Abstract The purpose of this paper is to provide a theistic account of the nature of mathematics. Theism, or the belief in the existence of God, is taken as axiomatic. Two common ways of viewing the nature of mathematical knowledge are discussed. The first view is that mathematics is a human creation. The second is that God created mathematics, and humans discover it. I propose a new position that unifies the two: God and the creations of God, are co-creators of mathematics.
What is Mathematics? There are a wide variety of sensed and imagined objects which humans conceptualize as being mathematical. Algorithms, problem-solving heuristics, mental arithmetic, symmetry, a right triangle, proof, famous mathematicians, the elliptical orbits of the planets, currency, etc... These actions, objects, properties, and ideas make up only a small portion of the objects that people conceptualize as being mathematical. Efforts to define the nature of mathematics run the risk of emphasizing some aspects of mathematics while deemphasizing others. The field of mathematics consists not only of pure mathematicians reasoning about abstract structures, but also applied mathematicians applying and developing mathematics through their work while solving real world problems. Proponents of the new mathematics reform took a formalist stance and conceived mathematics as the study of abstract structure, whereas other mathematicians such as Morris Kline criticized these positions and argued that mathematics is “a tool for understanding the natural world” (Phillips, 2014, p. 540). Debates about the nature of mathematics point to the difficulty of pinning down the complex field.
Discovery or Creation? The debate over whether mathematics is discovered or created has a long history (Ernest, 1999). Though compelling arguments have been made for both perspectives, the debate is still ongoing and has implications for fields such as theoretical physics (e.g. Jannes, 2009). Many people believe that mathematics exists everywhere in creation. Recently a pre-service elementary school teacher in one of my courses wrote, “The pattern of numbers can be seen in everything ever created”. In addition to the pattern of numbers, geometric ideas seem to be reflected in nature. Take circles for example. Circular objects exist everywhere in perception: the full moon, a lover’s eyes. If mathematics exists everywhere in nature, then to many people, it seems clear that mathematical ideas (e.g. circle) existed in the mind of God before they were made manifest in physical creation. Brown (2008), a modern Platonist, wrote “Mathematical objects are perfectly real and exist independently of us” (p. 12). We gain access to these objects through “the mind’s eye” (p. 14). Other philosophers have vehemently opposed Platonic positions on the nature of mathematics. Platonist views are diametrically opposed to naturalistic or constructivist viewpoints. Von Glasersfeld (1995) wrote,
For a constructivist it was obviously impossible to think of numbers and geometrical forms as God-given. Nor could one accept the Platonic view of pure forms that float about as crystals in some mystical realm beyond experience. One would have to investigate their genesis as abstract entities in an experiential domain. (p. 16)
Hersh (1997), while acknowledging that many mathematicians were Platonist, provided a humanistic account for the reality of mathematics and criticized the notion that mathematics could exist in the mind of God or in a Platonist realm. He argued that mathematics is a human creation. Mathematical objects are social entities. When invented by humans, mathematical objects become imbued with unique properties. Other philosophers of mathematics have taken similar views.
Ernest (2002) presented a social constructivist view of mathematics. Mathematical knowledge is socially constructed through a subjective/objective cycle. Individuals subjectively create knowledge, and it is validated inter-subjectively by mathematicians so that it becomes objective knowledge accepted by wider communities. Publication is a key component of this process. This objective knowledge can then inspire more individual thought as it is subjectively reconstructed, and this may lead to further subjective creations, which may then become objective taken-as-shared knowledge. Because of the cyclic and social nature of this process, objective knowledge is seen as tentative and subject to revision.
The idea that mathematical knowledge is tentative and subject to revision (Lakatos, 1986) rather than absolute seems to be contrary to an absolutist perspective in which humans discover true mathematics as it exists in the mind of God. The notion of objectivity as being what is socially accepted is appealing and problematic. It is problematic because of the traditional conceptions of objectivity. People typically grow up believing that objectivity refers to what is true independent of human subjective experience. When “Heaven and the Mind of God are no longer heard of in academic discourse” (Hersh, p. 12), a new definition of objectivity seems necessary. Apparently academicians are no longer concerned with truth per se, rather objectivity. Truth is the reality of God. What is true then becomes meaningful only in religious contexts of which the mind of God is postulated to exist. Truth can replace the word objective as it has previously been understood.
In this paper, I take a theistic approach to the nature of mathematics. Rather than starting from the assumption that God is not needed to explain the nature of mathematics, I start from the assumption that God exists and any position on the nature of mathematics would need to delineate the role God plays. I consider if the Bible provides evidence that numbers or other mathematical objects exist in the mind of God. I then describe how I turned to prayer when rationality proved inadequate to answering the question of whether mathematics is discovered or created.
The Theistic Axiom My belief in God is a conscious commitment I have made in light of personal experiences—experiences in which the existence of God seemed undeniable. The commitment could be called faith. Though I have been influenced by a variety of spiritual traditions, I take a Christian perspective in this paper. I do not claim this perspective is representative of a perspective that all Christians share. It is necessarily a personal perspective. According to this perspective, reason is not the only source of knowledge. A Christian may also acquire knowledge by appeal to God’s authority through prayer or the reading of Biblical scripture. Personally, I believe that my accomplishments are made possible because of my trust in God. I do not always make purely rational decisions. I am more confident if my deliberations are attuned with my heart. I identify with the author of Proverbs 3:5-6.
Trust in the LORD with all thine heart; and lean not unto thine own understanding. In all thy ways acknowledge him, and he shall direct thy paths.
What needs to be transmitted to the reader is that the belief in God is central to my reality. I would be deceiving my self and others if I did not recognize that any position I take will necessarily be influenced by this belief. Mathematicians may take as axiomatic what cannot be proven, but is accepted as true. Due to my deeply held beliefs I must adopt the theistic axiom: God exists. Now that the existence of God is taken as given, we can get to the business of answering the age-old question: Does mathematics exist in the mind of God to be discovered by humans? Or is mathematics a human invention used to make sense of God’s creation? I now consider biblical evidence for and against each position. I then introduce and describe a third position which does away with this dichotomy: God and the creations of God, are co-creators of mathematics.
The Bible and Mathematics Is mathematics the creation of man or God? In order to lend support for or evidence against either position, a Christian may consider what the Bible might bring to bear on the issue. Consider Isaiah 55:8-9.
For my thoughts are not your thoughts, neither are your ways my ways, saith the LORD. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts.
Do we dare maintain that the creator of our universe would think mathematically as mere mortals do? To attribute mathematical thinking to God would be to project our own humanity onto God’s nature. It may be an insult to trivialize the thoughts of God in this way. One may interpret the passage from Isaiah as supporting the view that mathematics is a human creation, rather than a creation of God. However, an alternative interpretation would be that God too thinks mathematically, just in a much more powerful way than man.
References to number appear frequently in the bible. In the book of Leviticus, God lays out detailed commandments for the Israelites. Many of these commandments depend on number. Also consider John of Patmos’s recollection of an angel’s measurement of New Jerusalem in Revelation 21:15-18.
And he who talked to me had a measuring rod of gold to measure the city and its gates and walls. The city lies foursquare, its length the same as its breadth; and he measured the city with his rod, twelve thousand stadia; its length and breadth and height are equal. He also measured its wall, a hundred and forty-four cubits by a man’s measure, that is, an angel’s.
Apparently measurement is not only a human activity, but also an activity of angels. It appears that both human and divine beings think mathematically. One might conclude that both the angel’s and man’s knowledge of mathematics has a basis in the mathematics of God. But are the biblical references to number sufficient evidence to concede that mathematics is a creation of God? Perhaps God is simply communicating with humans in a way that we can understand and not in a way characteristic of the thought used to form the universe.
I did not consider the biblical references to number to be strong enough evidence to have conviction that God created mathematics. It could be that God created mathematics, or it could be that God thinks in ways far beyond and different from human mathematical thinking. Also, the scholarly arguments for Platonist or humanist views can be compelling. When one has reasoned deliberately, but is still unable to intellectually assent to a particular position, there are other means to obtain knowledge. In face of this uncertainty I turned to prayer. I prayed that God would reveal which position was true, so that I could confidently take the position with certainty. The next morning when I awoke, the problem and its solution were in my mind. Mathematics is not solely the creation of God or man, but is a co-creation. God made us in his image. God is a creator. Humans are creators too. God and the creations of God are co-creators of mathematics.
The Unified Position It is conceivable that mathematics is a divine creation that humans discover. But it is also conceivable that mathematics is a human creation used to make sense of existence. It is tempting to believe that one must make a choice (especially when told they must make a choice for an assignment). Faced with such a dilemma I decided to side-step rationality in favor of a different approach to knowledge: prayer.
As a Christian, my desire is to know and speak the truth. Every decision we make in life is made in the face of uncertainty. When our rationality fails to provide us with enough information to make a reasoned decision, we can consult the ultimate authority through prayer. If there is a true answer to this question of the nature of mathematics, then God knows the answer. I prayed that God would reveal that answer to me. I prayed that I wanted to speak truth, and thus if it be God’s will, he would reveal to me if mathematics is created by God or man. I prayed this prayer in the evening, then in the morning when I awoke, a new answer was in my mind. God and man are co-creators of mathematics.
Humankind was created in the image of God (Genesis 1:27). One of the ways we are like God is that we are creators. Although God made the universe, humans also continually shape creation. While this is most evident in the physical products of our existence such as our homes and highways, ideas are also products of human creation. These ideas are created in our mind but are made manifest and shared with other people through stories or other forms of communication. Humans are not only shaped by, but they also shape existence. Mathematics, though it exists in the mind of God also exists in the mind of humans. The question then becomes, which mathematical ideas are the creation of man? And which are the creations of God?
The Integers and Infinity It is said that Leopold Kronecker once remarked, “Dear God made the integers, all else is the work of man.” But would God have a reason for creating the integers? Perhaps in the design of electrons and protons God needed a way to reason about what we now call positive and negative charge. Or in order to create molecules the idea of number needed to be constructed to keep track of the protons, electrons, and neutrons involved. But this is mere speculation and I do not claim to have any insight into whether integers were imagined in the mind of God. The problem of distinguishing the mathematical ideas of God and man appears difficult.
Consider that mathematicians have developed a unique way to compare infinities. One way to determine which of two sets of numbers are greater is by making a 1 to 1 correspondence with objects from each set. If, after pairing all possible objects, it is determined that one set had no objects unpaired but the second set had several remaining unpaired objects, then we could conclude that set had more objects than set . This strategy for comparing sets is the intuitive basis for the formal mathematics used by set theorists to compare the cardinalities (size) of infinite sets. Using such conventions, set theoreticians beginning with Cantor have come to many interesting conclusions about infinity. For example: 1) The set of even numbers is equinumerous to the set of natural numbers. 2) The set of rational numbers is equinumerous to the natural numbers. 3) The real numbers are more numerous than the integers. These results are non-intuitive and seem absurd. Some mathematicians, including Kronecker, criticized Cantor’s ideas.
Because of the non-intuitive nature of these ideas, my first inclination is to conclude that God has not ruminated about infinite set cardinality. However, this may be a sign of my own egotism and tendency to anthropomorphize. Recall Isaiah 55:8-9 states that God’s ideas are far beyond our own. Perhaps the very fact that the ideas of infinite set theory are non-intuitive suggests that they are candidates for the mathematical constructions of God. There is some evidence that Cantor believed that these ideas were indeed divine (Dauben, 2004).
The Nature of Mathematics If God used mathematics in the creation of the universe, then it is conceivable that humans’ ability to mathematize is a direct result of living in a mathematical universe. However this is not the only possibility. Perhaps social constructivism needs to be extended to include global or universal constructivism by which all knowledge is socially constructed by individual consciousnesses in the universe. Perhaps God constructed the notion of number, and then humans did as well. Both human and divine beings think mathematically but the mathematical ideas of God do not direct the mathematical thinking of humans. Of course the mathematics we construct will be undeniably different from that constructed by a divine being, but it may also share some similarities (as school mathematics resembles pure mathematics). To have a shared understanding of mathematics with a divine being would be quite a meaningful event for a human, especially if one was cognizant of this shared understanding. It is also possible that individual construction of mathematical knowledge is made possible by God. Could it be that God blesses the minds of humble mathematicians with a mathematical insight that could not be achieved otherwise?
Conclusion Working from the assumption that God exists, I was led through prayer to the conclusion that God and humans are co-creators of mathematics. Hersh (1997) stated that “Heaven and the Mind of God are no longer heard in academic discourse” (p. 12). In fact, most scholarly work is conducted under the assumption that God is irrelevant. As Nietzsche bluntly put it, “God is dead. God remains dead. And we have killed him” (1977, p. 97). The absence of God in scholarly discourse can be attributed in part to the success of modern science. Rather than believe that rainbows are a sign of God’s promise never to again flood the earth, science provides a naturalistic explanation in terms of water’s refraction and reflection of light. If God is no longer needed to explain physical phenomena, then why should God be required to understand the genesis of mathematical knowledge? Personally, rationality alone was insufficient for deciding if mathematics is created or discovered. I could not achieve conviction through reason. Reason cannot provide us with true knowledge of reality (von Glasersfeld, 1989). Prayer can provide a believer insight into knowledge not accessible by reason. The choice to commit to such knowledge is faith. Absolute conviction about the true nature of reality can only be obtained as a gift from God.
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