And now as summer (United States edition) reaches its closing months I plunge into the fourth of my A To Z mathematics-glossary sequences. I hope I know what I’m doing! Today’s request is one of several from Gaurish, who’s got to be my top requester for mathematical terms and whom I thank for it. It’s a lot easier writing these things when I don’t have to think up topics. Gaurish hosts a fine blog, For the love of Mathematics, which you might consider reading.
Arithmetic is what people who aren’t mathematicians figure mathematicians do all day. I remember in my childhood a Berenstain Bears book about people’s jobs. Its mathematician was an adorable little bear adding up sums on the chalkboard, in an observatory, on the Moon. I liked every part of this. I wouldn’t say it’s the whole reason I became a mathematician but it did made the prospect look good early on.
People who aren’t mathematicians are right. At least, the bulk of what mathematics people do is arithmetic. If we work by volume. Arithmetic is about the calculations we do to evaluate or solve polynomials. And polynomials are everything that humans find interesting. Arithmetic is adding and subtracting, of multiplication and division, of taking powers and taking roots. Arithmetic is changing the units of a thing, and of breaking something into several smaller units, or of merging several smaller units into one big one. Arithmetic’s role in commerce and in finance must overwhelm the higher mathematics. Higher mathematics offers cohomologies and Ricci tensors. Arithmetic offers a budget.
This is old mathematics. There’s evidence of humans twenty thousands of years ago recording their arithmetic computations. My understanding is the evidence is ambiguous and interpretations vary. This seems fair. I assume that humans did such arithmetic then, granting that I do not know how to interpret archeological evidence. The thing is that arithmetic is older than humans. Animals are able to count, to do addition and subtraction, perhaps to do harder computations. (I crib this from The Number Sense:
How the Mind Creates Mathematics, by Stanislas Daehaene.) We learn it first, refining our rough instinctively developed sense to something rigorous. At least we learn it at the same time we learn geometry, the other branch of mathematics that must predate human existence.
The primality of arithmetic governs how it becomes an adjective. We will have, for example, the “arithmetic progression” of terms in a sequence. This is a sequence of numbers such as 1, 3, 5, 7, 9, and so on. Or 4, 9, 14, 19, 24, 29, and so on. The difference between one term and its successor is the same as the difference between the predecessor and this term. Or we speak of the “arithmetic mean”. This is the one found by adding together all the numbers of a sample and dividing by the number of terms in the sample. These are important concepts, useful concepts. They are among the first concepts we have when we think of a thing. Their familiarity makes them easy tools to overlook.
Consider the Fundamental Theorem of Arithmetic. There are many Fundamental Theorems; that of Algebra guarantees us the number of roots of a polynomial equation. That of Calculus guarantees us that derivatives and integrals are joined concepts. The Fundamental Theorem of Arithmetic tells us that every whole number greater than one is equal to one and only one product of prime numbers. If a number is equal to (say) two times two times thirteen times nineteen, it cannot also be equal to (say) five times eleven times seventeen. This may seem uncontroversial. The budding mathematician will convince herself it’s so by trying to work out all the ways to write 60 as the product of prime numbers. It’s hard to imagine mathematics for which it isn’t true.
But it needn’t be true. As we study why arithmetic works we discover many strange things. This mathematics that we know even without learning is sophisticated. To build a logical justification for it requires a theory of sets and hundreds of pages of tight reasoning. Or a theory of categories and I don’t even know how much reasoning. The thing that is obvious from putting a couple objects on a table and then a couple more is hard to prove.
As we continue studying arithmetic we start to ponder things like Goldbach’s Conjecture, about even numbers (other than two) being the sum of exactly two prime numbers. This brings us into number theory, a land of fascinating problems. Many of them are so accessible you could pose them to a person while waiting in a fast-food line. This befits a field that grows out of such simple stuff. Many of those are so hard to answer that no person knows whether they are true, or are false, or are even answerable.
And it splits off other ideas. Arithmetic starts, at least, with the counting numbers. It moves into the whole numbers and soon all the integers. With division we soon get rational numbers. With roots we soon get certain irrational numbers. A close study of this implies there must be irrational numbers that must exist, at least as much as “four” exists. Yet they can’t be reached by studying polynomials. Not polynomials that don’t already use these exotic irrational numbers. These are transcendental numbers. If we were to say the transcendental numbers were the only real numbers we would be making only a very slight mistake. We learn they exist by thinking long enough and deep enough about arithmetic to realize there must be more there than we realized.
Thought compounds thought. The integers and the rational numbers and the real numbers have a structure. They interact in certain ways. We can look for things that are not numbers, but which follow rules like that for addition and for multiplication. Sometimes even for powers and for roots. Some of these can be strange: polynomials themselves, for example, follow rules like those of arithmetic. Matrices, which we can represent as grids of numbers, can have powers and even something like roots. Arithmetic is inspiration to finding mathematical structures that look little like our arithmetic. We can find things that follow mathematical operations but which don’t have a Fundamental Theorem of Arithmetic.
And there are more related ideas. These are often very useful. There’s modular arithmetic, in which we adjust the rules of addition and multiplication so that we can work with a finite set of numbers. There’s floating point arithmetic, in which we set machines to do our calculations. These calculations are no longer precise. But they are fast, and reliable, and that is often what we need.
So arithmetic is what people who aren’t mathematicians figure mathematicians do all day. And they are mistaken, but not by much. Arithmetic gives us an idea of what mathematics we can hope to understand. So it structures the way we think about mathematics.