Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Pre·fac·er [pruh-fes-er] (noun)

A preface for professors. Or for professional mathematicians, or students with enough mathematical background to understand the rambling in this section, or curious students with no mathematical background, or high school teachers, or anyone who finds themselves thinking about mathematics often. . . or not.

This is not your standard “introductory” mathematics text. It is simultaneously more introductory and more advanced than most individual books you are likely to encounter, and as such, it is something of an experiment.

What Kind of Experiment?

This book will be extremely easy to misunderstand if one comes to it expecting a mathematics textbook, although it shares many features in common with mathematics textbooks, and could indeed be used as one. To understand the goal and structure of this book, I must first coin a term that our lexicon is presently lacking: pre-mathematics. By “pre-mathematics” I do not mean those tiresome non-subjects such as “pre-algebra” and “pre-calculus” that we inflict on unsuspecting students. Rather, I will use the term to refer to the entire set of ideas, confusions, questions, and motivations that occupy the minds of the inventors of mathematical concepts, and which drive them to define and examine one species of mathematical object rather than another.

For instance, the definition of the derivative and the various theorems that follow from it are part of mathematics proper, and they can be found in any mathematics textbook that covers calculus. The reasons why the concept is defined the way it is, rather than any of the infinitely many other ways it could have been defined, as well as the processes of reasoning that would lead one to choose the standard definition over all other candidate definitions (in the absence of a preexisting mathematics textbook), are much less frequently given sufficient attention. It is this set of possibilities and processes of reasoning to which the term pre-mathematics refers. Pre-mathematics includes not only all of the possible alternative definitions of mathematical concepts that would lead to essentially identical formal theories, but also — perhaps more importantly — all of the blind alleys down which one would be led in attempting to invent the standard mathematical definitions and theorems from scratch. It is the conceptual heavy lifting that must be done to pull a mathematical concept into existence out of nothingness. Mathematics is the sausage; pre-mathematics is how the sausage is made.

That is the main topic of the book: the rarely discussed process of moving from the vague and qualitative to the precise and quantitative, or equivalently, how to invent mathematics for yourself. By “invent,” I mean not only the creation of new mathematical concepts, but also the more relevant process of learning how to reinvent bits of mathematics that were originally invented by someone else, in order to gain a deeper and more visceral understanding of those concepts than could be gained simply by reading a standard textbook. This is a process that we virtually never teach explicitly, yet one would be hard pressed to think of a more valuable skill that one could learn from any mathematics course. Learning how to (re)invent mathematics for yourself is of critical and fundamental importance in both the pure and applied domains. It includes pure questions like “How did mathematicians figure out how to define curvature in a way that lets them talk about it in seventeen dimensions, where we can’t picture anything?” and applied questions like “Given what I know, how should I build a model of the phenomenon I’m studying?” Such questions are often addressed in textbooks, briefly, as an afterthought, but it is dramatically less common to place these questions in the spotlight, on equal if not higher footing than the theorems and results themselves.

An honest description of the informal, messy creation process is the missing piece of the puzzle in our exposition of mathematics at all levels, from elementary school to the postdoctoral level, and its absence is one of the primary reasons that our subject so often bores even the most attentive students. The elegance and beauty of our subject cannot be fully appreciated without a visceral understanding of the pre-formal conceptual dance by which mathematical concepts are created. This process is not nearly as difficult to explain as it may seem, but doing so requires a radical shift in the way we teach our subject. It requires that we include in our textbooks and lectures at least some of the false starts, mistakes, and dead ends that a normal human mind would first need to experience before arriving at the modern definitions. It requires that we write our textbooks as narratives in which the characters often get stuck and don’t know what to do next. This book is a quirky, flawed, deeply personal attempt to outline what I believe are some of the core concepts and explanatory strategies of pre-mathematics: strategies that professional mathematicians use every day but rarely discuss openly in their textbooks and courses.

This highlights an important point. While a proper emphasis on pre-mathematics requires a radical change in how we teach mathematics, it does not require a change in how professional mathematicians think about mathematics. Pre-mathematics is their bread and butter. It is the language in which they think, since they are by definition the ones who create — or if you prefer, discover — the subject. In that respect, the content of this book is not novel. It is only novel insofar as it places under the spotlight all the content that is usually hidden behind the wall of formal proofs and pre-polished derivations (in the “unfriendly” textbooks) or behind cartoons and largely unexplained statements of fact (in the “friendly” ones). But at no point along the continuum between the friendly introductory books on the one hand and the awe-inspiring Grothendieck-style monographs on the other do we accurately represent the creation process in a pedagogically useful way.

It is impossible for any book to explore all of the pre-mathematics of a given concept before proceeding to its mathematics, and I do not attempt this impossible task. Rather, I attempt to construct a pre-mathematical narrative that leads from one concept to another, starting from addition and multiplication, proceeding immediately on to single-variable calculus, then backward through the (more advanced!) topics that we commonly think of as its prerequisites, and finally on to calculus in spaces of finitely many or infinitely many dimensions. A large amount of mathematics is found in this narrative, which is why it would not be a mistake to use this book as a mathematics textbook. However, once the pre-mathematics of any given concept has been developed at length, the mathematics itself often turns out to be startlingly straightforward, so we prefer to focus primarily on the former. That is not to say that the book contains an exhaustive and complete discussion of each of the topics it covers. Far from it! Rather, it is my estimate of everything that is missing, in terms of information, motivation, and where we should really begin teaching the subject. The book is a dirty proof of concept, not a polished diamond. I hope that it will start a conversation, but it is by no means the final word on any topic.

Further, it is important to be clear about what I am not criticizing. The problem of pedagogical foundations is fundamentally different from the problem of logical foundations, though they are implicitly conflated in most textbooks. I do not intend to criticize the logical starting point of the field, by which I mean choosing our logic to be first-order predicate calculus and our theory to be ZFC, NBG, or your favorite axiomatization of set theory.¹ What I want to criticize is the pedagogical starting point of the field, which is all that the vast majority of members of our society ever come into contact with.

Though for a brilliant critique of the standard set-theoretic approach to logical foundations, see Robert Goldblatt’s spectacular book Topoi: The Categorial Analysis of Logic.

Why Has Pre-mathematics Been Neglected?

Given the ubiquity of pre-mathematical reasoning inside the minds of professional mathematicians, it is worth asking why it so rarely appears in textbooks and journal articles. There are surely multiple reasons, but I believe that the primary culprit is professionalism. Though pre-mathematics is fundamentally important to understanding our field, it is structurally banned from any discussion that demands professionalism, including (but by no means limited to) all mathematical work in academic journals. Why? Because precise pre-mathematics is not formal. It is (by definition) the set of hunches, guesses, and intuitions that lead to the development of a formal mathematical theory in the first place, and the only precise, honest way to explain imprecise thought processes is with informal arguments expressed in informal language: language that accurately conveys to the reader that we are not 100% sure our intuitions are on the right track, and that we are always (to some extent) exploring in the dark. Such informal language is not simply dumbing things down. It is a precise manner of describing the chains of reasoning by which new mathematical concepts are created. And without a firm understanding of how mathematics is created, one’s understanding of the subject will be crippled in comparison to what it could have been otherwise.

To be clear, this is also not a criticism of formal expositions of mathematics or of the concept of the formal proof. But formal proofs do not spring into existence fully formed, nor (more importantly) do the formal definitions of the mathematical concepts on which they are based. An overly formal description of informal thought processes misleads the reader by providing evidence of nonexistent principles, and in doing so tricks the reader into believing that their failure to realize how A follows from B must be a deficit in their own knowledge, when in fact it is often a lack of perfect precision in the underlying pre-mathematical reasoning itself. Full disclosure requires that we offer informal descriptions of that which is informal. Professionalism has its place, but fundamentally its function is to censor honesty, and it has redacted pre-mathematics almost entirely out of existence.

What I Wanted the Book to Be, from the Start

This book grew out of an attempt to explain a subset of the universe of mathematics in as honest and unpretentious a way as possible, making sure at each stage to give away the secrets of our trade. At every step I attempt to separate necessary deductions from historically contingent conventions; I emphasize that the often intimidating words “equation” and “formula” are just code for “sentence”; I try to make it clear that all of the symbols in mathematics are just abbreviations for things that we could be saying verbally; I try to engage the reader in the process of inventing good abbreviations; I always attempt to make clear the distinction between what other textbooks actually do and how they could have done it; I attempt to present each derivation not in the standard post hoc, cleaned up form that reflects nothing about the thought process that led to it, but in a way that makes clear at least some of the blind alleys that most of us would be tempted to wander down before finally arriving at the answer; I try to explain everything as deeply as I can without sacrificing the coherence of the narrative; and I vowed that I would burn the book before I ever allowed myself to say “memorize this,” even once. There are many things I wish I had done differently, but at very least, the book is full of all the things mentioned above.

I also try to explain the strange dance our field does on the border between structured necessity and unrestrained anarchy. This is something that we virtually never explain to students, so I emphasize it whenever possible. Here’s what I mean. On the one hand, there’s the anarchy. We are free to use whatever axioms we please, even an inconsistent set. Defining and playing with an inconsistent formal system is not illegal, it’s boring. For instance, “dividing by zero” is not illegal, and every mathematics professor knows that. We are perfectly free to define a symbol by the property for all a, and many analysis books do just that, in a section on what is usually called “the extended real number system.”² But if you insist on defining the above symbol, then the algebraic structure you’re examining cannot be a field. You want to insist on saying it’s still a field? That’s perfectly fine, but then you can only be talking about a “field with one element.” You want to insist that there’s still more than one element, or that fields, by your definition, have at least two elements? That’s fine too, but then you’re working in an inconsistent formal system. You want to do that? Fine. But now any sentence is provable, so there’s not much to do.

Though they usually write ∞ instead of for obvious reasons. I’m using to remind us that the following argument is not a problem with “infinity,” it’s a problem with the boredom that starts to corrupt our mathematical universe when we assume that the additive identity has a multiplicative inverse.

It is important to emphasize that even when we hit rock bottom like this, we still have not done anything illegal. Rather, we’ve made the discussion boring. Every mathematician knows that, at least in the choice of what to study, there are no laws in mathematics. There are only more or less elegant and interesting mathematical structures. Who gets to decide what counts as elegant and interesting? Us. QED.

On the other hand, there’s the structured part of mathematics. Once we finish with the “anything goes” stage in which we say exactly what our assumptions are and what we’re talking about, then we find that we have conjured up a world of truth that is independent of us, about which we may know very little, and which it is our job to explore.

Needless to say, when we fail to inform students of this most fundamental point about anarchy and structure, we completely mislead them about the nature of mathematics. For whatever reason, we almost never tell them about this odd interplay between anarchic creation and structured deduction. I’m convinced that this is one of the things that make so many students feel as if mathematics is a kind of totalitarian wasteland full of undefined laws that no one tells you about, and in which you always have to be afraid of accidentally doing something wrong. That’s certainly how I always felt in high school, before the story I told in the first preface. This is one of the things I try to remedy in this book.

The Book Decides It Wants to Be About Something Else, Too

As much as I wanted to explain general things like the big-picture structure of mathematics, I eventually wanted to get around to explaining the ideas that are taught in the standard textbooks, so that the book might actually be helpful to students on Earth. To do this, I needed to build a narrative that somehow had to arrive at many of the standard textbook definitions before I could explain the mathematical arguments that spring from them. However, because of the goal of the book, I promised myself that I would not introduce these definitions in the standard way, which is usually to say “such and such is defined this way,” often out of the blue, or at best with a few pages of motivation, either conceptual or historical, followed by a huge conceptual leap into the mathematical definition itself. In swearing-off this practice, I found that I had placed myself under a rather large set of constraints. The problem can be summarized as follows:

Assume you don’t know anything about mathematics except the basics of addition and multiplication. Not necessarily the algorithms for performing them, but you know what phrases like “twice as big” mean, and you get the gist of both operations. You’re living in a world before textbooks. How could you discover even the simplest parts of mathematics? As a specific case, how would you figure out that the area of a rectangle is “length times width”?

It would be a non sequitur to answer this question by talking about how area is defined in measure theory, or by talking about axioms, or Euclid’s fifth postulate, or how the formula A = ℓw doesn’t hold in non-Euclidean geometry. It is not a question about rigor, and it is not a question about history. It is a question about creating something. The question is about how to move from a vague, qualitative, everyday concept to a precise, quantitative, mathematical one, when there’s no one around to help you or do it for you.

I was originally asked the above question by one of my closest friends, Erin Horowitz. Around the time I started writing this book, we would occasionally have many-hours-long sessions in which we would talk about mathematics. She doesn’t have a mathematical background, but she’s extremely curious, and always interested in knowing the “whys” of things. We would talk about formal languages, Taylor series, the idea of a function space, or any other crazy things we felt like talking about. One day she asked me the above question about how mathematical ideas are created. It wasn’t a hard question once it was phrased that way, using the area of a rectangle as a test case, and I basically just gave her the simplest argument I could think of, which is the argument about area that you’ll find in Chapter 1, in the section called “How to Invent a Mathematical Concept.” After I yammered for a bit, she asked why we’re never taught things like this in school. She completely understood the short argument, and so could anyone. Here’s the weird part: the argument involves solving a functional equation.

There are very few courses in mathematics departments that focus just on functional equations. I’m not certain that there should be more, but it is a rather confusing fact once one realizes it. After all, every mathematics undergraduate encounters plenty of differential equations, as they should, and they inevitably encounter integral equations as well, but the area of mathematics devoted to studying and solving general expressions involving unknown functions has been largely neglected by history. Despite the fact that it is one of the oldest topics in mathematics, we tend not to hear about it very often. In his monumental work Lectures on Functional Equations and Their Applications, J. Aczél laments that “through the years there has been no systematic presentation of this field, in spite of its age and its importance in application.”

Surprisingly, I started to discover that functional equations are enormously helpful in explaining even the simplest of mathematical concepts, as long as the ideas are presented in the right way.³ Here’s how. You don’t use the term “functional equation,” and in fact, if at all possible, don’t even use the word “function.” Most people have had bad enough experiences in math classes that it’s easy to scare them and shut off their natural creativity by using a lot of orthodox mathematical terminology. Instead, you say something like this:

Not functional equations in the full generality of Aczél’s monograph, but in a somewhat informal guise analogous to the calculus we teach before we teach analysis.

We’ve got a vague, everyday concept that we want to make into a precise mathematical concept. There’s no wrong way to do this, because we’re the ones who get to decide how successfully we performed the translation. However, we want to cram as much of our everyday concept into our mathematical concept as we possibly can. We start by saying a few sentences about our everyday concept. Then we come up with abbreviations for those sentences.⁴ Then we mentally eliminate all the possibilities that don’t do everything we asked them to. We can rinse and repeat if we want to, putting more and more vague, everyday information into abbreviated form, and then mentally throwing away everything that doesn’t behave like that. Occasionally, just by writing down examples, we can slowly become convinced that the precise definition we’re looking for has to look a certain way. We may not end up with just one possibility, and even if we do, we may not know when we’ve found the only one, but that doesn’t matter. If there’s more than one candidate definition that does everything we want it to, we can just do what mathematicians do all the time without telling you, and pick the one we think is prettiest. What counts as “prettiest”? That’s up to us.

At this point they’re writing down a functional equation without knowing it.

In short, as crazy as it might sound, I believe that informal mathematical arguments involving functional equations not only provide a way to better explain where our definitions come from at all levels of mathematics, but also that such arguments offer a kind of anti-authoritarian pedagogical style that engages the reader in the process of creating mathematical concepts in a way that is unheard of in introductory courses and textbooks. Surprisingly often (though certainly not always) the rarely discussed pre-mathematical practice of passing from a vague qualitative concept to a quantitative mathematical one turns out to involve the use of functional equations. In Chapter 1, we use this idea to “invent” the concepts of area and slope, arriving at the standard definitions not by simply postulating them, but by deriving them from qualitative correspondence with our everyday concepts. This is a simple illustration of what pre-mathematical pedagogy might look like, but it is only an example, and there is certainly room for improvement. In the meat of the book, we proceed to “invent” a large amount of mathematics this way, sometimes by an informal use of functional equations, sometimes not, but always making clear what we’re trying to do and how else it could be done.

How This Might Help

To see how an emphasis on pre-mathematics differs from the standard approach, let’s look at a specific example of how current teaching practice backs itself into a corner. Consider the problem faced by a teacher or an introductory textbook in attempting to explain where the definition of slope comes from. On the one hand, you want to motivate the idea. On the other hand, you eventually want to arrive at the conventional definition, , or as they say in introductory textbooks, “rise over run.” All of differential calculus rests on this formula plus the idea of a limit, so there could hardly be a more important concept to convey to students. Teachers and introductory textbook writers face the following problem. They might be able to think of some set of postulates that would single out “rise over run” as the unique definition satisfying all the postulates, but the proof of this would surely be too complicated for an introductory class, and it would probably just confuse everyone ten times more, so they just introduce “rise over run” as the definition of slope, possibly with a bit of motivation beforehand. Given the situation, this seems like a completely reasonable thing to do.

However, I believe that in this case and others like it, we’re confusing many more students than we realize, and turning them off of mathematics. When I first heard the definition of slope in high school, it did nothing but speed the process of demotivation for me. Introducing the concept like this (i) leaves open infinitely many questions, (ii) makes any reflective student feel as if they are missing something, and (iii) implicitly suggests that it is their own fault for not understanding it. The students are indeed missing something, but it is not their fault; they are missing something because it is being deliberately hidden from them, and it is being hidden by the best intentions of their teachers. In my own experience, I felt something like “I couldn’t invent any of these definitions on my own, from first principles, so there’s something I don’t understand about all of them.” I certainly didn’t put the feeling into those words at the time. All I thought explicitly was “I don’t understand this stuff.”

Years later, when I found myself explaining mathematics to others, I would always try to make the point that we could define slope to be “3 times rise over run,” or “rise over run to the fifth power,” or even “run over rise,” and we could go on to develop calculus using any of these definitions. All of our formulas would look slightly different (or possibly very different, depending on which definition we chose), and we might have to state some familiar theorems in a slightly different or even unrecognizable form, but the essential content of the theory would be identical, however ugly and unfamiliar-looking it might prove to be. An analogous story holds for any mathematical concept. I’ve yet to explain this to anyone without being asked why this isn’t explained in courses and textbooks. I don’t know. It should be.

Burn Math Class: A Mathematical Creation Story

What am I supposed to publish? L. J. Savage (1962) asked this question to express his bemusement at the fact that, no matter what topic he chose to discuss and no matter what style of writing he chose to adopt, he was sure to be criticized for not making a different choice. In this he was not alone. We would like to plead for a little more tolerance of our individual differences.

—E. T. Jaynes, Probability Theory: The Logic of Science

Writing a book is an emotional experience. In the course of preparing this book for publication, I was lucky to have two wonderful editors, T. J. Kelleher and Quynh Do, who were both extremely helpful throughout. I primarily dealt with Quynh for most of the publication process. She showed unending patience in helping me improve what I can only imagine was a very difficult book to edit, and though we did not always agree, her comments made the book tremendously better than it would have been otherwise. After mentioning one’s editors, it is customary to say “any remaining flaws in the book are my own,” but the customary phrase is far too mild.

Even in its final form, the book will inevitably contain numerous instances of the following sins: typos, hyperbole, poorly worded sentences, repeating myself, contradicting myself, sounding too arrogant, sounding too insecure, saying “I’ll never do X!” and then promptly doing X, saying “I’ll never do X!” and then later doing X (but not promptly), Easter eggs no one will find or understand, unintentionally alienating or offending innocent readers, experimenting with the medium in ways that some will find distracting, too many prefaces, too many digressions, too many dialogues, too few dialogues, too much meta-commentary, the use of arcane Greek and Latin words despite having made fun of them (and the people who use them) for being more pretentious than is necessary, and at least one unforgivably large error, most likely resulting from an accidental copy/paste of a random paragraph into a completely different part of the book. . . ad infinitum.

This is my first book. It was built on a scaffolding of my flaws. Writing a book is something I never thought I would do, and I was taken completely by surprise when it started happening. I wrote this book over a four-month period in the summer of 2012, in a euphoric flurry of coffee, eyestrain, sixteen-hour days, forgetful meal-skipping, and loving every minute. Writing had never been so fun. I was 25 at the time. Since then, I feel like I’ve become a different person. Parts of the book now hurt to read. When a book is created in the manner described above, it will inevitably be shot through with certain defects that no amount of editing or polish can hide.

Most of these flaws are accidental, but others are present by design. When an error is simply a misstep, there is no harm in hiding it. When we fix a typo in sentence N, sentence N + 1 is not harmed as a result. The same principle holds for sloppy wording or unnecessary repetition, and (although many such missteps surely remain in the book) this is the type of error that one should attempt to fix.

However, in some cases, an error is not a misstep but a stairstep. It is something without which we never could have arrived where we are. Removing the N^th stair from a staircase harms the steps after it, whether that staircase is a narrative or a mathematical argument. Certain rare ideas require flaws in order to be properly conveyed. My goal is to let the reader in on the secrets of the creation process, both of mathematics and of books themselves, and the process of creation cannot be accurately represented in a spotless manner. If there is a single unifying theme that ties together all the quirks of this book, it is full disclosure. Full disclosure in the sense of complete openness and honesty, not only about the process of mathematical creation, but also about the process of writing a book, as well as the emotional experience of returning to something one has written after a long absence and realizing that some of its flaws run too deep to ever be excised. The thought of a person taking time out of their lives to read this book makes me so happy that I have no desire to hide anything from them. I want to show them everything. All of it. Inevitably, this results in a rather unusual book.

I hope to convince you in what follows that the reason why so many members of our society never come to love or understand mathematics is that we have been communicating the subject entirely wrong. That does not mean I know how to do it right! This book may turn out to be a colossal failure, but if I’m certain of anything, it is that mathematics deserves better than the methods we currently use to teach it, at every level. This book is my personal attempt to right a few of these wrongs by writing the book I always wanted to read. Ready to have some fun? Me too. Let’s begin.