Mathematics- you either hated it or loved it in school. “1+1=2” is a given, we know it to always be true. Except it is not! There are actually instances of where “1+1=1” and an entire field of mathematics has been developed to prove this paradox. I was asked if there was a Biblical reason for mathematics and yes, of course there is. Without an orderly, intelligently designed world, there would be no use for mathematics. Therefore, in this article we will discuss the philosophy of numbers, the practical application of mathematics and the Biblical foundation for all of it.
Mathematics has been defined as “the study of relationships among quantities, magnitudes and properties, and also of the logical operations by which unknown quantities, magnitudes, and properties may be deduced” (Microsoft Encarta Encyclopedia) or more simply “the study of quantity, structure, space and change” (Wikipedia).
Historically, mathematics was regarded as the science of quantity, whether of magnitudes (as in geometry) or of numbers (as in arithmetic) or of the generalization of these two fields (as in algebra). Some have seen it in terms as simple as a search for patterns.
All that we know of the universe around us (except for the understanding of living things) is a result of the accurate application of mathematics. Knowledgeable people have used mathematics to try to explain the variance in behavior of living things, but to no avail.
We will first explore mathematics itself, its development from simple counting of objects to abstract hypothesis dealing with generalities instead of specifics (fuzzy logic). Then we will deal with how the knowledge of mathematics helps us to identify the one and only Intelligent Designer of the universe and all that is in it- God Himself.
I will be approaching the subject from both the fallible and incorrect evolutionary perspective of millions of years (MYA) and the (Biblical perspective) until I can merge them both into an agreeable timeline for both.
Our ancestors (Adam and Eve) would have had a general sensibility about amounts, and would have instinctively known the difference between, say, one and two wooly mammoths. The intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of “two” might have taken some time.
Early Babylonian and Egyptian records and the Hebrew Bible indicate that length was first measured with the forearm, hand, or finger and that time was measured by the periods of the sun, moon, and other heavenly bodies. When it was necessary to compare the capacities of containers such as gourds or clay or metal vessels, they were filled with plant seeds (I hope they were not using mustard seeds) which were then counted to measure the volumes. So here we have length (or distances), time, and volume. The only critical measurement missing at this point in time is mass or weight. Now, I am sure that they knew that a chicken egg was not as heavy as a cow, but they just did not have a particular method to measure the difference at first.
Early man (Adam and Eve after the Fall) kept track of regular occurrences such as the phases of the moon and the seasons. Some of the very earliest evidence of the thinking about numbers is from notched bones in Africa (supposedly dating back to 35,000 to 20,000 years ago). However, we have no actual idea that this was what was actually being done- nobody was there). It is simple counting and tallying rather than mathematics as such.
Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and it is often referred to as the ‘Cradle of Civilization’. The Sumerians developed the earliest known writing system – a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets – and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics
Sumerian mathematics initially developed largely as a response to bureaucratic needs when their civilization settled and developed agriculture (possibly as early as the 6th millennium BC) for the measurement of plots of land, the taxation of individuals, etc. In addition, the Sumerians and Babylonians needed to describe quite large numbers as they attempted to chart the course of the night sky and develop their sophisticated lunar calendar
Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand and the five fingers on the other hand. I do no know about you but it certainly seemed like from high school through my college math classes every one of the instructors would have us to a set of problems in our typical base10 and then make us resolve the problems in either base12 or base60. Cruel I called them and useless was the exercise.
It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 – in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc).
The Babylonians also developed another revolutionary mathematical concept, something else that the Egyptians, Greeks and Romans did not have, a circle character for zero, although its symbol was really still more of a placeholder than a number in its own right.
The early Egyptians settled along the fertile Nile valley as early as about 6000 BC, and they began to record the patterns of lunar phases and the seasons, both for agricultural and religious reasons. The Pharaoh’s surveyors used measurements based on body parts (a palm was the width of the hand, a cubit the measurement from elbow to fingertips) to measure land and buildings very early in Egyptian history, and a decimal numeric system was developed based on our ten fingers.
The Rhind Papyrus, dating from around 1650 BC, is a kind of instruction manual in arithmetic and geometry, and it gives us explicit demonstrations of how multiplication and division was carried out at that time. It also contains evidence of other mathematical knowledge, including unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series. The Berlin Papyrus, which dates from around 1300 BC, shows that ancient Egyptians could solve second-order algebraic (quadratic) equations.
It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BC (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy
Practical problems of trade and the market led to the development of a notation for fractions. The papyri which have come down to us demonstrate the use of unit fractions based on the symbol of the Eye of Horus, where each part of the eye represented a different fraction, each half of the previous one (i.e. half, quarter, eighth, sixteenth, thirty-second, sixty-fourth), so that the total was one-sixty-fourth short of a whole, the first known example of a geometric series.
As the Greek empire began to spread its sphere of influence into Asia Minor, Mesopotamia and beyond, the Greeks were smart enough to adopt and adapt useful elements from the societies they conquered. This was as true of their mathematics as anything else, and they adopted elements of mathematics from both the Babylonians and the Egyptians. But they soon started to make important contributions in their own right and, for the first time, we can acknowledge contributions by individuals
The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed by about 450 BC, and in regular use possibly as early as the 7th Century BC. It was a base 10 system similar to the earlier Egyptian one (and even more similar to the later Roman system), with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the desired number. Addition was done by totaling separately the symbols (1s, 10s, 100s, etc) in the numbers to be added, and multiplication was a laborious process based on successive doublings (division was based on the inverse of this process).
To some extent, however, the legend of the
6th Century BC mathematician Pythagoras of Samos has become synonymous with the birth of Greek mathematics. Indeed, he is believed to have coined both the words “philosophy” (“love of wisdom”) and “mathematics” (“that which is learned”). Pythagoras was perhaps the first to realize that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers. Pythagoras’ Theorem (or the Pythagorean Theorem) is one of the best known of all mathematical theorems
Perhaps the most important single contribution of the Greeks, though – and Pythagoras, Plato and Aristotle were influential in this respect – was the idea of proof, and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Older cultures, like the Egyptians and the Babylonians, had relied on inductive reasoning, using repeated observations to establish rules of thumb. It is this concept of proof that gives mathematics its power and ensures that proven theories are as true today as they were two thousand years ago. This laid the foundations for the systematic approach to mathematics of Euclid and those who came after him.
GREEK MATHEMATICS – PYTHAGORAS
The over-riding dictum of Pythagoras’s school was “All is number” or “God is number”, and the Pythagoreans effectively practiced a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male.
The holiest number of all was “tetractys” or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.
However, Pythagoras and his school – as well as a handful of other mathematicians of ancient Greece – was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement. Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.
He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”). Written as an equation: a2 + b2 = c2. What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular) shape.
GREEK MATHEMATICS – PLATO
Although usually remembered today as a philosopher, Plato was also one of ancient Greece’s most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BC, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”
He demanded of his students accurate definitions, clearly stated assumptions, and logical deductive proof, and he insisted that geometric proofs be demonstrated with no aids other than a straight edge and a compass. Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”).
Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies, and its famous Library soon gained a reputation to rival that of the Athenian Academy. The patrons of the Library were arguably the first professional scientists, paid for their devotion to research. Among the best known and most influential mathematicians who studied and taught at Alexandria were Euclid, and Archimedes.
During the late 4th and early 3rd Century BC, Euclid was the great chronicler of the mathematics of the time, and one of the most influential teachers in history. He virtually invented classical (Euclidean) geometry as we know it. Archimedes studied for a while in Alexandria. He is perhaps best known as an engineer and inventor but, in the light of recent discoveries, he is now considered of one of the greatest pure mathematicians of all time. A mathematician, astronomer and geographer, he devised the first system of latitude and longitude, and calculated the circumference of the earth to a remarkable degree of accuracy. .
The simple but efficient ancient Chinese numbering system, which dates back to at least the 2nd millennium BC, used small bamboo rods arranged to represent the numbers 1 to 9, which were then places in columns representing units, tens, hundreds, thousands, etc. It was therefore a decimal place value system, very similar to the one we use today – indeed it was the first such number system, adopted by the Chinese over a thousand years before it was adopted in the West – and it made even quite complex calculations very quick and easy.
The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier than in China. There was a pervasive fascination with numbers and mathematical patterns in ancient China, and different numbers were believed to have cosmic significance. The main thrust of Chinese mathematics developed in response to the empire’s growing need for mathematically competent administrators. It was particularly important to the Chinese to solve equations – the deduction of an unknown number from other known information – using a sophisticated matrix-based method which did not appear in the West until Carl Friedrich Gauss re-discovered it at the beginning of the 19th Century.
Not North American Indians. But from the sub-continent of India. Mantras from the early Vedic period (before 1000 BC) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century AD Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre). The Indians were also responsible for the earliest recorded usage of a circle character for the number zero. The brilliant conceptual leap to include zero as a number in its own right would revolutionize mathematics.
The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India from the 8th Century onwards made significant contributions towards mathematics. They were able to draw on and fuse together the mathematical developments of both Greece and India. One important contribution was algebra, and they introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, they helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other than just the specific problems previously considered. used mathematical induction to prove the binomial theorem. A binomial is a simple type of algebraic expression which has just two terms which are operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x + y)2.
During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as “How many angels can stand on the point of a needle?” The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education. The equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized. The use of decimal fractions and decimal arithmetic was developed.
17TH CENTURY MATHEMATICS
The invention of the logarithm in the early 17th Century by John Napier (and later improved by Napier and Henry Briggs) contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy. It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and Newton could never have performed the complex calculations needed for their innovations without it. The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Descartes is also credited with the first use of superscripts for powers or exponents.
Two other great French mathematicians were close contemporaries of Descartes: Pierre de Fermat and Blaise Pascal. Fermat formulated several theorems which greatly extended our knowledge of number theory, as well as contributing some early work on infinitesimal calculus. Pascal is most famous for Pascal’s Triangle of binomial coefficients, although similar figures had actually been produced by Chinese and Persian mathematicians long before him.
So there we have a brief summary of the major developments in mathematics that has led us to the point we are today. However we still have not discovered why or how ‘1+1=1.” Stay tuned that is coming up. People tell me that long articles need to be shortened to keep the people’s attention. I am sure that is due to Common Core and all the drugs we give hyperactive students instead of spanking them to get them to sit still.
You will notice that in the above descriptions that various belief systems or “religious” systems and philosophy played a very important part of the development of mathematics.
In their world-views, Christian and non-Christian differ at fundamental points about just about everything. Surely, (the paradox notwithstanding) the world-views cannot affect mathematics. This, finally, is a neutral area, where Christian and non-Christian can agree. Both know that 1 + 1 = 2. How could religious differences ever affect it?
The “neutrality postulate” says that the knowledge and structure of a science—for example, mathematics—is not influenced by religious belief. At least science ought not to be influenced by religious belief. To be more forthright, secularists would say true scientific knowledge remains the same whether or not God exists.
What differences have arisen in mathematics in connection with religious belief? Differences have arisen over arithmetical truth, over standards for proof, over number-theoretic truth, over geometric truth, over truths of analysis, over mathematical existence-not to mention the long-standing epistemological disputes over the source of mathematical truth.
It may surprise the reader to learn that not everyone agrees that ‘1 + 1 = 2′ is true. If with Parmenides one thinks that all is one, if with Vedantic Hinduism he thinks that all plurality is illusion, ‘1 + 1 = 2′ is an illusory statement. On the most ultimate level of being, 1 + 1 = 1. No that is not the paradox we have mentioned above- that is just some gobbledygook where the Vede’s have broken into about 7 different schools of thought , but the commonality is the ”Universal Oneness of the World.” In other words, there may be many of us but we are all one which coming from a group who cannot keep themselves together is another paradox.
What does this imply? Even the simplest arithmetical truths can be sustained only in a worldview which acknowledges an ultimate metaphysical plurality in the world—whether Trinitarian, polytheistic, or chance-produced plurality. 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.
From the start, mathematics is plunged into the metaphysical problem of unity and plurality, of the one and the many. Without some real unity and plurality, ‘1 + 1 = 2′ falls into limbo. 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. From this process comes the raging debate between secular and Biblical scientists, both doing the same research and complementing each other’s experiments, but both ignoring the contributions of the others.
Mathematicians do not always agree about which proofs are valid. Intuitionists do not accept the proof by reductio ad absurdum (proof an assertion by deducting a contradiction from its negation).[i] Hence they will not accept some proofs that others will accept. The differences between intuitionists and the others have religious roots in the fact that these intuitionists will not accept as meaningful an appeal to the fact that God knows the truth about the matter, whether or not we do.[ii] For them some sense truth has its ultimate locus in the human mind. Mathematics is “only concerned with mental constructions”[iii]
The intuitionists also provide the most convenient example of how religious differences can lead to disagreement over number-theoretic truth. Consider the statements
A: Somewhere in the decimal expansion of pi there occurs a sequence of seven consecutive 7’s.
B: There are infinitely many primes p such that p + 2 is prime.
No man knows whether either A or B is true. Nor is there any known procedure which, in a finite amount of time, can assure us of obtaining a definite yes-or-no answer. For the intuitionists, this means that A and B should not be considered as either true or false.[iv] It makes no sense to talk about truth or falsehood so long as we have no way of checking. On the other hand, the Christian, on the basis of I John 3:20 (“God is greater than our hearts, and he knows everything”), Psalm 147:5, and other passages, is likely to feel that at least God knows definitely whether A or B is true. Our own limitations set no limits to His knowledge (cf. Isa. 55:8-9; Ps. 139:6, 12, 17-18).
If the evolutionists believe that the universe came from multi-billions and billions of random processes, then how do they explain the natural order found in so many things around us? How do they explain the natural order of mathematics and its application to so many areas of our lives if it developed as a part of random processes?
Most modern textbooks do not even attempt to offer an explanation for addition’s existence. Throughout my schooling, not one of my textbooks ever explained where addition originated or why it works. I eventually concluded that addition, along with all other math facts, is an eternal, self-existent truth and this was while I was an atheist.
Mathematicians throughout history have developed various theories to explain the origin and consistency of addition. Some have speculated that addition exists by sheer chance. Others have claimed man created addition and addition works because man designed it to work.
The verse “For by him all things were created” (Colossians 1:16 NIV’84) tells us where addition originated. It tells us God created all things. The word all includes everything, even math. This does not mean God created the symbols 1 and 2. Man developed those symbols. However, those symbols represent a real-life principle called addition that is embedded in everything around us—a principle God created.
Throughout history, cultures have used different symbols to represent quantities. For instance, the Romans used Roman numerals (I, V, X, etc.) instead of our current Arabic system (1, 5, 10, etc.). However, man has never created anything in math. He has merely developed different symbols and systems to represent the orderly way things add. Addition originated with God.
It is important to note that not everything follows our standard rules of addition. Two water droplets, when added together, form one larger water droplet (1 + 1 = 1). There is the paradox we have mentioned before. The mathematical field of fluid mechanics was developed to help solve the many different conundrums arising from how fluids react with each other. This presents a huge problem for those who look at math as an independent fact, but makes sense when we look at math as a way of describing the principles God created and sustains. God has different, though equally consistent, principles for governing liquids than He does solids. The apparent contradiction in how liquids combine reminds us that, no matter how well we think we have things figured out, God’s laws and universe are more complex than we can imagine!
Just looking at a sunflower, we can tell that the sunflower was carefully designed by a wise Creator. Evillutionists will however, say it is was the result of billions of years of random chance changes that caused the beautiful symmetry of the flowers.
The seeds in all sunflowers–be they large or small–are arranged according to two patterns. When we use math to examine these patterns, we observe that, regardless of how many seeds the sunflower contains, the number of seeds will be distributed between the two patterns in approximately the same mathematical proportion–a proportion that enables sunflowers to hold the maximum number of seeds and reproduce quite efficiently![v] Math, however, allows us to see God’s design at a new level, revealing the care God took with each aspect of His creation.
Seeing the amazing way God designed the sunflowers should remind us that we can trust Him to take care of the details of our lives.
First, an understanding of 3-dimensional geometry can give us perspective as we read Genesis 6:15 and Revelation 21:16. We realize the immensity of Noah’s Ark (450 feet long, 75 feet wide, and 45 feet high) and of the New Jerusalem (1400 miles wide, long, and high). We can also emphasize God’s precision in the blueprint for the ark and His majestic creativity in the design of the New Jerusalem.
A second example is making a simple reference to Proverbs 17:10 when introducing the study of inequalities. The writer of proverbs gives value to one reproof of a wise man that is greater than one hundred lashes to a fool. Many other examples are available, including Judges 16:30, Luke 15:7, and John 12:43.
Third, we can use a lesson on sequences as an opportunity to talk about the Fibonacci sequence. There is a 4-minute video available at http://disclose.tv/action/viewvideo/67897/Fibonacci_numbers___The_Fingerprint_of_God. This video portrays our God as one who values order and as one Who has repeatedly left a clear identifying mark on creation. I am amazed at how often this sequence appears.
Fourth, Jesus’ parable of the debtor from Matthew 18:21-34 can be used as an illustration of ratios, proportions, and unit conversion. Bible scholars believe that 1 talent = 75 pounds of gold (http://www.sundayschoolresources.com/biblestoryactivities2.htm). Then, 10000 talents = 750000 pounds of gold, which is 12000000 ounces. Recently, gold was valued at $1776 per ounce; so in today’s figures, 12000000 ounces would equal $21,312,000,000. Certainly, this parable teaches a valuable lesson on forgiving one another, but it should also remind us that God has made provision for us to be forgiven from a sin debt that we could never repay.
Finally, I would like to share some thoughts from the book Mathematics: Is God Silent?, by James Nickel (Ross House Books, 2001):
“Is there a connection between mathematics and evangelism? It has been shown that non-numerical mathematical methods such as set theory, modern abstract algebra, topology, and mathematical logic can be applied to the task of Bible translation. These mathematical formulations are powerful enough to deal with the structured relationships found in the complexity of linguistic structures.” (277-278)
We should continuously evaluate ourselves, not just in how we teach our content, but in how we link math and the Bible.
[i] Arend Heyting, “Disputation,” in Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics: Selected Readings (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1964), p. 61
[ii] “The . . . point of view that there are no non-experienced truths and that logic is not an absolutely reliable instrument to discover truths, has found acceptance with regard to mathematics much later than with regard to practical life and to science” (italics mine). Luitzen E. J. Brouwer, “Consciousness, Philosophy, and Mathematics,” in Philosophy of Mathematics, p. 78. Note the correlation that Brouwer makes between “life” and “science” on the one hand (expressing a religious world-view) and mathematics on the other. Elsewhere he acknowledges his philosophical debt to Kant, “Intuitionism and Formalism,” in ibid., p. 69.
[iii] Arend Heyting, “Disputation,” in Philosophy of Mathematics, p. 61
[iv] Cf. Luitzen E. J. Brouwer, “Intuitionism and Formalism,” in ibid., p. 77, and Arend Heyting, “Disputation,” in ibid., p. 56, for intuitionistic discussion of questions similar to A and B.
[v] This proportion has been named the Fibonacci sequence and is also present in many other flowers and parts of nature. “In nature, outside influences distort this pattern somewhat–seeds are not all exactly the same size, and external forces such as pressure against other flower buds during seed development may affect the spirals. Nevertheless, math helps us see a general relationship God put into sunflowers that enables them to reproduce quite efficiently.”