What is math?
Most people’s conception of math was drilled into them during grade school.
In my experience, grade school math goes something like this: The teacher says that we need to calculate a thing. He then shows how to calculate that thing, with seven slight variations. Your homework is to calculate six of each of the variations. The test will have five of those seven.
After a decade of this, most walk away thinking that math is calculation. And because of the rote way the material was introduced, many get the impression that math is set in stone. If you perform a particular set of arcane, incomprehensible steps, you will be led to the mythical “right answer.” No other steps are allowed, and heaven help you if you don’t happen to remember the right steps for a particular problem. In that case there is nothing to be done but despair.
And, of course, they believe that all math has been handed down to them from on high, as wisdom from the ancients. It is imperturbable, impenetrable, impeccable.
But that is not what math is.
So, what is math?
Calculation is a useful tool, but it is definitely not what math is.
Math is a quest for understanding. And like any good epic fantasy series, it seems to never quite be finished.^{1}
And the understanding we mathematicians seek is an odd sort of understanding. The goal of science is to understand what is, to describe and understand the universe around them.
Mathematicians, on the other hand, seek to understand what must be.
After all, the questions a mathematician asks are not generally about things that could even exist. Have you ever seen a perfectly straight, infinitely thin line? Or an angle of precisely 90 degrees? But, if I have a perfectly flat triangle with a 90 degree angle, I know the side lengths have a certain relationship, .
And sure, we can count 37 cows, but do the cows care that there are a prime number of them? But 37 is prime, and so the 37 cows cannot be evenly split between more than one person.
I sometimes like to describe this by saying that I, as a mathematician, try to figure out what even God cannot do. Even an all-powerful God cannot create a perfect flat triangle with a 90 degree angle, whose side lengths do not obey the Pythagorean relationship. Neither could He evenly divide 37 cows between more than one person.^{2}
The basis for deciding what must be are the definitions and axioms of mathematics.
Definitions and axioms are different, but very closely related.
Definitions describe the things we talk about. For instance, a straight line (versus a curved one) might be defined as “a line which lies evenly with the points on itself,” as in Euclid.
Axioms describe what we can do with the things we’ve defined. These tend to be very basic, “obvious” things. For example, the axiom of symmetry says that “If , then .” In this example, you could see the axiom as something you can do (“You can switch the sides of an equation.”) or you could see the axiom as defining what two things being equal really means.
On top of this foundation, mathematics is built with logic. Given the definitions and axioms, certain conclusions follow as inescapable consequences. These conclusions we call theorems or lemmas or propositions.^{3}
Because mathematics is taught in such an authoritative way, it can appear that the definitions and axioms of mathematics are in someway intrinsic, that they have existence outside of the creation of man. It can feel like the axioms and definitions are part of the “what must be” that mathematicians are searching for.
To some extent that may be true, but I don’t think this is completely true, and it’s certainly not how math is done.
When you read a textbook, the most recent thinking of the definitions and axioms that are thought to be important are presented. But that hides to some extent the fact that it took hundreds, or even thousands, or years to decide that those axioms should be the ones to form the foundation of the rest of mathematics.
Math evolves. Math changes. The definitions and axioms we use today are not the same ones that were used by Newton.
Referencing Newton actually brings up a good example of how math changes.
Newton (and Leibniz) invented calculus around 1670. It immediately proved its use in solving any number of important questions in physics and mathematics.
But Newton’s calculus was not built on what we would today consider a rigorous foundation.
In order to explain their ideas, both Newton and Leibniz used some idea of “infinitesimals,” quantities that were infinitely small.
Infinitesimals can be very useful in an intuitive explanation of calculus. (I often use them informally when I teach calculus myself.) And so Newton and Leibniz’s proofs of their results were accepted, even though some were uncomfortable with the idea of an infinitely small quantity.
But as mathematicians delved deeper into the ideas of calculus, it became clear that the infinitesimal arguments weren’t quite complete. There were important theorems that could not be carefully proven because the foundations of calculus were not proven with sufficient rigor.^{4}
Thus, one of the major mathematical projects of the 1800’s was to prove the “soundness” of calculus, and make sure the foundations were correct.
This involved inventing new definitions. For instance, one of the key ideas of calculus is the limit. Informally, the limit asks, “As the inputs get close to a number, what do the outputs get close to?”
The intuition for limits is not difficult; you plug in numbers closer and closer to the one you want, and see if the outputs get close to some other number. But the careful definition for limit that we use today, the (epsilon-delta) definition, was not introduced till the 1820’s by Augustin-Louis Cauchy.
Mathematics is not static, and the axioms and definitions we use are not necessarily natural, sitting there for us to find. As we seek deeper understanding, we often come to a point where we realize our earlier understanding was incomplete, or even incorrect, and we seek to fix the foundations. This has occurred over and over and over again to get to our “fixed” modern ideas of mathematics.^{5}
To summarize, mathematics is a quest for understanding what must be. But the very concepts we try to understand are not set in stone. The objects of mathematics are defined by people, and as we understand them better, the definitions and axioms we base our understanding on change.
In the next post I want to talk more in depth about why these definitions change, and how and why mathematicians come up with new definitions.
This post was mostly about the philosophy of math, which is quite a bit different than my normal post. But as we’ll see in a few weeks, Gödel’s incompleteness theorem is so weird that it is impossible to talk about it without discussing the philosophy of math. Gödel’s theorem puts a fundamental limit on mathematicians’ quest for understanding.
- I’m looking at you, George R. R. Martin… ↩
- Well, at least without taking the King Solomon approach and cutting the 37th cow in half! ↩
- Usually “theorems” are bigger, more important conclusions, while “lemmas” are littler conclusions that are needed along the way to show the theorems are true. Propositions can go either way. On the other hand, sometimes lemmas end up being more important than the theorems. ↩
- More recently, mathematicians have come up with rigorous methods to talk about infinitesimals, for instance the hyperreal numbers. However, infinitesimal methods are no longer considered standard. ↩
- Even the work on calculus done in the early 1800’s was not final. The “Riemann” integral, which was the formalization of the integral by Riemann, is what is taught in high schools and early college math. But at the graduate level, we use the “Lebesgue” (Luh-bayg) integral instead, which was introduced in the early 1900’s. Both are rigorous approaches to the integral, but the Lebesgue integral makes a few key lemmas and theorems much easier to prove. The basis of the Lebesgue integral is less intuitive at first, but easier and more powerful in the end. ↩