Street-Fighting Math: down and dirty guide to approximation and problem-solving

Street-Fighting Mathematics looks like a fun read: it's a Creative Commons-licensed math textbook that teaches approximation and "down-and-dirty, opportunistic problem solving." It's based on a MIT course taught by the author, Sanjoy Mahajan.
In problem solving, as in street fighting, rules are for fools: do whatever works--don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.

In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge--from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool--the general principle--from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest.

Street-Fighting Mathematics (Thanks, Musicman! via Submitterator)

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  1. Funny you should mention street fighting. When the US and Israel decided that the WWII model of grand battles was giving way to low-level asymmetric urban warfare they went looking for the Masters of Street Fighting. Apparently the Champ was some Puerto Rican street gang warlord. His philosophy was that you can’t go by pure logic because your opponent can do that too as well as deduce what you’ll be thinking. So everyone gets stuck in an infinite regression and nothing really happens.
    So what you have to do is take actions at random that have your goals as their strange attractor.

  2. I do this sort of thing all the time. Unit conversions (knowing how many seconds are in a year, for example) and one-significant-digit math can answer a lot of what-if questions handily.

    People look at me as if I’m some kind of freak for being able to come up with even an approximate value for some weird-ass thing, like how thick a layer of rubber piles up on the side of the road due to tire wear.

  3. The CC download is available from the MIT Press link that Cory gave at the bottom of the post but it took me a minute to find — and MIT Press didn’t help by including a separate link to a TOC and sample chapters (huh?) and by saying that the book “will appear” online.

    Anyway, the download is under “related links” in the left-hand column on the MIT Press page. Direct link to the PDF here: http://mitpress.mit.edu/books/full_pdfs/Street-Fighting_Mathematics.pdf

    1. Thanks for the direct link to the .pdf. I glanced at the publisher’s site but never saw it. Guess I just wasn’t looking closely enough.

  4. You can’t go by pure logic because your opponent can do that too as well as deduce what you’ll be thinking. So everyone gets stuck in an infinite regression and nothing really happens. So what you have to do is take actions at random that have your goals as their strange attractor.

    I’m reminded of a scene from Smilla’s Sense of Snow … about halfway into the novel (if memory serves), Smilla (who finds herself at the periphery of some murderous mystery) deliberately reveals herself to the bad guys … the narrative explains that sometimes, when you’re hunting reindeer (I think it was reindeer — Smilla’s mother was a native Greenlander), they won’t show themselves until you show yourself first.

    It seems counter-intuitive, but apparently you can get the drop on your intended prey by confusing them with irrational behavior.

  5. My favorite graduate math course — and maybe my favorite math course ever — was on mathematical modeling. It taught methods for approximating and simplifying those real-world problems that don’t line up to be nice and exactly solvable.

    I majored in physics. I quickly realized that once you step outside those idealized, “neglect the mass of the pulley” physics problem sets, the math gets impossible very quickly. And I mean really impossible, not just hard — you can write the differential equation, but you can never get an exact solution.

    But you can simplify things by looking at cases where something is very heavy or very light, or looking only at what happens in a very short or long time, and so forth. These are probably the “easy cases” that Mahajan talks about.

    Then you can try to patch together these cases by figuring out what must be happening at the boundary between short and long time, or the boundary between heavy and light things, etc.

    There are mathematically rigorous ways of doing this, using lots of calculus and infinite series and weird-looking Greek letters, so your work can look properly impressive to other scientists and mathematicians. But all you’re really doing is a more sophisticated back-of-the-envelope calculation, being a little more explicit about what you’re assuming and when your calculation applies.

    It’s fun stuff. I love solving these kinds of problems, whether I’m doing it the formal way or the informal way.

  6. “There are mathematically rigorous ways of doing this, using lots of calculus and infinite series and weird-looking Greek letters, so your work can look properly impressive to other scientists and mathematicians. ”

    Spoken like a physicist rather than an engineer. :)

    Mathematical precision is impossible, but in most of engineering the math and science lets you get REALLY damn close to predicting an outcome. All the fancy stuff serves a purpose beyond simply laying out the assumptions and looking impressive. It reduces your margin of error to a level where you can confidently say “good enough. now let’s make it twice as strong and we can be sure no one will die.”

  7. I love analogies but what does someone from MIT know about street fighting? I think it has come down to selling books with whatever means necessary.

    one rule of street fighting is to end it as fast as you can by attacking vulnerable areas.

    so I should attack my math problem in the vulnerable areas and solve them as soon as possible?

    1. “so I should attack my math problem in the vulnerable areas and solve them as soon as possible?”

      Actually, yeah, that’s a pretty good description of what the course (and, I assume, the book) is all about.

  8. aw, puleeeze, dear physicist, drop that analytical chauvinism. after all, all your fancy analytical functions have to be developed into a series somewhere, sometime. if i can solve it numerically, the problem is possible enough.

  9. So under the guise of getting away from mathematical rigor, the author instructs people on how most mathematicians that I know essentially work. Sneaky.

  10. I find that I do this a lot when tutoring. Even when students must learn very precise methods, teaching them a way to get a quick and dirty answer is good. Spend a minutes or two getting a half-way decent approximation gives them a tool to self-assess their precise answer.

    I don’t think anyone should ever do a complicated problem without always having in mind what a reasonable answer will be.

    Its also usually simple enough to do, even when you’re dunk and you stumble across a great deal on plastic balls on the internet and you quickly need to figure out how many you need to buy in order to fill up your friend’s apartment while he’s out of town.

  11. I’m not sure if this is “street-fighting” but you can tackle many simple maths word problems that have two variables by considering the real-life situation and doing something definite like asking:
    (a) Does doubling one variable halve the other? (constant product)
    (b) Does doubling one variable double the other?(constant ratio)
    (c) Does doubling one variable increase but not double the other?(constant difference)
    (d) Does doubling one variable decrease but not halve the other? (constant sum)
    This usually gives you equations you can solve.

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