Chapter 3

The Universal Language

Truth & the Transcendent

by Larry L. Zimmerman

There are two directions of mathematical inquiry. It can either penetrate into the other sciences, making models, maps, and bridges for reasoning; or it can mind its own business, …

… Earth’s crammed with heaven,
And every common bush afire with God;
And only he who sees, takes off his shoes—
The rest sit round it and pluck blackberries.
—Elizabeth Browning1

Descriptions of Mathematics

From the inability of non-theistic assumptions concerning the nature of mathematics to account both for its unity and its involvement in the natural sciences, comes surrender in the battle to define mathematics. James R. Newman delineates the problem when he says,

There are two directions of mathematical inquiry. It can either penetrate into the other sciences, making models, maps, and bridges for reasoning; or it can mind its own business, cultivate its own garden. Both pursuits have been enormously fruitful. The success of mathematics as a helper to science has been spectacular.2

Then, as always, comes the unanswerable. “How,” Newman asks,

is this universality to be explained? Why has mathematics served so brilliantly in so many different undertakings—as a lamp, a tool, a language; even in its curious, ape-like preoccupation with itself? What, in other words, is the mathematical way of thinking?3

Caught up on the same twin horns, von Neumann wrote, “There is a quite peculiar duplicity in the nature of mathematics. One has to realize this duplicity, to accept it, and to assimilate it into one’s thinking on the subject.”

“Peculiar duplicity.” Nothing epitomizes both the problem and the despair of solutions better than this phrase. Von Neumann goes on in his surrender, “This double face is the face of mathematics, and I do not believe that any simplified, unitarian view of the thing is possible without sacrificing the essence.”4

Herman Weyl agrees, saying,

We are not surprised that a concrete chunk of nature, taken in its isolated phenomenal existence, challenges our analysis by its inexhaustibility and incompleteness. However, it is surprising that a construct created by the mind itself, the sequence of integers, the simplest and most diaphanous thing for the constructive mind, assumes a similar aspect of obscurity and deficiency when viewed from the axiomatic angle.

Weyl then surrenders:

The ultimate foundations and the ultimate meaning of mathematics remain an open problem; we do not know in what direction it will find its solution, nor even whether a final objective answer can be expected at all.5

The pessimism of mathematicians like von Neumann and Weyl has not deterred others from tendering possible definitions, or at least descriptions, of their discipline. Newman lists several of these attempts.

Felix Klein describes it as the science of self-evident things; Benjamin Peirce as the science which draws necessary conclusions; Aristotle, as the study of ‘quantity;’ Whitehead, as the development of all types of formal, necessary, and deductive reasoning; Descartes, as the science of order and measurement; Bertrand Russell, as a subject identical with logic; David Hilbert, as a meaningless, formal game.6

Is it any surprise that Newman, considering the range of these differing and sometimes contradictory responses, throws up his hands and concedes, “… any definition of mathematics, however elaborate or epigrammatic, will fail to lay bare its fundamental structure and the reasons for its universality.”7

The Theistic Description of Mathematics

There is another description of the character of mathematics, one which flows from theistic, creationist presuppositions. It is possible that mathematics is an entity always existing in the mind of God, which for us is the universal expression of His creative and sustaining word of power. It is clear that the answers to the two questions plaguing mathematicians come easily out of this definition, judgments notwithstanding on how “simplified,” “unitarian,” “elaborate,” or “epigrammatic” it is. It accounts not only for the unity found in mathematics, but also for its “peculiar duplicity,” as delineated by von Neumann and many others.

What happens when the implications of this theistic description of mathematics encounter the special revelation of God in the Bible and His general revelation in the universe? Do the implications square with reality? As C.S. Lewis pointed out,

The credibility will depend on the extent to which the doctrine, if accepted, can illuminate and integrate the whole mass. It is much less important that the doctrine itself should be fully comprehensible. We believe that the sun is in the sky at midday in summer not because we can clearly see the sun (in fact we cannot) but because we can clearly see everything else.8

The thinking of A.N. Whitehead on this topic parallels Lewis’. “In the study of ideas,” wrote the great mathematician,

it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions.9

The Word of God in the Creation

One of the major themes of the Bible is that in both the Scripture and the creation, God is revealed by His word. Many Christians, while trumpeting their belief in the Bible from cover to cover, including the covers, ignore the whole counsel of the Word of God on the word of God.

According to the Bible, God spoke all things into existence; the Scriptures have it that “the worlds were created, beautifully coordinated and now exist at God’s command; so the things that we see did not develop out of mere matter.10By the word of the Lord the heavens were made, and by the breath of His mouth all their host … for He spoke and it was done; He commanded, and it stood fast.11 The poet begins Psalm 148 by exhorting heavens, heights, angels, hosts, sun and moon, stars of light, highest heavens and the waters above the heavens, to praise the Lord, “for He commanded and they were created. He has also established them forever and ever; He has made a decree which will not pass away.12 In his next stanza, the psalmist invokes praise from the creation under the skies, from “the earth, sea monsters and all deeps” to “young men and virgins, old men and children.” As he’s listing the components comprising the “lower” creation, he suddenly interjects the phrase, “fulfilling His word.13 Not only was creation by the word of God, He presently “continues to uphold the universe by His mighty word.14

Echoing the commands of its Creator, the creation itself speaks, not in the syntactical structure used in the Bible, but in “utterances” nonetheless—in its own language—as Psalm 19 proclaims.

The heavens are telling the glory of God; and their expanse is declaring the work of His hands. Day to day pours forth speech, and night to night reveals knowledge. There is no speech, nor are there words; their voice is not heard. Their line [sound] has gone out through all the earth, and their utterances to the end of the world.15

The work of His hands “pours forth” a message for which men are held accountable.

For ever since the creation of the world, His invisible characteristics—His eternal power and divine nature—have been made intelligible and clearly visible by His works. So they are without excuse …16

Many Christians live as though Romans 10:18 did not follow verse 17; verse 18 clearly states that nature’s voice is the “word of God” mentioned previously, whose hearing engenders faith. And yet this word is evidently one and the same with the “gospel of peace” mentioned in the immediately preceding passage. The word of God which runs through creation implicitly is the same word which runs through the Bible explicitly. How could it be otherwise, since God is one?

This truth is beautifully expressed in Psalm 147:

He sends forth His command to the earth; His word runs very swiftly. He gives snow like wool; He scatters the hoarfrost like ashes. He casts forth His ice as fragments; who can stand before His cold? He sends forth His word and melts them; He causes His wind to blow and the waters to flow. He declares His words to Jacob, His statutes and His ordinances to Israel.17

The poem moves without break or pause from the implicit word in creation to explicit “words,” “statutes,” and “ordinances.”

A similar stanza is found in Psalm 119, which links smoothly the natural law-word in verses 89 and 90 with the preceptual law-word in verses 91 and following.

Forever, O Lord, Thy word is settled in heaven. Thy faithfulness continues throughout all generations; Thou didst establish the earth and it stands. They stand this day according to Thy ordinances, for all things are Thy servants. If Thy law had not been my delight, then I would have perished in my affliction. I will never forget Thy precepts …18

The Bible speaks precisely and therefore must be the primary revelation of God to man; the creation speaks generally and must be secondary, but “primary” and “secondary” do not mean “different.” Even though the glass of creation is “darker” than that of the Scriptures, the word of God forms a unity; the general revelation brings a message to mankind from God which supplements the message of the Scriptures.

James Olthius is right when he says,

… I will argue that creation can only be understood in relation to God’s word which brought creation into being and which continues to uphold it. It is this word for creation, as God’s dynamic plan for creation, which is incarnated in Jesus Christ and which is inscripturated … God’s word may not—as is generally done in evangelical circles—be limited to the scriptures, nor is it the mysterious, dynamic power of the Wholly Other, a la Karl Barth. God’s word dynamically structures creation, is experienced in terms of creation—even though it may not be identified with creation—and is the very condition for its existence.19

The Thought Behind the Word

Obviously God did not create everything He could have created. Out of His thought, He evidently chose and then spoke. Since His word, unlike ours, is only quantitatively different from His thought, His creative word of power resulted in precisely those entities, seen or unseen, which were in His thoughts. Just as a well-written book is bigger than the sum of its words, opening our mind’s eye to the author’s thinking, so the word of God in nature broadens our vision of the powerful mind of the One Who “by wisdom founded the earth; by understanding established the heavens, and by whose knowledge the deeps were broken up and skies drip with dew.20 So, anthropomorphisms notwithstanding (I mean by them what God means by them), His thoughts can be known truly, though not completely.

Edward Everett’s enthusiasm may have gotten the better of him and his eloquence sound overstated now, but he was no ignoramus. The first American to win a doctorate at Gottingen, he wrote, “In the pure mathematics we contemplate absolute truths which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.”21

The theistic description of mathematics, then, would lead us to expect the deepest scientific probes into the micro or macro cosmos to reveal a language fabric in which are woven the forces and relationships governing the tangible creation. This language fabric itself should be suggestive of an intellectual antecedent, an orderly, powerful, infinite universe of thought, seen by 19th century economist Walter Bagehot to be “a region different from our own … a terra incognita of pure reasoning [which casts] a chill on human glory.”22

And this is exactly the case. “Transcending the flux of the sensuous universe,” wrote Keyser, “there exists a stable world of pure thought, a divinely ordered world of ideas, accessible to man, free from the mad dance of time, infinite and eternal.”23

Most mathematicians, however, while acknowledging the existence of an underlying structure of thought in mathematics, claim it is spun out of their own minds. For Kline, “mathematics does appear to be the product of human, fallible minds.” “[It] is not a structure of steel resting on the bedrock of objective reality but gossamer floating with other speculations in the partially explored regions of the human mind.”24 This leads Kline to join in Bertrand Russell’s paean to man’s mind:

Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape the dreary exile of the natural world.25

Francis Schaeffer pointed out that the man who ignores his Creator sooner or later elevates his God-given thinking process to shrine status, bestows on it the title of “Reason,” and bows down in worship before it. Surely Russell has penned a marching hymn for this religious order, but because of “common grace,” there is truth here in which the Christian also can rejoice. He need only change Russell’s word “created” to “discovered.”

Another stanza on the same theme was composed by Jourdain.

Out of the striving of human minds to reproduce conveniently and anticipate the results of experience of geometrical and natural events, mathematics has developed … and then it appeared that there is no gap between the science of number and the science of the most general relations of objects of thought.26

Again, this is factual except for the presuppositional code word, “developed.” The Christian can “desecularize” the statement by substituting, “been revealed.”

In any case, it is obvious that no matter what the philosophical grid through which they filter information, and no matter what else they would label “mathematics,” eminent mathematicians, past and present, recognize the “ordered cosmos” of pure mathematical thought behind nature. Wrote Pierre Duhem in The Aim and Structure of Physical Theory, “… it is impossible for us to believe that this order and this organization [produced by mathematical theory] are not the reflected image of a real order and organization … ”27

Another example is given by E.T. Bell, who points out

… when abstractness and precision are attained, a higher degree of abstractness and a sharper precision are demanded for clear understanding. Our own conception of a ‘point’ will no doubt evolve into something else yet more abstract. Indeed, the ‘numbers’ in terms of which points are described today dissolved about the beginning of this century into the shimmering blue of pure logic, which in its turn seems about to vanish in something rarer and even less substantial.28

Foundations of the Thought Behind the Word

Bell is alluding to explorations into the foundations of mathematics, motivated by the maelstrom Goedel created in the areas of completeness and consistency. At least five schools of thought about foundations are extant. The logicist, intuitionist, formalist, set-theoretic, and non-standard analytic vie for support, causing Kline to claim there are now many mathematics.29 He does not really mean this, of course, since he also says,

Whereas in science there have been radical changes in theories, in mathematics most of the logic, number theory, and classical analysis has functioned for centuries. They have been and are applicable.

“The power of mathematics,” he adds, “cannot be abandoned while foundational issues are thrashed out.”30 What he is really saying is that there are many theories of the foundations of mathematics.

If mathematics is God’s thinking, it is to be expected that mathematicians would be able to venture only so far toward ultimate foundations without facing contradictions and uncertainty. One has only to read a little history to realize this is exactly the situation. Kline says, “the recent research on foundations has broken through frontiers only to encounter a wilderness.”31 His response parallels Bertrand Russell’s testimony: “The splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze … ”32 Kline adds, “The tragedy is not just Russell’s.”33

The theist suspects that research into origins and axiomatics is always destined for mazes and wildernesses. He wonders why, if mathematics is built from cultural inventions constructed by the mind of man, the inventors are unable to lay its foundation.

Correctly proclaiming mathematics as “a bold and formidable bridge between ourselves and the external world,” Kline must add, “it is tragic to have to recognize that the bridge is not anchored in reality or in human minds.”34

The Testimony of Physics

Physicists, on the trail of the smallest sub-atomic particle, are becoming more and more convinced that beyond the tiniest, there is something even less tangible than force, holding everything together. Smaller than neutrons, electrons, or protons, smaller than the quarks of which the former are composed, is the gluon, which “glues” the quarks together.35 Yet even more basic to universal solidarity than gluons is a power which can be described only as pure thought. Gluons, it seems,

need something to interact by. That turns out to be ‘instantons,’ [which are] solutions to mathematical equations but … have no materiality. Instantons are mathematical, but have a physical effect: in their presence the gluons feel forces. So nothing can affect something. [Instantons are] mathematical beings that teeter on the edge of reality and affect the behavior of material objects …36

Whether theorizing about instantons proves fruitful or not, it appears that scientists on the outer, deeper limits are being confronted by a powerful, intangible cohesive structure, describable only in mathematical terms. Dr. Fritj of Capra says,

At the subatomic level, the solid material objects of classical physics dissolve into wave-like patterns of probabilities, and these patterns, ultimately, do not represent probabilities of things, but rather probabilities of interconnections … subatomic particles have no meaning as isolated entities, but can only be understood as interconnections … Quantum theory thus reveals a basic oneness of the universe. It shows that we cannot decompose the world into independently existing smallest units. As we penetrate into matter, nature does not show us any isolated ‘basic building blocks’ but rather appears as a complicated web of relations [Werner Heisenberg called them ‘events’] between various parts of a whole.37

Capra’s group at Berkeley

… makes much of [John S.] Bell’s theorem, a hypothesis apparently confirmed by experiments at their lab in 1972. Bell’s theorem assumes that separate parts of the universe may be connected at a fundamental level, and that things once connected remain attached over distance by some unknown force traveling faster than the speed of light. Bell, a Swiss physicist, called this force ‘that-which-is’—a distinctly mystical label—and suggested that space, time and motion are all forms of this unobservable connection.38

The Christian recognizes immediately the phrase “that-which-is” to be a common grace reflection of the One who claimed “I AM THAT I AM.

The title of Capra’s lecture series, “The Tao of Physics: Reflections on the Cosmic Dance;” Newsweek Magazine’s considering “Physics and Mysticism” news of the week; Einstein’s feeling that “The most beautiful and profound emotion one can experience is the sensation of the mystical; it is the source of all true science;”39 these give expression to scientists’ awareness that the thought waves they encounter in nature presuppose a thinker. For the scientist, a contentless mysticism results; to the Christian, there comes the assurance that his faith in the God by whose powerful word all things hold together has rational consequences.

Philosophy

Some mathematicians downgrade not only mysticism but philosophy in general as having nothing to do with mathematics. Bertrand Russell defined “… philosophy as an unusually ingenious attempt to think fallaciously.”40 In the opinion of Henri Lebesque, “… a mathematician, in so far as he is a mathematician, need not concern himself with philosophy—an opinion, moreover, which has been expressed by many philosophers.”41

Courant and Robbins think

… Creative minds forget dogmatic philosophical beliefs whenever adherence to them would impede constructive achievement. For scholars and laymen alike it is not philosophy but active experience in mathematics itself that alone can answer the question: What is mathematics?42

In response, we could ask: “How does one know his ‘active experience’ is in mathematics if one has not established what mathematics is?” A more fundamental problem is that all of these statements are philosophical—yes, even “dogmatically” so. No person, no scientist, operates in a philosophical vacuum. Alfred North Whitehead was right when he said, “No science can be more secure than the unconscious metaphysics which tacitly it presupposes … All reasoning, apart from some metaphysical reference, is vicious.”43 Facts are never just facts; they come to life and dance to the tune the scientist plays, and his choice of “music” is governed by his philosophical filtering system. These mathematicians try to utilize their specialized philosophical grids to strain out philosophy itself.

One is reminded of the more general statement made by C.S. Lewis, of which the preceding may be considered a special case:

… no account of the universe can be true unless that account leaves it possible for our thinking to be a real insight. A theory which explained everything else in the whole universe but which made it impossible to believe that our thinking was valid, would be utterly out of court. For that theory would itself have been reached by thinking, and if thinking is not valid that theory would, of course, be itself demolished. It would have destroyed its own credentials. It would be an argument which proved that no argument was sound—a proof that there are no such things as proofs—which is nonsense.44

To claim that philosophy has nothing to do with mathematics is like claiming that cooking has nothing to do with food. The picture one gets is of a gourmet chef dedicating his life to the preparation of delectable morsels but starving to death because he refuses to eat them.

As might be expected, Galileo was one scientist who recognized the philosophical basis of all truth, including mathematical. He wrote,

Philosophy is written in that vast book which stands forever open before our eyes, I mean the universe; but it cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometric figures, without which means it is humanly impossible to comprehend a single word.45

For an elementary treatment of the “language” about which Galileo is speaking, see Anthony Ravielli’s little book, An Adventure in Geometry. Since Ravielli imputes personality to nature, the Christian reader needs mentally to replace the author’s word “nature” with the words, “God” or “creation,” depending on the context.

Mystery

Assuming mathematics to be God’s thought, the Christian would be surprised not to encounter mystery there.
Contrary to E.T. Bell’s belief that “so long as there is a shred of mystery attached to any concept, that concept cannot be mathematical,”46 no Christian shrinks from mystery. Assuming mathematics to be God’s thought, the Christian would be surprised not to encounter mystery there. In fact, when one examines the bewildering results of human probes into the foundations of mathematics, it is obvious that mathematicians are encountering not just shreds of mystery, but shrouds. And the Christian, knowing something of the character of his Creator, is confident this mystery will not be arbitrary, as Whitehead feared. Again, the Christian’s expectation is realized; his is the realistic system of belief. As Os Guinness put it, “Rationalism is the opposite of absurdity, not mystery.”47 Guinness’ observation on faith could be applied here: “We cannot explain it … But because of the evidence neither can we explain it away.”48 The great Christian mathematician and philosopher, Blaise Pascal, summed it ably: “The last step that Reason takes is to recognize that there is an infinity of things that lie beyond it.”49

Suppose you receive a letter written on paper in ink. The paper is torn and worn, the ink smudged, and your eyesight is none too good anyway. Also, some pages are missing, among them the beginning and ending of the letter. Finally, it is written in a foreign language. You dig out a magnifying glass and begin to piece together the message.

The paper and ink represent the tangible creation, stricken by the Fall; the arrangement of the words on the paper, the force holding creation together; the glass is the Bible, or a biblically-saturated mind; the message you receive initially is Browder’s Mathematics II or Mathematics III (see pp. 33–34); and the message the writer really had in mind when he wrote, and of which you can discern parts as you meditate and study, Mathematics IV. Subject to refinement and generalization, the mathematics discovered in level IV is not the whole picture, but it is, in Schaeffer’s words, “true truth.”

Platonism

Recognizing the transcendent aspect of mathematics, Kurt Goedel evidently believed that “only a thoroughgoing Platonic realism” can supply a definition of mathematical or logical truth.50 Newman describes Galileo as being convinced, “in the true Platonic tradition, that the mathematical models which led to observations were the ‘enduring reality, the substance, underlying phenomena.’”51

Though there is an element of similarity between the platonic and the theistic view of the nature of mathematics, the differences are significant. The latter posits the existence of the infinite; an abnormal world, due to sin; the importance of mathematics in obeying God’s command to take dominion over the earth, subduing, and replenishing it; and most important, a personal and ethical deity who creates ex nihilo.52 These concepts are foreign to Platonism, which allows for categories but not for meaning. As Kline observes, Plato “wished not merely to understand mathematics but to substitute mathematics for nature herself.”53

Browder’s Mathematics I and II are also accounted for by the theistic description of mathematics; entrée to the “divinely ordered world of ideas” is gained by clues from the tangible, creational expression of that thought. Professor Synge saw it as “a dive from the world of reality into the world of mathematics; a swim in the world of mathematics; a climb from the world of mathematics back into the world of reality, carrying a prediction in our teeth.”54

We need to be cautious, however, about the use of the word “reality.” Christians believe the world of God’s creation thought and speech is no less real than the physical world; perhaps, in the light of 2 Corinthians 4:18, even more real, because it will outlast the physical and will not change while we sleep. To signify mathematics as “the language of creation” is obviously realistic. As Kline says, “The essence of any modern physical theory is a body of mathematical equations.”55 Even Russell once stated that though logic and mathematics are “not the book [of nature] itself,” they are “the alphabet of the book … ”56

The only view of mathematics which allows for a speaker of this language is the theistic view. Though a person may have faith that his mind is structuring nature mathematically, the facts seem not to support this belief, which causes its adherents to resort to mystical analogies for shoring-up. One such was purveyed by Richard Hughes, who first states his assumption: “Science, being human enquiry, can hear no answer except an answer couched somehow in human tones.” He then goes into his analogy:

Primitive man stood in the mountains and shouted against a cliff; the echo brought back his own voice, and he believed in a desembodied spirit. The scientist of today stands counting out loud in the face of the unknown. Numbers come back to him—and he believes in the Great Mathematician.57

Hughes is referring to men like Sir James Jeans who said, “The Great Architect of the universe now begins to appear as a pure mathematician,”58 and Sir Arthur Eddington, who “continued to look upon God as the First Cause, the ultimate raison d’etre.”59

Hughes’ analogy is faulty in at least two respects. First, “primitive man” received back as many different echoes as there were shouters; but in mathematics, the same “numbers” come back, no matter who the “counter” is. Most important, as history demonstrates conclusively, it is simply not the case that the mathematician counts and numbers come back. Rather, “numbers” come winging out of the unknown (universe) and force the mathematician to count, his voice blending into a harmonious chorus with those of his colleagues.

Truth & the Transcendent

“How do you know that God created mathematics?”

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Footnotes

  1. Elizabeth Barrett Browning, “Aurora Leigh,” BK VII, The Complete Poetical Works of Mrs. Browning (Boston: Houghton-Mifflin, 1900), p. 372.
  2. Newman, “Commentary,” TWOM, vol. 3, 1830.
  3. Newman, TWOM, 1830.
  4. von Neumann, vol. 1, 2.
  5. Weyl, 219, 220.
  6. Newman, TWOM, 1830.
  7. Newman, TWOM, 1830.
  8. C.S. Lewis, 114.
  9. A.N. Whitehead, Adventures of Ideas (New York: The Free Press, 1967), p. 72.
  10. Hebrews 11:3 (Williams Translation).
  11. Psalm 33:6, 9 (New American Standard Bible).
  12. Psalm 148:1–6 (NASB).
  13. Psalm 148:7–12 (NASB).
  14. Hebrews 1:3 (Williams).
  15. Psalm 19:1–4 (NASB).
  16. Romans 1:20 (Williams).
  17. Psalm 147:15–19 (NASB).
  18. Psalm 119:89–91 (NASB).
  19. James Olthius, “Return to Creation: Theme for sabbatical study,” Perspective, Newsletter of the Association for Advancement of Christian Scholarship, vol. II, No. 6, Sept./Oct. 1977, p. 12.
  20. Proverbs 3:19, 20 (NASB).
  21. E.T. Bell, Mathematics: Queen and Servant of Science (New York: McGraw Hill, 1951), p. 21.
  22. TWOM, vol. 1, 75.
  23. Keyser, 57.
  24. MIWC, 431.
  25. Quoted in MIWC, 470. Toward the end of his life, Rus- sell repudiated most of what he once believed about the nature of mathematics. This particular passage he labeled “largely nonsense; partly for technical reasons and partly from a change in my general outlook on the world. Mathematics has ceased to seem to me non-hu- man.” See Russell’s My Philosophic Development (New York: Simon and Schuster, 1959), p. 211.
  26. Jourdain, 112.
  27. Pierre Duhem, The Aim and Structure of Physical Theory (New Jersey: Princeton Univ. Press, 1954), p. 26.
  28. MOM, 15.
  29. MTLC, 269.
  30. MTLC, 333.
  31. MTLC, 276.
  32. Russell, My Phil. Dev., 212 (See end note 25 above).
  33. MTLC, 230.
  34. MTLC, 332, 333.
  35. See “Glueballs,” Scientific American, July 1981, pp. 82–84.
  36. Axiom Hunt,” Science News, Vol. 113, No. 15, 15 April 1978, p. 228.
  37. Fritjof Capra, The Tao of Physics (Boulder: Shambala, 1975), p. 68. At the time, Capra was doing research in theoretical high energy physics at the Lawrence Berkeley Laboratory, Univ. of California.
  38. Kenneth Woodward, “Physics and Mysticism,” Newsweek, 23 July 1979, p. 87.
  39. Quoted by Woodward, above.
  40. Bertrand Russell, Unpopular Essays (Simon and Schuster, 1950), p. 45.
  41. Quoted in Robert A. Bonic et al., Freshman Calculus (Lexington, Mass.: Heath, 1976), p. 347.
  42. Courant and Robbins, xix.
  43. Whitehead, Adventures, 154.
  44. C.S. Lewis, 20.
  45. Galilei Galileo, Opere: Il Saggiatore, Question 6, quoted in The History of Science, Crombie, 295.
  46. MOM, 357.
  47. Os Guiness, In Two Minds (Downer’s Grove; IV Press, 1976), p. 105.
  48. Guiness, 105.
  49. Pascal’s Pensees, trans. by H.F. Stewart (London: Rout- ledge, 1950), p. 31.
  50. Nagel and Newman, TWOM, vol. 3, 1694.
  51. Newman, “Commentary,” TWOM, vol. 2, 731.
  52. See All Truth is God’s Truth, by Arthur F. Holmes, (Grand Rapids: Eerdmans, 1977), p. 39.
  53. MTLC, 16.
  54. Quoted by M.R. Kenner, “Mathematical Education Notes,” American Mathematical Monthly, Oct. 1961, Vol. 68, No. 8, p. 799.
  55. MIWC, 320.
  56. Quoted by Alan Wood in “Cautionary Notes,” Russell’s Philosophy, appended to My Philosophical Development, B. Russell, p. 277.
  57. TWOM, vol. 1, 537.
  58. James Jeans, The Mysterious Universe, New Revised Ed. (New York: Macmillan, 1932), p. 165.
  59. MIWC, 229.

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