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This was originally posted on my weblog, 3 September 2001. I’m reposting it here temporarily.
Heard on NPR on Saturday: Some joker named <a href=“http://www.crispinsartwell.com”>Crispin Sartwell</a>, spouting off about mathematics. Sartwell is apparently the kind of university professor who doesn’t mind using the word “shiznit” when it’s called for (which is when, exactly?), the kind of critic of culture who is happy to reel off for you lists of famous things that suck. He’s a professor of philosophy at the Maryland Institute College of Art, and publishes occasional pieces in places like the L.A. Times Book Review and the Philadelphia Inquirer. He’s the kind of character that I dislike even when I agree with him, but sort of respect even when he’s puking up the utterest bushwa.
The piece I heard on NPR was apparently published in the L.A. Times on August 21 or so. (It’s also <a href=“http://www.crispinsartwell.com/meds/math.htm”>here on his website</a>, and a followup is <a href=“http://www.crispinsartwell.com/meds/whatmath.htm”>here</a>.) It’s an interesting piece. Sartwell stumbled up against one of mathematics’ deepest, darkest secrets: that it’s very difficult indeed to carefully define what most people think of as the most central concept of mathematics, Number. Yes, there are elaborate schemes involving sets of sets (”two” = {{}}, and so on), but as Sartwell correctly observes, these are at best unsatisfying, and at worst laughable. Other options aren’t very enticing: Empiricism is likely to make you look like a fool at cocktail parties, and Platonism only avoids this fate by association with a famous name.
So it looks like mathematicians are S.O.L., exposed as hallucinators or charlatans, their whole reason for existing having a tenuous hold on existence itself. Well, then, Sartwell, says,
Mathematics is a sort of necromancy or pagan religion. It has no basis in fact or in theory. It is concerned entirely with entities of which it has no clear conception.
[…]
Unless and until mathematics can give a better account of itself, we must regard it as so much gobbledygook, and we must protect our children from such superstitious nonsense. Mathematics represents just the kind of loose, empty thinking that characterizes so many of the ephemeral fashions in education these days.
Well, of course, he’s joking, though there were apparently kajillions of math teachers who couldn’t tell (not terribly surprising). One more quote before I start knocking down these straw men (this one’s from the followup):
This time I’m pretty sure he’s not joking.
All right, Crispin, my man. Here’s the straight poop. Number is not the central object of mathematics. It isn’t even the central object of arithmetic, which is what most people think of (wrongly!) when they think of mathematics. It’s the central object of checkbook-balancing, baseball statistics, and idiot savants. The central objects of mathematics are assertions. Propositions, equations, theorems, proofs, they’ve got lots of names, but at root they’re all assertions or lists of assertions. (The basic form of an assertion is “If A, then B”. To get more interesting things, nest if-then statements within each other.) These are the things we study. Take an example from my own research: “Finite representation type ascends from a local ring to its completion.” It’s not such a hard exercise to massage this statement into a nested list of if-then statements; I’ll let you think about it for a while.
The point, though, is that whatever else it may be, that isn’t in any way interpretable as a statement about numbers. The existence, or non-, of anything between six and eight hasn’t the slightest relevance, any more than a wordless definition of the word “word” is relevant to writing literary criticism. So whatever else is wrong with mathematics, shaky foundations in the world of checkbook-balancing don’t affect the truth or falsity of my theorems, or make their proofs any less clear, precise, or convincing.
Thanks for worrying about me, though. Now back to your own shiznit.
This was originally posted on my weblog, 3 September 2001. I’m reposting it here temporarily.
Heard on NPR on Saturday: Some joker named <a href=“http://www.crispinsartwell.com”>Crispin Sartwell</a>, spouting off about mathematics. Sartwell is apparently the kind of university professor who doesn’t mind using the word “shiznit” when it’s called for (which is when, exactly?), the kind of critic of culture who is happy to reel off for you lists of famous things that suck. He’s a professor of philosophy at the Maryland Institute College of Art, and publishes occasional pieces in places like the L.A. Times Book Review and the Philadelphia Inquirer. He’s the kind of character that I dislike even when I agree with him, but sort of respect even when he’s puking up the utterest bushwa.
The piece I heard on NPR was apparently published in the L.A. Times on August 21 or so. (It’s also <a href=“http://www.crispinsartwell.com/meds/math.htm”>here on his website</a>, and a followup is <a href=“http://www.crispinsartwell.com/meds/whatmath.htm”>here</a>.) It’s an interesting piece. Sartwell stumbled up against one of mathematics’ deepest, darkest secrets: that it’s very difficult indeed to carefully define what most people think of as the most central concept of mathematics, Number. Yes, there are elaborate schemes involving sets of sets (”two” = {{}}, and so on), but as Sartwell correctly observes, these are at best unsatisfying, and at worst laughable. Other options aren’t very enticing: Empiricism is likely to make you look like a fool at cocktail parties, and Platonism only avoids this fate by association with a famous name.
So it looks like mathematicians are S.O.L., exposed as hallucinators or charlatans, their whole reason for existing having a tenuous hold on existence itself. Well, then, Sartwell, says,
Mathematics is a sort of necromancy or pagan religion. It has no basis in fact or in theory. It is concerned entirely with entities of which it has no clear conception.
[…]
Unless and until mathematics can give a better account of itself, we must regard it as so much gobbledygook, and we must protect our children from such superstitious nonsense. Mathematics represents just the kind of loose, empty thinking that characterizes so many of the ephemeral fashions in education these days.
Well, of course, he’s joking, though there were apparently kajillions of math teachers who couldn’t tell (not terribly surprising). One more quote before I start knocking down these straw men (this one’s from the followup):
This time I’m pretty sure he’s not joking.
All right, Crispin, my man. Here’s the straight poop. Number is not the central object of mathematics. It isn’t even the central object of arithmetic, which is what most people think of (wrongly!) when they think of mathematics. It’s the central object of checkbook-balancing, baseball statistics, and idiot savants. The central objects of mathematics are assertions. Propositions, equations, theorems, proofs, they’ve got lots of names, but at root they’re all assertions or lists of assertions. (The basic form of an assertion is “If A, then B”. To get more interesting things, nest if-then statements within each other.) These are the things we study. Take an example from my own research: “Finite representation type ascends from a local ring to its completion.” It’s not such a hard exercise to massage this statement into a nested list of if-then statements; I’ll let you think about it for a while.
The point, though, is that whatever else it may be, that isn’t in any way interpretable as a statement about numbers. The existence, or non-, of anything between six and eight hasn’t the slightest relevance, any more than a wordless definition of the word “word” is relevant to writing literary criticism. So whatever else is wrong with mathematics, shaky foundations in the world of checkbook-balancing don’t affect the truth or falsity of my theorems, or make their proofs any less clear, precise, or convincing.
Thanks for worrying about me, though. Now back to your own shiznit.