Numbers, Mathematics, Formulas, and Geometric* Designs
Preliminary Considerations
Note:
The following is incomplete. I had to make the choice
whether to devote fulltime study to the subject for several
months, or to present it in preliminary form here. I chose the
latter. I will continue to work on it, so if you check back from
time to time, there may be additions and subtractions.
When I approached this
subject, I had no idea of its complexity. I believe that I have
fairly grasped the other worldview areas that I have summarized
and written about, but a more comprehensive summary of
mathematics would require more study that I can give at this
time. Therefore, I present this material as somewhat
superficial, but needed to address this important area to
complete our review of worldview areas. I refer students and
readers who want to go further in mathematics to the references,
especially James Nickel, Vern Poythress, and Lloyd Jones.
Attention readers:
Any who may
wish to send in suggestions on this worldview, are happily
invited to do so. email epayne7@comcast.net

I am using “mathematics
and geometry” to include trigonometry and formulas for the laws
of physics and chemistry. While the processes of these areas
function according to God’s design regardless of man‘s
understanding, a system of numerology is a tool for man to
learn, work, and advance his application of these areas.
I am using geometry to include all shapes in the universe. These
areas are central even to the arts which are considered by many
to be the most subjective of disciplines, as music is dependent
upon mathematical scales by lengths of strings and art is
dependent upon symmetry, proportion, and spatial relationships.
A short study of
mathematics for the purposes of summary principles has perhaps
been the most rewarding of any worldview area for me.
Everything in the material universe involves a process and a
precision to which mathematics may be applied. Perhaps, many
Christians do not understand the prevalence and complexity of
mathematics because the systems that we use are so familiar and
easy to use. I was once asked, “What is a worldview in
mathematics?” At that point, I could not answer other than to
what end mathematics were used: to glorify God and advance His
common grace (which are extremely valid, but only the smallest
tip of the iceberg).
Recent history in the
United States provides a slightly deeper understanding of the
theories of mathematics. In the 1960s, an attempt was made by
educators to teach mathematics on bases other than ten, which
had been used for centuries. That attempt was a disaster, as
will be reviewed below.
Scripture is full of
mathematics: one God in three Persons; six days of creation in a
seven day week; time measured in days, months, and years; twelve
tribes of Israel and twelve disciples of Jesus; and all the
numbers of imagery of heaven and of prophecy. These is even a
book of Numbers! These numbers are both measurements and
symbols. It is an area of study that is virtually inexhaustible.
But, I will try to present enough in summary to introduce
readers to an understanding of the link of mathematics with The
Creator and His Design, as well as the selective and arbitrary
nature of mathematical convention whose simplicity of operation
belies its underlying complexity.
1. Mathematics, as any
area of study with Biblical first principles. reflects the
greatness, beauty, unity and complexity, power, strength, order,
symmetry, and vastness of God Himself.
While creation of a vast universe is a demonstration of His
power and immensity, mathematics is an attempt to find the order
of His design and to demonstrate its underlying complexity.
The processes that
numbers represent never change, but the theories that underlie
them and their method of practical application may change
greatly by one‘s culture and preference.
Pythagoras worshipped numbers because they never change (from
his perspective). He never knew the Great “I AM” who “is the
same yesterday, today, and forever. But, he recognized numbers
and geometric designs as a great constant in the universe. Two
and two are always four by practical demonstration (but not
necessarily by philosophical agreement, as we will see below).
Object of worship by
pagans. Because
Pythagoras and his followers believed numbers and shapes to be
the great pattern and constant of the universe. They literally
worshipped “counting” numbers. Here is a brief look at their
worship.
The number one,
they argued, is the generator of numbers and the number of
reason; number two is the first even or female number, the
number of opinion; three is the first true male number, the
number of harmony, being compose of unity and diversity;
four is the number of justice or retribution, indicating the
squaring of accounts; five is the number of marriage, the
union of the first male and female numbers, and six is the
number of creation… The holiest of all was the number ten,
or the tetractys, for it represented the number of the
universe, including the sum of all possible dimensions.”
(Nickels, page 2223)
With this pagan
reasoning in mind, the reader here can begin to see that many
philosophers (literally, searchers of wisdom) understand a
complexity and majesty in numbers that the average person does
not grasp. This search and the attempt to understand the
correspondence of man’s mind to a numerical and geometrical
universe occupied the entire lives of many philosophers of
mathematics (below).
But, Pythagoras false
religion was crushed by the very Theorem that bears his name:
the sum of a measurement of one on each side of a rightangle
triangle is two. The square root of two is not a whole number or
a whole number ratio! Tradition says that Hippasus, who
discovered this fact, was thrown overboard from a ship because
he pointed it out! (Nickel, page 21)
Plato, too, merged his
beliefs with mathematics and geometry. “Mathematical objects
(e.g., triangles, circles, etc.) were a part of Plato’s
impersonal world of abstract and perfect ideas, and therefore
fused with his religious philosophy” (Nickel, page 30). “Because
Plato saw the physical world in terms of shadows, his few
applications of geometry to the real world were merely playful
gestures, fanciful pastimes, and intellectual cogitations”
(Nickel, page 31).
While the modern mind
may perceive this sort of belief to be simplistic and fanciful,
it is far closer to the reality of The Unchanging God, than is
the monism of Hinduism, the chance and random forces of
humanists and evolutionists, and the ancestor worship of the
Chinese and Japanese. Indeed, one can posit that mathematics and
geometrical designs reveal a greatness and prevalence about God
that is not revealed in any other way.
Fibonacci numbers and
ratios— extraordinary!
Fibonacci numbers are identified by adding two sequential
numbers, then that sum is added to the previous :
1,1,2,3,5,8,13,21,34,55,89... What is interesting is that any
number (other than 1,1,3), divided by its previous number is a
number that approaches 1.618, no matter how far the numbers are
projected up the scale. This number 1.618 is called phi (f )
Then, there are
diagrams that have the same ratio. Take a square of any size;
from the midpoint on any side and use a line from this point to
either of the opposite corners, and bring the arc down to an
extension of the side that the line is on. The ratio of one side
of the square to the extended line defined by the arc is the
Fibonacci ratio.
Then, there are all
sorts of flowers, breeding animals, geometric designs in
animals, and many other natural designs that may be described by
Fibonacci numbers and ratios. In fact, there are so many that
there is a Fibonacci Quarterly publication and
website (see below), Indeed, these Fibonacci systems are widely
prevalent throughout God’s creation.
Fibonacci numbers and
forms defy any explanation of pagans.
Why should these designs be so prevalent in the universe? If
evolution was the product of chance, how did these numbers
become so widespread in the universe?
2. Cosmology and first
principles. As
I have read and meditated on mathematics, the foolishness and
incoherence of the humanistic, evolutionary worldview has become
more apparent. Nothing in this universe exists without complex
order and design. Even to postulate that such complexity could
come from a Big Bang, random order, and chance defies any
reasonable argument from first principles. Somehow, the
humanists must come up with the creation of atoms, the most
basic unit of the universe. And, they must postulate the
creation of an infinitesimal number of atoms that compose the
universe. Just postulating a theory of evolution is starting a
great distance from the real origin of the universe. While
mathematically atheists might postulate order from randomness,
the complexity of the universe is not possible given
any time frame that evolutionists presently believe.
Even to allow for
chance demands an underlying structure.
If I come to a crossroads, whether it has two, three, or five,
or more open avenues for choice or chance, the avenues
themselves exist because of some prior order or design. If one
goes to a roulette wheel to bet on his “chances,” there is a
structure in which the “chance” takes place. (If the “house” has
rigged the bets, then there is even less “chance” of winning.)
Thus, even chance is not an endless number of choices, but only
one of a few!
Christians have not
recognized, much less taken advantage of the philosophical
arguments available to them.
In many places on this website, I have stated that all religions
and philosophies start with first principles or presuppositions
(axioms)— one‘s position of faith. If one only “knows the
Bible,” he is lost in a philosophical argument and throws away
his power of reasoning with unbelievers. He actually assumes the
worldview of the pagans to argue against them! Now, I am not so
naïve to believe that learning to think logically and rationally
about first principles will necessarily and suddenly overcome
the opposition, but they can be painted into a corner that makes
their position seem ridiculous by any agreedupon philosophical
standards.
Why are nontheists
always depending upon or reasoning from infinity?
Evolutionists posit endless periods of time for their increasing
complexity of genetics by chance, yet under any system,
theoretical or pragmatic, time (by definition) is limited. Even
those measurements of “billions” of years by certain isotopes,
still posit a definite period of time. Thus, by their own
system, naturalists cannot claim endless periods of time.
They still measure time in one way or another! Thus, they
deny their own system with any claim to endless periods of time.
3. Mathematics and
worldview in the United States.
One of the best ways to
begin to understand the philosophical basis of mathematics is
its teaching in American schools. From before 1900 to 1960, “old
math” with basic skills, as simple as, multiplication flash
cards, later algebra, geometry, and trigonometry through high
school.
Then, about 1960 came
the “New Math,” learning by the “application of mathematical
laws… from first grade to college… based upon Set Theory… New
Math eventually self–destructed because no one but
mathematicians can learn math that way.” This attempt was called
a “debacle.”
In 1975, “Reform Math”
which the “practice of basic skills are de–emphasized in favor
of ‘self–paced’ and ‘constructed’ learning“ was begun.
Assessment is based on portfolios, projects, rubrics,
observation, and self–reflection, as well as written tests” (the
“real world”). “Reform Math places great importance on the
selfesteem of the student. Reform Math has stumbled in the eyes
of many educators because of its lack of emphasis in developing
basic skills in a timely fashion.”
“Chaos has resulted for
both educators and students due to the everchanging goals and
teaching methods used over the past 50+ years.”
(Quotes are from the
“mathnasium” website below.)
4. The means and ends
of mathematics are not morally neutral.
All human beings in society use mathematics: to balance check
books, to buy and sell objects, to know how long is a distance
between two cities, to determine time and schedules, and many,
many other operations. All these activities involve choice based
upon moral values.
For example, what are
the moral values of balancing a checkbook? Thou shall not
steal. Whenever I have failed to balance a checkbook, I
usually end up with an overdraft. I have stolen from the bank.
Now, the bank will not allow that overdraft to continue. They
charge me an exorbitant fee to correct that error. The amount of
that fee must be deducted from other purchases that I would
make, so I made new decisions based upon some priority of
purchases.
All choices of priority
are based upon values or moral choices.
Should I buy food to fee the family? Should I put gasoline in
the car? Should I pay the electricity bill? If I “bounced”
several checks, I might not even be able to tithe to my church
this week, as I had planned. These simple examples do not even
begin to describe the centrality of mathematics to our everyday
lives. We hardly think about these things. I could spend several
boring pages on other implications, even that eventually the
bank could take legal action against me. That could have severe
social implications for my job, my time, huge expenses, and my
reputation.
Thou shall not kill.
Complex mathematics, like calculus, is used for building bridges
and skyscrapers. If those structures are not built properly,
great loss of life can (and has) occurred.
Statistics and
numerical grading.
Our modern social and political concerns are based upon a
maniacal reliance upon statistics. Several times an hour, the
“news” channels spew forth some social or political statistic.
Private and government decisions that affect millions in their
freedoms and their pocketbook are based upon these statistics.
Almost all schools “grade” their students by a numerical number,
as though that really represented their knowledge, wisdom, and
willingness to work! See and quote Postman…
TechnopolyI.
Selection of
“significance” numbers and “normals“: arbitrary.
A “normal” may be abnormal and an abnormal may be normal in
individual instances.
5. The very possibility
of mathematics necessitates that man’s mind correspond to the
regularity of the universe.
In a “chance” universe as
postulated by many atheists, agnostics, and humanists, they can
explain neither the regularity of the universe nor the
correspondence of man’s mind to those “laws” that they observe
and apply so effectively.
In fact, perhaps the
best illustration that mathematics has a profound underlying
philosophy is the voiced perplexity of mathematicians from the
beginning of time to understand and explain why mathematics
corresponds to man’s mind and to the universe.
The more honest of
these mathematicians state that fact quite clearly.
“The questions
of the ultimate foundations and the ultimate meaning in
mathematics remain an open problem; we do not know in
what direction it will find its solution, nor even
whether a final objective answer can be found at all.”
(Herman Weyl, Philosophy of Mathematics and Natural
Science, quoted in Nickel, Mathematics, page
3)
I wanted
certainty in the kind of way in which people want
religious faith. I thought certainty is more likely to
be found in mathematics than elsewhere. But I discovered
that many mathematical demonstrations, which my teachers
expected me to accept, were full of fallacies, and that,
if certainty were indeed discoverable in mathematics, it
would be in a new field of mathematics, with more solid
foundation than those that had hitherto been thought
secure. But as the work proceeded, I was continually
reminded of the fable about the elephant and the
tortoise. Having constructed an elephant upon which the
mathematical world could rest, I found the elephant
tottering, and proceeded to construct a tortoise to keep
the elephant form falling. But the tortoise was no more
secure than the elephant and after some twenty years of
very arduous toil, I came to she conclusion that there
was notation more that I could do in the way of making
mathematical knowledge indubitable. (Bertrand Russell,
The Autobiography of Bertrand Russell, quoted in
Nickel, Mathematics…, page 196.)
The problem of
the logical and epistemological foundations of
mathematics has not yet been completely solved. This
problem vitally concerns both mathematicians and
philosophers, for any uncertainty in the foundations of
the “most certain of the sciences” is extremely
disconcerting. Of all the various attempts already made
to solve the problem, none can be said to have resolved
every difficulty. {Paul Benacerraf and Hilary Putnam
(Editors), Philosophy of Mathematics: Selected
Readings (Cambridge, England: Cambridge University
Press, 1983, 2^{nd} Edition), page 41.]
The conflict
between empiricism and rationalism reflects some tension
in the traditional views concerning mathematics, if not
logic. Mathematics seem necessary and a priori,
and yet it has something to do with the physical world.
How is this possible? How can we learn something
important about the physical world by a priori
reflection in our comfortable armchairs? … Mathematics
is essential to any understanding of the world and
science is empirical, if anything is— rationalism
notwithstanding. (Stewart Shapiro, The Oxford
Handbook of Mathematics and Logic (Oxford: The
Oxford University Press, 2005), page 4.)
Other quotes
will be added here. They are numerous.
The unregenerate
(nonChristian) is able to use mathematics because God has
structured his thinking according to His design of the universe
and because nothing else will “work” except that which
corresponds to this design.
Further, his ability to work with mathematics in this way is
directly because of God’s Common Grace to all mankind and the
image of God that continues even in Fallen man.
The unregenerate
scientist assumes the first principle of the constancy and
predictability of the universe.
Otherwise, there would be no reason to investigate because every
experimental result would be different. Thus, he assumes that
some very large intelligent being both designed the universe and
structured its continuing constancy. In essence, he has assumed
God, as only the character of God is “the same yesterday, today,
and forever.” This expected structure is the same as that of
language which is actually just symbols that carry thoughts from
one person to another. Mathematics allows a common language of
function within the universal design. See
Regeneration.
6. The laws and
formulas that man devises are his own conception of creative
design. For
example, the distance (D) that an object falls in a certain
amount of time (t),is represented by the formula, D=½ gt^{2}.
But, the speed of a falling object according to gravitational
attraction is conditioned by an assumption of being a sea level
and in a vacuum. Most places on earth are not at sea level and a
vacuum exists nowhere for this formula. Thus, while the formula
is quite empirically valuable in its usefulness, its application
must be modified in reality. Thus, it is a precise, but not
perfect, “law.” The constancy is God’s design; the law is man’s
design to make that constancy useful.
7. Mathematics within
nature is virtually ubiquitous.
From the orbits and gravitational attraction of the huge bodies
of the universe to the subatomic particles that fly at the speed
of light around the nucleus, the universe is observable and
predictable in its regularity that we call natural laws. These
natural laws are quantifiable in mathematical precision. This
universality of regularity defies any possible explanation other
than a Designer, the Almighty God of Biblical Christianity.
8. The whole is
different than the sum of its parts: the fallacy of composition.
In chemistry two poisons, sodium and chloride, combine to form
an ingredient that few people on earth would eat without its
flavoring— stable salt. The members of a team may be the best
players in the league individually, but if they do not work
together, they will not likely win many games. In history, the
Scottish Highlanders were individually better fighters than the
British troops, but they were easily routed by the discipline of
the English army as a whole. (Many, many other battles of
history were won or lost for the same reasons! These examples
come from Gordon Clark’s book, Logic, pages 1214.)
The Trinity would be an
exception to the whole being greater than the sum of its parts.
While this argument would require extensive Scripture quotes, I
will simply state here that The Trinity is not greater than the
Father, Son, or Holy Spirit. Omnipotence, omnipresence, and
omniscience can be neither more or less than it is.
9. Two plus two may be
something other than four?
There is a Hindu philosophy that “everything is one.” That is,
1+1=1. There is not two or higher number.
Now, that fallacy is
known to the youngest child who knows that the child who gets
two cookies has gotten “more” than the child who got one. But,
make no mistake, this idea of Hinduism is so seriously believed
that one is willing to base his whole life and eternity on it.
Now, one of the tests
of truth is pragmatism. As soon as the Hindu goes to the market
place, he will have to abandon his monism. Actually, perhaps,
there is no position more rationally inconsistent, as his own
thinking is certainly not my thoughts, nor yours, nor anyone
else’s.
“The ‘agreement’ over
mathematical truth is achieved partly by the process, described
elegantly by Thomas Kuhn and Michael Polanyi, of excluding from
the scientific community people of differing convictions.
“Radical monists … are not invited to contribute to mathematical
symposia.” (Poythress, “Mathematics…”— my emphasis).
And, this “theft” from
the Biblical worldview is, perhaps, the strongest philosophical
argument for the Christian to learn. Every nonBiblical argument
must borrow from Biblical Christianity to make its own argument.
The astute and properly taught Christian will break down his
opponent’s argument to that theft. For example, there is no
example in the billions of natural and manmade disasters and
explosions in which order has come out of chaos. Why do we give
any credibility to chaos theory? The Big Bang? The ability of
the mind to think rationally and logically?
10. The statement that
“mathematics is neutral” is an ethical statement in itself.
How does one
get from “what is” to “what ought to be?” The statement itself
is a first principle (axiom, basic presupposition, statement of
cosmology, etc.) that precedes “proof.”
And, evolutionists and
atheists are not consistent.
When I was in the 4^{th} grade, the existence of the
universe was postulated at one billion years. Today, it is
postulated at about 13 billion years. So, changing their periods
of time is philosophical dishonesty. If science is truth, how
can it change to the radical degree of 1300 percent in my
lifetime? To posit that we have greater knowledge and better
instruments today is also dishonest. How do we know that future
science will not demonstrate that the longer periods of time are
actually themselves in error. Using this reasoning of scientists
that the future will bring better methods of measurements, we
could even arrive at the age of the universe being precisely
consistent with Bishop Ussher’s 6000+ years!
11. God has used
several different bases in the Bible.
There are the Ten
Commandments. The numbering of genealogies in Genesis and
elsewhere are on the basis of hundreds (centuries). The
governance of Israel was based upon tens, hundreds, and
thousands (Exodus 18). Prophecies in Revelation concern 1000
years. And, these are only a few examples. God created the earth
in 6 day and rested the seventh, which makes a week based upon
seven days, equivalent to a lunar cycle, and a year (that is one
more day than an equal number of weeks and lunar cycles).
But, then there were
twelve tribes of Israel and twelve disciples. Correspondingly,
there are twentyfour elders around the throne in John’s
Revelation (Chapter 4). A day is based upon 24 hours of 3600
seconds each. Lunar cycles are 28 days and a year is one day
short of 13 lunar cycles.
*Note: Any symbolic
interpretation of these numbers is far beyond our concern here.
12. The English system
is a confusion of bases.
Weights and volumes are based upon sixteen. Length is on the
base of 12 inches to the foot.
13. Negative numbers do
not exist in God’s creation.
Negative numbers are only
useful in calculation. Negative numbers only appear where a zero
base has been set arbitrarily, for example, on the Fahrenheit
scale of temperature, below freezing is negative, but on the
Kelvin scale there are not negatives as zero equals absolute
zero.
14. Infinity.
How can infinity be
numbered? There is no end to either positive or negative
numbers, yet by numbers this infinity can be segmented, that is,
counted. Further, each segment, for example, 1 to 2, can be
decimally for fractionally divided infinitely! Yet,
experientially whole numbers work precisely, as in buying three
oranges or a dozen or one hundred. Infinity is not really a
concept that man’s mind can grasp, as everything that he
encounters on a tangible basis can be measured approximately or
exactly.
15. Numbers are linear,
not circular.
Numbers may be plotted along any line except one that is
circular, that is, meets its own starting point.
16. God the Three in
One answers the problem of the One and the Many.
Monism vs. pluralism.
14. Mathematics has its
own limitations.
Trying to reach a destination by segments, as the tortoise
chasing the hare in 1/2 increments.
The Greeks not only
limited their own developments, they retarded those who came
after them.
Thus, the return of the Scholastics to Greek thought could not
have led to the scientific revolution. The Reformation was
necessary. Their gods were too small.
The Greeks also proved
the limitations of pure rationalism.
Understanding reality and working within it requires both
rationalism and materialism, the physical and the mental
You could even
say that this true pragmatism refutes any possibility of
“epiphenomenalism,” that is, the brain existing without
the mind… what?
Modern times.
Moderns must adopt the theistic worldview before proceeding with
modern science.
___ Set theory.
Discussed by Plantinga
http://www.leaderu.com/truth/1truth10.html
Biblical Chronology and
numbering . God
the three in one. God created in seven days… He planned history
(time) according to exact dates. He numbers the hairs of our
heads. Ten Commandments. Seventy generations. Forgive 70x7. Book
of Numbers.
Mathematicians, perhaps
in a way that not other experts do, realize that the universe
has a oneness that cannot be explained. Floyd Jones CDs from
Megahistory conference and put his book in References.
Biblical chronology is
reliable!
The problem of motion.
Mathematicians,
starting with the Greeks, were concerned with motion. As was
Thomas Aquinas, if motion, then a first mover. The wave
structure of matter is the latest theory that tries to tie
together all matter in the universe. See various references in
the References Bookmark folder.
Chaos theory?
Worship of numbers
today. Science,
sociology, psychology, etc. See Postman on making everything
into numbers.
“The one and the
many is perhaps the basic question of philosophy. Is unity
or plurality, the one or the many, the basic fact of life, the
ultimate truth about being? If unity is the reality, and the
basic nature of reality, then oneness and unity must gain
priority over individualism, particulars, or the many. If the
many, or plurality, best describes ultimate reality, then the
unit cannot gain priority over the many; the, state, church, or
society are subordinate to the will of the citizen, the
believer, and of man in particular. In the one is
ultimate, the individuals are sacrificed to the group. If the
many be ultimate, then unity is sacrificed to the will of
many, and anarchy prevails.” [R. J. Rushdoony, The One and
the Many (Thoburn Press, 1978), page 2, n2— italics his.]
“The one
refers not to a number but to unity and oneness; in metaphysics,
it has usually meant the absolute, the supreme Idea for Plato,
the universe for Parmenides, Being as Such for Plotinus, and so
on. The one can be a separate whole, or it can be the sum
of things in their analytic or synthetic wholeness … The many
refers to the particularity or individuality of things; the
universe is full of a multitude of beings; is the truth
concerning them inherent in their individuality, or is it in
their basic oneness. If it is their individuality, then the
many are ultimate and the proper source of authority, and we
have philosophical Nominalism. If it is their oneness, then the
one is
ultimate, and we have Realism. According to Realism, universals,
which are terms applicable to all the universe and can be called
real “second substance,” are aspects of the one Idea and exist
within it. Egyptian, much Greek, and medieval scholastic thought
has bee “Realistic.” For “Nominalism,” abstract or general terms
have no real existence and are mere names applied to aspects of
reality; reality belongs to particulars, actual physical
particulars, so that the truth of being is simply that
individual things exist. Truth is not some abstraction
concerning particular things but is simply the fact of
particularity.” (Rushdoony, The One…, pages 23.)
Here is primarily a
philosophical (metaphysical) problem, and secondarily, an
ethical or worldview problem. It is philosophical, as Rushdoony
has discussed, for what one understands as Ultimate Reality.
(Ultimate Reality should really be capitalized, as It really
defines Who or What a person worships and from Who or What one
defines ethics— what is right and what is wrong. Ultimate
Reality is thus one’s God.)
For nonChristians,
their ethics is simply those unexamined, accumulated “oughts”
over their lifetime from parents, teachers, and others which
they find convenient and pleasurable to guide their own lives.
For Christians, their ethics is also usually unexamined, but at
least they hear sermons and read other Christians which may
cause some degree of implementation into their lives.
References
http://www.engineering.sdstate.edu/~fib/.
Website on
Fibonacci numbers
http://www.framepoythress.org/poythress_articles/1974Creation.html
Creation and
mathematics by Frame
http://www.framepoythress.org/poythress_articles/1976Biblical.htm
A Biblical view of
mathematics
http://nrich.maths.org/public/viewer.php?obj_id=2591
A simple article on
base theory and the abacus
http://www.mathnasium.com/history.htm
A brief history of
mathematics in the United States since 1900. It has some
discussion of the philosophical approaches.
http://mathworld.wolfram.com/
“The web’s most
extensive mathematics resource” Hundreds of articles on all
areas of mathematics and geometry.
Nickel, James.
Mathematics: Is God Silent? (Vallecito, California: Ross
House Books, 1990)http://www.chalcedonstore.com/xcart/product.php?productid=2463&cat=0&page=1
Postman, Neil.
Technopoly. Available at Amazon.com
