Mathematics for Human Flourishing

Why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish? – Dr. Francis Su

At the 2017 Joint Mathematics Meetings Dr. Francis Su, the outgoing president of the MAA, gave the retiring MAA presidential address titled “Mathematics for Human Flourishing.”

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Francis Su. Photo Credit: Kate Awtrey, Atlanta Convention Photography

Elsewhere I have mentioned a previous talk by Francis Su on the “Lesson of Grace in Teaching.” In this most recent address, Dr. Su gives another amazing lecture on the nature and purpose of mathematics. Dr. Su is a great example of a Christian striving to better his discipline and to honor God by serving as an advocate in the public square.

Dr. Su begins by quoting the philosopher Simone Weil, “Every being cries out silently to be read differently.” What follows is an insightful commentary on the desires we all have as human beings and where those desires intersect with the discipline of mathematics. He ends by returning to reference Simone Weil.

She had found a path through struggle to virtue.
She understood that mathematics is for human flourishing.
The mathematical experience cannot be separated from love!
The love between friends who play with a mathematical problem.
The love between teacher and student working to help each other flourish.

The love of a community like the Mathematical Association of America working with each other towards a common goal: through the knowledge and virtues wrought by mathematics, to help everyone flourish.

Here is the full text of the talk: “Mathematics for Human Flourishing.”

Here is a great summary of the talk and a link to the MAA Facebook page where a video of the talk can be found. 

 

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The Role of Mathematical Aesthetics in Christian Education

by R. Scott Eberle

Scott Eberle has a Ph.D. in Math Education and currently serves as a missionary in Niger, working to spread the Gospel message through Christian education. Scott works to build up Christian leaders and educators in Niger who approach mathematics through a distinctly Christian perspective. You can follow Scott and his family at nigerministry.tumblr.com.

Josh Wilkerson invited me to contribute something on the aesthetics of mathematics from a Christian perspective. I’d especially like to discuss how such a seemingly abstract idea has application in Christian education.

Detail of the Mandelbrot Set in the plane of complex numbers
Detail of the Mandelbrot Set in the plane of complex numbers

Mathematical Aesthetics

Mathematics has been considered an aesthetic subject from antiquity. The Greeks considered mathematics to be the highest form of aesthetics because of its perfection. The Pythagoreans and Platonists considered mathematical concepts to have a real, mystical existence in some perfect realm.

Throughout history, mathematicians and philosophers have continued to claim that mathematics is beautiful for a variety of reasons. For example, whereas the Greeks saw beauty in the ontology of mathematics, the French mathematician Henri Poincaré saw beauty in its epistemology. Because of the way we teach mathematics, many students believe there is always one hard-and-fast method for cranking out the answer to any mathematical problem. But as mathematicians know, true mathematical problems require a great deal of creative intuition to solve. Poincaré pointed out that mathematicians rely on aesthetic-based intuition to distinguish fruitful paths of mathematical inquiry from dead ends. He wrote, “It is this special aesthetic sensibility which plays the rôle of the delicate sieve” (1908/2000, p. 92).

Today, nearly all mathematicians continue to recognize the aesthetic nature of mathematics (Burton, 1999). The British mathematician John Horton Conway went so far as to claim, “It’s a thing that non-mathematicians don’t realize. Mathematics is actually an aesthetic subject almost entirely” (Spencer, 2001, p. 165). The reason the general population doesn’t realize that mathematics is an aesthetic subject is probably due to the mechanical way in which we frequently teach mathematics. School exercises are often artificial, simplistic, and have only one right answer. There is nothing creative or aesthetic to see in the average math lesson.

Conway’s claim that mathematics is almost entirely aesthetic is a bold one. But actually, modern mathematics can be seen to be an aesthetic subject from its foundations to its methods to its end results. This is especially true since mathematics’ divorce from physics in the 19th century as it became a purely abstract study, inspired by, but independent of, the natural world.

  • Foundations: Mathematics rests on a foundation of axioms and definitions. But these are chosen, not deduced. Mathematicians choose definitions and axiomatic systems based on criteria of logic, relative completeness, consistency, mutual independence, simplicity, connectedness, and elegance. These criteria are partly aesthetic in nature.
  • Methods: As Poincaré pointed out, mathematicians rely on a certain aesthetic sense to guide their explorations. Paths that seem particularly elegant often prove to be the most successful. In 1931 Gödel destroyed earlier hopes of purely mechanical methods of generating mathematical theorems and proofs, making the fundamental role of intuition even more necessary. Modern researchers are beginning to understand that intuition is not a fuzzy feeling, but rather a rigorous source of insight. Robert Root-Bernstein (2002) makes a powerful argument that all scientific thought occurs first as an aesthetic intuition, and is then confirmed by verbal logic. Therefore aesthetics guides our mathematical exploration and is the basis for our mathematical reasoning. But we often show only the final algorithmic logic to our students.
  • Results: Mathematicians don’t often discuss aesthetics explicitly, but when they do, they usually point to theorems and proofs, which they insist should be elegant. The American mathematician Morris Kline observed that “Much research for new proofs of theorems already correctly established is undertaken simply because the existing proofs have no aesthetic appeal” (1964). Mathematicians especially appreciate results which are surprisingly simple or have significant connections or visual appeal. Such results are said to be beautiful. The Mandelbrot Set, for example, is beautiful partly because its definition is surprisingly simple and partly because it has great visual appeal. It is interesting to note that criteria such as significant connections indicate that beautiful results will be among the most useful and important. Criteria such as surprise suggest that beautiful results may be important for insight and understanding, and therefore also for education.

So mathematics is seen to be aesthetic “almost entirely.” At this point, some would say the discussion is merely philosophical and has no real world implications. Indeed, most mathematicians give aesthetics little explicit thought unless questioned about it. It is perhaps for this reason that many educators have not picked up on the importance of aesthetics in mathematics.

Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)
Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)

The Christian View

Throughout history, most mathematicians have been Platonist, at least in practice. We tend to think that mathematical ideas are discovered, rather than invented. In more recent times, some have questioned this, claiming that mathematics is simply the brain’s way of understanding how the universe is structured, and mathematics could be very different for an extraterrestrial species. (See, for example, Lakoff & Núñez, 2000.) Others disagree, pointing out how mathematics inexplicably predicts new discoveries. Of course, all agree that certain things, such as notation, conventions, and choice of axioms, are man’s invention. But where do the beautiful results we admire come from? The Greeks cannot be said to have “invented” the Pythagorean Theorem. Most would agree they (and other cultures) “discovered” it.

Most Christian theologians, from Augustine (354 – 430 AD) onwards, as well as Christian mathematicians, have agreed with a Platonist perspective, believing that mathematics is in the mind of God, and we discover these eternal truths. Mathematics cannot be part of Creation because it is not a physical part of nature—it is a collection of abstract ideas. One does not physically create abstract ideas, one conceives them. And God must have always known these ideas, so they have always been part of his thoughts. Mathematics preceded Creation and is untouched by the Fall. It is perfect and beautiful and contains awe-inspiring ideas, such as Cantorian infinity, which is part of God’s nature but not part of our physical universe. However, we ourselves are fallen, so our understanding and use of mathematics is imperfect.

Some modern Christian thinkers have proposed other possibilities similar to those of Lakoff and Núñez, making mathematics a human activity, or only one of many possible systems of mathematics in the mind of God. Nevertheless, all Christians affirm that mathematics is not independent of God. Even if there are other possible systems of mathematics, the one we know is the one God chose for us as good, and it has always been known by God. It is not some arbitrary invention. I like to think that when I am studying mathematics, I am studying the very thoughts of God, that mathematics is part of God’s attributes. God did not “create” love; God is love (1 John 4:8). Likewise God did not “create” one and three; God is one Being in three Persons. God did not “create” infinity; God is infinite. And so on.

But whatever position you take, whatever the ontology of mathematics, it should not surprise us that mathematics is beautiful, because God is beautiful. Mathematics is indeed “an aesthetic subject almost entirely.” Mathematical beauty and usefulness is a mystery only if we do not believe it comes from God. (See, for example, the classic article by Wigner, 1960.)

Euler's Identity, relating five fundamental constants and three basic operations, is often called the most beautiful result in mathematics (Wells, 1990).
Euler’s Identity, relating five fundamental constants and three basic operations,
is often called the most beautiful result in mathematics (Wells, 1990).

Education

Though aesthetics is part of the very foundation of mathematics, it is largely neglected in math classrooms. As mathematician Seymour Papert pointed out, “If mathematical aesthetics gets any attention in the schools, it is as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force which makes mathematical thinking function” (1980, p. 192). However, an increasing number of researchers (including myself) have been noting important consequences of mathematical aesthetics for how we teach mathematics at all ages.

The interested reader can turn to researchers such as Nathalie Sinclair to see how modern research has been discovering the importance of aesthetics in mathematics education. Aesthetics is a “way of knowing” mathematics prior to verbal reasoning and should be an important part of our mathematics classrooms. Indeed, Sinclair (2008) has found that good math teachers tend to use aesthetic cues in their teaching implicitly, though they may not realize it. For example, teachers who reveal a “secret weapon” or present a surprising fact or note simpler ways to express certain solutions are modeling a useful aesthetic to their students. In my own research (Eberle, 2014), I have found that even elementary school children come with their own aesthetic ideas and use them in valid mathematical ways when given the opportunity to do open-ended math problems. And this is true of all children, not just those that are gifted in mathematics. Children’s initial aesthetic ideas are far from those of mathematicians, but through experience they are refined. Educators from John Dewey to the present day have argued that aesthetics is important for all of education, and now we are discovering how this is true for mathematics.

Nathalie Sinclair (2006) has proposed that mathematical aesthetics has three roles in education:

  1. Aesthetics gives intrinsic motivation to do mathematics. This is in contrast to the extrinsic coaxing we often use with students. Instead of “sugar-coating” math problems by placing them in artificial contexts, we should allow students to explore the natural symmetry and patterns found in every branch of mathematics. I sometimes challenge teachers to see how many patterns they can find in the “boring” multiplication table. They are usually very surprised. Students can also engage in mathematics in a natural way by pursuing projects they themselves have suggested. Such genuine contexts are highly motivational. (See these posts by Josh Wilkerson for a Christian perspective on this idea.)
  2. Just as with mathematicians, aesthetics guides students to generative paths of inquiry. When allowed to explore freely, children use their own aesthetics to find valid mathematical insights, though this may take time. Students need opportunities to pursue their own ideas and conjectures.
  3. Aesthetics helps students to evaluate their results. Often math is presented as black-and-white with only right and wrong answers. But if students are allowed to do more open-ended inquiry or project-based mathematics, they can use their growing sense of aesthetics to evaluate the solutions found.
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry

Christian Education

As Christian educators, we should realize that God gives common grace and we should always be open to learning from the best results of secular research, filtered through the worldview shaped by our faith. Throughout history, Christians have often been at the forefront of recognizing the importance of aesthetics. God gave us our ability to appreciate beauty and patterns for a reason, and what is math if not the study of patterns (Hardy, 1940)? We Christians should be among the first to recognize the importance of educating the whole child, even in mathematics, and embracing research showing the importance of allowing aesthetics to have a deep role in education, including our mathematics instruction.

Even more importantly, we should be careful not to make a sharp dichotomy between “secular” knowledge and “spiritual” knowledge. Mathematics is often taught as if our faith had nothing to do with the knowledge we are learning. Though it is wrong to artificially “spiritualize” every lesson, at the very least Christian students should understand the relationship between their faith and their studies. One way to do this is to let students know that math is not just a series of arbitrary algorithms and heuristics to be memorized, but a rich, creative, beautiful subject to be explored and appreciated. And when students see some of the beauty of the subject, we can lead them to reflect on the Source of that beauty. Indeed, we are doing a great disservice to Christian students if we lead them to believe that a subject that is in the mind of God is somehow boring or ugly.

I have to admit I am distressed sometimes by certain popular views of mathematics. I remember reading one author who wrote that mathematics was part of Creation, and as such, the author seemed to believe mathematics was purely arbitrary, as if there were no special reason God created 2 + 2 to be 4. I often come across this idea that math is not understandable, a result of learning by rote. All we can supposedly do is grit our teeth and memorize the mysterious methods. This author’s solution was to teach students to plug away at exercises and learn to praise God every time they correctly found God’s answer, and be thankful that God, in his faithfulness, had not changed the answer in the meantime. I fear that such instruction will not generate praise for God but rather fear of mathematics. My hope is that we can learn instead how to teach that mathematics is a deep, joyful, meaningful, beautiful subject. It is a reflection of God’s nature.

Flower with spirals in Fibonacci sequence Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5
Flower with spirals in Fibonacci sequence
Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5

Conclusion

For Christians, mathematical aesthetics must not be an optional extra-credit topic, but must rather be at the very foundation of our mathematics teaching. As Christian educators, aesthetics should guide our understanding of mathematics, inform the way we teach, and be a goal for our students’ learning—and this from the youngest ages. Just as students learn to appreciate poetry or music, Christian students should learn that mathematics is beautiful, and why.

References

Burton, L. (1999). The practice of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121-143.

Eberle, R. S. (2014). The role of children’s mathematical aesthetics: The case of tessellations. The Journal of Mathematical Behavior, 35, 129-143.

Hardy, G. H. (1940). A mathematician’s apology (1967 with Foreword by C. P. Snow ed.). Cambridge, UK: Cambridge University Press.

Kline, M. (1964). Mathematics in Western Culture (Electronic version ed.). New York: Oxford University Press.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.

Poincaré, H. (2000). Mathematical creation. Resonance, 5(2), 85-94. (Original work published 1908)

Root-Bernstein, R. S. (2002). Aesthetic cognition. International Studies in the Philosophy of Science, 16(1), 61-77.

Sinclair, N. (2006). Mathematics and beauty: Aesthetic approaches to teaching children. New York: Teachers College Press.

Sinclair, N. (2008). Attending to the aesthetic in the mathematics classroom. For the Learning of Mathematics, 28(1), 29-35.

Spencer, J. (2001). Opinion. Notices of the AMS, 48(2), 165.

Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37-41.

Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1).

A Triune Philosophy of Mathematics

by Dusty Wilson, Highline College

The following is from a talk given by Dusty Wilson at the ACMS Conference this past May. It is with his gracious permission that I am sharing it here. Here is a link to a previous talk given by Dusty on “Where does mathematics come from?”

Abstract

What is mathematics and is it discovered or invented? The Humanist, Platonist, and Foundationalist each provide answers. But are the options within the philosophy of mathematics so limited? Rather than viewing and describing mathematics in a mutually exclusive manner, each of these approaches includes components of truth from a greater triune philosophy of mathematics. This paper will briefly outline existing philosophies and then introduce an inclusive triune paradigm through which to explore fundamental questions about mathematics.

1 Introduction

My parents were hippies who were leery of traditional education. So out of a desire to both protect and also encourage questioning they put me into alternative public schools. These weren’t edgy enough and so they allowed me to homeschool junior high into high school. This put me on a fast track and I began community college during what would have been my junior year of high school. I jumped right into calculus and worked my way through differential equations. After two years I transferred to The Evergreen State College to continue my alternative education with an interdisciplinary liberal arts degree studying political science, literature, and mathematics. With such an eclectic background, I didn’t have a clear direction following my bachelor’s degree so I went on to graduate school in mathematics thinking, “If this doesn’t work out, I can do something else later.” While a graduate student I was given the opportunity to teach and my career path suddenly became clear. I was hired by Highline College right out of graduate school where I became the youngest tenured faculty member in College history.

While this makes me sound smart, it really means that I had a lot of growing to do as an educator and colleague. But I was in a supportive environment and by my eighth year I was firmly established as a teacher, in service, and in professional growth, and I generally felt that I knew my professional direction. In 2008 I attended a talk by a colleague [2]. The talk itself was on polling and statistics and not related to this paper. However in the midst of the lecture my coworker said, “I think math was invented by people, not discovered.”

Is math discovered or invented? In all my non-traditional education as well as traditional community college and graduate studies and then continuing into the first eight years of my professional work, I don’t once recall having asked myself the question, “Where does math come from?” But while this was the first time these ideas had ever registered in my mind, I have come to realize over the last seven years of study that I had subconsciously adopted a framework for understanding mathematics. As John Synge said, “[E]ach young mathematician who formulates his own philosophy — and all do — should make his decision in full possession of the facts. He should realize that if he follows the pattern of modern mathematics he is heir to a great tradition, but only part heir.” [9, pp. 166] This certainly encapsulates my mathematical journey.

As I have come to have “full possession of the facts”, I’ve learned that there are three main ways to explain the origin of mathematics. Within each of these broad categories there is a spectrum of nuance. Others have written compelling descriptions of this, but allow me to outline using broad strokes so that I may synthesize the field. The three broad views are as follows.

Foundational philosophies: Mathematics is developed from axioms and definitions using logic

Humanistic philosophies of mathematics: Mathematics is invented by humans who are the source of math

Mathematical Platonism Also called ’mathematical realism’, this view holds that mathematics exists ‘out there’ to be discovered; perhaps owing its existence is to God, but perhaps not

While some readers may recognize or be able to articulate their philosophy of mathematics, others may resonate with my story in that I was years into my career as a professional without realizing that I even had a view. I believed mathematics devoid of presupposition without even having the vocabulary to articulate my own presupposition about the field. So as I clarify the basic views available for later synthesis, I encourage you to ask yourself where these views match your training, intuition, and pedagogy.

2 The Foundational Philosophies

The first paradigm is that math is logic — this is the basis of the foundational philosophies: intuitionism, logicism, and formalism. If you do research on the philosophy of mathematics, these three views are described over and over again to the point that they nearly define the field.

The intuitionists such as Kronecker and Brouwer held that humans create the axioms of logic/mathematics and that we then manipulate these axioms to construct the theorems of mathematics in a constructivist manner. Because it stems from our work, the intuitionism shows existence by demonstrating a formula/algorithm/recipe to explain how each entity may be constructed. Because of this, intuitionists rejected proof by contradiction as well as the existence of an actual infinity. For them the source of mathematics was decidedly human. Or as Kronecker famously wrote, “God made the integers, all else is the work of men.” [11, pp. 19]

figures

The logicists movement was begun by Frege, reached its height with Russell and Whitehead, and concluded with Godel. They felt that the axioms of logic were self-evident truths that were known intuitively to the logician. They accepted the rules of logic apriori. In their effort to make solid their foundation, they held that some axioms were self-evident that are not so evident. Certainly the axiom of choice is on this list. Of the foundational camps, logicism was the most fully developed. For the logicist, the source of mathematics was beyond the human experience, self-evident, and discovered (albeit by a select few).

The formalists led by Hilbert were perhaps the largest group. They did not concern themselves with the source of the axioms but worked from these using every clever device they could devise. They had no issue with contradiction or infinity. Hilbert referred to math as a meaningless game. [1, pp. 21] The formalist didn’t have a strong opinion about where mathematics comes from; after all, it didn’t matter anymore than the source of Chess or Monopoly.

3 Mathematical Humanism

The second paradigm is mathematical humanism: all mathematics is somehow human in nature/origin. Unlike the foundational philosophies, the subcategories are not as clearly defined. In part this is because mathematical humanism is more current and thus hasn’t had as much time to mature. The spectrum within mathematical humanism that I will discuss ranges from a biology-brain model, to language, and ends with social constructivism. Of these, the idea that math is a language is probably the oldest while social constructivism seems most dominant among educators.

According to authors Lakoff and Nunez, our ability to perform abstract reasoning is biological. [8, pp. 347] Mathematics is ultimately grounded in experience. [8, pp. 49] It is effective because mathematics is a product of evolution and culture. [8, pp. 378] Mathematics doesn’t have an independent existence. It is culture dependent and only exists through grounding metaphors. [8, pp. 3 356, 368] Consequently the philosophy of mathematics is the realm of cognitive science and not the domain of mathematicians. [8, pp. xiii] Where does mathematics come from? For these philosophers, the source of mathematics is biological and evolutionary and thus serves only an evolutionary purpose. . . which is to say it has no intended purpose.

Perhaps the most commonly held humanistic philosophy of mathematics is summed up in the phrase, “Mathematics is the language of science.” This originates with Galileo who wrote: “[The universe] cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it.” [3, pp. 4] Today this phrase is most often used outside of university math departments because it defines mathematics through its applications and universities produce pure mathematicians (more akin to the formalists of the foundational movement). The basic premise of the view is that mathematics is invented as a way to describe discoveries in the natural world. Math isn’t monolithic and unchanging because language changes. The strength of this view is that it seems to explain the perceived transcendence and beauty in math by tying it back to science.

Mathematics is something people do according to Reuben Hersh. [5, pp. 30] The philosophy of mathematics is the study of what mathematicians do. [5, pp. xii] The emphasis of social constructivism is on practice. As a practice there has been an evolution of mathematical knowledge. [5, pp. 224] This extends even to including proof itself. [5, pp. 6] As such, mathematics is a social construction. It draws on conventions of language, rules, and agreement in establishing truths. Mathematical knowledge and concepts change through conjectures and refutations. The focus is on creation rather than the justification knowledge. [5, pp. 228]

4 Mathematical Platonism

The third paradigm is mathematical Platonism (or mathematical realism) and is loosely based on the Plato’s theory of forms and divided line. There is less explicitly written supporting Platonism (and much against). However many mathematicians are Platonists although not aware of it. Some know it and are reluctant to admit it because it seems mystical. Unlike the foundational philosophies and mathematical humanism, there is less written on the subtlety and nuances of Platonism. Thus the spectrum that I am about to describe is of my own creation (that is, unless I happened to get it from some abstract realm).

As given in Principia, Russell practiced what I dub, “finite mathematical Platonism.” He began from a short list of self-evident “discovered” axioms. Then mathematics was built/created from these few eternal building blocks. This is a similar approach to that taken by Euclid. Finite Platonism gels nicely with the axiomatic method. It probably isn’t a stretch to claim that the opti- 4 Mathematical Platonism/Realism Finite Axioms Countable Uncountable (All Truth) (Kitchen Sink) Figure 3: How much lies within the Platonic realm? mism following the “discovery” of Newton’s laws stemmed from this same view: namely that the universe could be described by just a few simple laws. Today physicists are searching for the theory of everything . . . a few simple statements to describe all. In this view, very little is required of the mathematical form (which only contains a few statements) and much of the human mathematician (who massages the few givens into the body of mathematics). So where does mathematics come from? Well it begins with a few eternal truths and then is created by the mathematician.

realism

The most widely held version of realism holds that all mathematical truth is found “out there” (in the Platonic form). This includes triangles, pi, and the golden ratio. It also includes every real number (yes, I know they are uncountable) and proofs big and small (think the proof of Pythagorean theorem vs. that of Fermat’s Last Theorem). That is every true mathematical statement (regardless of how useful or elegant) can be discovered in the form. One author, Lakoff, calls this the romance of mathematics. [8, pp. xv]

Finally, the philosopher Alvin Plantinga affirmed what I call an uncountable mathematical Platonism. As a theist, he holds that math exists in the mind of God. He sees all mathematical entities as uncreated necessary beings whose existence is affirmed by God’s nature. For Plantinga, God affirms the existence of all propositions, states, and possible worlds. But God affirms the truth of only some. [10, pp. 143] This is a kitchen sink view. That is everything exists “out there” to be discovered—true and false. But not all is true.

5 Mutually Exclusive Models?

In my reading, most authors want you to choose one and only one paradigm. They often use an elimination argument to justify their position. For example, a typical humanist’s argument might be summarized: formalism is dead and Platonism requires God. Thus the only option remaining to us is humanism. The problem is that this assumes that (a.) the discussed options are disjoint, (b.) that all the options are being considered, and (c.) that the premises are correct. I am going to focus on the first assumption: namely that the options are disjoint.

At first glance, Platonism, humanism, and the foundational views seem mutually exclusive (disjoint). If math is discovered “out there” then it can’t originate within us. If it comes from within us, then it isn’t a game we play, and certainly a meaningless game sounds nothing like the eternal truths of an 5 Humanistic Philosophies Mathematical Platonism/Realism Foundational Philosophies Intuitionism Logicism Formalism Biology & Brain Finite Axioms Language Countable (All Truth) Social Construction Uncountable (Kitchen Sink) Figure 4: A synthesis of views ideal realm. But perhaps this is a false trichotomy.

Consider the often told Indian parable of blind men trying to describe an elephant. One blind man feels a serpent, another a tree, and a third a spear. While these seem very different, we know that the legs, trunk, and tusks of an elephant are all part of the same animal. Could our seemingly mutually exclusive views of mathematics simply be appendages of a greater and more inclusive truth?

What I am proposing is a triune philosophy that envelops and includes much of a wide swath of the paradigms discussed. Whereas before, we saw three world views, each with its own nuances, now we envision mathematics on a higher dimension.

trinity math

The key to this view is quite simple. Namely that the center of each of the three views represents the strength of the position. I dare say that many would agree that mathematics is a logical language we speak to describe abstract or immaterial truths.

To understand this view, it is insightful to think about what each paradigm sees as its greatest ideological adversary. We see this by comparing the center of each edge with its opposite vertex.

Example 1 Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history.

Authors Lakoff and Nunez explain their view and make it very clear who/what they view as their opposition. “Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! We create it . . . Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history.” [8, pp. 9]

They take on Platonism very directly. They write, “Human beings can have no access to a transcendent Platonic mathematics, if it exists. A belief in Platonic mathematics is therefore a matter of faith . . . There can be no scientific evidence for or against the existence of a Platonic mathematics . . . therefore human mathematics cannot be part of a transcendent Platonic mathematics, if such exists. [8, pp. 4]

Whether you accept their argument or not, it should be clear that they see the primary counterargument to their biology & brain explanation for the origin of mathematics as what they dub “The Romance of Mathematics.” As they write, it’s the stuff of movies like 2001, Contact, and Sphere. But while it initially attracted them to mathematics, they are now more enlightened. [8, pp. xv]

Example 2 Formalism vs. mathematics as the language of science—the debate between pure vs. applied mathematics.

The distinction we make between pure and applied mathematics is relatively recent. Are we standing on the shoulders of mathematicians or physicists – a good argument can be made for both. Prior to 1900, one can make the broad generalization that there was some pure mathematics but no pure mathematicians. But around the time when the foundational philosophies were being developed, this distinction was drawn. As G.H. Hardy wrote, “Pure mathematics is on the whole distinctly more useful than applied.” [4, pp. 134] Taking this one step farther, the father of Formalism, David Hilbert is quoted as saying, “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” For the formalist, mathematics was certainly a language. However, it wasn’t a language intended to communicate information outside of mathematics. Rather than being the language of science, mathematics was the language of mathematics.

The author Morris Kline wrote that most mathematicians have withdrawn from the world to concentrate on problems generated within mathematics. They have abandoned science. [7, pp. 278] Today mathematicians and physical scientists go their separate ways . . . mathematicians and scientists no longer understand each other. [7, pp. 286] Under the influence of formalism and the other foundational philosophies, mathematicians no longer speak the language of science.

Example 3 Mathematics is fallible and a social construction.

The social constructivists reject the narrow definition that math is logic. For example, the humanist Reuben Hersh is concerned with the edifice that remains in university mathematics departments. He believes that his philosophy recognizes the scope and variety of mathematics, fits into general epistemology and philosophy of science [note, science and not mathematics], is compatible with practice – research, application, teaching, history, calculation, and mathematical intuition. He also rejects certainty and indubitability as false and misleading. [5, pp. 33]

The opposition is clear: It’s the foundational philosophies (primarily formalism as the most dominant view). Proof in particular is the opponent of this view. Hersh writes, “The trouble is, ‘mathematical proof’ has two meanings. In practice it’s informal and imprecise. Practical mathematical proof is what we do to make each other believe our theorems. Theoretical mathematical proof is formal. It’s transformation of certain symbol sequences according to certain rules of logic.” [5, pp. 49] The only reason to believe in mathematics is—it works! [5, pp. 213] There is no infallibility. [5, pp. 215]

Hersh wants to redefine mathematics as fallible and a social construction. As such he must take on the establishment. And the power brokers in mathematics hold the foundational view that math is logic and as such is pure and unchanging.

Before further sharing what I call the triune philosophy of mathematics, it’s important to recognize that ideas have consequences, and that this remains a truism in the philosophy of mathematics as it is elsewhere. The way we answer “Where does math come from?” impacts research and education. Given my own background as an educator, I’d like to say few words on education.

Example 4 The philosophy of mathematics and its influence on education.

Nicolas Bourbaki is the collective pseudonym under which a group of mathematicians wrote a series of books with the goal of grounding all of mathematics in set theory. Their approach is similar to that of the formalists. The manifesto of Bourbaki has had a definite and deep influence. In secondary education the new math movement corresponded to teachers influenced by Bourbaki. ”The devastating effect of formalism on teaching has been described … [5, pp. 238]” through books like Why Johnny Can’t Add. [6]

Today we can see the influence of each major paradigm through the competition between teaching through a discovery method, cooperative learning, and skill based manipulations. Given the massive fiscal investment in mathematics education in the U.S., finding the perfect pedagogy is somewhat of a holy grail. But what if our issue is having too limited a view on mathematics? While I don’t claim to be an expert on human cognition or learning, I postulate that a pedagogy that incorporated aspects of all three major philosophies would be more attractive to the next generation of students. If you will, it’s almost a philosophical parallel to teaching to multiple learning styles.

6 A Synthesis of Views

If the three major branches of the philosophy of mathematics are not mutually exclusive, it is possible that a broader, more inclusive, philosophy of mathematics exists. Is mathematics invented or discovered – yes. I’m proposing a view that incorporates the strengths of each paradigm but that comes with some ambiguity – what I am calling, “A triune philosophy of mathematics.”

Just to clarify, this essay isn’t intended to end a discussion but rather to begin a conversation. What is good? What arguments are logically sound? What passes the experiential sniff test? This conversation is going to force us to go much deeper into the details than this paper has allowed. And as Whitehead and Russell learned full well, the devil is in the details.

triune

For me, this was the image that first opened my eyes to a triune philosophy of mathematics. It incorporates the greatest strengths of each paradigm inside a single figure. There is a common practice of mathematics between philosophies. That is, our old friends from calculus and algebra haven’t changed—the integral and derivative are calculated the same way whether discovered, invented, or based on the axioms of logic.

One powerful aspect of this model is that it gives a place to look for counterarguments. That is, the center of each edge of the original triangle is strong while its vertices are potential weaknesses. As we look to certain places to find counter examples to prove/disprove a mathematical claim, this gives us a direction to look to substantiate/discount philosophical arguments.

Bear in mind that the distinctions I am making are tentative. That is the vertices of the new solid triangle (triune math) could shift to include more/less of the gray triangle. For that matter, one might argue that these are not triangles at all, but that there are many more sides on each figure. But while I acknowledge that this is a legitimate objection and requires serious consideration, it isn’t my hypothesis.

For the humanist, the source of truth must come from within the cosmos. The Platonist says math resides outside the material world. The foundationalist says that math is from self-evident axioms and doesn’t bother to justify their existence. Going back to the parable of the blind men and the elephant, there was a clear source for higher knowledge (namely the elephant). If this triune philosophy more fully describes the nature of mathematics, then it too is likely grounded in a greater rationale.

I believe mathematics is firmly grounded in the triune God of orthodox Christianity. Following in the footsteps of Kepler, Newton, Euler and countless others, I believe that there are aspects of mathematics that go beyond the physical world:

  1. mathematics is logical and self-existent because it is part of the nature of a logical and self-existent God
  2. humans create and speak the language of mathematics as image bearers of one who walked among us
  3. we can discover eternal transcendent truth because the spirit of God speaks to each one of us

For some the very mention of God may be enough to discredit this whole triune philosophy of mathematics. For others, the selection of a specific God may be too much. I accept this critique but challenge you: Is there any existing philosophy of mathematics that fully describes the marvel and practice of mathematics? If not, could there be a greater elephant in need of description? If so, what is its size and shape?

Conclusion

So far as I know, this triune philosophy of mathematics is a new idea (perhaps discovered, perhaps invented). This essay marks the first time it has been shared in print. It’s quite possible that I will soon find out the importance of tenure as this could be the last essay I ever write. Jests aside, I anticipate next steps in two directions. The first is in answering the likely objections that this paper will receive. The second is in fleshing out the details wherein the truth most likely lies.

Is it worth it? Yes, ideas have consequences and we have gone too many years under the allusion that mathematics is a field devoid of presuppositions. This introduction to a triune philosophy of mathematics should bring this out in the open. Something needs to change in mathematics and I propose that it is how we view and understand where mathematics comes from.

References

[1] E. Bell. Mathematics: Queen and Servant of Science. Bell, London, 1952.

[2] H. Burn. Polling: When mathematics meets the real world. In Highline College: Science Seminar, 2008.

[3] G. Galilei. The Assayer. unknown, 1623.

[4] G. Hardy. A Mathematician’s Apology. Cambridge, Cambridge, 1940.

[5] R. Hersh. What is Mathematics, Really? Oxford, New York, 1997. 10

[6] M. Kline. Why Johnny Can’t Add. Random, New York, 1974.

[7] M. Kline. Mathematics: The Loss of Certainty. Oxford, New York, 1982.

[8] G. Lakoff and R. E. Nunez. Where Mathematics Comes From. Basic, New York, 2000.

[9] J. Nickel. Mathematics: Is God Silent? Ross House, Vallecito, 2001.

[10] A. Plantinga. Does God Have A Nature? Marquette, Milwaukee, 1980.

[11] H. M. Weber. Obituary for leopold kronecker. unknown, 2:5–31, 1891/2.