I recently spoke to a mathematician who said he only liked reading fiction.

“Fiction and math?” someone asked.

“No,” he said, “I count math as fiction.”

…

If you ask people, “Is mathematics fiction or nonfiction?” you get wildly different responses. Some people don’t even understand the question—of *course* it’s nonfiction! Yet many others, including those who do math for a living, are not so sure.

But there is an even more crucial point to address first.

**What is math?**

This is a surprisingly tricky question. Mathematics is *really *old. But modern math includes entire fields that have little to do with their ancient predecessors.

Many people think of numbers when they think of math. And around four thousand years ago, math could have rightly been called “the science of number.” The Babylonians developed a system of practical arithmetic. Egyptians used tables full of numerical information to build the pyramids. Other ancient societies, in the Arab world, China and India, developed number systems as well. What we would recognize as regular math– rules of addition and multiplication, as well as abstract formulas– began with the ancient Greeks.

”The science of number” doesn’t include geometry, however, which was also invented by the Greeks. How could “1” or “27” capture circles, triangles, or parallel lines? With the development of geometry, math transitioned from “the “science of number,” to “the science of number and shape.”

Fast forward to calculus, invented simultaneously by Gottfried Leibniz in Germany and Isaac Newton in Britain. Calculus is about motion and change: it’s the stuff of rockets, baseballs and waterfalls. The definition of math stretched to “the science of number, shape, and motion.”

Today math includes wildly different-looking fields such as game theory, topology, combinatorics, and operations research.

So what do they all have in common?

Here is one suggestion:

Patterns.

Give a kindergartener four apples, and she may count “1, 2, 3, 4.” She might do the same thing with four crayons or four of her friends. Number is a pattern that people sense in the world. But it also exists independently in our minds. A person can think of the number 7 without picturing seven spoons or seven marbles. The same goes for seemingly real-world principles like population growth, rectangles, and acceleration. We can study them by just thinking about the patterns that they represent. In the words of award-winning mathematician Keith Devlin, *mathematics is the science of patterns*.

Thinking logically about patterns turns out to be extremely useful. Mechanics, in physics, depends entirely on the use of calculus. Encryption technology in computers was created with number theory. In a practical sense, the formulation of patterns allows us to make predictions. When we know a pattern, even of something as simple as the turn of seasons, we know what to expect in the future. The more precisely the pattern is formulated, the more precise the prediction. Mathematics is handy for this.

Sometimes, though, the reverse occurs: we think about patterns that don’t seem to apply to anything concrete at all. No one has ever seen a nine-dimensional space. But the pattern of a 9-D space is something people can study. Calculating, say, the distance between two points in 9-D is no more complex than calculating distance in 3-D.

So math is incredibly useful—even irreplaceable—in science and technology. Yet it seems that we can dream up math that has nothing to do with the real world. The actual “location” of math suddenly becomes unclear: whether it exists in the world or it exists in our minds.

Which brings us back to our original question: Is math fiction?

Let’s figure that out in the next post.

One might ask what is the definition of ‘pattern’ and in what sense does a pattern exist? The answer to that is not at all clear. To define ‘pattern’ vaguely though, just as a working definition, you might say something like “a repetition of a certain event or phenomenon” or “a tendency of some entity to exhibit certain behavior” or some such type of statement.

The point being that using any such type of definition, patterns appear to be the property not just of mathematics, but of many other sciences and fields of study too. Physical sciences examine the ‘laws of nature’ which are in a sense patterns. Psychology studies the patterns of human thought and instinct and the like. Linguists may study the patterns of speech and commonalities that are found across languages etc etc. In fact, I would say any study assumes there is some generality that recurs and is thus capable of being studied–thus any field of study appears to be pattern-based!

So my main point is that I believe saying mathematics is the study of patterns is too broad; that doesn’t really describe the character of what the mathematician engages in. It is definitely true that mathematics studies patterns–but I think in order to define mathematics the question should be: ‘what sort of patterns?’ What unique characteristic do the patterns of math possess, as opposed to those used and studied in other fields?

And I think this line of reasoning leads you to a specific word: ‘quantity’. This may sound circular, since people associate and interchange the word ‘quantity’ with ‘number’ and ‘mathematics’. But I think to throw these words around lightly and equate them is not being true to their meanings, and is not a precise categorization. If I had to describe the relationship I would say “math is the study of quantity, and number is a type of quantity” (I freely admit that I am influenced by Aristotle in this regard). So what then is quantity? Am I just replacing one empty word with another? The truth is that I am not equipped to fully answer this question, as I think defining math is, as you mention, really not at all an easy task. I think this is a topic that should be revisited and refined over time. But to just give a basic idea, I think quantity has to do with the abstract relationships between things that can be measured, i.e., that possess some sort of magnitude. I think this idea can be fleshed out and added to, but thats my intuition on the heart of math.

Obviously this is my own very biased opinion, and I’d like to hear your thoughts on this matter, whether to bolster your position, or poke a hole in my claim, or to explore a new facet in this discussion. Okay, rant over.

LikeLike

Thanks for commenting! I actually considered using your definition of math in this post (“Mathematics is the ability of the mind to think and formulate in quantitative terms”), but I thought it would be too difficult to explain.

Part of why math is so hard to define is that it is so “close” to our basic mechanisms of thought. If you define concepts in terms of other concepts, you eventually regress to undefinable terms. So it’s possible that “quantity”, or “abstraction”, or whatever, is a word that we can’t define well in terms of other words. (I’ve heard that Aristotle discussed this but I don’t know which concepts he said were undefinable.) In that case, your definition is as good as it’s going to get.

Anyway, I think you make a good point. “The science of patterns” is an incomplete definition, and this topic certainly warrants further discussion.

LikeLike

SB:

The fields you mentioned all do have patterns, but the thing that makes something a pattern is repetition with some regularity. To the extent that math is a study of patterns, it is a quantification of those regularities. In other words, it is the human ability to quantify, which you distinguished well, applied to patterns– that’s what makes math the science of patterns. This is my gut reaction to what you wrote.

LikeLike

TL and EL:

I think EL’s comment is a nice blend of both of our ideas, in such a way as to preserve the essence of each idea. Its also a good expansion/peirush on the phrase “science of patterns”.

As to Aristotle, I think your right about ‘infinite chains of definitions’. At some point you have to have axiomatic terms that are clear to the human mind; I think this is called in Aristotelian terms a ‘primary intelligible’. You could just say we know what we mean by quantity, but I believe some basic description can be given for the term, and perhaps after that, we couldn’t go back further. further investigation required…

But one subtle point can be seen from comparing all our descriptions/definitions:

1) the ability of the mind to think and formulate in quantitative terms

2) the quantification of regularities; the human ability to quantify.

3) the study of quantity

My focus in the previous post was to explain WHAT is studied in math, or what math is the study OF. Whereas the first two approaches emphasize the dependency on human beings; namely, that math is no more than a certain mode of thought. It would seem that such an idea is more in line with classifying in what sense math exists, rather than defining what it strives to do, what is the material with which it works.

In a sense, there is always human ability injected into any study, since we are the thinkers of the study and it is our minds which create theories etc. But still it seems more appropriate to say ‘human ability’ about math since “we can dream up math that has nothing to do with the real world”; whereas in physical sciences, this would be absurd–the reality of a physical theory is dependent solely on whether it describes our universe. If it fails in this regard, we wouldn’t call it a theory at all. Not so for math, not so at all. I am reminded of a quote of Einstein: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

Perhaps I’ve encroached upon the topic of your second post, so I apologize! But I just think its a very subtle point to distinguish between the existence of math and its object, and perhaps I’ve done just that very recently.

LikeLike

“It would seem that such an idea is more in line with classifying in what sense math exists, rather than defining what it strives to do, what is the material with which it works.”

Can you explain what you mean here?

LikeLike

Also, I may lift that Einstein quote for my next post.

LikeLike

Ah, I forgot to mention your definition of “science of patterns”. That clearly seeks to define the material of math, and its object of study.

LikeLike