1. Mathematics: Questions, Answers, and Wonder by James Nickel B.A., B.Th., B.Miss., M.A. Copyright  2008 www.biblicalchristianworldview.net

2. Starting points It is crucial that we start right in our study of any aspect of knowledge. If we do not interpret knowledge in relationship to the Biblical God, then we will only have pseudo-knowledge. Copyright  2008 www.biblicalchristianworldview.net

3. Starting points Psalm 36:9, “In Thy light, we see light.” Without God’s light, we walk in darkness. Copyright  2008 www.biblicalchristianworldview.net

4. Starting points If we start with God, then we shall see the discipline under study in truth (see light). If we start with man, then we shall only see surface truth (walk in darkness). Copyright  2008 www.biblicalchristianworldview.net

5. Starting points Every point of view is a view from a point. Copyright  2008 www.biblicalchristianworldview.net

6. Starting points The Biblical Christian presupposes that we can know truly on the basis of the Bible, God’s verbal and written revelation. This knowledge is not exhaustive (e.g., the Bible does not reveal knowledge about quadratic equations), but it is knowledge that brings a true perspective to all aspects of the human endeavor, eternal and temporal. Copyright  2008 www.biblicalchristianworldview.net

7. Starting points Colossians 1:15-17 (Weymouth), “Christ is the visible representation of the invisible God, the Firstborn and Lord of all creation. For in Him was created the universe of things in heaven and on earth, things seen [visible] and things unseen [invisible], thrones, dominions, princedoms, powers—all were created, and exist through and for Him. And He is before all things and in and through Him the universe is a harmonious whole.” Copyright  2008 www.biblicalchristianworldview.net

8. Starting points Colossians 2:3 (Weymouth), “In Him [Christ] all the treasures of wisdom and knowledge are stored up, hidden from view.” Copyright  2008 www.biblicalchristianworldview.net

9. Consider 2 + 2 = 4 Copyright  2008 www.biblicalchristianworldview.net

10. Consider 2 + 2 = 4 Where do the symbols 2, +, =, and 4 come from? Man 2 + 2 = 4 (Hindu-Arabic) II + II = IV (Roman) What about the underlying process of addition? Copyright  2008 www.biblicalchristianworldview.net

11. Two Key Questions Question 1: Where does mathematics come from? Question 2: Why does mathematics work? Copyright  2008 www.biblicalchristianworldview.net

12. Question 1: Origins Where does mathematics come from? Man, starting from man, gives conflicting answers. Mathematics is the “by-product” of time + chance + matter. Man generates mathematics truths. Copyright  2008 www.biblicalchristianworldview.net

13. Biblical Answers By Him (that is, Christ) “all things were created.” All things includes what is visible and what is invisible. Behind the symbols 2 + 2 = 4 there is a truth principle that is imbedded in the created reality. Copyright  2008 www.biblicalchristianworldview.net

14. Biblical Answers Created reality includes: –Man’s mind: what goes on “inside” (his ability to think mathematically). –The physical world: what goes on “outside” (the patterned structure of creation that can be modeled using mathematics). Copyright  2008 www.biblicalchristianworldview.net

15. Counting Think about all the things in creation that can be counted. Counting is the foundational root of the mathematics tree. Copyright  2008 www.biblicalchristianworldview.net

16. Counting Counting is a way in which man, made in God’s image (imago deo), can organize (name) the creation. Counting is a “tool of dominion.” Copyright  2008 www.biblicalchristianworldview.net

17. Addition of Counting Numbers Addition is repeated counting (counting forward). 1 + 1 + 1 + 1 = 2 + 2 = 4 Copyright  2008 www.biblicalchristianworldview.net

18. Multiplication of Counting Numbers Multiplication is a shortcut way to illustrate repeated addition (there is much more to multiplication than this, though). 4  3 = 3 + 3 + 3 + 3 = 12 4 + 4 + 4 = 12 Copyright  2008 www.biblicalchristianworldview.net

19. Subtraction of Counting Numbers Subtraction is counting backward. 4 – 2 = 4 – 1 – 1 = 2 Copyright  2008 www.biblicalchristianworldview.net

20. Division of Counting Numbers Division is a shortcut way to illustrate repeated addition (again, there is much more to division than this). 12 ÷ 3 = 12 – (3 + 3 + 3 + 3) = 0 You subtract 3 a total of 4 times to get 0. Hence, 12 ÷ 3 = 4 Copyright  2008 www.biblicalchristianworldview.net

21. Exponentiation Involving Counting Numbers Exponentiation is a shortcut way to illustrate repeated multiplication (again, there is much more to division than this). 10 3 = 10  10  10 = 1000 Copyright  2008 www.biblicalchristianworldview.net

22. Extraction of Roots of Counting Numbers Extraction of roots is a shortcut way to illustrate repeated division. Copyright  2008 www.biblicalchristianworldview.net

23. Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Addition and Multiplication Subtraction Division Extraction of roots Copyright  2008 www.biblicalchristianworldview.net

24. Tool of Dominion The real number system is an intellectual tool of dominion. It enables the mathematician and the scientist to explore and quantify the ordered patterns of creation. Rational Numbers Irrational Numbers 2 0  e -18 Copyright  2008 www.biblicalchristianworldview.net

25. Structure of Creation God’s creation is replete with objects that can be: –Counted and –Measured. Mathematics, starting with arithmetic (counting), describes the numerical structure of the physical creation. This creational structure finds its source in the Creator. Copyright  2008 www.biblicalchristianworldview.net

26. Foundations The infinite, personal, living, and triune God created the physical universe, a marvelous structure that reflects the principles of … … number, counting, and addition (and ultimately, every branch of the mathematics tree). 2 + 2 = 4 because of Who the Creator is. Copyright  2008 www.biblicalchristianworldview.net

27. Question 2: Applicability Question 1: Where does mathematics come from? The source of structured order in creation is the Creator. Question 2: Why does mathematics work? Copyright  2008 www.biblicalchristianworldview.net

28. Why Does Mathematics Work? “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  2008 www.biblicalchristianworldview.net

29. Why Does Mathematics Work? Morris Kline, “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  2008 www.biblicalchristianworldview.net

30. Why Does Mathematics Work? “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  2008 www.biblicalchristianworldview.net

31. Why Does Mathematics Work? “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 (New York: Dover Publications [1959] 1981), p. ix. Copyright  2008 www.biblicalchristianworldview.net

32. Why Does Mathematics Work? “The eternal mystery of the world is its comprehensibility.” Albert Einstein Copyright  2008 www.biblicalchristianworldview.net

33. Why Does Mathematics Work? When man rejects fundamental Biblical truths, he is left with the option of mystery, “I do not know why mathematics works.” Copyright  2008 www.biblicalchristianworldview.net

34. Biblical Answer Colossians 1:17 (Weymouth), “And He [Christ] is before all things and in and through Him the universe is a harmonious whole.” There is a harmony, a resonance, in the created order. Copyright  2008 www.biblicalchristianworldview.net

35. Biblical Answer Addition reflects the predictable way in which you can count objects. Without predictable order, it would be impossible to do any mathematics. What if 2 + 2 = 4 worked some of the time? Why learn your addition or multiplication tables if they are not consistent? Copyright  2008 www.biblicalchristianworldview.net

36. Biblical Answer Mathematics is consistent because God holds all things in harmony in and through the person of Christ. Jeremiah 33:25-26, “If I had not established My covenant with day and night and the fixed laws of heaven and earth [including all the laws of mathematics], then I will reject the descendents of Jacob and David My servant.” Copyright  2008 www.biblicalchristianworldview.net

37. Biblical Answer Christ is the mediator of redemption and creation. As we do mathematics in Biblical Christian fashion, our work should continually remind us of God’s faithfulness to the word of His covenant in both redemption and creation. Copyright  2008 www.biblicalchristianworldview.net

38. Biblical Answer Mathematics works (2 + 2 = 4) because the infinite, personal, living, and triune God holds all things in harmony in the person of God the Son, the Lord Jesus Christ. Copyright  2008 www.biblicalchristianworldview.net

39. Biblical Answer The harmony between the physical creation and the human mind exists because of the doctrine of Creation. Man can think mathematically and the physical creation reflects mathematical patterns because God created both. The doctrine of Creation reconciles what is irreconcilable when one’s thinking is based upon man-centered starting points. Copyright  2008 www.biblicalchristianworldview.net

40. Questions Answered Question 1: Where does mathematics come from? The source of structured order in creation is the Creator. Question 2: Why does mathematics work? Mathematics works because of the resonance that exists between the physical creation and the human mind because God created both. Copyright  2008 www.biblicalchristianworldview.net

41. An Example of Harmony The Golden ratio captures the essence and the spirit of creational harmony. height base base/height = Golden Ratio = Copyright  2008 www.biblicalchristianworldview.net

42. Golden Ratio Sometimes this ratio is referred to as the ratio of the “extreme to the mean.” Greek letter “phi.” Copyright  2008 www.biblicalchristianworldview.net

43. Mean/Extreme Ratio a b a + b Copyright  2008 www.biblicalchristianworldview.net

44. Mean/Extreme Ratio Applying the Quadratic formula Copyright  2008 www.biblicalchristianworldview.net

45. Amazing Connection 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … What is the pattern? Copyright  2008 www.biblicalchristianworldview.net

46. Fibonacci Sequence Named after the Italian mathematician Leonardo Fibonacci (1170-1230). He introduced the work of al ‑ Khowarizmi and the Hindu-Arabic positional/decimal system to Christian Europe. Copyright  2008 www.biblicalchristianworldview.net

47. Fibonacci Sequence He discovered this sequence by studying the population of rabbits and the genealogy of bees. More creation connections: seeds of a daisy head, pine cone and pineapple scales, seeds of a sunflower. Copyright  2008 www.biblicalchristianworldview.net

48. Question 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … What happens to the ratio F n+1 /F n as n gets larger and larger? Copyright  2008 www.biblicalchristianworldview.net

49. Answer nF n+1 /F n n 11/1 = 17 21/13  1.615 22/1 = 28 34/21  1.619 33/2 = 1.59 55/34  1.618 4 5/3  1.67 10 89/55  1.618 58/5 = 1.611 144/89  1.618 613/8 = 1.62512 233/144  1.618 Copyright  2008 www.biblicalchristianworldview.net

50. Limits Copyright  2008 www.biblicalchristianworldview.net

51. Johannes Kepler (1571-1630) “Geometry has two great treasures: one is the theorem of Pythagoras; the other the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” Copyright  2008 www.biblicalchristianworldview.net

52. Thinking God’s Thoughts Kepler also stated that when doing mathematics, he was “thinking God’s thoughts after Him.” Copyright  2008 www.biblicalchristianworldview.net

53. The Golden Rectangle Begin with a SquareDivide the Square into two equal partsDraw the radius of a circleExtend the base line Draw a portion of a circleComplete the Rectangle, which is Golden 2 11 2 By the Pythagorean Theorem: 1 2 + 2 2 = 1 + 4 = 5 The length of the red line is  5 The yellow rectangle is also a Golden Rectangle! 2 Copyright  2008 www.biblicalchristianworldview.net

54. Why? Copyright  2008 www.biblicalchristianworldview.net

55. 1 2 3 4 5 6 End with the perfection and wondrous beauty of a Logarithmic Spiral! Start with a Golden Rectangle Copyright  2008 www.biblicalchristianworldview.net

56. Point O is the center of the Spiral The Center Point of the Logarithmic Spiral A B C D O Draw Diagonals AB and CD Amazing! M N Copyright  2008 www.biblicalchristianworldview.net

57. Marvelous Unity in Diversity O J   K   The Logarithmic Spiral is also called an Equiangular Spiral for this reason. Copyright  2008 www.biblicalchristianworldview.net

58. Fractal Geometry Logarithmic Spiral! Copyright  2008 www.biblicalchristianworldview.net

59. Chambered Nautilus Logarithmic Spiral in God’s creation. Note how mathematics conforms exactly with the handiwork of God. Copyright  2008 www.biblicalchristianworldview.net

60. Conclusion Man can think mathematically and the physical creation reflects mathematical patterns because of a common Creator. Mathematics is an unique tool of wonder. Studying mathematics in a Biblical way will lead us to better worship and adore our Faithful God! Copyright  2008 www.biblicalchristianworldview.net