Transcript Philosophical Principles of Mathematics

Seeing God in Creation Through
Mathematics
by
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1011 Nickel,
James
B.A., B.Th., B.Miss., M.A.
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Thesis
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The infinite, personal, and Triune God
revealed in Scripture is the foundation
for understanding and doing
mathematics.
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Language
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Sir Oliver Graham Sutton (1903-1977), “The
universe, both as a whole and in its
microstructure, suggests that in neither aspect
can it be treated merely as an enlarged or
diminished version of the world which we
know through our senses. The ultimate secrets
of nature are written in a language which we
cannot yet read. Mathematics provides a
commentary on the text, sometimes a close
translation, but in words we can read because
they are our own.”
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Intelligence
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Heinrich Hertz (1857-1894), “One
cannot escape the feeling that these
mathematical formulas have an
independent existence and an
intelligence of their own, that they are
wiser than we are, wiser even than their
discoverers, that we get more out of them
than was originally put into them.”
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Reality
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Godfrey H. Hardy (1877-1947), “I
believe that mathematical reality lies
outside of us, and that our function is to
discover or observe it, and that the
theorems which we prove and which we
describe grandiloquently as our
‘creations’ are simply notes of our
observations.”
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Truth
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Philip E. B. Jourdain (1879-1919), “Some
philosophers have reached the startling
conclusion that Truth is made by men, and that
Mathematics is created by mathematicians, and
that Columbus created America; but common
sense, it is refreshing to think, is at any rate
above being flattered by philosophical
persuasion that it really occupies a place
sometimes reserved for an even more sacred
Being.”
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Proximate and Ultimate
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Proximate: close to (what we experience
and observe).
Ultimate: far away (God, the
transcendent One).
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Logos
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Genesis 1:1, “In the beginning God
created the heavens and the earth.”
John 1:1-3, “In the beginning was the
Word, and the Word was with God, and
the Word was God. He was in the
beginning with God. All things were
made through him, and without him was
not any thing made that was made.”
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Logos
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Colossians 1:15-17, “He [Christ] is the
image of the invisible God, the firstborn
of all creation. For by him all things were
created, in heaven and on earth, visible
and invisible, whether thrones or
dominions or rulers or authorities—all
things were created through him and for
him. And he is before all things, and in
him all things hold together.”
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Church History
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Athanasius (ca. 293-373) stood contra
mundum (against the world) of Arian
heresy (that purported Christ as not being
fully God).
In this context, he recognized the
significance of a fully rational and wise
God creating a fully rational and good
(interconnected and interacting) creation.
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Against the Heathen
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After citing John 1:1, Athanasius described the
universe as a divine hymn: “For just as though
some musician, having tuned a lyre, and by his
art adjusted the high notes to the low, and the
intermediate notes to the rest, were to produce
a single tune as the result, so also the Wisdom
of God, handling the Universe as a lyre, and
adjusting things in the air to things on the
earth, and things in the heaven to things in the
air, and combining parts into wholes and
moving them all by His beck and will,
produces well and fittingly, as the result, the
unity of the universe and of its order …”
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Against the Heathen
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“He produces as the result a marvelous
and truly divine harmony.”
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What is Mathematics?
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“The spirit of genuine mathematics, i.e., its
methods, concepts, and structure–in contrast
with mindless calculations–constitutes one of
the finest expressions of the human spirit. The
great areas of mathematics–algebra, number
theory, combinatorics, real and complex
analysis, topology, geometry, trigonometry,
etc.–have arisen from man’s experience of the
world that the infinite, personal, Triune, and
Sovereign God has created and currently
sustains.”
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What is Mathematics?
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“These branches of mathematics,
constructively developed by man made
in the image of God [imago deo], enable
man to systematize the order and
coherence (the unity in diversity … the
proximate one and the many) of creation
mediated to us by the Creator and
upholder of all things–the logos and
wisdom of God revealed in the person of
the Lord Jesus Christ.”
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What is Mathematics?
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“This systematization not only gives man
a tool whereby he can take effective
dominion over the creation under God in
Christ, but also gives man the experience
and enjoyment of a rich intellectual
beauty that borders the sublime in its
infinitely complex, yet structured
mosaic.”
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Mathematics: Two Complementary
Contexts
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Practical, as a tool of dominion
(personal, business, scientific).
Beauty (seen via the study of its
symmetry and structure); mathematics
unveils fresh and exciting vistas into the
nature of truth, beauty, and goodness.
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The Mystery of the Comprehensible
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Albert Einstein (1879-1955), “At this point an
enigma presents itself which in all ages has
agitated enquiring minds. How can it be that
mathematics, being after all, a product of
human thought which is independent of
experience, is so admirably appropriate to the
objects of reality? ... The eternal mystery of
the world is its comprehensibility.”
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The Mystery of the Comprehensible
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John Von Neumann (1903-1957), “The
relationship of mathematics to the natural
sciences is quite peculiar, but this is the
most vitally characteristic fact about
mathematics.”
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The Mystery of the Comprehensible
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Nicholas Bourbaki (pseudonym for a group of
20th century French mathematicians),
“Mathematics appears thus as a storehouse of
abstract forms ... and it so happens–without our
knowing why–that certain aspects of empirical
reality fit themselves into these forms, as if
through a kind of preadaptation.”
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The Mystery of the Comprehensible
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Eugene Wigner (1902-1995), “The
enormous usefulness of mathematics in
the natural sciences is something
bordering on the mysterious. There is no
rational explanation for it.”
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The Confidence of the
Comprehensible
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The fact that there is a correlation
between the subject (man) and the object
(creation) is no mystery for the Biblical
Christian.
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The Confidence of the
Comprehensible
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The infinite, personal, Triune, and
sovereign God revealed in Scripture
made the universe.
This same God made man to live in the
universe.
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The Confidence of the
Comprehensible
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This same God gave man the Bible,
verbalized, propositional, and truth
revelation, to reveal to man what he
needs to know.
In the Bible, God gives comprehension
to correlate our knowledge.
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The Confidence of the
Comprehensible
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The reason why the Biblical Christian
has no problem with the “mystery of the
comprehensible” is because the all-wise
God, being the ultimate in reason, made
both things, the world outside of man and
the world inside of man, the subject and
the object … He is the creator of both.
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The Confidence of the
Comprehensible
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Since the Biblical God is the creator of
the physical universe that reflects
mathematical properties and the human
mind that can think in mathematical
terms, the fact that the world is
comprehensible is what we would expect!
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The One and the Many
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The Triune nature of God (One and the
Many) forms the foundation for
understanding the proximate “one and
the many”–and mathematics is adept at
unveiling this “unity in diversity.”
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Universals and Particulars
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Scientific law (a unity derived from a
variety of observations).
Scientific law reduces particulars into a
universal.
Scientific law is often written in
mathematical form.
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Number: Unity in Diversity
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Number is unity (i.e., the concept of
“oneness,” “twoness, “threeness,” etc.)
Note: Numerals are symbols of number.
This unity is revealed through diversity
(one coin, two coins, three coins, four
hens, five dogs, etc.).
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Number: Unity in Diversity
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Without a philosophy that confidently
justifies the connection between unity to
diversity, number is in limbo.
That the world is comprehensible (in
terms of mathematics) is a mystery only
for those who reject God’s word is that
starting point for knowing anything.
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Unifying Elements
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Why are there so many unifying
elements in mathematics (functions,
vectors, etc.), interlacing what initially
appears to be completely diverse ideas
invented independently by different
people at different times in different parts
of the world?
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Trigonometry
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Trigonometry was developed by
mathematicians of Alexandria, Egypt (ca.
200 BC) and starts with ratios of the
sides of a right triangle.
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Trigonometric Functions
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y = sin(x)
In the 16th century, mathematicians
developed analytical or coordinate
geometry.
The graph of this function is periodic or
wave-like in nature.
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Trigonometric Functions
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3.5
3
2.5
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1.5
1
fx = sinx
0.5
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-5
-4
-3
-2
-1
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-0.5
-1
-1.5
-2
-2.5
-3
-3.5
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Fourier Series
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The sum of trigonometric functions.
In the 18th century, Joseph Fourier (17681830) uncovered this series in his study
of heat.
From right triangle ratios to the dynamics
of heat!
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Stunning Connections
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Music (sound waves)–oscilloscope.
Electromagnetic spectrum.
Meandering rivers.
Tides: ingoing and outgoing.
Branch of a tree swinging in the breeze.
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Stunning Connections
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The swell of the ocean when the sea is
smooth.
Rotation of galaxies.
Alternating current: ebb and flow.
Radio waves (AM & FM).
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Stunning Connections
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Spring and bob motion, studied by
Robert Hooke (1635-1705).
John Harrison (1693-1776) used the
theory developed by Hooke to invent the
first practical marine chronometer.
This clock enabled man to “conquer” the
longitude.
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Super Law
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Isaac Newton (1642-1727) connected
motion celestial with motion terrestrial
with one law, described mathematically
as the inverse-square law:
Gm 1m 2
F
2
D
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Inverse Square Law
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Law of Universal Gravitation.
Electricity: Coulomb’s law.
Intensity of light and sound waves at a
distance.
What unity in diversity!
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Golden Ratio
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 = extreme/mean ratio.
Divide a line segment into two sections,
one larger than the other.
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Golden Ratio
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x x 1

1
x
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If we set
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  1.618
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Stunning Unity in Diversity
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Infinite nested radical.
  1 1 1 1
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Stunning Unity in Diversity
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Infinite continued fraction.
  1
1
1
1
1
1
1
1
1
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Stunning Unity in Diversity
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Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 134, …
Related to how large populations grow
 often reflected in God’s created order.
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Fn 1
lim

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n 
Fn
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Simultaneous Inventions
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Why are there so many simultaneously
produced identical mathematical
“inventions”?
Calculus (Newton & Leibniz),
Non-Euclidean Geometry (Gauss,
Lobachevsky & Bolyai), and vector
analysis (Hamilton and Grassman).
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Simultaneous Inventions
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It is like a group of artists who
independently all find themselves
painting the same gigantic picture; and
some are painting the same parts at the
same time!
God, in His providence, orchestrates
these historical unities in diversities.
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Language, Logos, and Unity
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Johannes Kepler (1571-1630), “The chief
aim of all investigations of the external
world should be to discover the rational
order and harmony which has been
imposed on it by God and which He
revealed to us in the language of
mathematics.”
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Language, Logos, and Unity
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Stanley L. Jaki (1924-), science historian,
notes that in Kepler we find the “heroic
groping of a great man of science with
facts, with ideas, with perspectives, and
not least with the need to arrive at a law
which enabled the prediction of planetary
positions with the greatest possible
accuracy.”
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Language, Logos, and Unity
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Kepler saw harmony in the universe, a
harmony derived from his commitment
to God the Creator, not the sovereignty
of number (as the ancient Pythagorean’s
posited).
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Language, Logos, and Unity
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Harmony is only possible if one can discern
patterns (unity) in observations (scientific and
mathematical).
Einstein, regarding Kepler’s approach as a
basic principle of scientific work, comments,
“Knowledge cannot spring from experience
alone but only from the comparison of the
inventions of the intellect with observed fact.”
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Language, Logos, and Unity
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Mathematics is a language.
By learning this language you will be
able to better sense the “language fabric”
that sustains the universe (the logos of
God in Christ).
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Language, Logos, and Unity
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Vern Poythress (1946-), “The created
world, as result of God’s speech, bears
within it from top to bottom a kind of
quasilinguistic character ... through
God’s act of creation, things in the world
themselves become wordless voices to
the praise of God.”
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Language, Logos, and Unity
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Larry Zimmerman, “It is possible, that
mathematics is an entity which always exists in
the mind of God, and which is for us the
universal expression of His creative and
sustaining word or power.... So we would
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.”
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Language, Logos, and Unity
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“This language fabric should itself be
suggestive of an intellectual antecedent,
an orderly, powerful, infinitude of
thought, a ‘terra incognita of pure
reasoning’ which ‘casts a chill on human
glory.’”
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Language, Logos, and Unity
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Richard Feynman (1918-1988), “To
those who do not know mathematics it is
difficult to get across a real feeling as to
the beauty, the deepest beauty, of
nature.”
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Language, Logos, and Unity
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Larry Zimmerman, “Without the knowledge of
the patterns of God’s speech in the creation, we
are powerless to replenish the earth and will
instead be subdued by it .... the lyric of the
‘music of the spheres’ is mathematics, a
knowledge of which unveils not only vistas of
beauty and power unsuspected before but also
an order, symmetry and infinitude which stuns
and awes the beholder.”
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Thesis
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The infinite, personal, and Triune God
revealed in Scripture is the foundation
for understanding and doing
mathematics.
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Implications
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Resolves the mystery of
comprehensibility.
Justifies the “unity in diversity” that we
see all around us.
Frames mathematics where its picture of
wonder, beauty, and power directs the
observer to worship, in transcendent awe,
the infinite, personal, and Triune Creator
and Sustainer of all things!
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