[2] Kurt Gödel, “Russell's Mathematical Logic”, p. 449.

[3] At least in a practical way, if not with formal precision.

[4] In fact, Berkeley was one of the few to point to the deeper problems of calculus. But mathematicians were hardly going to worry about the objections of a philosopher. They had work to do developing the implications of the calculus.

[5] In Freud’s case, his psycho-sexual theory, as it evolved out of the Oedipus complex. In Russell’s case, his conviction that the entire world could be reduced to symbolic logic.

[6] Freud, when confronted with a new set of ideas by his young successor Jung, was unwilling to give up his patriarchal authority stance, or to open the floodgates to the “black mud of occultism.” Russell chose to ignore the three blows he received from the unconscious: (1) his mystical insight that feeling and intuition was primary; (2) the paradox of the set of all sets; and (3) his loss of love for his first wife Alys.

[7] Russell, was never comfortable with the demands that his feelings and intuition placed upon him, and retreated into a world where logic was omnipotent. Interestingly, in social issues such as his fights for women’s rights and for pacifism, Russell had the courage of his convictions, even to the point of being jailed as a pacifist during the First World War. Eventually he even broke from his passionless first marriage and conducted a long-term affair with Lady Ottoline Morrell. He received the Nobel Prize in literature in 1950 for his philosophic writings, and lived nearly a hundred years. However, he was never able to confront the inadequacy of his attempt to reduce mathematics to logic. Though he began his philosophical life as a Kantian, he gradually divorced himself further and further from feeling and intuition, until eventually he came to believe in the omnipotence of logic.

[8] Freud born in 1856, Russell in 1872, Jung in 1875.

[9] Which was to be complemented by a proof by mathematician Paul Cohen in 1963. This will all be discussed later in this chapter.

[10] Including continued thought on the continuum hypothesis.

[11] C. G. Jung, Memories, Dreams, Reflections, p. 27. Jung’s problem was not so much that he had no mathematical ability as that he saw too deeply into what mathematics actually is. I had a similar experience in the 2nd grade, when a homework assignment introduced the concept of zero. There were a number of problems where zero was added or subtracted from various numbers. The answer was, of course, always the same number. Most of the other children in the class just regarded this as still another rule to be memorized and experienced no more difficulty than with any other such incomprehensible rule. But I sat alone in my room that night, staring at the problems, in tears at their seeming senselessness. How could I add something to a number and the number remained unchanged? And then an understanding burst forth, and I had the first mystical experience of my life. The immensity of the concept of nothingness overwhelmed me. The realization that mathematicians were brilliant enough to be able to capture that immensity in a symbol awed me. I determined on the spot to be a mathematician.

¹² I.e., the integers: 1, 2, 3,…

[13] Jung’s colleague Marie-Louise von Franz has extended this work in her Number and Time (Evanston: Northwestern University Press, 1974).

[14] C. G. Jung, “Synchronicity: an Acausal Connecting Principle,” par. 870.

[15] Marie-Louise von Franz, Number and Time, p. 13.

[16] Quotation by Leopold Kronecker in Bell, Men of Mathematics, p. xv.

[17] Tobias Dantzig, Number and the Language of Science (New York: MacMillan Company, 1954), p. 3.

[18] In fact, the Australian Aborigines actually limit themselves to “one” and “two”, then use composites of “one” and “two” to make up numbers up to “six”. For example, “three” is “two” and “one”, “four” is “two” and “two”, “five” is “two” and “two” and “one”, “six” is “two” and “two” and “two”. They count in pairs, so that they wouldn't be likely to notice if two pins were removed from a heap of seven pins, but would instantly recognize if only one pin had been removed. See Tobias Dantzig, Number and the Language of Science, p. 14.

[19] C. G. Jung, Collected Works, Vol. 10: Civilization in Transition, par. 776. Jung’s emphasis.

[20] 1729 equals both 12[3] + 1[3], and 10[3] and 9[3]. Anecdote from Robert Kanigel, The Man Who Knew Infinity (New York: Charles Scribner's Sons, 1991), p. 312.

[21] See Wolfgang Pauli, “The Influence of Archetypal Ideas on the Scientific Theories of Keeler,” in C. G. Jung and Wolfgang Pauli, The Interpretation of Nature and the Psyche (New York: Pantheon Books, 1955). Also see Charles R. Card, “The Archetypal View of Jung and Pauli,” Psychological Perspectives #24 & #25 (Los Angeles: C. G. Jung Institute, 1991)

[22] Quotation by Wolfgang Pauli, in Charles R. Card, "The Archetypal Hypothesis of Wolfgang Pauli and C.G. Jung: Origins, Development, and Implications", in K. V. Laurikainen and C. Montonen, eds., Symposia on the Foundations of Modern Physics, 1992 (Singapore: World Scientific Publishing Co., 1993), p. 382.

[23] Kurt Gödel, "What is Cantor's continuum problem?", p. 254.

[24] The power set is all the combinations of the natural numbers, taken one-at-a-time, two-at-a-time, on-and-on.

[25] Which remember is the same size as all fractions or rational numbers.

[26] Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, p. 269.

[27] Though as we will see later, Gödel ultimately hoped to prove the continuum hypothesis to be false within a broader mathematics.

[28] Kurt Gödel, "What is Cantor's continuum problem?", p. 259-60. Note that mathematician Paul Cohen later proved the reverse, so that Gödel's conjecture did, in fact, turn out to be true; i.e., within set theory, Cantor's continuum hypothesis is undecidable.

[29] Or the axiom of choice, which Gödel proved to be equivalent to the continuum hypothesis.

[30] In 1977, Paris and Harrington discovered a “numerically simple and interesting proposition, not depending on a numerical coding of notions from logic, which is undecidable. See Kurt Gödel’s Collected Works, Vol 1, p. 140.

[31] Kurt Gödel, "What is Cantor's continuum problem?", p. 260.

[32] Kurt Gödel, "What is Cantor's continuum problem?", p. 261. Note that this is almost exactly what Jung meant when he discussed psychological truth. In such cases, reference to physical reality for determining truth or falsity is beside the point.

[33] Kurt Gödel, "What is Cantor's continuum problem?", p. 256.

[34] Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, p. 269.