Addition. Subtraction. Multiplication. Division. These are the basic mathematical operations one typically learns in elementary school. The first four cover the ways you can “combine” numbers. We also begin to learn about two new kinds of numbers: Negatives (thanks, subtraction) and Fractions (thanks, division).
After sufficient comfort with these concepts, one may begin to learn about equations (Algebra) and shapes (Geometry), and the equations that define particular kinds of shapes (Trigonometry). Basically, this is where you learn everything that your TI-85 calculator already knows how to do.
For some, this is as far as the mathematical road goes. For others, the future engineers or the analytically precocious, they take the on-ramp to Calculus in late high school or starting in college. Here one learns new tools for studying a broad category of continuous functions. Anyone jonesing for yet more math can sign up for Differential Equations and Linear Algebra. If you want to hop on the AI train, better add a course in Statistics and Probability, too.
You might think that anyone who has sat through all these courses from elementary school through college has seen most everything there is to know about math. But they haven’t. They’ve only seen a tiny fraction.
There’s an infamous mathematical “puzzle,” known as Fermat’s Last Theorem. It can be stated as follows:
Consider the algebraic equation xn + yn = zn in variables x, y, and z.
There are no integer values of x, y, and z such that this equation is true if n is greater than 2.
Let’s take a moment and digest this. It’s simpler if we first consider the case where n = 2, though it should be noted that the theorem does not apply in this case. For n = 2 we have the more familiar equation:
x2 + y2 = z2
Or, using variables from high school algebra:
a2 + b2 = c2
This is the good old Pythagorean Theorem, which describes how the lengths of a right triangle relate to each other.
Obviously, this equation has integer solutions. They are the “Pythagorean triples,” such as (3, 4, 5) — i.e., 32 + 42 = 52.
But the mathematician Pierre de Fermat figured out in 1637 that if you generalized that power of 2 to any other integer (such as 3 or 10 or 102,038,234), there are no integer values of a, b, and c that can make the generalized Pythagorean equation work. For example, there are no integer values for a, b and c that make the following equation (n = 5) work:
a5 + b5 = c5
Actually, Fermat famously claimed to have figured it out, without actually providing any proof. So it’s more accurate to say that he had a good hunch. And this hunch turned into one of the most infamous unsolved problems in the history of mathematics for the next 300 years.
The integer part is key, because it’s known from other results in mathematics that any algebraic equation such as xn + yn = zn will have some solutions.
In all of the many mathematics courses described above, one learns to recognize certain kinds of problems and then apply the correct formula to solve it. Got a word problem where Susie is trying to keep track of apples? Better whip out your arithmetic. Need to solve a quadratic equation? Quadratic formula. BAM! Need to find the length of some side in a triangle? Use this handy trig identity! Need to integrate some function? Use one of these handy integration tricks one learns in Calc II!
But this isn’t actually how most mathematics—that is the mathematics practiced by actual mathematicians—is actually done. There’s no magic formula or clever trick one learns in Algebra I or Algebra II that will allow you to prove Fermat’s Last Theorem even though it’s technically just an “algebraic” equation.
So how then, do you go about proving it?
How much? Well the proof of Fermat’s Last Theorem, first given in 1994 by Andrew Wiles, is over 120 pages long. And very little of it would be recognizable to Fermat, who first wrote down the problem. In fact, nothing in it would be recognizable to anyone whose run the gauntlet of courses I described above, as most of the material taught in those courses predate Fermat.
That’s right folks, all that math you learned is over 300 years old. Stale AF. Imagine if your history classes in college all stopped in the 1600s.
Andre Weil (not Andrew Wiles, though I admit it’s confusing) was born in 1906, his sister, Simone, 3 years later. He would go on to become one of the most celebrated mathematicians of the last century, living to the ripe age of 92. Simone died in her thirties, with little acclaim in her lifetime, though posthumously she’d become celebrated in her own right as “a moral genius in the orbit of ethics.
These two siblings are at the center of a new book called The Weil Conjectures by Karen Olsson. Olsson is a novelist, but this book is not fiction, or at least, it is made clear where she veers into speculation, employing a clever mechanic again and again: “Let us suppose Simone is thinking…”.
The book is a contemplation on the nature of genius, relationships, and math, and what can really be conveyed about the latter between someone who is steeped in mathematics (Andre) and someone who is not (Simone).
And it is. But it’s not like any other language we have. It is not a language for describing things out there in the world. The subjects of mathematics live mostly in our minds. And, with time, they are compounded, spreading into internal spaces that do not yet exist. This means that translating mathematics to another language—say, conversational English—is highly problematic. The only reasonable translation of most abstract mathematical concepts is into yet another abstract mathematical concept.
Simone Weil, we learn, is a tortured soul. A young French woman with a penchant for getting ill, who yearns to experience the deprivation of those who live at the bottom of society. At the same time, she yearns to understand exactly what it is that her mathematically-minded brother is up to. (In fact, if one learns anything about Simone from Olsson’s book, it is that she was in a near constant state of agonized yearning for something). She wonders what good any of that abstraction can do if it cannot be appreciated by more than a handful of other mathematically-minded people in the world.
Olsson is never able to explain exactly what Andre is up to either—she is no more able to escape the language barrier than is Simone. But she does, through a tumble of exquisite words, give the reader a sense of why. At one point Olsson writes:
I wonder whether mathematicians and fiction writers might be people for whom the lure of alternative worlds is particularly strong. And then I wonder whether this is just a natural consequence of abstract representation itself, that once you start putting words or numbers on paper you are already beginning to piece together a kind of parallel universe, which you then want to access, discover, flesh out.
Its purpose is tautological, but not anymore so than the purpose of many other creative endeavors.
It exists to propagate itself. But not anymore so than you or I.
Just as history may be divided into different sub-disciplines: ancient history, U.S. history, European history, and so on, mathematics is also divided into various branches. The primary branches include Analysis, Geometry/Topology, and Algebra. Some other branches are: Logic, Number Theory, Combinatorics, and Probability (but not statistics, which is a real world application of probability theory). These are all branches of what is often called Pure Mathematics, in that these are not topics studied to some greater end (in contrast to Applied Mathematics, which is studied because it is useful in some other discipline).
Analysis includes everything related to the study of functions, especially continuous functions. The Calculus sequence taught in high school and college is contained within analysis as a very narrow subset.
Geometry, and its cousin Topology, is the mathematics of spaces and objects (including shapes) within those spaces. It includes everything you learned in high school geometry, but also the concepts needed to investigate the 10-dimensional spaces of string theory.
Algebra is the mathematics of the structures and patterns that underlie equations. It greatly generalizes what most of us tend to think of as “algebra” (and is often called “abstract algebra” to avoid confusion).
For example, in this more general algebra the “commutative law” of multiplication—that order doesn’t matter in multiplying to numbers (i.e., 2*3 = 3*2 = 6)—may fail to hold. This failure of commutativity in some cases leads one to an entire new sub-branch of algebra known as “non-commutative algebra”, which just so happens to be highly applicable to the bizarro world of quantum mechanics.
Number theory is the study of the numbers you know about (integers, rational numbers, irrational numbers) as well as numbers you don’t (algebraic integers, for example). It turns out that some of the hardest mathematical problems in the world are problems that can be stated in a way that you or I could easily understand.
Goldbach’s Conjecture, for example, is just this:
Every even number can be expressed as the sum of two prime numbers.
Compare this with an infamous problem from Topology known as the Poincare Conjecture:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
One looks much simpler than the other (well, one looks intelligible and the other one looks mostly like some kind of Seussian gibberish). But the Poincare Conjecture has been proven, while the Goldbach Conjecture has not, despite being around for centuries.
Combinatorics is all about the different ways you can combine, or order, some group of objects. This is the branch of mathematics where some really, really big numbers (like, ungodly large) start to show up—as solutions to all the possible ways you could do something seemingly pointless like drawing lines been a set of dots.
Logic concerns the most fundamental concepts of mathematics. The very rules for how you define any mathematical object in the first place and the rules for manipulating it. This is the land of infinities (there are multiple kinds, it turns out) and mathematical paradoxes, such as The Barber’s Paradox:
The barber is the one who shaves all those, and those only, who do not shave themselves. Question: Does the barber shave himself?
Of course things start to get really interesting when you find yourself at the intersection of various branches. For example, Analytic Number Theory is at the intersection of Analysis and Number Theory. It uses information about continuous functions to solve problems in Number Theory, such as how many prime numbers there are up to a given number N. On the other hand, Algebraic Number Theory is at the intersection of Algebra and Number Theory, and uses the algebraic properties of various number systems (such as whether or how certain numbers can be factored into “primes”) to attack problems in Number Theory.
There’s also Algebraic Geometry, a vast and difficult field, that allows one to reframe problems in geometry in terms of purely algebraic equations (and vice versa). Andre Weil was instrumental in helping to develop algebraic geometry. Taken even further, one can use techniques from algebraic geometry to attack problems in Number Theory, yielding a new sub-branch called Arithmetic Algebraic Geometry. Arithmetic algebraic geometry is at the heart of Andrew Wiles approach to proving Fermat’s Last Theorem.
To be certain, many who find algebra and calculus boring might very well find the greater mathematical universe to be just as uninteresting. Just as there are many people who might not be bothered to look up from their game of Candy Crush even as their spaceship first touches down on Mars.
But I suspect there are at least some people who would find a tour of the broader mathematical universe a worthwhile experience, if not downright fascinating. To these folks, I say:
Despite her attempts to draw connections between mathematics and fiction, Olsson does note at one point, “It’s probably best to leave analogies between math and writing in the semi-obscure territory.”
Because though doing mathematics is very much an act of creation, it’s a highly constrained sort of creation. The worlds, thusly constructed, are elegant, but austere. And they are, in the end, world’s without people to populate them. To get lost in a mathematical “story” is to travel to an alien landscape. To thrill in the unfamiliarity.