Transcript Philosophical Principles of Mathematics

Mathematics: Locus of Reality
By James Nickel, B.A., B.Th.,
B.Miss.,
M.A.
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Differentiate
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There are facts of mathematics and there
are philosophies of the facts of
mathematics.
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Practical Teaching
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Must be based upon sound principles.
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Line upon line.
Precept upon precept.
Simple to complex.
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It may be surprising to realize that many
teachers do not teach using sound
principles.
Arithmetic is not easy to teach!
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Practical Teaching
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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.
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Issues in Philosophy
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“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.”
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Remo J. Ruffini, “The Princeton Galaxy,” interviews by
Florence Heltizer, Intellectual Digest, 3 (1973), 27.
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Issues in Philosophy
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“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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Issues in Philosophy
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“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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Issues in Philosophy
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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.”
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Morris Kline, Mathematics and the Physical World, p. ix.
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Issues in Philosophy
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“The eternal mystery of the world is its
comprehensibility.”
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Albert Einstein, Out of My Later Years, p. 61.
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Important Questions
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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.
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Miracles Abound
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“It is difficult to avoid the impression that
a miracle confronts us here.”
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Eugene Wigner, Symmetries and Reflections: Scientific
Essays, p. 229.
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Miracles Abound
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“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.”
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Erwin Schrödinger, What is Life? The Physical Aspects of the
Living Cell, p. 31.
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Denying reality
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These eminent scientists and
mathematicians will not posit the Biblical
God as an answer.
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Creator of the human mind with its ability to
think mathematically.
Creator of the physical world that reflects
mathematical principles.
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No answers
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Without the Biblical God as the
foundation, there are no answers as to
“why mathematics works.”
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Consequences
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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).
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Consequences
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The logic of the human mind has no
relationship to physical reality.
The appeal to mystery is a leap into
irrationality.
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Reality
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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.
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Reality
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Hence, to suppress the reality of meaning,
mathematicians/scientists invoke
probabilistic concepts (the sun may rise
tomorrow).
But probability belies a “hint” of meaning.
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Reality
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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.”
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What is mathematics?
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Do we create mathematics autonomously?
Or, do we discover and classify
mathematical patterns in a world of preestablished harmony?
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Important Questions
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“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?”
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Leslie A. White, “The Locus of Mathematical Reality: An
Anthropological Footnote,” The World of Mathematics,
4:2348.
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Creation vs. Classification
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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.
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Pre-established Harmony
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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?
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Locus of Reality
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Where is the locus of reality, including
mathematical reality?
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Man?
Matter?
Biblical God?
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Starting Points
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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.”
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Glenn R. Martin (1935-2004)
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Starting Points
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What is the ultimate foundation for
reality? for knowledge?
God, matter, or man
The starting point will always be assumed
to be autonomous.
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Starting Points
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“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.”
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Nancy Pearcey, Total Truth, p. 41.
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Pseudo-starting Points
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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.”
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NB. Man is still the determining factor.
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True Starting Points
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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.
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Minds matter
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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.
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Key Questions
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The key issue in mathematics.
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Does the mind of man create the world out of
chaos?
Or, is there a pre-established, God-created,
and covenantally-ordered world?
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Key Answer
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“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.”
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Rousas J. Rushdoony, Systematic Theology, 1:199.
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Autonomous Man
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“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.”
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Autonomous Man
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“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.”
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John W. N. Sullivan (1886-1937), “Mathematics as an Art,”
The World of Mathematics, 3:2021.
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The Meaning of Number
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Every culture has its own mathematics
because of its world view or faith.
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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.
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The Meaning of Number
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“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.”
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Oswald Spengler, “Meaning of Numbers,” World of
Mathematics, 4:2315.
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Ancient Greeks
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For the ancient Greeks, number was
merely and only positive integers.
The ancient Greeks were radically hostile
to the idea of infinite.
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Irrational numbers.
Infinite processes.
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Christendom
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In Christendom, infinity was not viewed
with hostility.
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God, the Creator, is personal and infinite.
Medieval scholastics in the High Middle Ages
embraced the rudiments of infinite processes
(the real number continuum).
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Real Number Continuum
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“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.”
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Real Number Continuum
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“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;”
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Real Number Continuum
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“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.”
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Tobias Dantzig, Number: The Language of Science, pp. 245246.
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Christendom
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This world view released the ideas
necessary that resulted in the viable birth
of modern science.
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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.
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Analysis
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According to Oswald Spengler (18801936), faith, like mathematics, is culturecentric.
Hence:
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Is there one, true Faith?
And, resultantly, is there one, true mathematic
(or philosophy of mathematics)?
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The One and the Many
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“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.”
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The One and the Many
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“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.”
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The One and the Many
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“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.”
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The One and the Many
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“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.”
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Vern Poythress, “A Biblical View of Mathematics,” The
Foundations of Christian Scholarship, p. 161.
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The One and the Many
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Is “oneness” or unity ultimate?
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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!) .
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The One and the Many
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Is “manyness” or diversity ultimate?
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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!
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The One and the Many
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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.
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The Locus of Reality
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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.”
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The Locus of Reality
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Mathematics, subsumed by this foundation,
becomes:
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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.
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The Locus of Reality
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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.
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The Locus of Reality
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all things must be defined in relationship to
Him, or else we have a false definition.”
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The Locus of Reality
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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.
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