There was a flurry of discussion recently on campus about "critical thinking," and how we sell that idea to prospective and current students. This was prompted by a recent report arguing for the importance of the humanities and social sciences (which I found really frustrating in ways that are neither surprising nor important for this post). This eventually led to a meeting on Monday this week to discuss this sort of thing, in the course of which one of our Deans mentioned an abandoned project to collect statements about the modes of thinking associated with particular disciplines, which I thought was the most interesting thing to come out of the whole business.

One of the examples she gave was from a mathematician, who described solving problems in math as a process of taking a big problem and breaking it down into smaller and smaller pieces, solving the pieces, then putting them back together. Which is not that different from the quick overview of physics I give students on the first day of intro mechanics, a coincidence that got me to thinking about what it means to think like a physicist.

That math description is probably fairly reasonable for scientific problem-solving in general-- the various scientific fields have a lot in common, and a generally reductive approach is one of those things. Scientists in general tend to look at a big problem as a bunch of little problems that can be individually attacked and then combined to provide some answer to the larger issue. This is arguably different from the approach common to some humanities disciplines, where they tend to view things somewhat more holistically, with big problems being intrinsically Big and interconnected in a way that defies attempts to break off pieces.

But I think there are some differences in fundamental approach that set the different fields of science apart from one another, though I find them kind of hard to articulate (thus the rambling equivocation in this thinking-out-loud blog post). At least, I have a sense from teaching tons of majors from other departments and from interactions with colleagues in other sciences that there are real differences between the approach I take and the approach they take, that I suspect are attributable to my background as a physicist.

I think it comes back to that reductive approach, which I suspect is taken to greater lengths in physics than many other sciences. By which I mean that physicists, even more than other scientists, are inclined to abstract things away to get to the simplest and most universal principles. Even when that means that we end up working mostly with idealizations and first-order approximations. There's a reason why the "Assume a spherical cow" joke is nerd-funny, after all-- physicists really are prone to approximating complex objects as spheres, and spheres as points.

Going along with that, though, is the realization that a lot of the time, an approximation is good enough. There's another terrible science joke I heard once, where a mathematician and a physicist are brought in for a psychology experiment. The mathematician is led into a room containing a bed and a chair, and told to sit in the chair, whereupon an exceedingly attractive individual of the appropriate gender enters, naked and ready for action, as it were, and lies down on the bed. The psychologist then explains the terms of the experiment: after five minutes, the chair will be moved half the distance to the bed. Five minutes later, it will be moved half the remaining distance, and so on.

"What?!?" the mathematician explodes in indignation. "I'll never get to the bed! That's outrageous!" The mathematician storms out in a huff. Then the physicist is brought in, and given the same explanation. "Sounds good!" says the physicist. "Let's get started!"

The surprised psychologist says "You do realize that you'll never get all the way there, don't you?" "Oh, sure," says the physicist. "But I'll get close enough for all practical purposes!"

(I did say it was a terrible joke, right?)

Like the "spherical cow" joke, that captures something of the physicist mindset. The reason why we treat cows as spheres is that it's often good enough for practical purposes. By the time you figure in measurement uncertainties and all the rest, the difference between the spherical-cow model and what you actually see probably isn't too far off. You can see a lot of this in action at Rhet's awesome Dot Physics blog, where he does a lot of simple calculations making these kinds of approximations, and gets pretty close.

Another variant of this is the "Fermi problem" game-- the image above comes from this page of Fermi problems. The basic idea is associated with the great Italian physicist Enrico Fermi, who used to pose seemingly unanswerable questions as a sort of game. The idea is to use general intuition and round numbers to try to find a quantitative answer, which usually turns out to be pretty good. A good Fermi problem answer doesn't use any exact numbers, but it's rare to find a situation where an educated guess is off by more than an order of magnitude. Throw a few such guesses together and some are bound to be a little high, others a little low, and you end up in the right general ballpark, sometimes even in the right section and row of seats.

Physicists seem to do a lot more of this kind of back-of-the-envelope estimating than even other scientists. To say nothing of things like dimensional analysis, another of the great no-knowledge-required tricks for arriving at answers-- Sean Carroll's explanation of "effective" field theories is a great example.

The textbook we use for introductory courses has a couple of sections that use these tricks-- there's a bit where they get a value for the speed of sound in a solid from dimensional analysis, and a couple of problems where they estimate things. I try to make a point of presenting these bits (though several of my colleagues just skip them) because I think they're important illustrations of the physics approach. Students (the vast majority of whom are engineers and chemists) invariably look at me like I've sprouted an extra head when I do dimensional analysis tricks, though, and whenever I assign a problem asking for an estimate, I'm all but guaranteed to get answers reported to all the digits that a calculator can muster, which misses the point.

But I've also had this happen even with other faculty from science and engineering departments. I've had several meetings where I've done some back-of-the-envelope toy model to check the plausibility of something or another, and get baffled stares from everybody else. Or arguments about how the round numbers I used weren't exact ("But we don't have 600 students in the first-year class. There are only 587 of them...") It was a real shock the first time that happened, because I've always thought of that as a general science trick, but I'm coming around to the idea that it's really more of a physicist trick. And maybe, if you're looking for an explanation of what it means to think like a physicist, specifically, that might be the place to look.

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A bit deeper (and much longer) description of "how to think like a rationalist": http://wiki.lesswrong.com/wiki/Reductionism_(sequence)

Amusingly enough: although I learned it from my father, who (now retired) taught physics, I think of a heavy reliance on estimates as an engineering trait. Partly it comes from areas of electrical engineering where the tolerances of your parts are several percent, but some of it was explicitly trained into me as well. I remember a required course when I was at RPI, called "Engineering Modeling and Design"; the final included a question along the lines of estimating the amount of water needed to coat the north face of a ski mountain.

More recently, I've run Fermi Questions events at the state level (in Rhode Island, so the scale isn't very large) of the Science Olympiad, and I'm always disappointed that there are students who want to use calculators. It's kind of missing the whole point.

This is arguably different from the approach common to some humanities disciplines, where they tend to view things somewhat more holistically, with big problems being intrinsically Big and interconnected in a way that defies attempts to break off pieces.

There are times when physicists behave like that, too: in cases where the coupling between elements of a system is too strong to be ignored. QCD is an example. The key point is that physics gives you tools to tell you when you have to consider the coupling between components of a system and when you can safely ignore them. People studying humanities fields have to decide at some point what interactions to consider and what to ignore, but that's always a judgment call that has to be made on a case-by-case basis. For instance, the internal politics of the Chinese Empire in the early 15th century probably had a bigger effect than the Norse settlements in Greenland (which were dying out around then) did on European settlement of the Americas in the 16th and 17th centuries.

"The reason why we treat cows as spheres is that it’s often good enough for practical purposes."

It must be very hard to see real cows from the top of the tower... I am afraid down here in the real world treating them as spheres is completely counterproductive for *all* practical purposes, that's the whole point of the joke after all.

http://en.wikipedia.org/wiki/Cattle

... and the point sails completely over Arv's head...

The old one about a mathematician, physicist, engineer, and accountant, which class of numbers, order of magnitude, number of digits, and what you would like the answer to be.

A spherical cow is much easier to tip over.

Hardly! These cows aren't just perfectly spherical, they're also frictionless and their mass is uniformly distributed. Just defining what it means to "tip over" isn't easy, and actually doing so may be impossible.

Chad Orzel wrote (July 9, 2013):

> […] that physicists, even more than other scientists, are inclined to abstract things away to get to the simplest and most universal principles.

Since simplicity and universality aren’t necessarily and always compatible, physicists (perhaps more than engineers) would rather compromise simplicity, than universality.

(Just what exactly is that fitting quote again? … &).

> The reason why we treat cows as spheres is that it’s often good enough for practical purposes.

The justification for using (or rejecting) some approximation _confidently_ for incidental “practical purposes” is to understand its limitations, too; being able, at least in principle, to judge under _any/universal_ circumstances whether it is (or it is not) “good enough”, and to quantify the error thereby incurred.

This way of thinking (and doing simple order of magnitude calculations) is why I like to have my upper administration have a physics (or at least a science) background.

This is amazing, and in a bad way. I learned dimensional analysis in HIGH SCHOOL as a simple way to check for an error in calculation. If you did the same math on the dimensions as you did on the numbers and if the resulting dimension is the dimension you were expecting, there's a good chance your numeric answer is right also. If not, the numeric result is almost certainly wrong. In engineering school I learned the value of BOTE calculations, and also of doing arithmetic to one or two digits in my mind - if I can keep track of the powers of 10, I can get the result faster than I can key it into a calculator. Such techniques were reinforced from the works of others, such as Bob Pease's engineering column and Richard Feynman's popular writings. Too many people, apparently even in science and engineering, don't see any value (or have a clue how to) double-check a calculation, even to the "first approximation" of guessing the order of magnitude of the answer.

I don't expect humanities students or graduates to be able to sum two and two (perhaps they got the answer from Orwell's "1984"), but I have much higher expectations of those in technical fields.

I always wanted to be a scientist or mathematician, but I settled for being an engineer.

Acknowledging that all generalizations are wrong, the tendency of students to over-rely on the numerical precision of their calculator results in a prevalent inability to identify a key stroke error due to a lack of 'close enough for practical purposes' mental estimate and compounds the error of failing to judge the relative precision of their measurement or input data. This problem unfortunately is not limited to students, but frequently extends into the public square where politicians, the media, and innocently ignorant citizens frolic about in their policy making and legislative agendas.