27 July 2007

checks for nothing, and why English is useful

Karl Fogel attempts to pay a bill for $0, because of course you have to pay bills for zero, because otherwise the companies that issue them keep sending them.

In this case, the bill was for the purchase of a book from the Mathematical Association of America, so he wrote a check for e+1 dollars. They didn't deposit it, because "check needs to be wrote out in U. S. dollars", as they put it.

I can see a more legitimate reason for rejecting the check. Usually, when writing a check, one puts, say, "3.14" in the little box on the right, and "Three and 14/100" on the line. The reason for writing out the value of the check in both figured and words is for redundancy. (Although then why don't we write "three dollars and fourteen cents" on the line? I suppose redundancy doesn't matter quite as much when we're talking about sub-dollar amounts.)

So you might say he should have written "e to the i π plus one dollars" on the line. But even that seems a bit suspect, because e, i, and π are themselves bits of mathematical notation. It seems that he really should have written something like

"The base of the exponential function, raised to the product of the imaginary unit and the ratio of a circle's circumference to its diameter, plus one"

for the number of dollars he wanted. Of course, this is the sort of thing that makes it obvious why having a compact mathematical notation is a good idea. I am not enough of a mathematical historian to have looked at the way things used to be written, but from what I understand this is the sort of thing they would have written five centuries ago, and I can't imagine working like that.

Unfortunately, the fact that we have such a good mathematical notation creates another problem -- people think that they can just put a bunch of symbols on a page and not explain what they mean by them, and that's "mathematics". Terry Tao, at his blog, has lots of writing advice; of particular interest in this discussion is his advice to take advantage of the English language. Here he gives a couple dozen ways to say that two statements are true, which are logically equivalent but have a wide variety of connotations. To take two examples of his examples at random, "P(x) is true. Unfortunately, Q(y) is also true." and "P is satisfied by x. Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct. I think that mathematicians as a whole are not sensitive enough to the connotations of their words; this is useful when doing formal mathematics but not so useful when trying to express the results of it. Perhaps we kneel too much at the altar of Bourbaki.

5 comments:

Anonymous said...

I agree...

I think meaning and context toward the mathematics shows an alternate method to understand the concepts and will be better for everyone. I think you are correct in meaning philosophy, as it's a 'why' clause to the data, meta-data, to the problem. Clearly, a philosophical facet.

MetaMath: The Search for Omega is a great example of context (via philosophy). It isn't filled with complex notation, but shows the results, and path that lead to the discovery of the halting probability.

Aaron said...

... but on the other hand, Bourbaki is the one [the many?] who started calling groupoids "magmas," which makes set theory sound a lot less like set theory and a lot more like playing with molten lava. And that's an altar I'd kneel at any day of the week. :)

Isabel said...

Aaron,

playing with molten lava sounds fun in theory. But I manage to get myself burnt around any sort of fire larger than the kind that comes out of my stove, so I think I'll continue to avoid set theory.

John Armstrong said...

Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct

I think the word you're looking for is "semantically".

Logicians give far too little thought to semantics. Bunch of anti-semantic b******s...

Aaron said...

P is satisfied by x. Similarly, Q is satisfied by y.

And don't forget that z satisfies R in a way that m never could. ^_^