Mathematics: Questions, Answers, and Wonder by James Nickel B.A., B.Th., B.Miss., M.A. Copyright 2008

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Mathematics:Questions, Answers, and

Wonder

byJames Nickel

B.A., B.Th., B.Miss., M.A.

Copyright 2008www.biblicalchristianworldview.net

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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.


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Starting points

• Psalm 36:9, “In Thy light, we see light.”• Without God’s light, we walk in darkness.


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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).


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Starting points

• Every point of view is a view from a point.


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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.


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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

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Starting points

• Colossians 2:3 (Weymouth), “In Him [Christ] all the treasures of wisdom and knowledge are stored up, hidden from view.”


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Consider

2 + 2 = 4


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Consider2 + 2 = 4

• Where do the symbols 2, +, =, and 4 come from?

Man2 + 2 = 4 (Hindu-Arabic)

II + II = IV (Roman)• What about the underlying process of

addition?


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Two Key Questions

• Question 1: Where does mathematics come from?

• Question 2: Why does mathematics work?


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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.


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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.


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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).


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Counting

• Think about all the things in creation that can be counted.

• Counting is the foundational root of the mathematics tree.


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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.”


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Addition of Counting Numbers

• Addition is repeated counting (counting forward).

1 + 1 + 1 + 1 =

2 + 2 =

4


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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


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Subtraction of Counting Numbers

• Subtraction is counting backward.4 – 2 =

4 – 1 – 1 = 2


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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


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Exponentiation Involving Counting Numbers

• Exponentiation is a shortcut way to illustrate repeated multiplication (again, there is much more to division than this). 103 = 10 10 10 = 1000


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Extraction of Roots of Counting Numbers

• Extraction of roots is a shortcut way to illustrate repeated division.


31000 10

3times

1000 100 10100 10 1

10 10 10

144444444444424444444444443

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Real Numbers

Real Numbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

Addition and Multiplication

SubtractionDivision

Extraction of roots


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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

85

e

5 2-18

23


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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.


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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.


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Question 2: Applicability

• Question 1: Where does mathematics come from?The source of structured order in creation is the Creator.



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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.


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• 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 …


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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.”


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• “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.


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• “The eternal mystery of the world is its comprehensibility.”

Albert Einstein


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• When man rejects fundamental Biblical truths, he is left with the option of mystery, “I do not know why mathematics works.”


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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.


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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?


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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.”


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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.


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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.


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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.


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Questions Answered

• Question 1: Where does mathematics come from?

The source of structured order in creation is the Creator.


Mathematics works because of the resonance that exists between the physical creation and the human mind because God created both.


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An Example of Harmony

• The Golden ratio captures the essence and the spirit of creational harmony.

height

base

base/height = Golden Ratio = 1 5

1.618033989...2


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Golden Ratio

• Sometimes this ratio is referred to as the ratio of the “extreme to the mean.”

ExtremeGoldenRatio

Mean

Greek letter “phi.”


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Mean/Extreme Ratio

aa b

b

a b aa b

b b bb b

a b

a + bb( 1)

b

b

b

2

2

1

1

1 0

Copyright 2008


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Mean/Extreme Ratio2 1 0

1 1 4( 1)

21 1 4

21 5

1.618, 0.6182

Applying the Quadratic formula


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Amazing Connection

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

• What is the pattern?


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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.


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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.


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Question

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

• What happens to the ratio Fn+1/Fn as n gets larger and larger?


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Answer

n Fn+1/Fn n Fn+1/Fn

1 1/1 = 1 7 21/13 1.615

2 2/1 = 2 8 34/21 1.619

3 3/2 = 1.5 9 55/34 1.618

4 5/3 1.67 10 89/55 1.618

5 8/5 = 1.6 11 144/89 1.618

6 13/8 = 1.625 12 233/144 1.618


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Limits

n 1

n n

Flim 1.618

F


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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.”


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Thinking God’s Thoughts• Kepler also stated that when doing

mathematics, he was “thinking God’s thoughts after Him.”


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The Golden Rectangle

5 1

Begin with a SquareDivide the Square into two equal partsDraw the radius of a circleExtend the base lineDraw a portion of a circle Complete the Rectangle, which is GoldenL 1 5

1.618W 2

2

1 1

5

2

By the Pythagorean Theorem:

12 + 22 = 1 + 4 = 5

The length of the red line is 5

The yellow rectangle is also a Golden Rectangle!L 2

1.618W 5 1

5 1

2


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Why?

2 1 525 1

2 5 1 22 5 15 15 1 5 1

2

5 1

4

2

1 52


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12

34

5

6

End with the perfection and wondrous beauty of a Logarithmic Spiral!

Start with a Golden Rectangle


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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

AMMB

CNND

Amazing!

M N


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Marvelous Unity in Diversity

O

J

K

The Logarithmic Spiral is also called an Equiangular Spiral for this reason.


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Fractal Geometry

Logarithmic Spiral!Copyright 2008


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Chambered Nautilus

• Logarithmic Spiral in God’s creation.• Note how mathematics conforms exactly with the

handiwork of God.Copyright 2008


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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!


Documents

Mathematics: Questions, Answers, and Wonder by James Nickel B.A., B.Th., B.Miss., M.A. Copyright 2008