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.

Vocational (machining, welding, plastics, ag) - Even if you aren't going to fix your own car or work in a factory, understanding the foundations of the most real parts of our civilization can't possibly be a complete waste of time. If I had to relinquish either my machine shop apprenticeship or my university engineering education, I would jettison the latter.

Language History - This is commonly called "Latin", but really learning some Old English and Latin and how English became to be like it is (throw in King James, Shakespeare, etc) turns out to be extremely useful in building solid communication skills. Many foreign speaking cultures now teach their kids English well enough to smoothly participate in the Anglosphere. Language history would help English speakers preserve an edge. For the same reason, I would mention enhanced grammar study as being much more useful than calculus but I understand that, unlike the colorful history of language, it's only slightly more interesting.

Art - Did you know that the BLS predicts that by 2024 "Arts, design, entertainment, sports, and media occupations" will increase by 4.1% (source)? Despite wasting so much of my life learning calculus I have enough math sense to notice that because the percent increase in total occupations is 6.5%, this is actually a per capita net loss. So we really should be focusing on engineering, right? Uh... No. That is projected to grow at only 2.7%. That's 50% more of a reason to support the arts over engineering. The US is a world leader in design, fashion, fine art, graphic art, digital art, performing art, movies, video games, photography, cuisine, typography, and sports. All despite calculus. We could reinvest in our waning art culture or leave it to other cultures to take over.

Music - And if American art has been a grand success, America has been to music what the 13th century Mongols were to Asia. Calculus is probably best learned well into adulthood if the need arises. Music is best learned when young. We neglect our true cultural legacy at our peril.

Home Economics - Are our real problems today a lack of people who can derive formulae for ballistics trajectories using 18th century techniques? Or is it that as a species we're all becoming depressingly unhealthy and fat? Empowering high scholars to make better decisions about food would surely pay society back far more than calculus. Or maybe offer kids the option of an hour of running or an hour of calculus lessons - obesity epidemic cured!

Personal Finance - How about helping hapless high school seniors out with the facts of life about debt before they take on those predatory student loans. And credit card debt, payday loans, adjustable rate mortgages, etc. To give them calculus instead is a shameful dirty trick.

Statistics - Proponents claim calculus is good mental exercise for later skills in technical fields that are essential. They also say the history of calculus is important for understanding modern technology. That's fine, but the warm up doesn't also have to be useless and disorienting. I have a lot of problems with statistics as it's normally taught (I had almost 2 solid years of that), but even if it's all completely bogus, it's still topical and essential for engaged discourse. Who knows, maybe if we treat it seriously we'll produce an Einstein type figure who will revolutionize the field and create a paradigm shift appropriate for the modern uses (e.g. quantum physics) which Bernoulli, Laplace, Gauss, and other early pioneers had no intention of contributing to.

Linear Algebra - "Think of the engineers!" cry the calculus apologists. Surely they need all this "math" just for good practice and it just might be useful. Bollocks. If we care about that, linear algebra is the way to go. If you're an engineer in 2016 and you're using Newtonian calculus way more than linear algebra, you're doing it wrong. In fact, if you're doing something serious and you're not using linear algebra to do your Newtonian calculus you're probably doing that wrong. Linear algebra has the delightful bonus property that it teaches itself to many adolescents (and adults) - "Hey, who'd like to make a 3d video game?"

Numerical Analysis - Less interesting than linear algebra but way more useful than calculus is numerical analysis. If we're compulsively fetishizing the cult of personality of Sir Isaac then numerical analysis is ideal.

Information Theory - "No, no, we need calculus because $BetterAlternative is too easy." If this is your feeling, that high school kids need to be tormented with a weird subject that is incomprehensible in any way to normal people, then information theory is ironically hard to communicate to students. Honestly, it's not a great idea to add information theory to a standard high school curriculum but it would be much better (useful/interesting) than calculus! If we could just broadly teach people that "password" is not a good password, it'd be a good trade.

Computer Science/Programming - It may not have been obvious in 1953 when the first transistorized computers were built that we, as a society, should immediately scrap calculus and instead focus on these miraculous new tools. But it is obvious now! You may never need to know how a linked list works, but if you're reading this, you're using a profusion of them right now. And that's esoteric computer science mumbo jumbo that may or may not be useful. The starkly obvious real-world potential utility of computers that goes untapped because of a lack of education is pathological.

Philosophy/Ethics - A common justification for calculus (here's one) is that it helps "teach people to think", including logic, problem solving, etc. I believe that if we have to disguise the study of philosophy as calculus (Newton and Leibniz did the reverse by the way) then that itself is terrible philosophy and proof that we have a problem. Just teach philosophy! It's worth it!

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.