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If we changed the way we count, could 2+2 fail to equal 4? If, for example, we started counting with zero, we might count X X X as 0, 1, 2 X's. Then 2 + 2 would equal X X X X X X, and when we counted the X's, we would count, "zero, one, two, three, four, five." So 2+2=5.

So, does this example show that 2 + 2 doesn't necessarily equal 4? Or, would we have to say that we were speaking another language when we truthfully say that 2+2=5?

October 15, 2005

Response from Daniel J. Velleman on October 15, 2005
I'd say we're speaking another language. In this language the numeral "2" means 3 and the numeral "5" means 6, so "2+2=5" means 3+3=6, which is true. But 2+2 is still equal to 4.

This isn't really a fact about mathematics, it's a fact about language. If we changed the names we use to refer to people so that "John Kerry" referred to George W. Bush, then in that new language the sentence "John Kerry is president of the U.S." would be true. But it wouldn't change the facts about who the president is. George W. Bush would still be president, we'd just be using a different name to refer to him. If you want to overturn the results of the last election, this isn't the way to do it.

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