This thread grows out of a recent thread in which we were discussing the nature of philosophical truth. Somebody claimed that mathematics provides a way to universal truths that are independent of human opinion. Another way I’ve heard it put, in a thread awhile back, is that in math everybody gets the same answer. I, personally, disagree with this statement and I provided a counterexample. Nobody responded, which is just as well given that it was tangential to the thread in question. Now I’d like to bring the topic up again in its own thread. I’d also like to mention that I was inspired by this excellent blog post.
My counterexample is this:
For those who don’t want to click, it’s a Calvin and Hobbes strip where Calvin’s dad is trying to teach him addition. The dialogue is this:
Dad: Here, maybe this will make more sense. I have eight pennies and I ask you for four more.
Calvin: I say forget it. You’re the one with a steady paycheck.
Dad: Just give me the four pennies. Good. How much money do I have now?
Calvin: Investments and all?
Dad: No. Just here on the table.
Calvin: Eight cents.
Dad: No, eight plus four is twelve. See? count them up.
Calvin [pointing]: But those four are mine!
Now when I first read this as a child, I assumed that the disagreement arose from Calvin being wrong while his Dad was right. Now, however, I realize that both of them can be right simultaneously. Calvin’s dad views this as an instance of addition taking place in the ring of integers. A set of eight pennies combines with a set of four pennies to become a set of twelve indistinguishable pennies. (8+4=12).
Calvin, on the other hand, views the operation as occuring in the non-abelian free group generated by the integers. He sees two different sets, his pennies and his Dad’s pennies. They combine into a single mathematical object, but remain distinct sets with different properties. (8+4=84)
Both Calvin and his Dad are entirely correct in their computations. They arrived at different answers because they assigned different meanings to the concept of “eight plus four”.
Of course one might respond by adjusting the claim slightly to say that everybody will get the same answer in math if they first meet and agree on the meaning of what is being said. That, however, basically takes the core out of the argument. Defenders of universal truth in mathematics would typically want to say that results in math don’t require any discussion of subjective, emotional issues such as how to interpret the pennies.
One might also respond that this issue is rather trivial. I don’t think that it’s trivial at all, but instead it’s at the heart of many major issues today. Many times in debates about government policy we hear people saying “You can use my money to _____________”. The thing is that some people view the money they pay in taxes as remaining their money while others view it as becoming the government’s money and no longer having any connection to the taxpayer that it came from. It’s really the same disagreement that Calvin and his Dad had, only with larger amounts of money.