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IN DEFENCE OF PLATONISM IN THE MATHEMATICS CLASSROOM

Andrew Schroter

University of Toronto

INTRODUCTION

Mathematics is a challenging school subject that calls for effective instructional strategies. It is also a rich field of endeavour that has occupied the minds of many philosophers. If I am to stand in front of my mathematics class and justify what we are doing, then the philosophy of mathematics—not just pedagogy—must play a role. But which philosophy of mathematics? In his book What is Mathematics, Really? (1997) Reuben Hersh argues that Platonism should be tossed out in favour of “humanism”. Yet while he is right to seek an approach that accords with mathematical practice and teaching, his proposal substitutes history and ideology in place of a satisfactory justification for mathematics. This, I contend, is not the best way to help students appreciate the subject. In this paper I will argue that Hersh’s humanism does not give a legitimate account of norms for mathematical practice; consequently, it is not an appropriate philosophy for teaching mathematics. Instead, teachers should not be afraid to embrace (a fallible brand of) Platonism.

My paper lies in the more general debate between Platonism and naturalism in mathematics. Hersh’s naturalistic humanism is allied with the social-constructivist philosophy of mathematics education of Paul Ernest (1989). On the other side stand mathematicians such as Martin Gardner as well as Platonist philosopher James Robert Brown, who has previously rejected Hersh’s ideas (2012). In this paper I will extend their views and build on my own experiences in mathematics education and the history of mathematics, with the aim of defending Platonism in the classroom.

CONTEXT AND THESIS

Platonism holds the following: Mathematical objects exist abstractly, independent of space, time, and human consciousness; mathematical truths are objective and real features of the universe, neither formal stipulations nor human creations; all propositions hence have a truth-value independent of human minds, which we can come to know through a priori intuition and deduction. Many associate Platonism with the “certainty” of mathematical knowledge. Platonism has tended to dominate among mathematicians, who often talk in terms of “discovery”. Hersh, though, holds that Platonism is wrong: History shows that mathematical knowledge is socio-culturally situated and changeable, so it cannot be certain. The Platonist’s intuition and abstract realm are “far fetched” religious relics, out of touch with natural science and the secular world (1997, 11). Most mathematicians and teachers are (or should be) embarrassed to be Platonists, yet don’t know any different; hence mathematical practice and pedagogy is shrouded in mystery and the tyranny of “right answers” (ibid.). Instead, Hersh proposes a “humanism” that “links mathematics with people, with society, with history” to undo the “damage” that Platonism has caused in education (ibid., 238). In this view mathematics is a social institution, and its objects are cultural-historical concepts, like money, war, or the Supreme Court (ibid., 14); mathematical concepts, while invented by humans, are “objective” in the sense of being inter-subjective “shared ideas” (ibid., 18); they have properties that mathematicians can discover, but such “discoveries” are not statements about the way the universe is.

Let me credit some of Hersh’s points. He is correct that a philosophy of mathematics must account for the historically malleable process of doing mathematics. His focus on education is also relevant. When I began my career I was more concerned with enthusiasm and technique than philosophy. But teaching experience has shown me that the philosophy of mathematics matters. Certainly, instructional strategies are important, but our beliefs about mathematics affect our teaching. We want our students to experience mathematics—the “aha” moment, the flash of insight and sense of beauty. As Ian Hacking says, that’s why there is a philosophy of mathematics “at all” (2014, 84). Hence, teachers must wrestle with what mathematics really is and form their pedagogy accordingly. Interestingly, though, many who champion the role of philosophy in mathematics education are anti-Platonists. For example, Paul Ernest, editor of the Philosophy of Mathematics Education Journal, says that teachers’ beliefs about mathematics are vital; but Platonism—only slightly better than “instrumentalist” rote-learning—is impeding progressive, student-centred education. He says teachers should forget timeless, abstract truths and instead facilitate the shared construction of knowledge (Ernest 1989). However, I must disagree. While Ernest and Hersh are right to emphasize the philosophy of mathematics in teaching, their anti-Platonist conclusion is unwarranted.

Indeed, despite his raising of important issues, Hersh’s answer to the question, “what is mathematics, really?”, is insufficient for the classroom. He wants mathematics to accord with history and pedagogy, but he hits the wrong target; authoritarian teaching is surely unfortunate, but the fault does not lie with Platonism itself. Hersh makes a caricature of Platonism so that his “humanism” can provide a seeming rescue. However, his approach is inadequate, because it substitutes naturalized history for rational justification of mathematics—an ideological overreaction that leaves us without a foundation. To the contrary, I will argue that a fallible brand of Platonism, with an enriched epistemology of proof and visualization, constitutes a philosophy of mathematics that teachers can unashamedly embrace.

1. History and Philosophy

When Hersh attacks Platonism for being unscientific, he means that it is not naturalistic. Hersh assumes naturalism: He holds that the natural world is all there is, which rules out the possibility of abstract entities. Naturalism, of course, is a popular view, but it tends to stumble over how to account for mathematics. Several naturalist philosophers have wrestled gamely with the subtleties required to do justice to mathematics, as detailed by Brown (2012). But Hersh declines to justify mathematics at all; he will only describe it, retreating to the stance of observer. Mathematics is then simply what mathematicians do in their mathematical lives: It is not about answers, it is about the questions mathematicians ask (1997, 18). He summarily dismisses the problem of justificationof mathematics (and many other problems) as unanswerable: “Some of today’s questions about cosmology, ethics, determinism, or cognition may be futile” (ibid., 20). To Kant’s question of how mathematics is possible, Hersh replies: “Why should your question have an answer?” Mathematics, he says, is possible, because it is happening. It is just an empirical fact (ibid., 21).

This is not adequate for a mathematics teacher. Students often ask: “Why do we have to learn this?” I try to train them to go further and ask: “How is it that we can know that?” Hersh can’t give us a good answer to either question. He abdicates justification in favour of mere explanation, which dodges the important questions of why a Platonist mindset has been so fruitful for mathematicians, and how we should do mathematics now. To motivate our mathematical knowledge and teaching we need norms, not “this is what mathematicians do”. Hersh gives us no norms and no justification for anything, including, ironically, his own ideology (see section 3).

Now, I share with Hersh a love of the history of mathematics; it helps make mathematics “pop” for students. But history, philosophy and mathematics are different disciplines. History is descriptive and explains what happened in mathematical practice; philosophy is normative and justifies our mathematics today. Of course we must account for history; but Hersh uses history as philosophy, thinking that all he must do to defeat Platonism is to point to history, which shows that mathematical knowledge is changeable (1997, 27). He approves (1997, 228) of Ernest’s social constructivism, a view partly inspired by the movement known as the sociology of scientific knowledge (SSK). A canonical example of SSK is Shapin and Schaffer’s Leviathan and the Air-Pump (1985), which describes the experimental programme of seventeenth century natural philosopher Robert Boyle. Although Boyle championed his experimental “matters of fact”, the authors show how those “facts” were actually quite constructed by Boyle’s social and literal “technologies” (i.e., persuasive techniques). Hence, there are no unchangeable “mere facts”; even the most basic knowledge is shaped socially. In fact, I agree, and appreciate the historical insights that SSK authors provide. However, we must not rest our philosophical case here; the conclusion doesn’t follow that there is no rational basis whatsoever for knowledge, no “oughts” for today. Data may well be relational, but it can still be representational of reality:These are not mutually exclusive. Something about Boyle’s air-pump constrained its results. So, just as the social construction of physical “facts” does not prevent them from possessing a core of mind-independent reality, the social nature of mathematical knowledge throughout history is compatible with the independent existence of mathematical objects that constrain what our knowledge can be. Taking refuge in history (even SSK) clinches nothing: History describes what the actors did and explains why they did it, but it can’t tell us about the status of their studied objects. Hersh’s reliance on historical explanations to carry philosophical weight causes him to swing the pendulum from authoritarian certainty to postmodern relativism, which is going too far.

The history and philosophy of mathematics can harmonize, but we must approach them differently. For example, in the eighteenth century, Euler made use of divergent series, which Cauchy subsequently banned in the name of rigour. But in the late nineteenth century, mathematicians founded a rigorous theory of divergent series. Thus G. H. Hardy (1949), speaking as a typical mathematician, observes that Euler should be vindicated as a “summability theorist” ahead of his time. Yet, historically speaking, I claim that this is incorrect: Euler’s definition of “sum” was a Euclidean-style description which attempted to capture the true nature of divergent series, while Hardy’s notion of “sum” is a Hilbert-style arbitrary stipulation—not at all the same concept. They weren’t doing the same thing, so Euler cannot be “vindicated” that way—if indeed he needs vindication.[1] While mathematical practices and beliefs do change, historians refrain from anachronistically evaluating one in terms of another. However, this does not mean that objects of Euler’s and Hardy’s study have no mind-independent existence. In fact, despite the incompatibility of their theories, from a philosophical standpoint they share some constraining aspects. Both seem limited by brute mathematical realities, such as the notion of addition. This kind of constraint does not apply to social institutions like “money” which involve an “explicit contractual agreement” on our part (Brown 2001, 142). Additionlies outside all scientific paradigms; it never changes and we can’t imagine it any other way. It is not that we lack the creativity; the best answer is that all mathematical theories—even if social conventions—are constrained at the level of reality, which according to mathematician Alain Connes is “sharply distinguished from the concepts of the human mind elaborated in order to understand it” (quoted in Hacking 2014, 201). Thus the methods of history and philosophy are in tension but not contradictory; we need both, but must do them “wearing different hats”. Hersh, though, is not clear about which “hat” he is wearing, so that his history is an outdated, “Whig” history: To convince us of mathematics’ changeability and uncertainty, he says that Euler’s use of divergent series was mostly mistaken but later theories “showedin what sense Euler was right” (1997, 44). He makes the same error as Hardy in thinking like a mathematician, not a historian—thus fumbling the very history he tries to use philosophically. In fact, the statements of Hardy and Hersh only make sense if one holds there is a “right” definition for sum.

But this is not surprising, as overwhelmingly, mathematicians cannot escape the feeling of discovering a reality out-there. Hardy, at least, was upfront: “317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way” (2012, 130).Leading mathematician Robert Langlands agrees that such ideas are “hard for a professional mathematician do without” (quoted in Hacking 2014, 256). Hersh, though, engages in a mental contortion; he rejects realism even as his statements about mathematics and its history seem to require it. Indeed, in his last chapter he Platonistically describes some beautiful theorems and proofs but forgets to inform us these require a big asterisk. This is telling: As Gardner (1997) points out, even Hersh seems to think that Platonism is the simplest way to talk about mathematics. Mathematical teaching needs this simplicity; mathematics is hard enough to teach without equivocation. Mary Leng puts it well: “Is it fair on mathematicians to interpret them as not really meaning what they appear to say?” (2007, 5). It is doubly unfair to teachers and the students whose mathematical experience we want to facilitate. We must properly account for the experience, not just describe it, as a historian. History doesn’t tell us what we should do in mathematics, and a naturalistic depiction of mathematics as a purely human creation will not suddenly motivate students to learn it more. As Brown points out (2012, 83), languages are human constructs that nonetheless can be quite daunting to learn. In my experience, many students find learning French just as much a hurdle as mathematics.[2] Nor is the answer to recast mathematics as a formal game of symbols, as in the “New Math” initiatives of the 1960s. Presenting mathematics as a “language” or “game” doesn’t give students a reason to learn it either. What we need, in fact, is to warrant our students’ experience of mathematical truth. With so much dividing the world today, we need a basis for agreement in the classroom that motivates and undergirds our knowing, not some vague notion of a shared mathematical life.

“Why should we learn this?” requires a good answer. But neither “You need to know it for the next course” nor “We are constructing our knowledge together” will do. Both are just technology—the rationalized achievement of some output. While Ernest (1989) states that “rote learning” is pure “instrumentalism”, I counter that forming a “problem solving community”, without having a proper epistemology or methodology to justify its output, is just as instrumentalist—putting the cart before the horse. Philosopher Jacques Ellul warned about education that is only technique,a mere production of useful citizens instead of an “unpredictable and exciting adventure in human enlightenment” (1964, 349). I love to solve puzzles, but if puzzle-solving is the ultimate end, I lose students if they can’t solve my puzzles. Instead, the “why” question needs an answer like: “Mathematics allows us to understand and explain what the universe is like and justify it to others”. That is enlightening. We need a meaning that transcends mere use.[3] The worldneeds students with critical capacities to seek what is true, not just solve problems.

2. Lakatos and Fallibilism

Hersh’s “humanism” cannot provide the norms we need. But, maybe Platonism can’t either. Perhaps Hardy and Connes suffered delusions and worked successfully despite their Platonism, not because of it. Hersh objects to Platonism’s abstract realm, its apparent misalignment with natural science, and its association with “certainty”, claiming that “the objections to Platonism are never answered” (1997, 12). I will answer him.

As a philosophically-minded teacher I’ve wanted to help my students appreciate mathematical proof and distinguish mathematics from science. I was impressed by the impassioned Platonism of figures such as Kurt Gödel[4] and strove to introduce such discussions in the classroom. But, questions remained: For example, are axioms “self-evident” statements or arbitrary stipulations, and how do they relate to “truth”? How should we understand the incompleteness theorems? For mathematician Gregory Chaitin, incompleteness means that there are an infinity of “irreducible” facts which can’t be proved and must be discovered. Hence, he says, mathematics should be more like physics: “Proofs are fine, but if you can’t find a proof, you should go ahead using heuristic arguments and conjectures” (Chaitin 2006, 199). His talk of physics seemed heretical—don’t we want certaintyand proof, not conjectures? But, indeed, the history of mathematics puts “certainty” into question. Upheavals (e.g., Russell’s paradox) abound. Must we dethrone mathematics from its special place? And how does it properly mesh with empirical science? Hersh is right to demand a coherent account of this.

One of his answers is to invoke Imre Lakatos’ Proofs and Refutations, a touchstone for many anti-Platonists. Lakatos attacks the idea that mathematics is certain. Dismissing formal mathematics as unrealistic, he shows us mathematics as it is frequently done. Mathematicians do not begin with fully-formed definitions and deduce consequences; they engage in gradual concept-formation. A proof seems certain until various counterexamples reveal our inadequate assumptions or definitions; then we improve the proof. It is fitting that Lakatos wrote in the form of a classroom dialogue. Our students’ mathematical experience would be enriched by explorations of this sort. Students should find and justify patterns, and critique the proposed justifications to form more sophisticated concepts. Also, it may be that the recapitulation of historical developments could help students understand some mathematical ideas better. Certainly, this is an interesting book for educators. But does it score points for Hersh?

Lakatos’ book is an argument against a static “certainty”. But is it really an argument against Platonism? How can both Hersh, a naturalist, and Brown (2008, 20), a Platonist, cite Lakatos with approval? The answer is that Hersh caricatures Platonism as entailing certainty, thus creating a straw man that he rails against. But fallibilism does not imply anti-realism. We may err, but that does not mean there is no truth of the matter. Perhaps a naive sort of Platonism would cling to certainty; indeed, a priori knowledge has often been thought to be “certain” because it is not open to empirical revision. However, a nuanced modern Platonism is quite compatible with Lakatos. This is what Hersh and his allies misunderstand.

What is Lakatos’ point? His targets are formalism and foundationalism (i.e., certainty), not realism. Lakatos shows that Euclidean-style formal proofs happen after the interesting mathematical activity—the heuristical fumbling of intuition and discovery (2015, 149). Most teachers would agree that “deductivist style hides the struggle, hides the adventure” (ibid., 151). Like Hersh, Lakatos laments authoritarianism in education, “the worst enemy of independent and critical thought” (ibid.,152 note 2). But what are the objects of our critical thought? I believe the interpretive key is found in Lakatos’ positioning of his work alongside empirical science. Lakatos decries tyranny in both(deductive) mathematics and (inductive) science, and states that his aim is to do for mathematics what Popper did for science (ibid., note 3). Popper had argued a parallel thesis—that inductivism is actually not the method of discovery in science; rather, that scientific discoveries occur by “conjectures and refutations”. Yet, crucially, Popper was a realist, holding that science has always been understood as the search for “truth”, whichis what gives it its great “liberalizing influence” (Popper 1963, 4). Therefore, since Lakatos explicitly desires to parallel Popper, the proper conclusion to draw from Lakatos is not anti-realism (hence anti-Platonism), but simply that mathematical activity is not always formal deductivism. Indeed, Lakatos writes as a realist when discussing Sediel’s “discovery” of uniform convergence (2015, 149). Proofs and Refutations is not really about ontology at all; Lakatos just says that our teaching and philosophy of mathematics should match actual practice, and I agree.[5]