It’s science if it bites back

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!”

33 Responses to “It’s science if it bites back”

  1. David Molnar Says:

    It’s my impression that one of the reasons we have such an emphasis on rigor in math (and by extension, theoretical computer science) is that our forebears were burned by “results” that turned out not to be true and other spectacular failures of intuition. Rigor is our way of cutting down on these unfortunate incidents. Now, as an empirical question, I wonder if we have fewer false theorems now than in the 1800s..?

    (or, cynical interpretation, we have
    roughly the same proportion, but the absolute number discovered has stayed the same, leading to the illusion of fewer problems?)

    Shai Halevi has an interesting proposal, by the way, for combining rigor and readability in cryptography. (Now there’s a science that bites back…) The main idea is to use an automated proof assistant for cryptographic proofs. Some parts of the proof are machine-checked, while the rest of the proof proceeds in the standard hand-verified style.
    http://eprint.iacr.org/2005/181

  2. Miss HT Psych Says:

    Immanuel Kant in his Critique of Pure Logic (and several other philosphers, as well) considered math to be THE science, and everything else can never hope to be in the same league (this includes physics, biology, chemistry, physiology and psychology). I wonder what your teacher would have said to that…

  3. Miss HT Psych Says:

    Oops… that should have said “Critique of Pure Reason”

    I highly recommend reading this… Kant is just brilliant (I don’t always agree with him, but I am continually impressed by his thought processes)

  4. Scott Says:

    Miss HT: Thanks for the suggestion! I confess that I haven’t tackled Kant yet, but you’ve reminded me that I kan’t put it off forever…

  5. Scott Says:

    David: “Results” that turn out to be false are a commonplace in every other science. And unless we formalize everything in ZF set theory, we’re never going to get rid of them completely in math either. So maybe we should just adopt the chemists’ and physicists’ model, and see shooting down wrong results as part of the fun!

  6. Anonymous Says:

    Perhaps in Math and TCS we don’t need to spend a lot of time doing experiments, so we can make up for it by writing things for rigorously.

    There’s also an additional advantage to emphasizing rigor. Not everybody has a talent for writing and explaining, and such a skill is hard to teach and also hard to objectively evaluate. However, everybody can sit down and write a rigorous proof if they work hard enough. It’s also easier for referees to demand complete proofs than to dmeand this intangible quality of good writing. Now, if someone has no writing talent I won’t trust them to know which details to skip and which to keep, and I’d rather they at least wrote down all of the details.

  7. Michael Mitzenmacher Says:

    It’s always nice to hear stories of high school teachers that inspire their students to reach great heights.

  8. Anonymous Says:

    – Now, as an empirical question, I wonder if we have fewer false theorems now than in the 1800s..?

    We do, for sure. Mathematicians started the great process of axiomatization and formalization in the XIX century, upon discovering that so many things they had naturally assumed were not true, starting with Euclid’s fifth postulate. Each generation would introduce more rigor than the previous one and come ahead for it… until Bourbaki that is, when finally top mathematicians decided to go all the way start from scratch and rewrite all mathematics from the foundations up in a formal way.

    What came out of that was a set of incomprehensible volumes that failed to prove or disprove any significant number of theorems. It took some time for people to realize this, but eventually a consensus arose that this last round of formalization was unnecessary and that there was a need to roll back to the just-before-Bourbaki standard, which is where we are still at now.

    Alex Lopez-Ortiz

  9. Anonymous Says:

    The problem of course is that in the standard modern picture, science is empirical.

    Currently the definition of science has became stuck on lab coats. If you wear a lab coat, you are a scientist, if you don’t, you aren’t.

    This is a shallow definition of what science is yet it seems widely accepted, even among scientists and mathematicians.

    The “experimental” definition of science excludes for example, astronomers, who are otherwise universally accepted as scientists.

    This is the stage when “experimentalists” pop up with the alternative definition of “empirical” or “connected to the real world”.

    Yeah, right, because 4+3=7 is such an entirely made up convention unsupported by actual facts. Or if you compute the integral of volume under a surface and then build a vessel with that shape the amount of water it holds is completely unconnected to what was computed using integral calculus.

    The fact that Witten has been able to propose deep conjectures in advanced mathematics using his physics insight (what some people call “theorems with physical proofs”), is testament that this connection to the real word is still there even at the deepest levels of modern mathematics.

    Alex Lopez-Ortiz

  10. Who Says:

    Scott,
    now you’re recovered you might enjoy looking at

    http://arxiv.org/abs/quant-ph/0511096
    A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

    Dorit Aharonov, Vaughan Jones, Zeph Landau
    26 pages

    “The Jones polynmial, discovered in 1984, is an important knot invariant in topology, which is intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et. al are heavily based on deep knowledge of TQFT, which makes the algorithm essentially inaccessible for computer scientists.
    We provide an explicit and simple polynomial algorithm to approximate…
    The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems…”

  11. Dave Bacon Says:

    Man I could go on about this subject for hours. Especially over beer. But instead I will make one silly comment.

    I think your comment
    “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.”
    is totally off mark. For example, how can this possibly be true of “A record of Permian subaqueous vent activity in southeastern Brazil” or “Fruitless specifies sexually dimorphic neural circuitry in the Drosophila brain” or “Cis–trans isomerization at a proline opens the pore of a neurotransmitter-gated ion channel”, just to cite a few recent Nature articles, possibly be achieved by your algorithm? ;)

  12. Anonymous Says:

    One would think that a subject like theoretical physics would be part of mathematics but in my brief encounters with statistical physics (as applied to CS theory problems) it seems that the methods are quite different from what one would find in a mathematics or CS theory. There seem to be 3 levels of results:
    1. Those based on some hypothesis (‘ansatz’) that is expected to lead to approximately plausible results (but may not be sound mathematically), and associated approximations (e.g. calculations based on partial sums of an infinite series that is not known to converge.)
    2. Exact: Those based on some similar ansatz(s) that may not be sound mathematically but whose consequences follow rigorously using mathematical proof (the ansatz can include the statement that a certain procedure leads to a plausible result in this case.
    3. Rigorous: Consequences follow rigorously without an ansatz.

    Often method 2 has led to interesting insights despite its lack of rigor at the start. Math might benefit by being more open about using such hypotheses.

  13. zoss Says:

    So Scott, would you be the one who has just recalled “the mind body problem” from the uwaterloo library? cause I was reading it and I was stuck for two months at page 195 until somebody recalled it and I had to finish it in a weekend — if it was you, thanks.

  14. Greg Kuperberg Says:

    Let me offer the analogy that mathematics is to science as floating currency is to backed currency. At first glance, backed currency (backed by gold or whatever) is the only credible kind. But as the currency is more and more widely used, its abstract properties become more and more important, until finally the currency is completely reliable and useful without backing.

    Likewise, science seems more important at first than mathematics, because it is backed by reality. But as you develop it, the abstract reasoning becomes more and more important, so that eventually it no longer has to be backed by reality to be interesting.

    To complete the analogy, many people have trouble believing in floating currency. They want to return to the unnecesssary “gold standard”. The same sort of people have trouble believing that mathematics done for its own sake is still much like science.

    Anyway, turning from idle philosophy to real research, I heard a talk on the paper quant-ph/0511096 that another commenter just mentioned. The paper seems reasonable, except that it should have a crucial disclaimer about the limitations of the main result.

    Namely, approximating the Jones polynomial of a braid is not the same as approximating the Jones polynomial of a knot. If you apply the main construction to more and more complicated presentations of any fixed knot, then the quality of the approximation decreases exponentially with the number of strands. This means that without new ideas, the algorithm is not useful to computational topology. Absent new ideas, the only clear use of the construction is as a model of quantum computation, and not as an algorithm to do anything specific.

  15. Scott Says:

    “I think your comment ‘We all know the recipe for getting a paper published in Science or Nature…’ is totally off mark.”

    Dave: Obviously I wasn’t talking about all Science and Nature papers, only some of the ones I’ve read! I wouldn’t recommend my recipe for papers about climate change or neurobiology, for example. As Lance suggested earlier, I think the long-term answer is to get members of the math/CS community on the editorial boards.

  16. Scott Says:

    “So Scott, would you be the one who has just recalled ‘the mind body problem’ from the uwaterloo library?”

    Nope, wasn’t me. I own a copy.

  17. Scott Says:

    Who and Greg: Unfortunately there are many interesting papers that I can’t discuss on this blog right now, since I’m on the STOC’06 program committee. :)

  18. Robin Hanson Says:

    IMHO, your chemistry teacher deserves all the hate you have for her, and far more. I see my children slowly losing their love for learning at the hands of similar minds.

  19. Robin Hanson Says:

    Philosophers do use lots of erasers by the way. So do litcrits. Something bites those scolars back, to make them erase. The question is what exactly that is, and how much we care.

  20. Scott Says:

    “IMHO, your chemistry teacher deserves all the hate you have for her, and far more. I see my children slowly losing their love for learning at the hands of similar minds.”

    Thanks, Robin. I wish I had a surefire answer for you. At all the schools I attended, I had a few great teachers and a lot of awful ones. On the other hand, friends who went to magnet schools almost always report better experiences than I had. I know the Thomas Jefferson High School for Science & Technology is in your area; I would’ve loved to go there.

  21. Kyle Says:

    If you haven’t confronted this chem teacher yet, now is as good a time as any.

  22. Anonymous Says:

    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…

    You’ve said things like this before, but I have to disagree. The people I imagine you think of as your “colleagues” are the people who can indeed re-derive proofs from scratch in their heads. But have some mercy on the rest of us non-geniouses! Or, if you like, have mercy on the poor graduate students who will read your paper before they are familiar with what you consider a “standard proof technique.”

    Now, obviously, there has to be some tradeoff here or else we do get to the Bourbaki-level point where we prove things from first principles. Clearly there is some happy middle ground. If you force me to pin it down, I would say that assuming material covered in an intro graduate course in an area is acceptable.

    PS: I feel the same way about papers in the physical sciences. If you have ever tried to replicate an experiemnt based on the description in a paper, you will appreciate what I mean.

    An additional point, rasied by David Molnar, is that errors do in fact creep in to our papers to a rather alarming extent. At least that appears to be the case in cryptography (with the “obvious” proof of OAEP [that was later shown to be wrong] being a prime example).

  23. Richard Cleve Says:

    Greg Kuperberg said…

    Let me offer the analogy that mathematics is to science as floating currency is to backed currency.

    That’s a beautiful analogy!

  24. Bram Says:

    Everything I’ve ever read by Wittgenstein seemed like gibberish to me. Rebecca Goldstein’s summaries of it seem like equal gibberish (although I assume there’s something to them, because her summaries of Godel are quite accurate).

    There’s a nascent movement for mathematicians to do all their work formally, as in, verifiable by computer. The mathematicians say this is a waste of time. The programmers think the mathematicians should stop whining. I personally think doing everything computer-verifiably would be an enormous boon, because it (1) would shorten the review process to almost nothing, or allow it to be skipped completely (2) would get rid of the mistaken results which still crop up from time to time, and most importantly (3) would allow one to write explanatory papers which, completely unburdened by the needs of formality, could provide some intuition, context, and commentary. These days the standard way to write a math paper is to take the proof, remove all the insight, and write up what’s left, and that’s a bad thing (not my witticism).

    Both of the claims you mention, “quantum computers are really just analog computers,” and “all cellular automata that aren’t obviously simple are Turing-complete,” happen to be false. I’m not sure if that was your point about them.

  25. Anonymous Says:

    – There’s a nascent movement for mathematicians to do all their work formally, as in, verifiable by computer.

    As automated theorem provers get better and have more powerful primitives, making a proof computer verifiable wouldn’t be any more painful than writing the paper in Latex to begin with.

    In fact it might turn out to be easier, as some rather cumbersome proofs can now be skipped, something along the lines: “the theorem prover agrees and the only proof I have is entirely pedestrian and unelightening, so let’s not waste each other’s time and skip it”.

  26. Scott Says:

    Bram: “Both of the claims you mention, ‘quantum computers are really just analog computers,’ and ‘all cellular automata that aren’t obviously simple are Turing-complete,’ happen to be false. I’m not sure if that was your point about them.”

    It was. The problem, in my experience, is that people who advocate these claims tend to do so in a slippery, unfalsifiable way. Every nontrivial theorem that goes against the claims is met by an excuse, rather than by a counter-theorem supporting them.

  27. Scott Says:

    “Everything I’ve ever read by Wittgenstein seemed like gibberish to me.”

    Tell me about it. Yet there’s clearly an audience ready to wolf down gibberish with mustard: Wittgenstein is far more respected these days than Russell, who repeatedly made the mistake of being clear.

  28. Bram Says:

    Scott, the claim about cellular automata is false in a way so spectacular that even wolfram couldn’t deny it. Unfortunately it will require a quite non-trivial proof to demonstrate that though.

    Basically, if you play around with wolfram’s automata for quite a while (as I have) it becomes fairly obvious that certain of them correspond to strictly weaker models of calculation than general turing machines (I’m guessing presburger arithmetic). One of them happens to tie for grungiest-looking on random inputs with the one which an employee of wolfram’s (most definitely not wolfram himself, despite the impression ANKOS may give you) demonstrated to be ‘np-complete’ (wolfram doesn’t seem to understand encoding issues, but we can gloss over that as an uninteresting detail, despite the existence of a different interesting-looking automata for which the encoding issues are fairly serious because it can’t carry information to the left). In ANKOS, wolfram decides to single out this automata to talk in his usual chest-thumping way of how you can build a computer in it. He then gives an example run of it with a small set of living cells in a dead arena, which becomes patterned fairly quickly in an interesting way. He goes on to brag about how he’s done this for two million different starting configurations, but none of them did anything interesting. It doesn’t seem to have occured to him that maybe none of them did anything ‘interesting’ because such ‘interesting’ initial patterns just plain don’t exist. Oh the irony…

    Now we just have to get a proof that the give automata can’t build general computers. We unfortunately don’t have such a proof, but it’s well worth looking for – such a result would be far more surprising and interesting than anything in ankos.

  29. Anonymous Says:

    Re: Wittgenstein

    The sentence “Mathematical propositions express no thoughts” is dumb in the same way that “approximating the Jones polynomial of a braid” is dumb if you assume that the sentence is about hair. Wittgenstein uses the word “thoughts” as a technical term much in the same way that mathematicians use the word “braid” to mean something quite different from its common usage. Using common words as technical terms is nice because they are often evocative of their actual meaning, but they have the unfortunate effect of misleading people outside the field.

    As to whether or not Wittgenstein is “dumb” in the larger sense, I guess
    one is left to their own “reasons of the heart”. Any such debate, whether debating the status of the Christian Trinity, the morality of engaging in duties prescribed to a different caste, or the ontological status of mathematical entities, only matters if you care about Christianity, caste systems, or ontology and epistemology. If I don’t care a whit about Christianity then I could care less about the status of the Trinity.

    Now I’m sure that philosophers get a lot wrong about mathematics, but it’s pointless to criticize their writing unless one learns their language first. I would also say that it’s rather rash to assume that philosophers are talking nonsense if you can’t understand their language. I don’t understand a word when people talk about “BRST cohomologies” and “ghost numbers”, but I’m in no position to say whether those people are making sense or not. I also know that I have little interest in understanding the topic, not for any defendable reason, but once again, for reasons of the heart. I would say the same to philosophers who talk about mathematics of course.

    – Homin

  30. Anonymous Says:

    On another note…

    I would say that Kant was on to something when he described mathematics as being part of the strange case of “synthetic a priori” judgments. The failure of Russell and Whitehead seems to point to mathematics not being “analytic a priori” judgments. The “synthetic a priori” explanation is nice because then mathematics provides us with new information that is necessarily true, and thus explains the predictive power of the natural sciences, which would otherwise be purely “synthetic a posteriori” judgments.

    – Homin

  31. niel Says:

    Yeah, right, because 4+3=7 is such an entirely made up convention unsupported by actual facts.

    The most basic mathematics is quite obviously devised with the purpose of talking about the real world. But is all mathematics equally suitable for talking about every aspect of the world? For instance, no-one worth listening to argues that 4+3 != 7. But it counting days of the week, asserting 7 == 0 is useful, whereas in counting apples it isn’t. So is 7 “really” equal to 0, or not?

    My current answer is that 7 and 0 are both ideas used to understand the world, including apples and the time-measuring standards of humans. What you might like to assert as axioms about them depends entirely on what structures you are interested in (or if one is unrepentently a pure mathematician, one’s sense of aethetics).

    Math seems to me a tool for naming and describing structures: analogies stripped from the objects which are the analogs, or without requiring that analogs even exist in the world. The fact that this discipline is useful for the sciences indicates that the world is somewhat orderly, or at least that humans aren’t entirely delusional. Conversely, if the world is structured, should it be so surprising that physics inspires structures, as well as the intuition necessary to describe them?

    Yes, mathematics “merely a convention” — in the same sense that money is “merely a convention”, and in the same sense as gravity is “merely a theory”. Just because something is a convention does not mean that it cannot “bite back”.

    Something “bites back” precisely when it is a useful idea. But is every useful idea necessarily scientific?

  32. niel Says:

    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.

    To this, we may apply the populist solution: if you want to see math in the Times rather than proto-scientific reruns, take it up with the Times and encourage others to do so also.

  33. Anonymous Says:

    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.

    Remembering Arnoldi, I can imagine she’d be tremendously smugly satisfied that all of your successes after Gr. 10 were due to her inducement; the fact of your impetus to achieve being solely to be her anathema would be a wholly inconsequential detail. Deliciously ironic.