Balanbesgue

Why is 1 + 1 = 2? At first, this seems like a stupid question. After all, 1+1 obviously can’t be equal to something like 0, right? But as any good law student knows, sometimes the most obvious ideas and assumptions are the ones in most dire need of scrutiny. As we shall see, when one asks whether 1+1 is equal to 2 or if it is equal to 0, one gets a very lawyerly answer: it depends.

To answer this question, we need to look to a set of rules that govern a large part of mathematics. These are called the Field Axioms. They’re 9* rules that determine what you can and cannot do with a list of symbols. In our case, these symbols are numbers like 1, 2, 4, 10, and so on.

Now you might be thinking, “This is ridiculous! Why do we have to get so technical about adding things up? After all, if I have 1 mango, and you give me 1 more, then I will clearly have 2 mangoes. Why must we be so pedantic?” Indeed, it seems like we’re going through a lot of trouble here to understand why 1 + 1 = 2 (or whether 1 + 1 = 0). I mean, we’ve brought in a list of 9* rules, the Field Axioms, to answer this question. Why do we need all these rules to answer such a seemingly simple question? I’ll answer this question below, but for now I will say this: why not? A good law student knows that intuition is no substitute for clarity of thought and precise language. For example, law students encounter the word “reasonable” a lot, but know that simply saying that their client’s conduct was “reasonable” will not guarantee them a victory in court. They need to dig deeper. They need to look at statutes, cases, and other materials that will shed lighton what “reasonable” means. They’ll need to answer questions like “what factors determine whether something is reasonable? What is considered unreasonable? When is reasonableness important to consider?” The law student knows that intuition is rarely enough—especially when so many words in legalese starkly depart from their common-sense meanings—this intuition must be supplemented with rigorous, sometimes painstaking, analysis of rules.

Now the moment you’ve been waiting for. Is 1 + 1 = 2, or is 1 + 1 = 0? Recall the mango example above. It seems painfully obvious that if I started with 1 mango, and you gave me another, I would clearly have 2 mangoes. But consider a different example. Imagine you enter a completely dark room to find your casebook, so you flick the light switch on, and the room lights up. In flicking the switch exactly one time, you caused the light to turn on. Afterward, you’ve found your casebook, and are now going to head to the library to do some reading, but you want to turn off the light first (because the environment is important!). In flicking the light switch once more, you’ve caused the light in the room to turn off. But just before leaving, you look back at the room, nowdark, and observe that this is exactly what the room looked like before you flicked on the light switch. In other words, it’s exactly what it looked like when you’ve hit the light switch zero times. In flicking the light switch once to find your book, and one more time after you found it, you caused the room to look the same as it would if you hadn’t flipped the switch at all. It’s as if by adding one flick to another flick, you obtain 0 flicks. 1 + 1 = 0. (The full proof can be found here/is reproduced below**).

Now we’ve encountered a dilemma. It does seem that 1 + 1 = 2 in some situations, and 1 + 1 = 0 in others, but maybe we need a systematic way of choosing which equation to use. This is where the Field Axioms come in. The beauty of the Field Axioms is that this list of nine rules gives you a systematic way of telling exactly when 1 + 1 = 2, and when 1 + 1 = 0. The rules show that the answer to the question, “What is 1 + 1?” is a lawyerly “It depends,” but in a very structured sense as opposed to an ad hoc sense.

I said above that the beauty of the rules is that they give us a systematic way of determining whether 1 + 1 = 0 or 1 + 1 = 2. But there is a deeper beauty: the rules allow both of these outcomes to coexist. Those who helped formulate the Field Axioms probably had more on their mind than how to count mangoes. They might’ve envisioned a world in which numbers like 1, 0, and 2 would be used for all kinds of human activities, and so there needed to be rules that governed how to add 1 + 1 in a way that made saying both “1 + 1 = 0” and “1 + 1 = 2,” in some sense, legal. Just like a lawyer who drafts policy wants to write their proposal in a way such that it can be applied far into the future and under different situations, the drafters of the Field Axioms might’ve wanted to write them in a way that would assist humans for a long time and through changing circumstances—from the time before money was invented when some of us might’ve bartered mangoes instead of paper, to the time when we had electricity and light switches, and hopefully beyond to the timewhen we’ve found a cure for cancer or invent teleportation.

Lawyers and mathematicians aren’t as different as you might think. Both analyze rules. Both can be extremely (sometimes insufferably) pedantic. Both can be highly creative. Finally, both often live by two simple words. “It depends.”

*Some mathematicians include within the Field Axioms something called the “axiom of choice” in addition to these 9 rules, thus totaling 10 rules. The reason why leads to a whole philosophical debate in mathematics that, unfortunately, is much beyond the scope of this blog.

**