than a single developmental path for all, there are a variety of paths which depend on our innate personality type; i.e. just what kind of person we are.

The most basic division among personalities is between introverts and extraverts. Introverts look inward to find their energy, their values, their direction. Extraverts look outwards. As an extreme, imagine a shy, retiring person for the introvert, an effusive used car salesman for the extravert.

Whether introvert of extravert, we all have to have a way of gathering information about the world and a way of processing that information. We can (1) gather it either through our senses or (2) through an access to inner knowledge which Jung termed intuition (a harder concept to describe to those who rarely experience it). Having gathered information, we have to process it in one of two ways: (1) we either think about it, categorize it, or (2) we evaluate it, we feel what it means to us.

In general, we pick one of those four "functions" of thinking or feeling, sensation or intuition, as our primary function, as the way we feel most comfortable with the world. If this is either thinking or feeling, which are ways of processing information, we tend to also develop a secondary function (or even secondary and tertiary), as a way of gathering the information: either sensation and intuition. Similarly if we are most comfortable with sensation or intuition, we also need a way to process that information and so we develop thinking or feeling to a lesser degree than our primary function.

One especially interesting concept in Jung's theory is that because we invest our primary function with so much of our energy and attention, its opposite function is necessarily condemned to be ignored and to sink from consciousness into the unconscious. If we are a thinker, our feeling becomes unconscious, and vice-versa. If we are a sensate, our intuition becomes unconscious. Jung referred to that function which has been pushed down into the unconscious as our inferior function. He felt that because that function is almost entirely unconscious, when we access it, we are opening the doorway into the unconscious. Thus our weakest point, our Achilles Heal, offers our greatest potential for change. Of course, learning about all this was still ahead of me.

More recently, Jungian analysts Mary E. Loomis and June Singer have developed a new personality test, called the Singer-Loomis Inventory of Personality (SLIP) in which Jung's four psychological functions are considered independent of each other, so that one might be a feeler with secondary thinking, or more significantly yet, one's type could change under different circumstances. Though this is a good correction to the way some of the more simplified attempts to apply Jung's theory, my own personal experience is that in most circumstances, Jung's original concept is more useful. Most people do have a superior and an inferior function and the knowledge of this helps both the therapist and the client. They may or may not have a fully developed secondary function, or may have progressed to a tertiary. They may even switch superior functions at some significant life point, but over most extended periods of time, Jung's original concept fits quite well. After all, Jung's purpose was to stress the variety of developmental paths, not to put people into little boxes.

Leibniz' Universal Logic Machine

Nearly four centuries were to pass before symbolic logic took its next big step forward. Since his death, Raymond Lull had come to be famed, not so much for his Ars Magna or his martyrdom, as for his legendary skill as an alchemist and magician. Though scholars differ sharply over whether there were actually two Raymond Lull's, one the martyred logician and one an alchemist, his followers, called Lullists or dreamers, certainly considered the two to be one and the same man.⁽⁸⁾ Seventeenth century Germany was certainly open to such hidden mysteries.

…Germany at this time was still recovering from the horrors and divisions of the Thirty Years' War, still backward and chaotic, its intellectual life often centering round secret societies of alchemists and Rosicrucians and the remnants of Renaissance magic.⁽⁹⁾

In 1666, Baron Gottfried Wilhelm von Leibniz [1646-1716] published his essay De Arte Combinatoria, in which the 20 year old Leibniz⁽¹⁰⁾ presented his plan for:

…a general method in which all truths of reason would be reduced to a kind of calculation. At the same time this would be a sort of universal language or scripts, but infinitely different from those projected hitherto; for the symbols and even the words in it would direct the reason; and errors, except for those of fact, would be mere mistakes in calculation.⁽¹¹⁾

This was to be a two-part project: (1) the creation of a universal language which could underlie all logical and scientific discussion (characteristica universalis); (2) the use of ideograms to represent the concepts of this universal language, together with a set of rules as to how they could be manipulated, hence a "calculus of reasoning" (calculus ratiocinator). Together, they would constitute a universal logic machine.⁽¹²⁾

Frances Yates tells us that "Leibniz was interested in Lullism and wrote a work De Arte Combinatoria based on adaptions of Lullism. . . . But the significant signs or characters of Leibniz's 'charactistica' were mathematical symbols, and their logical combinations were to produce the invention of the infinitesimal calculus."⁽¹³⁾ She goes on to explain that not only was Leibniz influenced by Raymond Lull's ideas, he tried to combine them with those from the classical memory tradition we discussed in chapter one. He intended the ideograms to "represent as nearly as possible reality, or the real nature of things."⁽¹⁴⁾

This desire to find the perfect symbols to represent concepts influenced his later mathematical discoveries, especially calculus. Though there is a great deal of argument as to whether Leibniz or Isaac Newton should be given priority for the discovery of calculus in mathematics, Leibniz's symbology is far superior and has come down largely unchanged into modern mathematics. We will see the power of a successful symbol system later when our discussion of logic advances to G. Spencer-Brown's Laws of Form.

Never one to fail tooting his own horn, Leibniz claimed that the idea came to him while still a schoolboy studying the ten categories⁽¹⁵⁾ into which Aristotle had divided all of reality. Leibniz decided that:

…Just as there are predicaments or classes of simple notions, so ought there to be a new genus of predicaments in which propositions themselves or complex terms might also be set out in a natural order.

…I worked on constructing such predicaments for complex terms or propositions. When, through my eagerness for this project, I applied myself more intently, I inevitable stumbled onto this wonderful observation, namely, that one can devise a certain alphabet of human thoughts and that, through the combination of the letters of this alphabet and through the analysis of words produced from them, all things can both be discovered and judged.⁽¹⁶⁾

It is also significant that, by the late 17th century, the concept of a mathematical variable was largely in place.⁽¹⁷⁾ The joint creation of calculus by Leibniz and Newton would have been impossible without mathematically explicit variables. So it is understandable that Leibniz, enamored both with mathematics and Aristotelian logic, would conceive of extending variables not only to mathematics, but to logic.

Though he thought completing this project would be a relatively simple one, in fact he bogged down when he got to actually dealing with the details. He was also never able to find patrons willing to finance this or a related projected to create a universal encyclopedia. The encyclopedia not only would bring together all knowledge, but each key notion would be assigned a symbol which could be readily committed to memory and which could be manipulated by his logical calculus. All he left were fragments, and they indicate a hopelessly complex approach that shares little with symbolic logic as we later know it.⁽¹⁸⁾ Still it took a genius such as Leibniz to make this leap of faith, regardless of whether he was able to actually produce a workable method. Twentieth century philosopher Stewart Hampshire remarks that:

Leibniz was perhaps the most universal genius of the modern world, comparable in insight with Newton, wider in range and lesser only in ultimate achievement.…Even now the whole of his work has not been published. He was the last man who could hope to master the whole range of modern knowledge, and to be an encyclopedia in himself.⁽¹⁹⁾

We should add that not everyone, either in his own time or ours, had the same high opinion of Leibniz. In Candide, French philosopher/author Voltaire [1694-1778] lampooned Leibniz as Dr. Pangloss, who used torturous logic to attempt to prove that, despite the ample evidence of suffering all around us, we actually live in the best of all possible worlds. Twentieth-century philosopher Bertrand Russell was another who considered Leibniz a tiresome windbag, who thought everyone around him doted on his every word.

When any young lady at the court of Hanover married, he used to give her what he called a "wedding present," consisting of useful maxims, ending up with the advice not to give up washing now that she had secured a husband. History does not record whether the brides were grateful.⁽²⁰⁾

Leibniz' pomposity notwithstanding, his concept of a universal logic machine was so far ahead of his time that it was almost two hundred years before anyone made a serious attempt at creating an actual symbolic logic.

Dream Work

Toward the end of my first session, the analyst told me that a lot of what we would be doing involved dreams, so he would appreciate it if I kept a journal to record my dreams. I told him truthfully that I didn't dream (or at least I didn't remember dreams). He said that was ok, but to keep a notebook in case I did happen to remember a dream. That night I remembered and recorded five dreams!

I can't tell you how moved I was by these initial dreams. I felt like a new world had opened up to me, a world which I knew from the start was as big or bigger than the physical world I knew. Within the first few days, I bought a paperback copy of Jung's Modern Man in Search of a Soul and swallowed it up in one gulp. I found the chapters on dream analysis and the stages of life especially compelling. The former showed me that there was a way to make sense of the wonderful images I was finding in the dreams that were coming to me in droves. Here is one of Jung's remarks that I noted at the time:

…the treatment of dream symbolism demands that we take into account the dreamer's philosophical, religious and moral convictions. It is far wiser in practice not to regard the dream-symbols as signs or symptoms of a fixed character. We should rather take them as true symbols--that is to say, as expressions of something not yet consciously recognized or conceptually formulated.

So my dreams were my dreams, portraits of me and me alone. And they were filled with symbols, which could not be reduced to simple explanations. This last point was especially telling for me. On this point, Jung added later on the same page:

It may seem strange that I would attribute an indefinite content to the relatively fixed symbols. But it is the indefinite content that marks the symbol as against the mere sign or symptom.

When I read the chapter on the stages of life, I felt like Jung was describing exactly the situation in which I found myself. He said that many people find themselves forced to make a major transition in mid-life, a transition marked by pain and despair because it is essentially a spiritual transformation. This made me realize that the misery I was experiencing was a necessary part of my personal metamorphosis to a later stage of life. Like a caterpillar forced to be confined within a cocoon, where its old rigid structure dissolves until nothing is left. Then a new creature--a butterfly--struggles to break through from the chrysalis, in order to fly! One quote I marked then made special sense to me:

The nearer we approach to the middle of life, and the better we have succeeded in entrenching ourselves in our personal standpoints and social positions, the more it appears as if we had discovered the right course and the right ideals and principals of behavior. For this reason we suppose them to be eternally valid, and make a virtue of unchangeably clinging to them.

George Boole's Logical Calculus

By the mid-19th century, mathematics was undergoing a sea-change. Where previously mathematics had been considered the "science of magnitude or number", mathematicians were coming to realize that their true domain was symbol manipulation, regardless of whether those symbols might represent numbers. In 1847, in a little pamphlet called The Mathematical Analysis of Logic, George Boole [1815-1864] clearly defined this new view in his presentation of his calculus of logic:

We might justly assign it as the definitive character of a true Calculus, that it is a method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation…It is upon the foundation of this general principle that I propose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis.⁽²¹⁾

This was the first time anyone had actually presented a fully developed symbolic representational system for logical relationships. These symbols could then be manipulated using an algebra⁽²²⁾ to determine whether complex logical relationships were true or false. Though Boole appeared to be unaware of Leibniz' goal of a "general method in which all truths of reason would be reduced to a kind of calculation," he took a major step toward achieving it. He developed these ideas still further in 1854, with his Laws of Thought.⁽²³⁾ There he presents his purpose as even more ambitious, no less than capturing the actual mechanics of the human mind.

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.⁽²⁴⁾

With some degree of hyperbole, Bertrand Russell once remarked that "pure mathematics was discovered by Boole in a work which he called The Laws of Thought."⁽²⁵⁾ In large part, Russell's overstated claim arose out of Russell's own attempt to reduce first all mathematics, then eventually all human thought, to logic. This later became Russell's own goal, which culminated in Russell and Whitehead's massive three-volume work Principia Mathematicia⁽²⁶⁾, which was named (perhaps through hubris) after Newton's most famous volume. We will return to Russell's attempt later. Boole was more realistic; even in the throes of his creation, Boole understood that there was more to the mind than logic. In a pamphlet she wrote about her husband's method, Boole's wife reports that he told her that when he was 17, he had a flash of insight where he realized that we not only acquire knowledge from sensory observation but also from "the unconscious."⁽²⁷⁾ In this discrimination, Boole was amazingly modern.

Boole's life can be seen as a triumph over the British class system of the 19th century, so faithfully recorded in the novels of Charles Dickens. Boole's father was a simple shopkeeper, which effectively cut the younger Boole off from the normal avenues of intellectual advancement open to those from a higher social class. So he advanced himself through arduous self-study of Latin and Greek, then considered the royal roads to intellectual accomplishment. He worked, meanwhile, as an elementary school teacher, one of the few intellectual occupations open to the lower classes (and treated accordingly). Too poor to aspire to any of the normal learned professions, Boole for a time studied to become a clergyman. But before he could complete his study, he was forced to establish his own school in order to earn enough money to provide for his needy parents.

There Boole found himself. Though he had never before studied mathematics, he found the available textbooks so wretched that he worked his way through the mathematical classics and began to write his own texts for the use of his students. That led him to develop original mathematical ideas which he submitted to the then newly established Cambridge Mathematical Journal. The editor, D. F. Gregory, was so impressed with Boole's work that they began a life-long friendship. This association brought Boole's work to the attention of other mathematicians who were more interested in the originality of his ideas than in his lowly birth. From that time on, his life became one of respect and honor, including an appointment as "Professor of Mathematics at the recently opened Queen's College at what was then the city of Cork, Ireland," as well as an honorary degree from the University of Dublin. Though he died far too young of pneumonia, he was one of those rare creative souls who have not only transcended their beginnings, but lived to hear the appreciation of their peers. It must be remarked, however, that few mathematicians were far-sighted enough to realize the signal importance of his creation of symbolic logic.⁽²⁸⁾

A New World Emerges from Within

Let your hearing stop, and let your mind stop. Let your spirit, however, be like a blank, passively responsive to externals. In such open receptivity on can Tao abide.

Chuang Tzu.⁽²⁹⁾

I came to the second session filled with excitement and with a notebook filled with dreams. My analyst had analyzed the Myers-Briggs test I had taken and was puzzled by the results. It showed that I was an outgoing person open to future possibilities, who though quite rational, trusted his emotions above his thoughts.⁽³⁰⁾ He asked if perhaps I had misunderstood how I should take the test, and had answered the way I would like to be as opposed to how I was. I told him "no", I had answered the way I was. Then he expounded a bit on what the test had showed, still doubting it fit me: dynamic, enthusiastic, skillful in dealing with people, gregarious, etc. I kept nodding my head in agreement: "yep", that was me alright. Eventually he accepted that's who I was, but remained puzzled as he had never had such an extraverted patient. I was later to find we were certainly a tiny minority among those attracted to Jungian psychology, who are overwhelmingly introverted. There are, however, an awful lot of intuitives among those interested in Jungian psychology.

Because I was an extraverted intuitive, the introverted sensate effort I put into dream work was exactly what I needed to open the door to the unconscious. In the process of recording dreams in detail, analyzing their content, making cross-references, and so forth, I was doing exactly the opposite kind of work from what was normally so easy and familiar to me. In most situations in life, I absolutely despised detail work, and spent little or no time looking inside myself. I remember walking along the street with a good friend who was an introvert. I guess I was uncharacteristically quiet, so he asked me what I was thinking about. I laughed and told him I wasn't thinking at all. And I wasn't! Where his mind was constantly filled with a dialogue on what was taking place in his life, my mind was to me as blank as the tabula rasa that behaviorists used to think we were born with. (I stress "to me" as there was undoubtedly plenty going on no matter whether I was aware of it or not.)

By working with my inferior function I experienced exactly what Jung's theory predicted: the smallest insights I found in my dream work were numinous⁽³¹⁾, magical! I felt the same excitement I used to experience with magic. I remember a dream I brought to that second session:

Sliding down a huge slope, riding on something almost like a light wave. I came skidding to a stop at the bottom. I think water greased the slide. I really enjoyed the ride and said "that was a hell of a ride." I look around and see I'm near the entrance to a large amusement park, where all the exhibits are done with light.

My analyst asked if, in the dream, I wasn't afraid of this new situation. When I told him "no", that I was excited with the possibilities, he shook his head, trying to get his bearings. For me, as an extravert intuitive, the possibility of a whole new world opening up for me, was more exciting than I could convey. For him as an introvert, the same experience would have been overwhelming and frightening. We started the process of getting to know each other better. Time and again in my journey, I was to need the insights he provided from the other side of the looking glass, from which he stared back at me.

1. The Art of Memory, p. 176.

2. See Charles Mackay, Extraordinary Popular Delusions and the Madness of Crowds (London, 1841), p. 112.

3. Etienne Gilson in Anne Fremantle, ed., The Age of Belief: The Medieval Philosophers (New York: Mentor Books, Houghton Mifflin, 1954), p. xii.

4. Edna E. Kramer, The Nature and Growth of Modern Mathematics (Princeton: Princeton University Press, 1970), p. 100.

5. W. L. Reese, Dictionary of Philosophy and Religion (New Jersey: Humanities Press, Sussex: Harvester Press, 1980), p. 319.

6. See Dictionary of Philosophy and Religion, p. 319.

7. Encyclopedia Americana, Vol. 17 (New York: Americana Corporation, 1970), pp. 834-835.

8. Paul Lacroix, Science & Literature in the Middle Ages and the Renaissance (New York: Frederick Ungar,1878/1964), pp. 180-181.

9. Stuart Hampshire, The Age of Reason (New York: Mentor Books, 1956), p. 142.

10. In his words "barely out of his school".

11. from De Arte Combinatoria, 1666. Quotation in Frederick David Abraham, Ralph H. Abraham, and Christopher D. Shaw, A Visual Introduction to Dynamical Systems Theory for Psychology (Santa Cruz: Aerial Press, 1990), p. III-21 .

12. Clarence Irving Lewis and Cooper Harold Langford, "History of Symbolic Logic", in James R. Newman, The World of Mathematics, Vol. 3 (New York: Simon and Schuster, 1956), p. 1861.

13. Frances Yates, The Art of Memory (Chicago: University of Chicago Press, 1966), p. 380.

14. Frances Yates, The Art of Memory (Chicago: University of Chicago Press, 1966), p. 384.

15. Also called predicaments: substance, quantity, quality, relation, place, time, situation, state, action, and passion.

16. G. W. Leibniz, "Preface to a Universal Characteristic (editor's title), in Roger Ariew and Daniel Garber (editors & translators), G. W. Leibniz: Philosophical Essays (Indianapolis: Hackett Publishing), pp. 6-7.

17. Though its full philosophical implications were still not yet understood.

18. See Roger Ariew and Daniel Garber, G. W. Leibniz: Philosophical Essay, pp. 10-18 for Leibniz' fragmentary notes.

19. Stuart Hampshire, The Age of Reason, p. 143.

20. Bertrand Russell, A History of Western Philosophy (New York: Simon and Schuster, 1945), p. 582.

21. Carl B. Boyer, A History of Mathematics (Princeton: Princeton University Press, 1995), p. 633.

22. In most ways identical in operation to ordinary mathematical algebra.

23. George Boole, An Investigation of the Laws of Thought: On Which are Founded the Mathematical Theories of Logic and Probabilities (New York: Dover, 1854/1958).

24. George Boole, An Investigation of the Laws of Thought, p. 1.

25. Quotation in Carl B. Boyer, A History of Mathematics, p. 634, among other sources.

26. Alfred North Whitehead & Bertrand Russell, Principia Mathematica (3 vols. Cambridge, U.K.: Cambridge University Press, vol. 1, 1910, vol. 2, 1912, vol. 3, 1913).

27. E. T. Bell, Men of Mathematics (New York: Simon and Schuster, 1965), pp. 446-447.

28. See E. T. Bell, Men of Mathematics, pp. 433-447 for most of the above biographical information. Also see Carl B. Boyer, A History of Mathematics, p. 636.

29. Ch'u Chai, The Story of Chinese Philosophy, p. 108.

30. In the terms of the test, I was an ENFP (Extravert-Intuitive-Feeling-Perceiving), with a strongly developed third function of Thinking. Hence my inferior function is sensation.

31. Numinous is a wonderful word coined by theologian Rudolf Otto in The Idea of the Holy (London: Oxford University Press, pb reprint, 1958). "Otto wanted a word that expressed the feeling of awe and mystery that we all experience at various times in our lives. Regardless of our religious convictions (or lack thereof), we invariably experience the collective unconscious as numinous. It might be numinous and frightening, numinous and nurturing, numinous and abstract, but always numinous. That is a sure sign that we are dealing with a more than human aspect of reality." (Robertson, Beginner's Guide to Jungian Psychology, p. 79.)

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