The Language of Mathematics Aiden Sagerman

THE LANGUAGE OF MATHEMATICS

LANGUAGE OF MATHEMATICS

Columbia’s Comparative Literature and to study literature, culture, and society with reference to material from several national traditions, or in combination of literary study with comparative study in other disciplines in the humanities and social sciences.” My interest is studying mathematics from a humanities and social sciences perspective—think history of mathematics, philosophy of mathematics, or sociology of mathematics. The CLS major seemed like a space that would support me doing the work I wanted to do. However, the major has strict application requirements, asking students to complete four semesters in each of two non-English languages by the end of sophomore year. I decided to ask for something different: the freedom to completely reframe my course of study by treating mathematics as one of these languages. This essay is an edited version of the letter I sent to the department to make my case.

I. Introduction

In order to study mathematics from the perspective of the humanities, you guage of mathematics. Mathematics, like any real language, has a grammar and ciently familiar with both that you may actually participate in mathematical discourse. Studying mathematics from the perspectives of literature, philosophy, and history therefore requires familiarity with mathematics in the same way that participating in a French literature class requires knowledge of French. And just as a French text must be translated to be understood by an English-speaking audience, mathematical thought must undergo a translation process in order to be understood by a broader academic audience. Since this is my area of interest, I believe mathematics should count as a language for the purposes of the Comparative Literature and Society application requirements.

II. Grammar

Before I discuss the grammar of mathematics, I would like to address an important distinction I will use throughout this essay. In mathematics, there is a distinction between mathematics itself and something called metamathematics. Mathematics proper studies mathematical objects—numbers, shapes, and their more abstract generalizations. Metamathematics, on the other hand, studies the process of doing mathematics: it looks at topics like logic, sets, and proof writing. This is equivalent to the difference between someone who studies the literature of a language and someone who studies the linguistics of that language. A literary theorist has a sociological understanding of the grammar of their language, and uses that understanding to produce writing. A linguist, however, works with a bigger and more precise picture of grammar. In the same way, a mathematician has a sociological understanding of mathematical grammar which they use to do math. A metamathematician, however, works with a bigger and more precise picture

of mathematical grammar. To see this in practice, we need to get a better understanding of what mathematical grammar really is.

There are two components to mathematical grammar: proofs and elementary mathematical objects. Proofs are the tool of establishing truth in mathematics. Just as a scientist might design an experiment to show that their hypothesis is true, a mathematician will construct a proof of a statement to show its truth. They are also the essential mode of mathematical discourse, in the same way that a conversation or literary text is a mode of linguistic discourse. Let’s look at an example. A mathematical statement which is true—that is, which has a proof—is called a theorem.

1 Theorems usually say something about basic verb to explain conjugation.

(Even Numbers). We say a natural number a is even if there exists a natural number b such that a = 2 · b.

In other words, an even number is twice some other whole number. Thus 4 is even, but 5 isn’t, as 4 = 2 · 2 and 5 does not equal 2 times any whole number. Now that we have an object, we can come up with a theorem about it.

1 I mean truth here in a sociological, rather than metamemathical sense; in more technical contexts, truth and provability are not necessarily equivalent. . Let a be an even number. Then a is even. As a reminder, a2 = a · a.

I will call this "Theorem 1," as is conventional. This theorem certainly looks true—we know 2 is even, so is 22 = 4; we know 4 is even, and so is 42 = 16; and we know 6 is even, and so is 62 = 36. For this theorem to be considered mathematical fact, though, we would want to prove it. Here's what a proof would look like. Proof. Suppose a is an even number. We want to show that a2 = 2c for some number c number, a = 2b for some number b. This means a2 = (2b)2, so

a2 = (2b)2 = 4b2 = 2(2b)2

So then if we just say that the number c is 2b2, then a2 = 2c.

Put simply, this is saying that if a is twice b, then a · a is four times b · b, which is b · b. This is, in essence, what a proof is: a process by which we take a mathematical theorem and show that from its assumptions, you actually know its conclusion is true.

What actually defines a proof is a little more complicated. In a metamathematical sense, a proof is a list of statements such that each statement is either an axiom of your logical system or follows (using some sort of convoluted set of rules for deduction) from the prior statements. A formalized version of the proof I did above, for instance, would

be a very long list of symbols like {,}, ∅, ∪, ⇒ which happen to follow a specific ruleset. In practice, though, that's not really what mathematicians use. While formal proofs do technically say why something is true, they are very hard for humans to read. As my current math professor Evan Warner likes to say, "proofs are sociological constructs"— something counts as a proof because mathematicians have decided it counts as a proof. A major part of learning mathematics is becoming comfortable with the rules of these sociological constructs.2

To see an example of this, consider the proof of Theorem 1. I will look at three different examples of ways we might prove this, and explain how mathema example is the proof I gave above. This —it follows standard mathematical conventions in both its language and structure.3 For our second example, let's consider the simpler explanation I gave

2 When I showed this to Professor Warner, he pointed out that there are also some inter- esting edge cases involving proofs which are sociologically still proofs, but provide almost no insight into why what they prove is actually true. These often take the form of, as he put it, “a sequence of random-looking algebraic manipulations” that just happen to give you the right answer. I think these are sort of comparable to sentences in a language that are technically grammatical, but are so hard to parse that you have to go through them repeatedly to figure out their meaning. 3 The one technical note her is that the word "number" would be substititued for the more after the proof. This would be a bit more borderline—while technically true, it is much less conventional in its structure and language, and would therefore not be appropriate in more rigorous contexts. For our third example, let’s say I wrote down a very long list of even numbers and their squares: 2 and 4, 4 and 16, 6 and 36, etc. all the way up to 3,456,278 and 11,945,857,613,284. This would certainly be quite convincing—if Theorem 1 doesn’t fail in almost 3.5 million cases, then how could it not be true? This would not, however, count as a proof in a mathematical context—a mathematician might object, for instance, that we don’t know if the theorem is true for 4, 000, 000 from the list alone.

The distinctions between these example proofs are equivalent to the rules of a grammar. Example one is like a well constructed sentence in a language—every speaker of the language will understand what you’re saying. Example two is like a poorly constructed sentence—while it gets the point across, it would not be appropriate for some forms of communication. And example three is a sentence that is so poorly constructed it actually does not make sense—while the speaker believes they are communicating their idea, they are actually failing to do so. Learning the grammar of mathematics means learning both how to construct a proof that makes sense on the most basic level, and how to construct a proof that is appropriate for engaging in mathematical discourse.

The examples I have provided are over whether something is or isn’t a valid proof. Even once one understands what counts as a proof, understanding what type of proof is applicable to a particular circumstance remains complicated. There is also a linguistic component: as we saw with examples one and two, the same idea expressed in slightly different language can appear to have differing levels of rigor. Proof writing is therefore a decidedly nontrivial skill, and is a requirement for mathematical education. ments the Columbia mathematics department mentions for participating in higher level classes is familiarity with proofs, which can come from Honors Math, Linear Algebra, Combinatorics, Number Theory, or a variety of other classes. While this requirement demonstrates the foundational nature of proofs for mathematical participation, it also highlights another key aspect of proofs: they don’t require any particular content. Linear Algebra, Combinatorics, and Number Theory have almost no content in common—to be reductive, they respectively deal with the idea of lines in many dimensions, complicated counting problems, and the properties they all somehow serve the same function: teaching basic proof writing. The same is true of Honors Math—Professor Warner has told us that building proof writing skills is just as much of a goal of the class as learning the content. This is what makes proof writing grammatical: it is a necessary structure to mathematical thought, but lacks any actual mathematical objects, in much the same way that grammar structures language but does not have inherent meaning without words.

The other key parts of mathematical grammar are sets and functions. Just as some metamathematicians study proofs, others—called set theorists—study sets and functions.4 Like proofs, sets and functions have a more sociological version—known as “naive set theory”—which is foundational in the same way that proof writing is. Naive set theory is sociologically determined in that it’s exactly as formal as mathematicians agree is necessary for doing math. In many ways, it has no resemblance to a formalized version of set theory. Since both sets and functions are fairly simple

(Sets). A set is a collection of objects.

(Functions). A function is a rule that takes objects from one set, called its domain, and matches them with objects in another set, called its codomain. Each object in the domain must match with exactly one object from the codomain.5

4 Technically speaking, set theory isn’t necessarily metamathematics, in that sets really are just mathematical objects like any other. However, as we will see, most of the objects in mathematics can be treated as sets. This means that set theory very often becomes metamathematical, as results about sets are often significant for the process of doing mathematics as a whole.

We generally write sets by putting brackets around the list of elements. For example, {1, 3, 5} is a set. So are {2}, {lampshade, squirrel}, and the set of all fractions. Functions sound complicated, but are much simpler when you look at some examples. For instance, the rule that takes a number x in and outputs x2 is a function. It takes in an object in its domain—the set of numbers—and spits out a number in its codomain. Generally, we write out a function named f by saying f(x) = and then the output for a given input x. For instance, in the case of the x2 function, we would write f(x) = x2 to show that an input x is matched with its square. Functions can also be visualized in a variety of ways, such as as charts and as graphs.

Almost every important object in math is a set or a function. Mathematics can (very roughly) be divided into three

5 Mathematicians do sometimes work with a more formal notion of functions than this definition, but it’s relatively rare. branches: algebra, geometry, and analysis. Intuitively, algebra is the study of number-system-like objects; geometry is the study of shapes and their generalizations; and analysis is the study of continuous processes.6 In mathematical practice, though, algebra is the study of sets that have particular types of functions associated with them; geometry is the study of sets of points in space, and generalizations thereof based on sets of points that are considered “close”; and analysis is the study of certain operations you can perform on functions. Sets and functions are very literally everywhere in mathematics. However, like proofs, they lack meaning on their own. While questions like “What actually counts as a set?” and “Do particular types of functions exist?” are meta-

6 Analysis is probably the trickiest to describe, as a lot of the words you’d use to characterize it have slightly different mathematical and colloquial meanings. By “continuous processes,” I mean processes where things get close together in a smooth, unbroken way. Other (slightly incorrect) characterizations include “calculus where you prove everything” and ‘the study of what happens when numbers get very close together.”

mathematically meaningful, they rarely have meaning in mathematics proper objects). Like proofs, there’s a basic intuition behind how sets and functions behave which is introduced in courses like Honors Math and is necessary for the study of mathematics. This why they are grammatical, rather than lexical.7

In summary, the grammar of math consists of foundational ideas which underlie all of mathematical thought, but which lack the substance needed to have meaning on their own. Just as some linguists study the grammatical structures of language, metamathematicians study these underlying structures in a rigorous context. For most mathematicians, though, an intuitive, sociologically-determined understanding is both necessary and sufficient. From a pedagogical perspective, the similarities between the grammar of mathematics and the grammar of a language continue to hold. While much of the introductory grammar of mathematics is taught through classes such as Honors Math,

7 It is also worth noting that set theory is not the only foundation for mathematics. It is, however, the one which is most convenient from both an intuitive and a historical perspective: sets are fairly simple to define (at least naively), and they’re what mathematicians have been working with for over 100 years, so it makes sense to keep using them. Similarly, the grammar of a language is sort of arbitrary—for example, I could decide to form plural nouns in English by adding a “t” at the end instead of an “s.” However, in practice, the particular sounds that build up grammatical structures have a historical and cultural basis—in other words, we’ve been using them for a while, so you can’t just arbitrarily shift them. students continue to learn parts of it throughout their education. When a student takes an analysis class, for instance, they learn the epsilon-delta argument, a type of proof used predominantly in analysis. Similarly, as a student progresses through a language, they learn the grammar required to express increasingly complex idea structures.

III. Lexicon

The lexicon of mathematics is easier to define: it consists of the many primary objects of mathematical study. It is also more straightforward to grasp concep- tually than the grammar, as it does not rely on a distinction between formal and sociological understanding. It is, however, harder to discuss concretely in an accessible way. To talk about an unfamiliar object, you need to give initial examples, a formal definition, and more examples and theorems to build intuition for the behavior of the formal definition. This challenge highlights a key difference between mathematics and other languages: whereas an element of the lexicon of a traditional language can usually be explained using words from another language, an element of the mathematical lexicon cannot undergo a direct substitution process. You can always read a French text in translation, although some of the meaning may be lost. But there is no concept of “algebraic geometry in translation,” since you can’t read an algebraic geometry textbook without already knowing some mathematics. This untranslatability actulally makes

mathematics a more compelling field of preparation for future study, as you can neither work with mathematics nor produce accessible work on it without heavy background knowledge.

This is not to say that mathematics is entirely untraslatable, as you can discuss it by way of analogy. The most elementary objects in algebra, for instance, are ing, a group is something that allows for addition and subtraction. For example, in the integers (positive and negative one can add and subtract; in the posi of a “word” in the mathematical lexicon. intuitive one, and takes the form of a list by adding other properties to the list, such as cyclic groups (groups which consist entirely of something added to itself a number of times) and symmetric groups (groups which consist of all the ways to order a list of things). There are also other types of “words” which relate to groups but are not directly derived from them, such as isomorphisms, a type of function which is used to say that two groups are in some way the same even if they look different. From these different objects, or words, a mathematician can begin to formulate statements about groups and their properties; these statements and their proofs are the “sentences” of the mathematical language.

An objection one might raise at this point is that these objects do not form a lexicon, but a discourse, as they are the building blocks of a discussion. What is important to understand, however, is that while these objects are relevant topics of study in lower-level courses, upper-level courses entirely presuppose their behavior. More concretely, a group is the object of study for Modern Algebra 1; in Modern Algebra 2, however, statements about groups will be assumed as fact without further discussion (in fact, one of the primary topics of Modern Algebra 2 is reducing more complicated than groups, to sipmle statements about groups8). This continues to build: an undergraduate Algebraic Number Theory course will likely assume statements from the undergraduate course; and a seminar in modern research in the subject will assume an understanding of all the prior topics. In other words, it is a tiered lexicon, with each level building on the next. Terence Tao, one of the preeminent living mathematicians, posits three “‘pre-rigorous’ stage,” the “‘rigorous’

8 Very technical note that Professor Warner pointed out: it is in some ways incorrect to say that fields are “more complicated” than groups. Really, because they have fewer rules, groups are more general than fields—in particular, every field is also a group. Since fields are more specific, some facts about them are actually much simpler, in that they exclude weird cases. In the context of Modern Algebra, though, this simplicity relationship roughly holds.

stage,” and the “‘post-rigorous’ stage.” In the language of my framework, the pre-rigorous stage refers to an informal type of mathematics which is fairly irrelevant for our purpose; the rigorous stage refers to the process of acquiring the basic grammar and lexicon of mathematics; and the post-rigorous stage refers to the stage after a student has acquired these skills, at which point, in Tao’s words, “[t]he emphasis is now on applications, intuition, and the ‘big picture.’” The existence of the post-rigorous stage is what makes the lexicon of mathematics into a lexicon—while it is initially of interest in its own right, it is eventually subsumed into intuition. This is similar to the process by which the rules of a language eventually become second nature—in a French literature class you are presumably expected to speak French. Unlike a normal language, however, this process does not only occur once—you can theoretically continue learning and internalizing new mathematical concepts right up until you reach cutting-edge research. In some ways, the lexicon of mathematics is considerably larger and more academically intense than that of a language, in that the acquisition process is much longer and more linear.

At some point, the distinction between learning the lexicon and engaging in mathematical discourse does start to become unclear. By the time you’re learning about very stance, you’ve probably crossed the line from language acquisition into scholarly contribution. The complexity here comes from the way in which the mathematics of the last 200 years or so has been layered on itself. Unlike science, where prior theories are frequently disproved, or the humanities, where new schools of thought appear all the time, the mathematics of the modern era has actually remained entirely relevant. Most of the mathematics you see in the required courses for a mathematics major actually predates WWII (although the way we write it is often a little more modern). There is also a large distinction between learning and producing mathematics. Whereas students in other disciplines are often asked to replicate a historical experiment or write their own paper on an important text from the history of their subject, mathematics students are rarely asked to actually reinvent a key idea (or produce any sort of novel research until graduate school). The presentation of mathematical content is often actively ahistoricized, apart from a quick reference to the name of whoever proved a theorem. This means that a lot of what would be considered a “literature” course in other disciplines is treated as more of a “language” course in mathematics: the act of gaining knowledge through production and engagement with foundational texts streamlined version of the theory. On a more practical level, the Comparative Literature and Society language requirements consist of two semesters of introductory courses, two semesters of intermediate courses, and one semester of a literature course for one language; and two semesters of introductory courses and two semesters of intermediate courses for a second lang-

uage. The tiered structure of the mathematical lexicon means that we cannot map the requirements of the Comparative Literature and Society application to those of a mathematics major directly, as mathematics students continue to learn the words of mathematics long after a language student would be However, we can still construct a comparison on the more grammatical level that is captured by Tao’s three levels of might be considered roughly equivalent terms of actual coursework, this would take the form of a calculus sequence and some sort of introductory proofs class, where students would begin to understand mathematical rigor. The intermediate courses would map to Tao’s rigorous phase; the direct comparison in coursework here are the Modern Algebra and Modern Analysis sequences, which presuppose basic grammatical understanding and teach more serious mathematical content and advanced proof techniques. And while, as I have erature course per se, it is likely comparable to something on the early side of Tao’s post-rigorous stage; in coursework, a 3000- or 4000-level course that assumes prior knowledge of algebra or analysis.9

IV. Communication

Mathematics, with its grammar and lexicon, therefore behaves much like a language. However, there is one last similarity which guides my interest in pursuing mathematics as a language: communication. Communication is, in some ways, the point at which the metaphor breaks down. A “real” language is a full system of communication—human languages have some sort of greeting, a way to express hunger, and a way to encode a variety of complex thoughts. Mathematics clearly doesn’t—to state the obvious, there’s no proof I can write that means “Hello.” But in the context of Comparative Literature and Society, languages aren’t valuable for their ability to contribute arbitrary ideas. Instead, the value of language is to communicate culture.

To illustrate this, let’s look at something that isn’t a language in this way: formal logic. By formal logic, I mean the sets of rules governing different methods of deduction rather than the discipline of metamathematical language. After all, “alphabet” and “formal language” Earlier on, I compared metamathematics to linguistics, as they both study the foundational behavior of linguistic concepts. However, there’s a key difference between the two: whereas linguistics is inductive—that is, based on concrete ob- servation—formal logic is deductive—that principles. More concretely: if I want to

9 We may rephrase this entire comparison in terms of the language of mathematics: Language courses and mathematics courses may both be considered partially-ordered sets with relations of dependence based on both knowledge and rigor. As posets under these dependence relations, these sets are not order-isomorphic (for one, the cardinality of the math poset is considerably greater). However, if we instead order the posets only by rigor-based dependence (and perform some sort of suitable quotient on the math one), we can construct an isomorphism.

tically grammatical, I would compare it to how speakers of the language would whether a sentence is logically true, I would look back at the axioms. Internally, then, formal logic has no culture, history, or people it references—logic developed to formalize mathematics, but does not have an inherent connection to mathematical practice. This holds true when we consider how logic interacts with other disciplines externally as well—while in theory, the “speakers” of formal logic are mathematicians, in practice, as we have seen, they speak a more informal version of the language. Formal logic is therefore empty: it communicates about itself.

The mathematical language I have described above, on the other hand, is far from empty. On the most basic level, it is the prerequisite to mathematical communication, in the same way that French language is an obvious prerequisite to communication in a French cultural context. However, you also need to speak the language of mathematics to engage with the philosophy, history, or sociology of mathematics. Take, for example, this essay: I am able to discuss mathematics and its parallels with language precisely because I already know how to speak in mathematics. And mathematics is also uniquely situated with regard to the sciences. While much of the grammatical knowledge of mathematics is cause mathematics is the language we use to describe the natural world, the mary, then, mathematics is a language not just because it looks like one with its grammar and lexicon, but because it is necessary to facilitate communication.