Is math a science? What about computer science? (A commenter on an earlier post repeated the well-known line that “no subject calling itself a science is one.”)

These are, at the same time, boring definitional disputes best left to funding agencies, and profound mysteries worthy of such intellects as Plato, Leibniz, and Gödel. In a recent comment on Peter Woit’s blog, the physicist John Baez — as usual — went straight to the heart of the matter:

“The problem of course is that in the standard modern picture, science is empirical, based on induction, and tends to favor a materialistic ontology, while mathematics is non-empirical, based on deduction, and tends to favor a Platonist/Pythagorean ontology… yet somehow they need each other! So, mathematics is not only the queen and handmaiden of the sciences – it’s the secret mistress as well, a source of romantic fascination but also some embarrassment.”

That 17 is prime strikes us as absolutely certain, yet there’s nothing in the physical world we can point to as the source of that certainty. (Seventeen blocks that can’t be arranged into a rectangle? Give me a break.) In that respect, math seems more like subjective experience than science: you might be wrong about the sky being blue, but you can’t be wrong about your seeing it as blue. Maybe this has something to do with mathematicians’ much-noted mystical tendencies: Pythagoras sacrificing a hundred oxen because the square root of 2 was irrational; Cantor naming infinite cardinalities using the Hebrew letter aleph, which represents the “infinite greatness of God” in Kabbalah; Erdös forswearing earthly pleasures to devote his life to the Book; Gödel updating St. Anselm’s proof of the existence of God; Penrose speculating that quantum gravity gives rise to consciousness. My favorite novel about mathematicians, Rebecca Goldstein’s The Mind-Body Problem, gets much of its mileage from this ancient connection. (For empirical types: according to a 1997 survey by Larson and Witham, ~40% of mathematicians say they believe in God, compared to 20% of physicists and 30% of biologists.)

And yet, if mathematicians are mystics during those rare late-night epiphanies when they first apprehend (or believe they’ve apprehended) a timeless thought of God, then they’re scientists through and through when it comes time to LaTeX that thought and post it to the arXiv. What makes me so sure of that? Mostly, that my 10th-grade chemistry teacher claimed the opposite.

To give you some background, this is a teacher whose hatred of curiosity and independent thought was renowned throughout the school district — who’d give her students detentions for showing up fifteen seconds after the bell — who’d flunk me on exams, even when I got the answers right, because I refused to write things like (1 mol)/(1 g) = 1 mol/g. Immediately after enduring her class, I dropped out of high school and went straight to college, picking up a G.E.D. along the way. For I had sworn to myself, while listening to this woman lecture, that the goal of my life was to become her antithesis: the living embodiment of everything she detested. Ten years later, I still haven’t wavered from that goal.

Which brings me to the term project in her class. We were supposed to interview a scientist — any scientist — and then write a detailed report about his or her work. I chose a mathematician at Bell Labs who did operations research. After I’d interviewed the guy and finished my project, the teacher ordered me to redo it from scratch with a different interviewee. Why? Because “mathematicians aren’t real scientists.” (To give some context, the teacher did accept a pharmacist, a physical therapist, and an architect as real scientists.)

Now, is it possible that my views about the epistemological status of mathematics are hopelessly colored by enmity toward my chemistry teacher? Yes, it is. But as far as I can tell, the refusal to count math and CS among the sciences has done some real damage, even outside the intellectual prison known as high school. Let’s consider a few examples:

The New York Times hardly ever runs a story about math or CS theory, but it runs the same story about cosmology and string theory every two weeks.

We all know the recipe for getting a paper published in Science or Nature: first gather up all your analytical results, and bury them in your yard. Then make some multicolored charts of Experimental Data, which suggest (at a 2σ level) the same conclusions you previously reached via the forbidden method of proving them true.

Philosophers like Wittgenstein have gotten away with saying arbitrarily dumb things, like “Mathematical propositions express no thoughts.” As my adviser Umesh Vazirani pointed out to me, the proper response to anyone who says that is: “Indeed, the mathematical propositions that you know express no thoughts.”

Many people seem to have the idea that, whereas scientists proceed by proposing theories and then shooting them down, mathematicians somehow proceed in a different, alien way. Which raises the question: what other way is there? Whenever I hear someone claim that “quantum computers are really just analog computers,” or “all cellular automata that aren’t obviously simple are Turing-complete,” I’m reminded that Popper’s notion of falsifiability is just as important in math and CS as in any other sciences.

Saddest of all, many mathematicians and computer scientists seem to reason that, because they can write their results up with something approaching Platonic rigor, it follows that they should. Thus we have the spectacle of math/CS papers that, were they chemistry papers, would read something like this: “First I took the test tube out of the cabinet. Then I rinsed it. Then I filled it with the solution. Then I placed it on the bunsen burner…” For whom are such papers written? The author’s high-school teacher? God? I would think it obvious that the goal of writing a math paper should be to explain your results in just enough detail that your colleagues can “replicate” them — not in their labs or their computers, but in their minds.

The bottom line, of course, is that math and CS are similar to biology and physics in the most important sense: they bite back. Granted, you might be sitting in your armchair when you do them, but at least you’re probably leaning forward in the armchair, scribbling on a piece of paper and willing to be surprised by what you find there.

This seems like an appropriate time to quote the distinguished American philosopher Dave Barry.

Here is a very important piece of advice: be sure to choose a major that does not involve Known Facts and Right Answers. This means you must not major in mathematics, physics, biology, or chemistry, because these subjects involve actual facts. If, for example, you major in mathematics, you’re going to wander into class one day and the professor will say: “Define the cosine integer of the quadrant of a rhomboid binary axis, and extrapolate your result to five significant vertices.” If you don’t come up with exactly the answer the professor has in mind, you fail. The same is true of chemistry: if you write in your exam book that carbon and hydrogen combine to form oak, your professor will flunk you. He wants you to come up with the same answer he and all the other chemists have agreed on. Scientists are extremely snotty about this.

And, since I can’t resist, here’s a classic joke.

The dean summons the physics department chair to his office. “You people are bankrupting us!” he fumes. “Why do you need all this expensive equipment? All the mathematicians ever ask for is pencils, paper, and erasers. And the philosophers are better still: they don’t even ask for erasers!”