Where Math Comes From
Math isn’t just about following rules. It’s about figuring out why they exist, and if there’s a way to break them.
The following is an excerpt from Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths by Eugenia Cheng.
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Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths
Why does 1 + 1 = 2?
One possible answer to this question is “It just does!” That is really a variation on “Because I say so!” an answer that has been frustrating children for generations. “Because I say so” means that there is an authority figure who makes the rules, that they don’t have to justify their rules but can make up any rules they want, and everyone else is a minion who just has to follow those rules.
It is quite right to feel frustrated by that idea. In fact, a strong mathematical impulse is to immediately want to break all the rules, or find sneaky situations in which those rules don’t hold, to show that the supposed authority figure doesn’t have quite as much authority as they think.
Math can seem like a world of rules you just have to follow, which makes it seem rigid and boring. By contrast, my love of math is some- what driven by my love of breaking rules, or at least pushing against them. I’m a bit sheepish about that as it makes me sound like an adolescent who never grew up. My love of math is also driven by my wanting to keep asking “Why?” about everything, which in turn makes me feel like a toddler who never grew up. But both of those impulses play an important role in advancing human understanding, and in particular mathematical understanding. Those impulses are an important part of the origins of math, which is what we’ll be looking at in this chapter. I’d like to stress that in normal life I am a very law-abiding person because I understand rules that are about holding a community together and keeping people safe. I believe in those rules. I don’t mind following rules that have a purpose; the rules I don’t believe in are the arbitrary ones that don’t seem to have a justification, or whose justification I don’t believe in. Rules like “You must make your bed every day” (which really isn’t to my taste) or “Never melt chocolate in a microwave” (it’s certainly easy to ruin it, but as long as you stir it every fifteen seconds without fail I’ve found it’s fine).
So I want to look at where the apparent “rules” of math come from, and indeed where math comes from at all. I’ll describe how it starts from small seeds and then grows to great heights in an organic way. The seeds are naive questions that any of us might pose, and that small children often pose innocently, like when they wonder why 1 + 1 is 2, rather than just being content to know that it is. Like any seeds, they need to be nurtured in the right way to grow. They need fertile soil, space to plant their roots, and then nourishment. Unfortunately our innocent questions are too often not nurtured in this way, but dismissed as “stupid” and tossed aside. But the difference between deep mathematical questions and innocent ones might only be the nurturing—that is, there is no difference. It’s the same seeds.
People who don’t like math are often put off by the apparently autocratic declaration that something is the right answer, without explanation. “One plus one just is two.” But wondering why something is true leads us to build strong foundations for mathematics, so that we can make clear and rigorous arguments. Some people find this clarity and reliability relaxing and liberating, while others find it restrictive and autocratic. A question like “Why does 1 + 1 = 2?” allows us to explore the idea that math doesn’t have clear right answers, but rather, different contexts in which different things can be true. This is going to lead us to explore where numbers come from in the first place, how we come to the ideas of arithmetic, and how we can then use those ideas in other mathematical contexts such as when we’re thinking about shapes. This touches on many important themes in how math is developed, starting with making connections between things, taking abstraction seriously, and then expanding our thought processes to encompass more of the world around us, little by little.
So rather than think about why one plus one is two, let’s go a little further and question whether it’s even true all the time.
Children seem to be natural seekers of counterexamples. A counter-example is an example showing that something isn’t true. Declaring that something is always true is like putting a boundary around something, and seeking examples that contradict that is like pushing against those imposed boundaries. It’s an important mathematical urge.
You can try prodding a child on one plus one by saying something like this: “If I give you one cupcake and another cupcake, how many cupcakes will you have?” But they might gleefully declare “None, because I’ve eaten them!” or indeed, “None, because I don’t like cupcakes.” I am always delighted when I see parents posting pictures of their children’s defiant answers online. One all-time favorite of mine was in answer to the question “Joe has 7 apples and uses 5 of them to make an apple pie. How many apples does he have left?” to which my friend’s child wrote “HAS HE EATEN THE PIE YET?” I enjoy answers that are arguably correct while definitely not being the answer that is supposed to count as correct. This shows an important aspect of mathematics, and the children’s thought processes are showing an important but underappreciated aspect of mathematical instinct, the instinct to push against unjustified authority.
Excerpted from Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths by Eugenia Cheng. Copyright © 2023. Available from Basic Books, an imprint of Hachette Book Group, Inc.
Dr. Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago and author of several books, including The Art of Logic in an Illogical World (Basic Books, 2018) and Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths (Basic Books, 2023).