Numbers, Mathematics, Formulas, and Geometric* Designs

Preliminary Considerations

Note: The following is incomplete. I had to make the choice whether to devote full-time 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.

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 22-23)

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 right-angle 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?

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 universeis 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 agreed-upon philosophical standards.

Why are non-theists 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.

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 self-esteem 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 ever-changing goals and teaching methods used over the past 50+ years.”

(Quotes are from the “mathnasium” website below.)

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.

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 (non-Christian) 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.

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 gt². 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.
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.
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 12-14.)

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.

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 non-Biblical 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 man-made 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?

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!

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 twenty-four 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.

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.
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.
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.
Numbers are linear, not circular.Numbers may be plotted along any line except one that is circular, that is, meets its own starting point.
God the Three in One answers the problem of the One and the Many.Monism vs. pluralism.
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 70×7. 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 Mega-history 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 2-3.)

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 non-Christians, 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.