Idea Transcript
Faith & Reason Honors Program
SENIOR THESIS
Name
Matthew J. Reeder
Thesis Title
Mathematics and its Relationship with Christianity
Thesis Director
Bro. Daniel P. Wisniewski, OSFS, Ph.D.
Year
2018
1. Abstract In this paper, I will discuss the debate of whether mathematics is created or discovered by examining this topic from the perspectives of mathematicians, philosophers, and theologians. Several philosophies regarding the nature of mathematics will be presented and analyzed, with some such as Platonism viewing mathematics as discovery, while others such as Fictionalism view it as invention. Before the theological perspective of mathematics is examined, I will present several different views regarding the relationship or absence of a relationship between religion and mathematics, and demonstrate that there is a positive relationship between the two. This will be done by examining the views of Francis de Sales, as well as an official Church document from Vatican II. This relationship will be used to provide evidence for mathematics as discovery. I will then discuss the parallels between the disciplines of mathematics and theology, and how faith and reason can lead to a deeper understanding of both. 2. Introduction In many ways, the fields of mathematics and theology seem at odds with one another. Theology centers around the idea of faith which cannot be empirically measured or calculated. Mathematics seems to exist on the other end of the spectrum, choosing to deal only with concepts that can be definitively shown and proven within the rules defined within the discipline. However, this does not mean that mathematics and theology are at odds or irreconcilable. I will examine the origins of mathematics and discuss the idea of mathematics being created by God and later discovered by man. I will also discuss the views of various theologians regarding how mathematics and theology exist in harmony with one another. 3.
Mathematics – Invented or Discovered, a Mathematical Perspective
The debate of whether mathematics was invented by man or discovered is an issue which has no clear answer, and whose answer could very well be some combination of invention and discovery. It must be examined from multiple perspectives, namely through the eyes of mathematics, philosophy, and theology. First, I will cover the main arguments made by mathematicians. In his article “Is Mathematics Discovered or Invented,” author Timothy Gowers begins by outlining what constitutes a mathematical discovery. The core component of a discovery is, by Gowers’ definition, finding something that was there independent of the discovery being made. For example, the observation of a new continent would certainly be a discovery, since it existed far before and independently of the observation. According to Gowers, discoveries must also have some significance, or they exist merely as observations. For example, Gowers believes that noticing that two words are anagrams is not significant enough to be called a discovery, even though it satisfies the requirement of prior existence, since words can be anagrams without anyone “discovering” so. The example he provides is the observation that “carthorse” and “orchestra” are spelled using the same letters and are therefore anagrams. This observation is, to Gowers, too trivial to be classified as a discovery. This definition of discovery
would then indicate that an invention is something that did not previously exist and was brought into existence by its creator (Gowers 4). To Gowers, an important characteristic of a discovery is the idea that if history was changed and someone else made the discovery it would remain the same (Gowers 6). Although it was stated above that mathematics is empirical and seeks definitive proof, the argument over whether mathematics was created or discovered does not follow that convention. Both sides of the debate deal heavily with how things “feel” or “seem” and no proof exists to validate one school of thought over the other. This does not mean they are unimportant, however after examining both sides, the debate will need to be examined from other perspectives. First, we examine the side of the debate favoring mathematics as discovery. Gowers considers several mathematical ideas that he believed show a preexisting mathematical reality, and the discovery of it. One example he gives is the quadratic formula, the method of finding roots of a quadratic polynomial, ax2 + bx + c, where a, b, and c are any numbers, and x is a variable. This method states that given any quadratic polynomial, the roots of the equation (the values that can be substituted for x to make the equation equal 0) can be found by substituting the values of a, b, and c into the quadratic formula, which is given below: . He argues that this is evidence for mathematical discovery since for any quadratic polynomial, this formula will give its roots. Although there are other equivalent representations for the quadratic formula, it would not be possible to define and invent a new version of it. Instead, the equation is a representation of the discovered relationship among a, b, and c (Gowers 6). As Gowers stated, “Whoever first derived that formula did not have any choice about what the formula would eventually be” (Gowers 7). Another mathematical concept that has convinced many mathematicians that their work consists of discoveries is the Mandelbrot Set. The Mandelbrot Set is formed by choosing a complex number, which is any number of the form a + bi where a and b are any real numbers and (Beachy and Blair 49), and then continuously applying the same rule to it over and over. The Mandelbrot Set is formed by selecting any two complex numbers. One, designated by z, will be the starting point of the set. The other, c, will be a constant used in the set’s formation. Then, the Mandelbrot Set is formed by taking z, and then replacing the value z with z2 + c. That value then becomes the new value of z. This process is then continuously iterated, and each new z value is graphed. (Davies 142). Many mathematicians feel that due to the incredible complexity and beauty present in the structure, that mathematicians discovered it. As the mathematician Roger Penrose explained it, “much more comes out of the structure than is put in in the first place. One may take the view that in such cases the mathematicians have stumbled upon ‘works of God’” (Davies 142-143). Figure 1 is an image depicting a small portion of the Mandelbrot Set. Figure 2 depicts a more complete picture of the Mandelbrot Set. The vertical line depicts the y-axis and the horizontal depicts the x-axis. When plotting a point z=a+bi from the set, the real part of the number, a, is plotted along the x-axis, while the complex part, b, is
plotted along the y-axis. Figure 1 only contains a small portion of the graph depicted in Figure 2, but interestingly enough, the same circular structure with smaller circles branching off from it can be observed in both figures, and this same figure seems to continuously manifest itself multiple times, no matter how far the image is zoomed in. The black of each region shows the numbers contained in the Mandelbrot Set, while the colored areas indicate how many iterations it takes to show the number is not a part of the Mandelbrot Set (University of Utah; Clark University).
Figure 1: The Mandelbrot Set (University of Utah)
Figure 2: Mandelbrot Set with axes (Clark University)
Many of the arguments in favor of mathematics as discovery deal with how the concept a mathematician developed made him or her feel. Gowers speaks about one of his own experiences working with a proof when he felt a clear sense of discovery rather than invention. He felt that there was a simpler way to prove a concept he was examining, so he set out to find it. When he had finished writing his proof however, he said that he felt as though he was not in control of where it was going, and that “the structure of the argument included many elements that I had not even begun to envisage when I started working on it” (Gowers 11). While feelings like this are hard to measure or quantify, they corroborate Gowers’ earlier criteria of discovery, which include the prior and independent existence of the concept, and some degree of significance. The feelings he described indicate a lack of control over the process, and therefore the prior existence of the concepts he studied. This seems to indicate that, in at least some cases, mathematical proofs are more discovery than invention, if it is permissible to accept how the mathematician felt regarding it. Another point raised when defending mathematics as discovery is the idea of the “unreasonable effectiveness” of mathematics in describing natural phenomena (Livio). Although this concept will be more fully examined in Section 5, even from a purely mathematical perspective, this concept provides strong support for the argument of mathematics as discovery. Essentially, “unreasonable effectiveness” is the idea that because of how well mathematics can describe the world, it cannot simply be a human construct, and could instead be the underlying order of the world being discovered by humanity. Livio suggests that there are two ways mathematics demonstrates its unreasonable effectiveness (Livio). The first kind of unreasonable effectiveness is called active effectiveness and refers to how mathematics is the “go-to” tool for describing scientific phenomena. As Livio puts it, “when scientists attempt to light their way through the labyrinth of natural phenomena, they use mathematics as their torch” (Livio). An example he cites for this form of effectiveness is Newton choosing to use mathematics to model change when he created the field of Calculus. The second kind of unreasonable effectiveness is passive effectiveness. This form of effectiveness refers to mathematics that is developed without an application which is later found to be applicable in describing something. An example of this form of effectiveness can be found in a form of geometry developed by Riemann in the 19th century, which was later used by Einstein for his theory of relativity (Livio). The fact that mathematics has proven to be useful, or even essential, in scientific discovery and research indicates that it is far more than a human invention. Examining this concept from a Christian worldview seems to point toward an intelligently designed universe, where mathematics is one of the tools science uses to glimpse the underlying order and structure of the world. As mathematician G. H. Hardy put it, “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations” are simply the notes of our observations” (Hardy 35). There are also many aspects of mathematics that seem to clearly be inventions, at least in part. These are areas or ideas in mathematics that do not fit the earlier description of a discovery,
generally by not fulfilling the discovery requirement of prior existence and the mathematician being pulled down a particular path to find it. An example of a mathematical concept that seems to be more of an invention is the construction of groups. A group is a mathematical structure consisting of a set, and an operation that will be performed upon the elements contained within that set, which satisfy a certain set of properties. For instance, the integers (…, -3, -2, -1, 0, 1, 2, 3,…) under the operation of addition is a rather elementary group. Once the basic concepts of a group are understood, mathematicians can arbitrarily combine potential sets with an operation, allowing them to form a new group. In many ways, this feels much more groups are invented rather than discovered since there are no rules pointing mathematicians toward a specific group other than the basic requirements needed to form a group. Instead, the ingenuity of the mathematician is employed. Gowers described something a similar personal experience when he created a Banach Space, a specific type of vector space (Heil), to disprove a theorem . The mathematician said when he created it, he had control over how he wanted to make and manipulate it (Gowers 11). This manner of speaking indicates creation over discovery but relies heavily on how well the words “create” or “discover” resonate with the mathematician when applied to the mathematical object. This example is interesting in that it seems to conflict with the idea of active effectiveness. According to Livio’s unreasonable effectiveness argument, this would in fact be evidence for mathematics as discovery, not invention. The conflicting ideas of Gowers and Livio show another example of the difficulty in debating invention and discovery. The example Gowers provided regarding Banach Spaces would be evidence for invention then, only if it satisfies the rather abstract criteria discussed earlier of eliciting a feeling of control and invention. An example Gowers believes is clearly invention rather than discovery is calculus. By this, he does not mean every individual concept or rule of calculus is invented. Instead, he is talking about the overall method and idea formed by gathering together all its concepts (Gowers 9). After examining both sides of the mathematical debate, it appears that both sides of the argument have valid points and examples. Many of the core principles of mathematics seem to be discovered, such as relationships between numbers and equations, for example the quadratic formula. Much also seems to be invented, such as groups and the collections of mathematical techniques, but many of the parts of mathematics that are arguably invented fall under the category of representations for concepts or techniques, both of which are free to vary based upon the inventor. While the concepts within a given discipline could be argued as discovery, the organization and collection into that discipline would be seen as invention. For example, there was nothing stopping Newton from choosing to exclude integrals from the discipline of calculus. It is possible to consider the different disciplines of mathematics, such as calculus or linear algebra, to be inventions. Looking at Gowers’ criteria for discovery, even if the techniques in a given discipline existed independently of the human mind, the disciplines seem to be inventions since there is no reason a concept must be included or excluded from one. Instead
mathematicians are free to organize and divide mathematical concepts into the disciplines they choose. Not all mathematicians agree with the approach of determining if something is invented or discovered by considering how it makes the mathematician feel. Mary Leng, a philosopher who studies the philosophy of mathematics, believes that mathematics is a human invention, but that there is a plausible reason for it seeming to be a discovery. In her writings, she examined situations similar to those above, where mathematicians felt they had little choice in the direction of their work and were making discoveries rather than inventions. However, she drew a very different conclusion than Gowers; according to Leng, there is a very important aspect of mathematical “discovery” that was not considered: the basic assumptions we base mathematics upon when the “discovery” was made. She concedes that when working to develop a theory in mathematics, mathematicians have little room when choosing what assumptions to make. According to Leng, the “discoveries” made by mathematicians are not fundamental truths, but rather results of creation (i.e., inventions). She argues that when discoveries are made, they are made in the context of the created mathematical rules governing the realm the mathematician works in. In other words, in the early days of mathematics, the concepts such as “integer” were invented, and the “discoveries” that have been made since are actually inventions since the concepts all mathematics is based on were invented. For example, any observations about adding numbers together would seem to be discoveries, but are not since numbers and addition are actually inventions. To Leng, mathematical discoveries are often “developed as solutions to problems we have set ourselves, where the constraints of the problems are enough to narrow down the range of options that could count as an appropriate solution (often pinning down a unique solution)” (Leng 63). Another example of how invention can be mistaken for discovery, according to Leng, is geometry. She believes that when geometry first began, it was invention rather than discovery since “Euclid’s axioms for geometry were intended to pin down truths about points and straight lines in physical space and were ‘discovered’ through examination of what had to be assumed in order to prove many other results believed to be true…” (Leng 64). Thus, the idea of geometric observations being discovery is flawed since it operates under the assumption that the foundations of geometry are discovery, which Leng would disagree with. Also, if these original ideas were changed, they could alter the future of the entire field of geometry, suggesting it is invention rather than discovery. One flaw in this idea is that it struggles to address how mathematics remains the same regardless of where in the world it is performed, and it also does not address why mathematics and its “invented” rules can be so effectively applied to scientific works. An option not considered by Leng however, is that math is created by something other than people. The possibility that mathematics is created by God would allow for mathematics to be discovered by man. This idea will first be examined through the writings of John Polkinghorne, who draws on both his mathematical and theological experience to develop this position.
Polkinghorne, a mathematician and Anglican priest, chose to approach this issue from another perspective, and in doing so finds evidence for mathematics being a discovery. In his paper “Mathematical Reality,” Polkinghorne examines where mathematical knowledge comes from, and how it can be described or quantified. He presents four arguments for mathematics being a discovery. His first argument for mathematics is based upon his observation that countless people have made mathematical observations, and the resulting mathematics is the same, regardless of where it came from (Polkinghorne 29-30). For example, no matter where mathematicians are in the world, mathematics stays constant to them. Addition does not change its meaning based on where in the world it is being performed. This observation relates back to the arguments presented by Gowers. If mathematics were an invention, multiple people would have the ability to create or mold it how they see fit, thus allowing for many different versions of it. Mathematics agreeing with itself, no matter where it comes from, seems to contradict Leng’s argument that mathematicians are free to create the rules of mathematics and points toward it being an ongoing discovery taking place all over the world. Polkinghorne’s second argument is that mathematical knowledge is infinite; for instance “the quantity of information contained in the set of all true propositions about the positive integers is infinite” (Polkinghorne 30). As an example, since there are infinitely many natural numbers, mathematicians can continue to make observations about each one, or some subset of them without end. He argues that this seemingly endless amount of mathematical knowledge available to humanity, combined with the limited intellectual abilities of humanity indicates that mathematical knowledge exists independently of humanity (Polkinghorne 30). His third argument is very similar to the argument by Gowers regarding discovery: the mathematical concept pushing the scientist down a certain path (Polkinghorne 30). Polkinghorne’s final argument for mathematics not being a mere human invention is based upon humanity’s ability to perform advanced mathematics. He argues that while some rudimentary mathematics could have been useful to our ancestors, much of the advanced mathematics today would not have been useful. He believes that the reason humanity’s mathematical abilities grew to the extent that they have today is that humanity evolved in a world filled with preexisting mathematical reality, and exposure to that reality drove humanity to develop in its ability to discover and make sense of the world’s mathematical order (Polkinghorne 31-32). This is a very difficult issue to examine from only a mathematical perspective, partially due to there not being a definitive answer or proof that can be followed. Since the question of the origins of mathematics remains, it is necessary to view the issue from other perspectives. In Section 5 of this paper, the issue will be re-examined by looking at not what mathematicians have to say, but by examining the writings of philosophers and theologians. By examining this issue with a Christian perspective, more evidence for the discovery of mathematics can be found. Adding the Christian viewpoint allows for the possibility for God creating mathematics along with the rest of the created universe.
Before this viewpoint can be applied to mathematics, the relationship between mathematics and religion must be examined. There are several possible forms the relationship between mathematics (or science as a whole) and theology can take. Some suggest that the relationship is either not present, or not beneficial. Therefore, if one of these negative relationships, or a complete lack of relationship, accurately describe the dynamic between science and religion, then examining mathematics from a theological perspective would not be beneficial. These potential relationships will be presented and examined, and through the insight of Francis de Sales and other theological sources, it will be shown that the relationship between science and theology is a mutually beneficial one, and therefore, examining mathematics through the eyes of faith can provide further evidence of mathematics being a discovery. 4.
The Relationship of Mathematics and Science to Theology
Although the issue being examined in this paper is relating mathematics to Christianity, the relationship between theology and science will be examined, since it is much more direct due to the resources available. Considering this relationship is also appropriate due to the extensive use of mathematics in the various scientific fields (i.e., physics). There are several potential ways theology and science can interact with one another. At one end of the spectrum, it is possible that they are at odds with one another, while at the opposite end they could bolster one another. There are also some options lying somewhere in the middle. The four general possibilities that will be examined here are those postulated by John Haught: Conflict, Contrast, Contact, and Confirmation (Haught 9). Throughout the course of this section, a case will be made supporting a positive relationship between the two. The views of Francis de Sales also seem to support a positive relationship. The concept of conflict between science and religion is not a new one. The governing principles of this viewpoint are: “Religion tries to sneak by without providing any concrete evidence of God’s existence. Science, on the other hand, is willing to test all of its hypotheses and theories against ‘experience’” (Haught 10). At first glance, this stance seems very plausible. One of the core concepts of science is testing and challenging ideas, while in the Catholic Church it may seem as though the tenants of faith are not questioned or examined, and must be accepted without empirical evidence. Further evidence can be found in the various clashes the Church and scientific minds have had throughout the years. Examples include the initial reactions of the Church to the scientific discoveries of Galileo and Darwin (Barbour 77). These conflicts can be interpreted as evidence of religion begrudgingly losing ground to ever-advancing scientific discoveries. Another reason that some believe science and religion cannot coexist, due to conflict between them, is the idea that the concepts in religion are “unfalsifiable” and the claim that religion does not submit itself to the same rigors as science. In other words, religion does not allow for its ideas to be cross-examined and disproven, whereas science welcomes it. Some also believe that “Science is ‘spiritually corrosive, burning away ancient authorities and traditions’” (Haught 10). Overall, this viewpoint is rather extreme and extremely pessimistic, but should still
be discussed to provide insight and an alternative viewpoint in comparison to the other potential relationships. The next potential form of relationship between religion and science proposed by Haught is Contrast, or as Barbour calls it, “Independence” (Barbour 84). Independence is perhaps the more suitable name, since the main concept of this viewpoint is that there is no conflict between science and religion because they deal with separate issues and do not have any overlap or interaction. According to this view, “Science is concerned with particular truths; religion is interested in explaining why we should seek truth at all” (Haught 15). While this approach provides a simple way to deal with apparent conflicts between the scientific and religious communities, it also seems to drastically simplify the relationship between the Church and science, and therefore between the natural and the supernatural. In fact, it denies the existence of any relationship at all, which seems to be a drastic oversimplification in an attempt to avoid potential conflict. This approach is not challenge-free however, since keeping religion and science separated in areas where some people would like to bring the two together can create issues. For example, some reference a combination of scientific evidence and the bible to explain the origins of the universe. There are of course several other ways science and theology can be combined where they should not be. One of these ways is called “Scientism,” which asserts that science is the only real source of truth available to humanity (Haught 15-16). In a way though, this is selfcontradictory, since there is no empirical evidence available that indicates that this is scientifically true. Independence may not be an accurate representation of the relationship between science and religion, but it is important to consider, because according to Haught it shows that science and religion are not pitted against each other as thought in the Conflict interpretation (Haught 17). It is also important to note that the avoidance of conflict is not the only possible motivation for this viewpoint, as it also serves to highlight the strengths and roles of science and religion by ensuring they can work without unnecessary conflict or interference from one another (Barbour 84). The final two relationships shift toward the idea of a relationship in which religion and science coexist and are beneficial to one another to some extent. First, Contact seeks to remove the barriers presented in Contrast and allow for some overlap between the two, since complete and total separation provides an inadequate representation (Haught 17). Confirmation extends this idea and suggests the opposite of Conflict, that instead of the two being opposed to one another, religion is wholly in favor of science and in harmony with it (Haught 21). One challenge that arises when utilizing this system of categorizing relationships is that it is sometimes challenging to determine where Contact ends and where Confirmation begins. The primary idea of Contact is that while religion and science have their differences, they are also undeniably connected, and have some form of relationship. “It insists on preserving differences, but it also cherishes relationship” (Haught 18). One way that science and religion have contact with one another is through assisting one another with issues that can arise. For example, religion complements science well when used to assist in moral or ethical debates over
scientific discoveries or issues. While science questions what humanity can do, religion questions whether or not scientific practices are ethical or appropriate. Another way that there is contact between religion and science is in the methods of questioning and learning they employ. For much of history, science has been viewed as purely empirical and data driven, while religion rests firmly outside this realm. However, Barbour asserts that in more recent times the division between the methods of science and religion have become blurred and somewhat removed. Scientific ideas are no longer limited to the observable and measurable. Instead, they are now free to come from sources demanding more creativity and analyzing ideas of what could be without physically observing them (Barbour 93). Similarly, although religion cannot be tested or observed in the same way as science, Barbour argues that “The scientific criteria of coherence, comprehensiveness, and fruitfulness have their parallels in religious thought” (Barbour 93). An issue that arises when examining these final two relationships may be accidentally losing their differences in an attempt to unite the two disciplines. A place where this can commonly occur is when the biblical creation story and the scientific origins of the universe are examined as though both religious and scientific sources were talking about the same physical events. As Ernan McMullin says, “‘What one cannot say is, first, that the Christian doctrine of creation ‘supports’ the Big Bang model, or, second, that the Big Bang model ‘supports’ the Christian doctrine of creation’” (as quoted in Barbour 91). Instead, while both seem to be describing the same event, they serve very different purposes. The scientific models seek to describe the actual physical beginning of our universe. On the other hand, the creation story in the bible does not seek to explain the events that physically occurred at the beginning, but to show the relationship present between God and the world (Barbour 91). The fourth possible relationship between science and religion, Confirmation, extends the ideas presented in Contact to the point where “…religion is in a very deep way supportive of the entire scientific enterprise” (Haught 21). Before continuing further, it is important to clarify what kind of science religion supports in the Confirmation relationship. It does not support science that seeks to become omniscient, as that would usurp the role religion plays in the relationship they share. Instead, it supports science’s goal to learn what it can, but without removing the relationship between the two fields. In this view, the pursuit of knowledge through science is good, but it also understands that humanity cannot claim to be omniscient through science, and that some things can only be known through religion and God. The religious worldview “…consistently nurtures the scientific quest for knowledge and liberates science from association with imprisoning ideologies” (Haught 22). One of the ways this relationship can be observed is in scientists accepting their work and mathematics as a means of describing the world. By doing this, scientists are operating under the assumption that the world is coherent and rational, an assumption confirmed by faith. In other words, religion allows for the foundation that all scientific knowledge is built upon (Haught 25). For these reasons, Confirmation appears to be the ideal form of relationship between religion and mathematics (and science as a whole), since it provides a deeper meaning to mathematical
discoveries, while supporting endeavors to apply mathematics and other scientific tools to better understand the world. But is this the relationship that exists between the two disciplines? 5.
Philosophical and Theological Views of Mathematics Assuming then, that the relationship between mathematics and religion is either Contact or Confirmation, examining mathematics from a theological perspective allows for the possibility of man discovering a mathematically logical universe created by God. The first Theological and Philosophical perspective that will be examined in this paper is from Saint Augustine of Hippo, an early Christian writer who lived around 400 A.D. Augustine’s argument covers four main questions regarding mathematics (Bradley 2): a. What is the nature of mathematical objects? (the ontological question) b. How do we obtain knowledge of them? (the epistemological question) c. What is the meaning of ‘truth’ in mathematics? How do we account for the certitude of mathematical truths? (the truth question) d. How do we account for the effectiveness of mathematics in describing the physical universe? (the effectiveness question) At first glance, the answer to the ontological question seems rather like the debate in Section 3, in that mathematical concepts are either already present and waiting to be discovered, or they are constructed (invented) by mathematicians. Examining this question from a Christian perspective, instead of a purely mathematical one allows for an argument in favor of mathematical discovery that could not be considered previously: the idea of mathematics as created by God. This is the viewpoint Augustine takes, stating So, just as there are true and unchangeable rules of numbers, whose orders and truth you said are present unchangeably, and in common to everyone who sees them, there are also true and unchangeable rules of wisdom… But when we begin to look above ourselves again, we find that numbers transcend our minds and remain fixed in the truth itself. (Bradley 3-4) An issue that arises when examining “the truth question” is what exactly does Augustine mean by “truth”? In fact, he uses “truth” and “Truth” to mean two very different things. To Augustine, a “truth” is an idea that must be true and needs to be consistent regardless of who is thinking about the given truth. This would then include mathematical truths and concepts. “Truth” (with a capital T) has more of a theological connotation originating from God than a logical one, and is described as “a kind of light, possessed by all who perceive the same truths at a given moment but not changed by any of them. It transcends our minds; in fact it rules them and is therefore independent of them” (Bradley 3). Examining Augustine’s writings with these definitions of truth in mind, it is evident that Augustine regards mathematics, which he calls “number,” very highly. He likens number to wisdom, calling them the same (Bradley 3). Augustine also strongly believes that mathematics is something humanity discovers rather than invents, and that it was created by God. He believed that all of mathematical knowledge must have come from God. As an example, he examined the concepts of infinity and one. Since we are capable of understanding infinity, but have no actual
exposure to it to base our understanding upon, the concept must have originated somewhere other than our observations. Similarly, we have no true experience with the concept of one since anything we observe can be split into multiple pieces. He asserts that the reason people are capable of comprehending ideas such as one or infinity, and mathematics in general is because God intended people to understand them and gave us the ability to do so when he created man (Bradley 4). This still leaves the question regarding effectiveness. However, in part, it has been addressed above. According to Augustine, mathematical truth comes from God and can accurately model and describe our world because God intended it to and intended for people to be able to use it for that purpose (Bradley 4). To further determine which of these relationships exist between theology and science, the views of Francis de Sales (1567-1622) are examined, along with the document of Vatican II Gaudium et Spes (1965). According to that document: “Therefore if methodical investigation within every branch of learning is carried out in a genuinely scientific manner and in accord with moral norms, it never truly conflicts with faith, for earthly matters and the concerns of faith derive from the same God” (Gaudium et Spes). The document then continues: “Consequently, we cannot but deplore certain habits of mind, which are sometimes found too among Christians, which do not sufficiently attend to the rightful independence of science…” (Guadium et Spes). It is clarified however, that when speaking of independence, it is important to always remember that all things, scientific or not, come from God. This makes it clear that it is not advocating for Barbour’s Independence or Haught’s Contrast views stated earlier. Gaudium et Spes also seems to caution against Scientism, stating Indeed today's progress in science and technology can foster a certain exclusive emphasis on observable data, and an agnosticism about everything else. For the methods of investigation which these sciences use can be wrongly considered as the supreme rule of seeking the whole truth. By virtue of their methods these sciences cannot penetrate to the intimate notion of things. Indeed the danger is present that man, confiding too much in the discoveries of today, may think that he is sufficient unto himself and no longer seek the higher things. (Gaudium et Spes) It is clear then that the Catholic Church does not believe in either the Conflict or the Contrast relationship model, since it is clearly supportive of the sciences, and promoting a relationship between science and theology. The Church is also careful however, to emphasize that each discipline has its own area of expertise, and therefore its differences. God created the world with the intention that man would utilize science to learn and discover, but also there are mysteries and questions that science cannot answer. Therefore, it seems as though the Church would support the relationship defined by Contact, since it is careful to both promote the similarities and the differences between the two fields. The Church’s views seem similar to those of Augustine, stating there are two distinct kinds of truth, both of which are valid, but also
distinct from one another, “faith and reason” and that the Church acknowledges that both have their own areas (Guadium et Spes). In the section of Gaudium et Spes titled “Some More Urgent Duties of Christians in Regard to Culture,” the positive relationship between the fields is emphasized by underlining the Christian duty of assisting the sciences by applying the Christian viewpoint to new scientific developments. It encourages dialogue and mutual understanding stating: “Let them blend new sciences and theories and the understanding of the most recent discoveries with Christian morality and the teaching of Christian doctrine…” (Guadium et Spes). It is clear that as science continues to develop, Christianity must develop with it and maintain dialogue with. Francis de Sales’ views of science and the pursuit of knowledge closely resemble those posed it the documents of Vatican II. Francis was explicitly supportive of sciences and the pursuit of worldly truth and knowledge. He also agreed with Augustine, supporting both truth and “Truth” in his writings on faith and reason, stating in his Catholic Controversy that they are “daughters of the same Father…. They can and must live together as very affectionate sisters” (as translated by Pocetto 1). His understanding of the truth is not exactly the same as Augustine’s however, as de Sales views both kinds as one truth. “Just as our eyes are receptive of different kinds of light, for example, the light of the sun and artificial light to see various objects, so our understanding is given the light of reason and the light of faith to arrive at truth which is indivisible” (as translated by Pocetto 1). It seems then, that de Sales also supports the Contact relationship, as he views both disciplines as being distinct but still able to coexist harmoniously. De Sales’ support of the sciences is also evidenced when he intervened on behalf of a priest by the name of John Baranzano, who was teaching controversial scientific discoveries of Galileo at the time. Baranzano was a scientist and Barnabite priest who was deeply involved in the scientific community, and was described as “a great mathematician, a great chemist and a great innovator, capable of writing against Aristotle and the greatest minds of Antiquity” (Pocetto 5). His passion for science negatively impacted his relationship with the Catholic Church however, when one of his works teaching ideas of Copernicus which had been recently banned by the Church. Francis, while not advocating the ideas Baranzano wrote about, interceded on his behalf to the Church and asked that Baranzano be allowed to return to his prior post (Pocetto 6). Because of his support for scientific work and the pursuit of truth, Francis de Sales seems to support Haught’s Contact relationship between science and religion. John Polkinghorne also links mathematics and religion by attributing the effectiveness of mathematics to God. He argues that mathematics being quite effective throughout so many facets of the natural world is evidence of it being far more than a human invention. He supports this claim by offering evidence; for example, every type of symmetry is present in art from hundreds of years ago yet these symmetries were only discovered in the 1800s (Polkinghorne 3334). While discussing the usefulness of math, he also discusses the beauty that many people find present in various mathematical equations and ideas. Such beauty could be viewed as evidence
that mathematics was initially created by God with order and aesthetics in mind (Polkinghorne 33-34). Another way to examine the philosophical debate is to reduce the argument down to the question “are mathematical objects real?” In other words, do the various concepts and objects discussed in mathematical courses exist, or are they simply invented to represent various scientific phenomena; are things such as numbers and sets actual objects, or are they inventions of the human mind? Did they exist before they were discussed by mathematicians; is their existence independent of human thought? The two primary philosophical views of mathematics are called Platonism and Nominalism, with Platonism supporting the idea that mathematical objects are real and Nominalism denying it (Balaguer “Platonism” 182, 187). The main points and arguments of both philosophies will be examined below. The main concept of Mathematical Platonism is that mathematical objects exist as abstract objects, and statements about them are true in the same way any other observation is true. The main argument of favor of Platonism utilizes the argument stated above about the usefulness of mathematics. According to Mark Balaguer, there are six key steps in proving Mathematical Platonism: 1. Our mathematical theories are extremely useful in empirical science—indeed they seem to be indispensable to our empirical theories—and the only way to account for this is to admit that our mathematical theories are true. Therefore, 2. The sentences of our mathematical theories—sentences like ‘3 is prime’—are true. Moreover, it seems that 3. Sentences like ‘3 is prime’ should be read at face value. (Philosophers would put this by saying that the logical form of ‘3 is prime’ is ‘a is F,’ where ‘a’ is a constant and ‘F’ is a predicate. Thus, the claim here that ‘3 is a prime’ has the same logical form as, e.g., ‘Mars is red.’ Both sentences just make straightforward claims about the nature of certain objects. The one makes a claim about the nature of Mars, and the other makes a claim about the nature of the number 3.) But 4. If we allow that sentences like ‘3 is prime’ are true, and if moreover we allow that they should be read at face value, then we are committed to believing in the existence of the objects that they are about. For instance, if we read ‘3 is prime’ as making a straightforward claim about the nature of the number 3, and if we allow that this sentence is literally true, then we are committed to believing in the existence of the number 3. But 5. If there are such things as mathematical objects (i.e., things that our mathematical theories are about), then they are abstract
objects. For instance, if there is such a thing as the number 3, then it is an abstract object, not a physical or mental object. Therefore, 6. There are such things as abstract mathematical objects, and our mathematical theories provide true descriptions of these things. In other words, mathematical platonism is true. (Balaguer “Platonism” 182–183) This proof for Platonism effectively takes the argument by usefulness of mathematics and expands upon it and can be summarized as follows: since mathematics can be used to accurately model scientific phenomena through its ideas, these mathematical ideas must be true. Therefore, the objects referred to in these ideas must be real objects. Since they are not concrete objects, they are abstract but still real objects. Finally, since the objects and ideas about them are real and true, Platonism is also true (Balaguer “Platonism” 182-183). But what does a realist mean by ‘abstract’? Using the example of natural numbers, an abstract mathematical concept can be described as follows: “They are eternal, or perhaps better, timeless. In like manner, mathematical objects do not have any causal relations with material reality, or anything else for that matter. Nothing that anyone or anything does can have any effect on the arithmetic properties of natural numbers…” (Shapiro 160). There have been many varieties of Nominalism, each with its own issue or argument against one aspect or another of Platonism. Therefore, rather than examining Nominalism as a whole, certain subcategories of Nominalism will be examined. Some say mathematical objects exist only mentally, and others go as far as to deny that mathematical objects are real at all (Shapiro 162). While there are many different objections to the proof for Platonism, the argument that will be examined here objects to the second step in the proof, that mathematical statements are true. This approach is a form of Nominalism more specifically referred to as Fictionalism. Fictionalism believes that the proof falls short because mathematical objects are not real, and therefore the statements about them cannot be considered true. Instead, Fictionalism believes that “our mathematical theories are not literally true for the same reason that, say, Alice in Wonderland is not literally true. Just as there are no such things as talking rabbits and hookah-smoking caterpillars and so on, so too there are no such things as numbers and sets and so on” (Balaguer “Platonism” 190). Objecting to the idea of mathematical objects means that Fictionalism has to overcome the challenge of explaining the usefulness of mathematics in science, which is no easy task. The first approach that Fictionalism adopted was to deny mathematics is a necessary part of science at all. To do this, two claims are made: 1) Mathematics is in fact not indispensable to empirical science. 2) The fact that it is applicable to empirical science in a dispensable way can be explained without abandoning fictionalism. (Balaguer “Platonism” 191)
While there has been some success in demonstrating the ability to express scientific concepts without mathematics, the first claim was highly controversial and has proven to be the biggest weakness of this argument. Because of this, a new argument for Fictionalism has been developed that accepts that mathematics is an essential tool of science but asserts that mathematics is independent of science. The new Fictionalist argument states that mathematics is simply used to describe scientific phenomena without actually being tied to them. For example, when describing the temperature of something as 40°, “we are using the numeral ‘40’ to help us say what we want to say about S (the object being 40°). In essence, what we’re doing is using ‘40’ as a name of a certain temperature state” (Balaguer “Platonism” 191). To new Fictionalists then, mathematics and science are separate, and while scientific ideas can be proven true, mathematics cannot be (Balaguer “Platonism” 191). The argument Fictionalism uses to show mathematical objects are not real is as follows: 1. Mathematical sentences like ‘4 is even’ should be read at face value; that is, they should be read as being of the form ‘Fa’ and, hence, as making straightforward claims about the nature of certain objects; e.g., ‘4 is even’ should be read as making a straightforward claim about the nature of the number 4. But 2. If sentences like ‘4 is even’ should be read at face value, and if moreover they are true, then there must actually exist objects of the kinds that they're about; for instance, if ‘4 is even’ makes a straightforward claim about the nature of the number 4, and if this sentence is literally true, then there must actually exist such a thing as the number 4. Therefore, from (1) and (2), it follows that 3. If sentences like ‘4 is even’ are true, then there are such things as mathematical objects. But 4. If there are such things as mathematical objects, then they are abstract objects, i.e., nonspatiotemporal objects; for instance, if there is such a thing as the number 4, then it is an abstract object, not a physical or mental object. But 5. There are no such things as abstract objects. Therefore, from (4) and (5) by modus tollens, it follows that 6. There are no such things as mathematical objects. And so, from (3) and (6) by modus tollens, it follows that 7. Sentences like ‘4 is even’ are not true (indeed, they're not true for the reason that fictionalists give, and so it follows that fictionalism is true). (Balaguer “Fictionalism”) The first steps of the argument seem plausible. Mathematical statements should be taken at face value, because as Balaguer argues, it is not reasonable to assume that statements as simple as that could have much more complicated meanings such as “if there were numbers, then 3
would be prime” (Balaguer “Fictionalism”). Similarly, the next two several seem logical, as if statements can be made regarding mathematical objects, it stands to reason that mathematical objects exist. The most important part of this argument is then step 5, which is the step which attempts to demonstrate mathematical objects do not exist. Balaguer admits that while there are several potential arguments supporting this idea, all of them have some degree of controversy. In fact, according to Balaguer, one of the stronger but more controversial argument for Fictionalism over Platonism involves Ockham’s razor, which states if there are two possibilities, the simpler of the two should be considered (Balaguer “Fictionalism”). Overall, both the arguments for Platonism and for Fictionalism have challenges that prevent either idea from clearly being superior to the other. One issue inherent to the idea of Platonism is how mathematicians can claim to make discoveries or observations about things that cannot be measured or observed in the same way as other scientific concepts, an issue which arises due to the abstract nature of mathematical objects (Chihara 135). As stated before, Fictionalism struggles due to the daunting challenge of either removing the need for mathematics in science, or at least reducing it to a system of representation for ease of scientific work and discussion. 6.
Intelligent Design and Other Theological Arguments for Mathematical Discovery
As stated in the previous section, expanding the discussion of invention versus discovery to include a theological perspective, in the context of a positive relationship between the sciences and theology, provides compelling evidence towards mathematics as discovery. This section will expand upon the ideas of the previous section and examine the question in ways more suited to a theological approach. This approach is very beneficial to the debate between discovery and invention since it allows for questions such as “why can humanity understand complex mathematics?” and “why can the world be modeled so accurately through mathematics?” Not asked or otherwise discussed thus far is “how did humanity develop the ability to perform mathematics at all?” and more specifically, “why are people able to delve so deeply into the complex and the abstract?” Mathematics, like man’s other traits, must have arisen through our evolution and development as a species. But unlike other traits, mathematics, especially advanced mathematics like calculus, does not seem to provide any benefit to the survival of early man. This has caused many scientists to wonder what could have caused people to develop the ability to perform mathematics such as calculus in the first place. “Why should our cognitive processes have tuned themselves to such an extravagant quest as the understanding of the entire Universe?” (Davies 149). Similarly, another mystery regarding man’s ability to perform mathematics is the apparent contradiction between the many years of education needed to perform advanced and abstract mathematics and the ability of shockingly young minds to make groundbreaking discoveries. For example, Paul Davies cites mathematicians such as Newton and Einstein who both made immense discoveries in their twenties (Davies 149). Both of these facts suggest that God intended humanity to discover and understand the mathematical nature of the world that He instilled in it.
This idea leads back to the question posed by Augustine regarding the effectiveness of mathematics in describing the world. When this question is examined theologically, there are two main possibilities. The first is that mathematics is only so effective since humanity attempts to describe the world mathematically. Davies explains this possibility by saying, “We are prone, he [Immanuel Kant] maintained, to project onto the world our own mental bias toward mathematical concepts. In other words, we read mathematical order into nature rather than read it out of nature” (Davies 150). By stating this, Davies argues that mathematics is used so prevalently to describe the world because man initially had success doing so. Once humanity found it could describe one aspect of the world with mathematics, man began to utilize mathematics wherever possible. Thus, the universe seems mathematically organized only because math became the way humanity decided to analyze and describe it. This idea is not completely unreasonable, because mathematics was found to effectively describe an aspect of nature, and so science continued to attempt to use it to model anything and everything scientific. The alternative is the viewpoint held by Augustine and by many Christians who practice mathematics: the idea that when God created the world, he created mathematics, and formed the universe on a mathematical foundation. Does the underlying mathematical order of the universe seem to point toward something more than random chance, that is, toward an intelligently designed creation? Due to the apparent order and structure of the universe, many find this notion convincing. When mathematicians and scientists formulate equations to represent physical reality, they are making two very important assumptions. The first is that the universe is contingent, and the second that it is intelligible (Davies 169), contingent meaning “it could have been otherwise, so that the reason why it is the way it is depends upon something else, something beyond itself” (Davies 163). Both characteristics are necessary because “without the contingency we would in principle be able to explain the universe using logical deduction alone, without ever observing it. And without the intelligibility there would be no science” (Davies 169). The fact that the universe is mathematically structured points toward something even stranger. Not only can the universe be described mathematically, but it seems in many ways to be arranged in a coherent and logical way. In some ways, simply describing how the universe came to be is not enough. What then, caused the chain of events that brought into existence this particular universe? This clearly evidences intelligent design; to a universe created with purpose by God. There is also more that can be learned from the nature of the universe. The nature of the universe can potentially reveal or confirm some of the characteristics of God. What does the universe have to say about the One who “breathes fire into the equations, and thus promotes the merely possible to the actually existing” (Davies 171)? When God created the universe, He could have chosen to create it in any way He desired. This, along with observations of His creation, shows some of the characteristics often attributed to Him in action. According to Davies, God’s rationalism and omniscience are evidenced in the selection of the ordering of the universe. God’s perfection and omnipotence are seen through His creation of the
universe He desired to form. Thus, God shows through the rational and mathematical order of the universe (Davies 172). 7.
Conclusion The idea of science and theology being in conflict with one another is not a new one, as seen in the conflict of the Catholic Church and Galileo. Likewise, however, the idea of harmony and support between the fields has also existed for centuries, such as in the views of Francis de Sales who believed that the Incarnation was “to teach us how to live… with and according to reason” (as translated by Pocetto 1). In his writings, it is evidenced that he supported the pursuit of truth and reason, and therefore the work of mathematicians and scientists. These sentiments are echoed in the documents of Vatican II, which re-emphasized the importance of reason and the pursuit of knowledge, as well as the role of the Church in guiding and assisting in these pursuits. Examining the issue of whether mathematics is invented or discovered from a mathematical perspective reveals compelling arguments from both schools of thought. Many concepts, when developed by mathematicians make them feel as though they are being guided toward one destination. Others allow the mathematician the freedom to develop the idea any way they see fit. Many of the arguments from a purely mathematical point of view rely too heavily on opinion and feeling, so other perspectives needed to be examined. Looking at the philosophical and theological arguments allowed for more compelling arguments for mathematics as discovery, most importantly the effectiveness arguments. The effectiveness of mathematics in science provides a compelling argument for discovery, especially when bolstered with the idea of mathematics being instilled into the design of the universe by God. It was only possible to augment the argument for discovery thanks to the idea of mathematics and Christianity having a positive relationship with each other, which was demonstrated thanks to the writings of Francis de Sales, and the document Gaudium et Spes of Vatican II. The theological perspective on the issue of discovery and invention not only breaks the deadlock between the schools of thought from a mathematical perspective, it provides a way for the two ideas to coexist. Therefore, the most reasonable possibility is that mathematics as a whole is created by God, and humanity discovers it and can use it to describe the world. However, mathematicians are able to construct and invent certain tools to aid in the discovery of the mathematical nature of God’s creation. The usefulness of mathematics and the acceptance of mathematics as a creation of God embodies perfectly Francis de Sales’ love of both faith and reason, and exemplifies the relationship that mathematics and faith have with one another.
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