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Zero is a number, and it is even

I've gotten into heated arguments over this at the coffee shop. Not a lot of them, but they become amazingly intense. Once, after some discussion, another patron told me I was arrogant for my insistence that zero is an even number. I told him that he was an idiot, and that it's a shame his parents didn't love him enough to keep him from eating lead paint chips when he was a child.

I am usually not that harsh. One of the baristas said it was because I just don't like being called arrogant. That wasn't the reason. I was irritated at someone confusing mathematical certainty with arrogance. There are worse things to be harsh about.

More recently, I  had said to another patron named Ted, who is studying math education, that his discipline is very important, because I have some students who don't even know that zero is an even number.

His response was that he didn't think zero was an even number, because he didn't think zero was a number at all. He insisted that since it represented no quantity, it couldn't be a number. Needless to say, heated and occasionally impolite debate ensued.

I know that Ted's original academic background is in the classics and philosophy, and there is a school of philosophical thought that rejects zero as a number. He  broached  the subject with a math professor one day. He asked what an instantiation of the concept of zero would be, and the professor answered that it would be the empty set.

You'd think that would have settled it.

But it didn't. I thought I would persuade him when I pointed out that without zero, the equation


would have no solution.

Then he asked what quantity x would be. I said a quantity of zero, and he said "HA! that's no quantity". Then he said that equation was equivalent to the equation 5=5.

At this point, I decided he was either jerking my chain, or was so committed to a belief that mathematics is philosophically invalid that he would not be persuaded.

It's not that I didn't have an answer to his counterargument though. Under his logic, the equation above has no solution. But how then are we to distinguish it from an equation like


which is unsolvable even with the introduction of other numbers. If you think of x as a quantity of apples, the first equation leads to the assertion "there are no apples", where as all you can say about the second equation is that there is no solution. In other words, there is a difference between a solution equal to nothing and no solution.

I'm glad I didn't make this argument to him though. I'm not good at following the teachings of Jesus, but I've found that obeying the one about not casting pearls before swine can save you a lot of heartburn. And he probably would have just  responded with one of those stupid proofs that 1=0.