from the zone of the mathemagician

What's the purpose of math?

It's something I'm sure everyone who has done a significant amount of math has thought about. You can't spend a lot of time doing something without thinking, at some point, “why am I doing this?” For most things the answer is pretty simple, you make food to eat it, you sleep to feel rested, you work to make money.

For math the answer is less obvious. If you want to, you can certainly live a full and happy life with no knowledge of mathematics at all. At the extreme end, there are people living deep in the Amazon jungle who have no notion of counting, and they are perfectly capable of everything they need to do to survive with some comfort. The vast majority of people in the world qualify as innumerate and aren't hampered by it at all, in fact, for a large amount of people being forced to “learn” mathematics in school is a major source of stress with little to no purpose. What math most people do need or use is fairly simple, such as counting, arithmetic, and Euclidean geometry. Occasionally a basic understanding of statistics comes in handy. Some people specialize into engineering and must have a solid understanding of calculus. Some people become physicists and must learn nearly as much mathematics as a mathematician, and a very small amount of people are professional mathematicians and eat because they do mathematics.

However, the material usefulness of math is not the main reason we do math. It's certainly not the main reason I do math, in fact it's been distinctly materially unuseful for me. Considering how many people are willing to become grad students and forfeit five or six years of their life to understand math, with almost no chance of becoming a professional mathematician, the situation is probably similar for most mathematicians. Physicists may find math useful for helping to understand the world, but physics is certainly not the purpose of math, or all mathematicians would be physicists and all mathematical theorems would be about physical things. Many people find math materially useful in investing or engineering, but they use only a sliver of the great body of mathematical knowledge.

Certainly, math is useful, and history shows that a significant amount of “useless” math does eventually become useful. But the usefulness of math cannot explain why it has been an enduring pastime across cultures and eons, or why it garners so much more respect than much more useful fields of science.

Well, you might think, maybe since math is a science (it is called a “formal science” rather than an “empirical science”), its purpose is the same as all sciences, that is: to understand the universe. And that is certainly an accurate description of the purpose of science, but it uses the vague term “universe”. Mathematicians hate vagueness, so this is certainly not satisfying. Maybe before that vagueness can be cleared up, a simpler question has to be answered.

What is math?

Despite what many people think, this is not at all a settled question. An entire subfield of philosophy, called the philosophy of mathematics, exists more or less to answer this question (as well as the first one I posed, in fact). The Stanford Encyclopedia of Philosophy has a plethora of fantastic articles about it, available at https://plato.stanford.edu/entries/philosophy-mathematics/ . I encourage everyone who finds this post interesting to read through some of it.

I would prefer not to take a stance in this article on the eternal (and increasingly politicized) questions of philosophy, I am merely pointing out how deep and hard to get at the questions I am posing are.

For the purposes of this article, the field of math is the study of mathematical objects. What a mathematical object is is left gloriously undefined so that you can fill in your own definition. The main property of mathematical objects that is important is that they can be described with perfect exactness, which is what differentiates them from normal everyday objects. In the field of math, and many fields besides, we use a broad range of techniques that have been lumped together into the term “mathematics”. The broad definition of this lump of techniques is that they use very precise statements and very restrictive standards of soundness to achieve certainty about mathematical objects and their properties. We call the use of precise statements and adherence to very restrictive standards of soundness “mathematical rigour”. A good example of rigour is one that may have confounded many people in calculus: you're not supposed to “divide the differentials”, meaning if you have an equation like this:

dx/dy = F

you're not supposed to leap from there to

dy/dx = 1/F

But in fact, under certain assumptions about F, x, and y, it is perfectly valid to do so! The only reason it's not rigorous is because the leap is unsubstantiated by a rigorous proof. Yes, indeed, it is true for numbers that a/b = c implies that b/a = 1/c, if c is not equal to 0, but just because something is true for numbers doesn't mean it is true for these different things called differentials, it requires proof.

Some math is more rigorous than others, but those mathematicians who throw aside rigour altogether in favor of seat-of-the-pants reasoning risk accidentally “proving” something that is in fact false, or worse, being called a physicist. Most of the hard part of learning higher mathematics is practicing rigorous proofs enough that you can skip them, that is, being sure enough that you *could* make a proof step rigorous that you don't need to write it out in full detail. There is nothing unusual about this; it's just like how an apprentice carpenter needs to pay more attention to es hands than a master carpenter.

Another thing we do in math, besides use very exact statements, is to use very exact shorthand. We use things called 'variables' to give things temporary names, and we often express general truths in terms of these variables, supposing the variables refer to something specific. We also often have notation for ideas or properties that show up frequently. It takes a long time to write out “the unique number x such that a*n will eventually become closer to x than any possible error term and stay within that error term from then on”, instead we write out the much more exact shorthand “x | ∀ ε > 0, ∃ N, n > N => dist(a*n,x) < ε” or the even shorter lim a_n. This is often called “formal language” and it is key. Finding good formal languages for accurately and conveniently expressing mathematical objects and properties is of key importance to the field of mathematics. For instance, while Newton may have been the first to come up with the ideas of calculus, his rival Liebnitz invented the convenient notation which we use today, and thus (in my view) is better regarded as the “father” of calculus.

So, we have made a stab at what the science of mathematics is and what it means to “do mathematics”. We can now go back to answering the main question of this article.

When you do mathematics, and you study it deeply, it is inevitable that you start to believe that there is something “there”. It certainly feels that, somehow, when we do mathematics we are really perceiving the mathematical objects, and rather than simply inventing them. Aside from the aforementioned usefulness of mathematics to physics, there is certainly no evidence for this besides the subjective experience. You cannot show me a real number line, but I can definitely picture one. Some mathematical objects are relatively easy to grok, although almost universally they have hidden depth. Others, such as higher-dimensional objects, can only be perceived like a half-remembered dream. Even if we might be able to vividly experience some aspect of a mathematical object — to viscerally *know* that something about them is true, like we know that water is wet — the real work of a mathematician is to prove it. One of my finest teachers once told me, it doesn't matter what you know, if you can't communicate it your knowledge is worthless. A good proof doesn't just tell you that something is true. There are plenty of very ugly proofs that are sound, and mathematicians hate them. A good proof is like a good poem, it not only tells you that something is true, it also evokes an image of the thing it's trying to prove, and through a series of statements that seem almost obvious once you glimpse the image, beckons the reader to grok.

This is not at all easy! In reality, mathematics is only useful for describing very simple things. Even a cup has far more properties than a typical mathematical object that mathematicians study. Mathematical objects can be defined in a book or two at most, whereas to describe every detail about the physical makeup of a cup to distinguish it from every other conceivable arrangement of matter-energy in the universe, supposing it were possible, would certainly require an entire library or more. The simplifications a mathematician does have for dealing with complicated objects are extremely coarse. For instance, homotopically, an idealized donut is indistinguishable from an idealized coffee cup. But homotopy is not at all easy to understand, in fact it is one of the more complicated things that mathematicians study. That should give you an idea of how hard it is to describe something fully with mathematical exactness. By requiring rigour, we restrict ourselves to simple things, and simplifications of complicated things.

So, TL;DR: what is the purpose of mathematics? To understand mathematical objects using the methodology of mathematics. What a mathematical object *is*, we don't really know, we just know that with practice we can sense them. The purpose of mathematics to achieve absolute certainty about something whose inner nature we can only catch glimpses of in our mind's eye. It is to grok simple things, rather than guess at complicated things.

It should be clear that this is not a purpose that most people would find worth pursuing, which is why most people aren't mathematicians. But I think most people should at least know what a mathematician does and why they do it.