IN DEFENCE OF PLATONISM IN THE MATHEMATICS CLASSROOM

Andrew Schroter

University of TorontoAndrewschroter @ gmail.com

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);

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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 PhilosophyWhen 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 justification of 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

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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

1 For Hardy’s use of modern definitions in summation of series, and views on Euler, see his Divergent Series (1949, 6-17). Contrast this with Euler’s own attempts to define “sum” to incorporate the use of divergent series, as discussed in Ferraro (1999). Their notions of “definition” and “sum of a series” are different, and historically speaking, we refrain from saying who is “correct” or closer to the truth: Interpreting historical actors as finding their way towards our triumphant present is an outmoded form of history known as “Whiggism”. However, a mathematics teacher is not a historian; the teacher can simultaneously appreciate history and hold an opinion on what we ought to do now.

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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). Addition lies 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 “showed in 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,

2 At least one credit in French is required to earn the Ontario secondary school diploma.

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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 world needs students with critical capacities to seek what is true, not just solve problems.

2. Lakatos and FallibilismHersh’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 Gödel4 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 certainty and 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 3 Unlike Wittgenstein, as discussed in Hacking (2014, 210).4 See Goldstein (2005).

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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”, which is 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

When Hersh demands that mathematics accord with science, he means natural science. This is why he relies so heavily on Lakatos; he thinks that once he establishes uncertainty, the only option is for the “diminished” mathematics to flee under the wing of natural science—to be “naturalized” and studied as just another human artifact. But his conclusion only follows if one is already a naturalist. His argument should only be compelling to the teacher who believes that mathematics is subordinate to (natural) science. But why would Popper’s science give us truth while Lakatos’ mathematics doesn’t? This is not the way we want our students to think about mathematics.

Instead, mathematics is reconciled to science by being independent yet parallel to it. Mathematics is a priori; science is empirical. Both are fallible, but fallible doesn’t mean fictional or artificial. Mathematics is surely free from empirical revision, but it is not quite “certain”; it is subject to a priori revision as our concepts become clearer. We can allow intuitions, like the regular senses, to be mistaken, without throwing away the realism and search for truth that makes mathematics worthwhile. Hardy said he thought of a mathematician as “an observer, a man who gazes at a distant range of mountains and notes down his observations” with the aim of discerning how to get from peak A to B (1929, 18). He might get the pathway wrong, but that doesn’t mean the peaks don’t exist or are cultural artifacts. Hardy describes precisely the sort of adventure we want our students to undertake.

The “side-by-side” conception also does better justice to how mathematics is used in “real life”, helping students connect experiences in their science and mathematics classes. As Pincock (2011, 1-5) remarks, we use mathematics to model the world. Some aspect of our model may be idealized, that is, “false” with respect to physical reality—such as when we approximate a discrete situation with a continuous variable. But the model nonetheless makes successful predictions. This supports the idea that mathematical entities have a kind of autonomy from the physical world. Yet they must have

5 Hanna (1997, 178) says that Lakatos over-dramatizes the idea that mathematical practice is all about proofs and refutations. Indeed, few historical examples display the rich back-and-forth of Euler’s polyhedron theorem. At any rate, to the degree that Lakatos has established fallibility, there is no conflict with Platonism.

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autonomy from the “social” or human world too. Hersh and Gardner have some history arguing this question: Gardner points out that surely two dinosaurs meeting two other dinosaurs would have made four dinosaurs long before humans were around to take note. Hersh replies that this fact about dinosaurs is empirical; the “two” and “four” are counting adjectives that give rise, in later socio-cultural contexts, to a shared concept of pure numbers in which 2 + 2 = 4 becomes a theorem (1997, 12). Gardner (1997) tries again by talking about Möbius strips’ properties being independent from the culture of topologists. Let me go further. Georges Lemaître used a mathematical model to posit a beginning of the universe (the “big bang”). Later calculations indicated that there should be background radiation left over from the explosion, if a real event. This radiation was finally discovered in 1965, by accident. Steven Weinberg explained why it took so long: “It is hard to realize that these numbers and equations we play with at our desks have something to do with the real world” (1977, 131-132). But of course, they do. The theorems of statistical mechanics and General Relativity are far more complex than 2 + 2 = 4, but they are quite relevant to events in the remote past, before we existed. Or, like dinosaur-numbering, was General Relativity also an empirical “adjective” before humans came along to construct the mathematical theory? If our mathematical models successfully represent the interactions of dinosaurs and primordial particles, perhaps there is more to them than a merely cultural intersubjective agreement.

I now address the question of intuition. Hersh protests the “impossibility of contact between the flesh and blood mathematician and the immaterial mathematical object” (1997, 12) making it “hard for a scientifically oriented person to defend” (ibid., 63). I admit that Platonic intuitions seem absurd to a naturalist. But Hersh’s own account of intuition, while dressed in neuroscience, doesn’t evade mystery either: “It’s the effect on the mind/brain of manipulating concrete objects” which “leaves a trace . . . in the mind/brain. That trace of manipulative experience is your representation of the natural numbers” (ibid., 65). The mind, or the brain? While most naturalists just see these as equivalent, such talk conveniently dodges the complexities of how belief-formation really relates to physical perception. In addition, Hersh makes a poor case for how his view is supposed to improve the classroom experience. He says the reason students “get a bad mark” is because their mental representations don’t match the teacher’s, whose “mental picture matches the one all the other teachers have” (1997, 65). This seems like it could lead to authoritarianism just as easily as his caricature of Platonism, because there is no justification (beyond majority-rule) for the teachers to have the mental picture they do. I argue that students should use their intuitions without equivocation, as guides to what may be true even as they critically test their consequences. Our intuitions are fallible, but they can grasp reality. We don’t have to throw cold water on that in our teaching.

To sum up: Yes, certainty is untenable. But if we give in to naturalism and relativism, we toss away the reason we do mathematics and confuse our students. A fallible Platonism lets us reconcile mathematics with science while preserving all that makes mathematics distinctive.

3. Ideology and TruthHersh repeatedly associates Platonism with things he doesn’t like, including elitism and conservatism (1997, 246) and any talk of God (ibid., 135). Although he admits he hasn’t studied the effect of Platonism on teaching, nonetheless he asserts that “elitism in education and Platonism in philosophy naturally fit together” (ibid., 238). Hersh classifies various philosophers as “right wing”/“left wing” and “mainstream”/“humanist”, and notes that the lefties are mostly on his side as humanists (ibid., 245). His explanation: Platonist mathematics is (allegedly) “unchanging” and political conservatism “opposes change” and is thus “oppressive”. Platonism is conservative, hence it is oppressive, so we

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should be humanists.

But Hersh contradicts himself here. He has already rejected norms for mathematical practice: Mathematics is just what mathematicians do, and the question of how it arises is one of neuroscience and psychology (ibid., 21). Accordingly, quandaries about ethics are likely “futile” (ibid., 20). Very well; if he could explain mathematics on a naturalistic basis, Platonism’s suitability for teaching would be undermined. But now he says: Conservatism and elitism are wrong, so we ought to adopt humanism as our philosophy of mathematics. He has switched gears, arguing against a caricature of Platonism based on the very ethics he has said aren’t up for debate. His choice of philosophy comes from ideological preference, leading to such unsupported statements as “Adoption by teachers of a humanist philosophy of mathematics could benefit mathematics education” (ibid., 238, emphasis mine).

Hersh’s survey of philosophers is unscientific and unconvincing. But the main issue is the “us vs. them” line he draws through the historical practice of mathematics. Should mathematicians be liberal or conservative? From the perspective of what is true, it shouldn’t matter. We don’t need ideology to motivate a mathematics classroom; we need a philosophy of mathematics that justifies what we say and do. Hersh says teachers are committed to a “shamefaced” Platonism (ibid., 42) because they fail to consider it critically. But many people believe in, say, “love” and “justice” without examining such concepts critically. Does it follow that they should be “ashamed” of these, too? No, we should add critical thinking to our beliefs to see if there’s a coherent, plausible warrant for that which is so compelling. Mathematical knowledge seems objective, a priori, and true. We sometimes get it wrong. Platonism does justice to these qualities, so it provides a strong basis for communal inquiry. A proper account of mathematical truth can bring us together, but ideology is divisive.

Indeed, elitism can take many forms; one person’s liberation is another’s oppression. Ndlovu (2013, 10) notes that social constructivist education has itself been criticized as a “secular religion” and “an elitist theory that has been most successful with children from privileged backgrounds”. Hersh’s proposals will excite those who share his ideology but alienate those who don’t. Yet if the search for objective truth drives our activities, we don’t need to fall back on divisive ideology. This is especially important in teaching. Academic mathematicians are adults who (presumably) choose their profession; it is conceivable that an ideological component might attach itself to their culture. But in schools, everyone has to study mathematics, some more willingly than others. The classroom must then be as inclusive as possible; we lay aside our differences and explore the universe with mathematics as best we can. Forms of Platonism have been embraced by theists (Gardner and myself) as well as atheists (Hardy and Brown). Exploration of “what is” transcends ideology; mathematical truths out there can be pursued “as a way of celebrating the universe” (Chaitin 2006, xii). We need to recover truth in a “post-truth” world.

Is it oppressive to hold that truth exists? Fallibilism is the key that defeats Hersh’s concerns. Fallibilism in the classroom is a microcosm of the history of mathematics. We can hold two things in tension: 1) A search for truth makes our investigations worthwhile and brings us together regardless of politics; and 2) fallibilism means that we can make mistakes and we have to work to justify assertions and arrive at understanding. (I have often been incorrect and have learned many things from my students.) Combined with a compassionate and inclusive pedagogy, this makes our classrooms challenging and safe, not oppressive. Let me illustrate. Hersh relates an anecdote he heard about Platonist teaching: “Teacher thinks she perceives other-worldly mathematics. Student is convinced teacher really does perceive other-worldly mathematics. No way does the student believe he’s about to perceive other-

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worldly mathematics” (ibid., 238). The real issue here is that mathematics is difficult and requires a caring teacher and a supportive community. Science was not my favourite subject, but it never “oppressed” me. The constraining physicality of ticker-tape and Popsicle-stick bridges was just difficult. I found it easier to manipulate mathematical objects in the laboratory of the mind. Many of my students do too, and many don’t. But for almost all, coming to terms with the constraining, compelling reality of mathematics helps them appreciate a subject they may have known previously as rote learning or unfocused busywork. In fact, for many, it is the very sense of discovering out-there truths, independent of us, that helps students love mathematics. At any rate—the validity of Platonism aside—it is by no means certain that replacing Platonism with humanism will increase students’ enjoyment of mathematics. Hersh hasn’t made this case.

We need a philosophy that warrants the investigations we undertake in the mathematics classroom. If Hersh were able to show that Platonism is wrong on philosophical grounds, we would of course have to abandon it and move on. But Hersh reveals only that he doesn’t understand Platonism and doesn’t like it.

4. Simplicity and ProofTeachers, then, need not shy away from embracing Platonism. But what should thoughtful Platonism look like in the classroom? I will sketch some implications.

A) Simplicity. The language of realism provides students and teachers with clarity. It distinguishes those who do mathematics from mathematical objects, clearing out the ambiguity and jargon in our lesson design. The effect of Hersh’s philosophy is to conflate mathematicians, their tools, and the entities they study, preventing us from getting a clear grip on what we are talking about. Why tell students one thing (“there are an infinity of primes”) but secretly disbelieve or relativize it? As Leng (2007, 5) says, “is it reasonable to suppose that, despite surface appearances, the truth of a claim such as ‘There are infinitely many prime numbers’ does not really require the existence of infinitely many objects?” Nothing is gained by applying such ideas to one’s regular teaching. Admittedly, these questions remain the subject of ongoing philosophical discussion, and thoughtful students may wish to delve into them. But until someone can decisively rout its account, Platonism provides invaluable straightforwardness in classroom discourse.

B) Proof for explaining and understanding. Hanna (1997, 179) notes that social constructivism and misconstrual of Lakatos have eroded the teaching of proof. Proof is nearly absent from the 1991 NCTM standards (ibid., 180) and is not connected to any course expectations in the Ontario curriculum documents (2007). Proof seems to have entered the crosshairs of those who blame Platonism for “authoritarian” teaching. But how can we deny our students the chance to see why something must be so, to find out just (as Hacking says) “by thinking” (2014, 84)? Exploration and problem-solving should coexist with proof, not replace it: “Exploration leads to discovery, while proof is confirmation” (Hanna 2000, 14). Proof is challenging, but I have seen its power to transform students’ conceptions of mathematics. Proof shows students the difference between mathematics and empirical science. As Hanna (2000, 19) discusses, students often approach mathematics with a muddled epistemology that doesn’t distinguish between inductive and deductive reasoning. We teach them the scientific method from a young age, yet fail to provide proper epistemological tools for mathematics. But Platonism allows us to reassert the independent and parallel status of mathematics toward the sciences. Admittedly, the role of rigorous proof is debated among mathematicians and philosophers; the rise of non-rigorous “experimental mathematics” has some wondering whether proof is “dead” (Hanna 1997,

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175). But at least some of this debate is about priority and prestige—that is, whether “theoretical” results are as “important” as rigorous proofs. (ibid., 177). Professional controversies notwithstanding, proof remains one of the key components of the mathematician’s repertoire; it is premature to discard it from mathematics teaching.

Furthermore, the worry about “authoritarianism” is misguided. Certainly, a teacher might offer an opaque proof whose power to convince is contingent on the teacher’s “authority”. But we can do much better. Many proofs “convince” (compel us to assent) but not all explain. Teachers should become collectors of explanatory proofs that “communicate ideas and generate understanding” (Hanna 1997, 176). The more transparent the proof, the better: These are the proofs that spark that unquantifiable mathematical experience, especially when students learn to construct them. “Proof conveys to students the message that they can reason for themselves, that they do not need to defer to authority. Thus the use of proof in the classroom is, if anything, actually anti-authoritarian” (Hanna 1997, 182, emphasis in original). Meanwhile, Ndlovu (2013, 9) notes that social constructivism has been criticized for tending to “epistemological relativism, where there exists no absolute truth . . . leading to ‘group think’ which tends to produce a ‘tyranny of the majority’”. Platonism gives us a strong epistemology based on cogent argument rather than tyranny.

I give some examples. Certainly, classic proofs (e.g., the infinity of primes) are essential. But we should also emphasize alternate proofs and uphold students’ creative approaches. Chaitin (2006, 98)

prefers the following proof of the irrationality of √2: If ab

= √2 then we must have a2 = 2b2. Now

factorize both a and b into products of primes. That is, write a=2p 3q 5r ... and b=2x 3y 5z .. . and square each factorization to get a2=22 p32 q 52 r .. . and b2=22 x 32 y 52 z .. .. But now the factorization of a2 contains an even power of 2, while the factorization of 2 b2 will contain an odd power of 2, so these numbers can’t be equal, giving a contradiction. This proof provides some fresh insight.

Here is another. Once, I demonstrated that 0.999... = 1 in a variety of ways, including letting 13=0.333 ... and multiplying both sides by 3. Afterward, a student showed the class the following:

Let α=0.999 .. .. Then 10 α=9.999 .. . 10 α=9 +α

9α=9 so α=1.

This assumes, of course, that you can shift an infinity of digits over when you multiply by 10, which opens another interesting discussion. The point is that the students did not just have to “take my word for it”. Empowering students to prove in their own way is the furthest thing from authoritarianism.

3) Visual epistemology. Platonism recovers a secure place for proof in the curriculum, but it does not entail dry formalism. If mathematical objects exist outside us and our task is to grasp them, then there are countless potentially valid ways of acquiring mathematical knowledge—including pictures, thought experiments, and intuitions. Of course, we must be careful to ensure we are not misled, but the ensuing discussion can be very rich. So-called “picture-proofs” are especially desirable in this regard, as they can impart great understanding, allowing students to see why a certain result must be true. An enriched

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epistemology opens itself to a variety of approaches and accommodates our students’ ways of thinking. It trains them to critically examine each justification for validity. Platonism hence offers a progressive approach to valid evidence in the mathematics classroom; indeed, Hacking (no Platonist himself) admits that it is “Platonism [that] provides the best story about how pictorial arguments can be so persuasive” (2014, 253). Here is an example of a picture-proof of the irrationality of √2(popularized by J. Conway, from cut-the-knot.org). (I have added some explanations to confirm the pictures’ meanings.)

We take two squares with integer sides, with one square’s area twice the other. (The pictorial equivalent of a2 = 2b2).

Then we place two copies of the smaller square in the opposite corners of the bigger.

Their intersection forms a square at the centre of the picture, while their union leaves two square corners uncovered. The intersection and the uncovered corners must have equal area.

But then these squares must have integer side lengths as well (as their sides arise from the subtraction of integer lengths). The fact that we can repeat this process indefinitely gives us a contradiction.

Thus we see that Platonism underwrites a clear and robust epistemology of proof, intuition and visualization. Social constructivism, on the other hand, aims to construct knowledge without securing the foundations.

CONCLUSION

Hersh is right that authoritarianism and rote learning can damage our students’ mathematical experiences. But these are symptoms of bad pedagogy and a lack of attention to the philosophy of mathematics; Platonism is not to blame. The modern form of Platonism embraces history and Lakatos and opens up a wide range of epistemological tools. Rather than engage with Platonism head-on,

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though, Hersh attacks a mere caricature, only to replace it with a view that undercuts the purpose and justification of mathematical activity. Certainly, psychology and the social construction of ideas play a role in all teaching. But it does not follow that we should abandon Platonism as our philosophy of mathematics for teaching. Instead, the need is for teachers to understand it better and let their classes be enlivened by what mathematics really is.

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