1. Mathematics: Locus of Reality By James Nickel, B.A., B.Th., B.Miss., M.A. copyright  2007 www.biblicalchristianworldview.net

2. Differentiate There are facts of mathematics and there are philosophies of the facts of mathematics. copyright  2007 www.biblicalchristianworldview.net

3. Practical Teaching Must be based upon sound principles. –Line upon line. –Precept upon precept. –Simple to complex. It may be surprising to realize that many teachers do not teach using sound principles. Arithmetic is not easy to teach! copyright  2007 www.biblicalchristianworldview.net

4. Practical Teaching There will be some slight variations on teaching methodologies. These variations are not gravely important. What does mark a fundamental difference is the philosophy undergirding the subject being taught. copyright  2007 www.biblicalchristianworldview.net

5. Issues in Philosophy “How a mathematical structure can correspond to nature is a mystery. One way out is just to say that the language in which nature speaks is the language of mathematics. This begs the question. Often we are both shocked and surprised by the correspondence between mathematics and nature, especially when the experiment confirms that our mathematical model describes nature perfectly.” Remo J. Ruffini, “The Princeton Galaxy,” interviews by Florence Heltizer, Intellectual Digest, 3 (1973), 27. copyright  2007 www.biblicalchristianworldview.net

6. Issues in Philosophy “But where and what are the physical agents that produce the effects mathematics describes? … There are no answers … Why does mathematics work? We are faced with a mystery …” copyright  2007 www.biblicalchristianworldview.net

7. Issues in Philosophy “The study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass, he has predicted and shown how to control radio waves which none of our senses can perceive, and he has discovered particles too small to be seen with the most powerful microscope.” copyright  2007 www.biblicalchristianworldview.net

8. Issues in Philosophy “Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explanation of this marvelous power is called for.” Morris Kline, Mathematics and the Physical World, p. ix. copyright  2007 www.biblicalchristianworldview.net

9. Issues in Philosophy “The eternal mystery of the world is its comprehensibility.” Albert Einstein, Out of My Later Years, p. 61. copyright  2007 www.biblicalchristianworldview.net

10. Important Questions Is mathematics only the product of the human mind (logic only)? If yes, then why is mathematics somehow related to physical reality? Over and over again, mathematicians couch the answers to such questions in terms of mystery, even miracle. copyright  2007 www.biblicalchristianworldview.net

11. Miracles Abound “It is difficult to avoid the impression that a miracle confronts us here.” Eugene Wigner, Symmetries and Reflections: Scientific Essays, p. 229. copyright  2007 www.biblicalchristianworldview.net

12. Miracles Abound “How can we, from the point of view of statistical physics, reconcile the facts that the gene structure seems to involve only a comparatively small number of atoms (of the order of 1000 and possibly less), and that nevertheless it displays a most regular and lawful activity – with a durability or permanence that borders upon the miraculous.” Erwin Schrödinger, What is Life? The Physical Aspects of the Living Cell, p. 31. copyright  2007 www.biblicalchristianworldview.net

13. Denying reality These eminent scientists and mathematicians will not posit the Biblical God as an answer. –Creator of the human mind with its ability to think mathematically. –Creator of the physical world that reflects mathematical principles. copyright  2007 www.biblicalchristianworldview.net

14. No answers Without the Biblical God as the foundation, there are no answers as to “why mathematics works.” copyright  2007 www.biblicalchristianworldview.net

15. Consequences Mathematical facts (1 + 1 = 2) are just brute facts. No fact has any meaning or any relationship to any other fact. No explanation for connectivity within mathematics and meaning in mathematics (other than mere pragmatism). copyright  2007 www.biblicalchristianworldview.net

16. Consequences The logic of the human mind has no relationship to physical reality. The appeal to mystery is a leap into irrationality. copyright  2007 www.biblicalchristianworldview.net

17. Reality There is only meaning and relationship in the universe if it is indeed a universe, a God-created universe consisting of unity (meaning) in diversity. To speak of law (scientific or mathematics) is to posit meaning. copyright  2007 www.biblicalchristianworldview.net

18. Reality Hence, to suppress the reality of meaning, mathematicians/scientists invoke probabilistic concepts (the sun may rise tomorrow). But probability belies a “hint” of meaning. copyright  2007 www.biblicalchristianworldview.net

19. Reality Even to speak in probabilities is impossible. Hence, there is no rational certainty that the sun will rise tomorrow. Any remote hint of meaning points to God and we must not let the “divine foot into the door.” copyright  2007 www.biblicalchristianworldview.net

20. What is mathematics? Do we create mathematics autonomously? Or, do we discover and classify mathematical patterns in a world of pre- established harmony? copyright  2007 www.biblicalchristianworldview.net

21. Important Questions “Do mathematical truths reside in the external world, there to be discovered by man, or are they man-made inventions? Does mathematical reality have an existence and a validity independent of the human species or is it merely a function of the human nervous system?” Leslie A. White, “The Locus of Mathematical Reality: An Anthropological Footnote,” The World of Mathematics, 4:2348. copyright  2007 www.biblicalchristianworldview.net

22. Creation vs. Classification Man does create in mathematics. How? Man creates the symbols and the formulas. But, these symbols and formulas express rational and meaningful connections (reality). Hence, when man “creates” mathematics, he is not creating reality, he is really classifying reality. copyright  2007 www.biblicalchristianworldview.net

23. Pre-established Harmony Does the mind of man create order out of a world of assumed chaos? Or, is there a pre-established world, a God-created world? copyright  2007 www.biblicalchristianworldview.net

24. Locus of Reality Where is the locus of reality, including mathematical reality? –Man? –Matter? –Biblical God? copyright  2007 www.biblicalchristianworldview.net

25. Starting Points Every point of view is a view from a point. “As finite creatures, we cannot escape the reality that every human conclusion is necessarily based on some starting point.” Glenn R. Martin (1935-2004) copyright  2007 www.biblicalchristianworldview.net

26. Starting Points What is the ultimate foundation for reality? for knowledge? God, matter, or man The starting point will always be assumed to be autonomous. copyright  2007 www.biblicalchristianworldview.net

27. Starting Points “Every system of thought begins with some ultimate principle. If it does not begin with God, it will begin with some dimension of creation – the material, the spiritual, the biological, the empirical, or whatever.... This starting point has to be accepted by faith.” Nancy Pearcey, Total Truth, p. 41. copyright  2007 www.biblicalchristianworldview.net

28. Pseudo-starting Points René Descartes (1596-1650), “I think, therefore I am.” Modern Science, “Matter is all that there is.” Postmodernism, “Neither the mind of man nor matter is ultimate; therefore, reality is up for grabs.” –NB. Man is still the determining factor. copyright  2007 www.biblicalchristianworldview.net

29. True Starting Points Biblical Christian, “God is, therefore I can think.” God is the Creator, therefore I can study His works. God is Triune, therefore I can determine relationships, patterns, and order in His creation. I am made in the image if God (imago Dei), therefore God’s works (including me) are connected in terms of a common Creator. copyright  2007 www.biblicalchristianworldview.net

30. Minds matter Is mathematics merely and only connected to mind and/or matter? Does man “create” mathematics in an autonomous fashion? Or, is mathematics an aspect of the reality that God has created? NB. Man cannot totally separate himself from reality even if that is his wish. copyright  2007 www.biblicalchristianworldview.net

31. Key Questions The key issue in mathematics. –Does the mind of man create the world out of chaos? –Or, is there a pre-established, God-created, and covenantally-ordered world? copyright  2007 www.biblicalchristianworldview.net

32. Key Answer “We must, moreover, hold that all knowledge is covenantal. As a result, all attempts by theologians and philosophers to arrive at a knowledge of God outside of His covenant and apart from His word are in reality denials of knowledge.” Rousas J. Rushdoony, Systematic Theology, 1:199. copyright  2007 www.biblicalchristianworldview.net

33. Autonomous Man “And it seems that the mathematician, in creating his art, is exhibiting that movement of our minds that has created the spatio-temporal material universe we know … The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.” copyright  2007 www.biblicalchristianworldview.net

34. Autonomous Man “It does not enable us to explore some remote region of the eternally existent; it helps to show us how far what exists depends upon the way in which we exist. We are the law-givers of the universe; it is even possible that we can experience nothing but what we have created and that the greatest of our mathematical creations is the material universe itself.” John W. N. Sullivan (1886-1937), “Mathematics as an Art,” The World of Mathematics, 3:2021. copyright  2007 www.biblicalchristianworldview.net

35. The Meaning of Number Every culture has its own mathematics because of its world view or faith. –A culture’s world view determines its particular form of mathematics. –Every culture has its own idea of numbers and sees reality in terms of it. copyright  2007 www.biblicalchristianworldview.net

36. The Meaning of Number “Every philosophy has grown up hitherto in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. The existence of numbers may therefore be called a mystery, and the religious thought of every culture has felt their impress.” Oswald Spengler, “Meaning of Numbers,” World of Mathematics, 4:2315. copyright  2007 www.biblicalchristianworldview.net

37. Ancient Greeks For the ancient Greeks, number was merely and only positive integers. The ancient Greeks were radically hostile to the idea of infinite. –Irrational numbers. –Infinite processes. copyright  2007 www.biblicalchristianworldview.net

38. Christendom In Christendom, infinity was not viewed with hostility. –God, the Creator, is personal and infinite. –Medieval scholastics in the High Middle Ages embraced the rudiments of infinite processes (the real number continuum). copyright  2007 www.biblicalchristianworldview.net

39. Real Number Continuum “Indeed, whether we use a ruler or a weighing balance, a pressure gauge or a thermometer, a compass or a voltmeter, we are always measuring what appears to us to be a continuum [what the real number system represents – JN], and we are measuring it by means of a graduated number scale.” copyright  2007 www.biblicalchristianworldview.net

40. Real Number Continuum “We are then assuming that there exists a perfect correspondence between the possible states within this continuum and the aggregate [set – J.N.] of numbers at our disposal;” copyright  2007 www.biblicalchristianworldview.net

41. Real Number Continuum “Therefore, any measuring device, however simple and natural it may appear to us, implies the whole apparatus of the arithmetic of real numbers: behind any scientific instrument there is the master-instrument, arithmetic, without which the special device can neither be used nor even conceived.” Tobias Dantzig, Number: The Language of Science, pp. 245- 246. copyright  2007 www.biblicalchristianworldview.net

42. Christendom This world view released the ideas necessary that resulted in the viable birth of modern science. –The God of Scripture is fully rational. –The universe is a God-created and thereby fully rational universe, an arena that can be studied empirically and unified quantitatively by number and laws relating number to number. copyright  2007 www.biblicalchristianworldview.net

43. Analysis According to Oswald Spengler (1880- 1936), faith, like mathematics, is culture- centric. Hence: –Is there one, true Faith? –And, resultantly, is there one, true mathematic (or philosophy of mathematics)? copyright  2007 www.biblicalchristianworldview.net

44. The One and the Many “It may surprise the reader to learn that not everyone agrees that ‘2 + 2 = 4’ is true. But, on second thought, it must be apparent that no radical monist can remain satisfied with ‘2 + 2 = 4.’ If with Parmenides one thinks that all is one, if with Vedantic Hinduism he thinks that all plurality is illusion, ‘2 + 2 = 4’ is an illusory statement.” copyright  2007 www.biblicalchristianworldview.net

45. The One and the Many “On the ultimate level of being, 1 + 1 = 1. What does this imply? Even the simplest arithmetical truths can be sustained only in a world-view which acknowledges an ultimate metaphysical plurality of the world – whether Trinitarian, Polytheistic, or chance-produced plurality.” copyright  2007 www.biblicalchristianworldview.net

46. The One and the Many “At the same time, the simplest arithmetical truths also presuppose ultimate metaphysical unity for the world – at least sufficient unity to guard the continued existence of “sames.” Two apples remain apples while I am counting them; the symbol ‘2’ is in some sense the same symbol at different times, standing for the same number. So, at the very beginning of arithmetic, we are already plunged into the metaphysical problem of unity and plurality, of the one and the many.” copyright  2007 www.biblicalchristianworldview.net

47. The One and the Many “As Van Til and Rushdoony have pointed out, this problem finds its solution only in the doctrine of the ontological Trinity. For the moment, we shall not dwell on the thorny metaphysical arguments, but note only that without some real unity and plurality, ‘2 + 2 = 4’ falls into limbo.” Vern Poythress, “A Biblical View of Mathematics,” The Foundations of Christian Scholarship, p. 161. copyright  2007 www.biblicalchristianworldview.net

48. The One and the Many Is “oneness” or unity ultimate? –If unity is ultimate, then you have no basis for understanding plurality, “manyness,” or diversity. –Everything is ultimately “the same” and all diversity is merely and only an illusion. –Hence, mathematics, which deals with plurality, is an illusion (it does not exist!). copyright  2007 www.biblicalchristianworldview.net

49. The One and the Many Is “manyness” or diversity ultimate? –If diversity is ultimate, then you have no basis for understanding unity, generality, law, and pattern. –Everything is ultimately “different” and all unity is an impossibility and non-existent. –Hence, mathematics, which deals with unity (results of calculations, 2 + 2 = 4, binding laws of number, functions, identities, properties, etc.) is non-existent! copyright  2007 www.biblicalchristianworldview.net

50. The One and the Many A solid balance in the relationship between unity and diversity, founded in the doctrine of the Triune God, is necessary for science and mathematics to flourish. This balance was achieved in Christian Europe and it was in Christian Europe that the quantitative (or mathematical) study of the Works of God engendered the Scientific Revolution, a self-sustaining enterprise. copyright  2007 www.biblicalchristianworldview.net

51. The Locus of Reality The issue in mathematics is ultimately and extensively a theological one. True mathematics must be built upon a metaphysical foundation of the balanced or equal ultimacy of the “one and the many.” This foundation is “rock solid” only in terms of the Biblical Christian doctrine of the Triune God, a revelation of the balanced or equal ultimacy of the “One and the Many.” copyright  2007 www.biblicalchristianworldview.net

52. The Locus of Reality Mathematics, subsumed by this foundation, becomes: –A way to think God’s thoughts after Him since we are studying His creational work, the fruit of His thought/speech (Word or logos), and thus furthering our knowledge of the wonderful works of God thereby. –A means towards establishing God ordained dominion-stewardship over the created reality. copyright  2007 www.biblicalchristianworldview.net

53. The Locus of Reality This is true not only in mathematics, but in every area of life. All things are understandable only in terms of the Triune God of Scripture. copyright  2007 www.biblicalchristianworldview.net

54. The Locus of Reality “If God be indeed the creator of all things, all things must be defined in relationship to Him, or else we have a false definition.” Rousas J. Rushdoony, Systematic Theology, 1:208. copyright  2007 www.biblicalchristianworldview.net

55. The Locus of Reality Any locus of reality that rests, not on the autonomy or ultimacy of the Biblical God, but on the autonomy or ultimacy of the human mind is falsifying reality. copyright  2007 www.biblicalchristianworldview.net