Calculus Is A Weird Anachronism

:date: 2016-08-09 19:18 :tags:

Here's a parody of a calculus problem for you.

dQ/dt = du/dt - di/dt + M

I don't know how to solve it but I know enough to know it's not really a proper calculus problem. In this equation Q is quality of life, u is the utility of calculus, and i is the investment one makes in developing a calculus proficiency sufficient for u. M is the intrinsic motivation to learn and be knowledgeable about calculus; mine is used up! Although this equation is quite silly it parallels all real world textbook problems involving calculus by distorting the situation into an absurd simplification.

I consider this equation a very dubious justification for the extraordinary emphasis placed on calculus in the educational system I was (and still am) a part of. For me, calculus was no small investment (i). Between high school calculus and a university engineering degree, I was studying calculus for about 3 solid years. And although I don't have a typical engineering career, I am horrified that I have put my calculus training to good practical use exactly zero times in my life (u). The reason for this, perhaps the reason I intuitively let myself not be as proficient at calculus as possible, is that I believe that calculus is never essential if you have access to a computer. And if you do have access to a computer (which is everybody reading this), calculus is actually irritatingly counter-intuitive because it implies some wrongish things about how to best model the world (an assumption of analog, for example). I'm not talking about using a computer instead of calculus to "get answers" the way a pocket calculator (app?) can do basic arithmetic. A pocket calculator does not replace the need to understand arithmetic but numerical solutions with a computer do obviate the need to understand calculus.

Hokey religions and ancient weapons are no match for a good blaster.

Apologists argue that calculus may not be super useful but that the equation above is close enough that a radical restructuring of education wouldn't be worth a mere quibble. The problem with this equation, however, is that it is missing a very important term. Here's the better version.

dQ/dt = du/dt - di/dt - dc/dt + M

Here c represents opportunity costs. In less economic terms, critics of calculus must answer the question, "What should we be teaching/learning instead?" There is no argument that calculus is useful. It is, just not very. This implies there are better ways to spend our time. There are, many. Besides potential engineers and physicists, I can't think of any reason for high school students to learn calculus that is better than the reasons to learn the following things.

Just as we have stopped beating kids with the lash (not sure about Texas), sometimes a society needs to accept that it's on the wrong track and abandon the cultural script that mindlessly proscribes suboptimal practices. Although there is gathering momentum for calculus reform, by speaking out against this educational hazing I'm doing what I can to break with our obsolete past. In an ideal world everyone would learn calculus - right after the thousand other worthier subjects.

Update 2018-05-11

I just discovered Computer-Based Math which seems like a Wolfram family project. Here's a talk by Conrad Wolfram about how weirdly out of touch today's math education is. His vision is slightly different than mine, but we both are in complete agreement that the current way math is taught is absurd.

Update 2020-04-14

More from the Wolfram family. Stephen Wolfram is either a foolish crank or should be hailed as the successor to Einstein. In this very long essay he demonstrates that he has no lack of serious work ethic when it comes to solving the mysteries of physics. Where you must give the guy respect is that the illustrations he produces are so clear and densely brilliant that its impossible to say he's an ordinary foolish crank. I have read his NKS and tend to think his chances of discovering something interesting are both tiny and far greater than anyone else's in theoretical math and/or physics.

And Stephen Wolfram proclaims...

Needless to say, people have thought that space might ultimately be discrete ever since antiquity. But in modern physics there was never a way to make it work — and anyway it was much more convenient for it to be continuous, so one could use calculus. But now it’s looking like the idea of space being discrete is actually crucial to getting a fundamental theory of physics.

Ahem, calculus. Seriously, go look at the images at that link and tell me this guy is an idiot.

Update 2020-12-28

Looking over the Wikipedia entry for Loss Function I find the following interesting argument apparently by founder of industrial engineering and that cranky Black Swan guy  —  probably not a contemporaneous collaboration.

W. Edwards Deming and Nassim Nicholas Taleb argue that empirical reality, not nice mathematical properties, should be the sole basis for selecting loss functions, and real losses often aren't mathematically nice and aren't differentiable, continuous, symmetric, etc. For example, a person who arrives before a plane gate closure can still make the plane, but a person who arrives after can not, a discontinuity and asymmetry which makes arriving slightly late much more costly than arriving slightly early. In drug dosing, the cost of too little drug may be lack of efficacy, while the cost of too much may be tolerable toxicity, another example of asymmetry. Traffic, pipes, beams, ecologies, climates, etc. may tolerate increased load or stress with little noticeable change up to a point, then become backed up or break catastrophically. These situations, Deming and Taleb argue, are common in real-life problems, perhaps more common than classical smooth, continuous, symmetric, differentials cases.

Update 2022-03-06

More Stephen Wolfram. And a quick reminder that he is smarter than you or me with respect to this topic. Here he is being interviewed by Lex Fridman and he talks about the critical deficiencies of calculus.

When people invented calculus 300 years ago, calculus was the story of understanding change, and change as a function of a variable. So people study univariate calculus, they study mulitvariate calculus  —  it's one variable, it's two variables, three variables, but who ever studied 2.5 variable calculus? Turns out, nobody. Turns out but what we need to have to understand these fractional dimensional spaces... they're spaces where the effective dimension is not an integer.