While I sympathize with the opening, because many neat things have been made/discovered without the person having any formal math knowledge like what the "U"-looking symbol in an equation stands for:
>"The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols."
I heavily disagree with the conclusion:
>"They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly.
>And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface."
I think the notion that they're unusable by the vast majority of people because of something fundamental about people is false. At some point in time reading and writing were not done by the vast majority of people, then they came into existence. Even as recent as 500 years ago, the vast majority of the population neither read nor wrote. Along came public education and that proportion flip flopped insanely fast such that the vast majority are capable of reading and writing (regardless of how good they are at it). Reading and writing were just as radical novelties as computing, just because something is a radical novelty doesn't mean most humans can't be proficient at it eventually.
I think we should teach everyone to use the command line. Well, not Windows' CMD.EXE, but bash preferably in a gnome-terminal. Here is a short quote expressing why I think this is a good idea:
>Linux supports the notion of a command line or a shell for the same reason that only children read books with only pictures in them. Language, be it English or something else, is the only tool flexible enough to accomplish a sufficiently broad range of tasks. -- Bill Garrett
I think we should teach everyone how to interact with the computer at the most general way--which means programming. Which means commanding the computer. Describing the inverse square law in terms of pictures and intuition isn't going to make it any more of a tool than some other method, people are still going to think of it as something one is taught. The only way to make it seem like a tool is to use it as a tool, this means programming for some purpose. Maybe a physics simulation. And the beauty of tools is why the software world has exploded with utility despite Dijkstra's depression at programmers' inability to program extremely well. The beauty of tools is that they can be used without understanding the tool, just what the tool is useful for.
The "Things Other People Have Said" at the end is more interesting than the essay.
I wonder what Dijkstra would think of it. My two best guesses are "This is just a continuation of infantizing everything" and "We did kill math, with programming." I think a lot of people's difficulties with symbolic manipulation are due to the symbols not having immediately interpreted meaning. Dijkstra seems to recommend fixing this by drilling symbol manipulation of the predicate calculus with uninterpreted symbols like "black" and "white". My own approach I have been using more and more is to just use longer variable names in my math, whether written or typed. It really seems like this simple step can be a tremendous aid in understanding what's going on.
Over the past couple of years I've realized just how important sheer memorization can be as I see almost everyone around me struggle with basic calculus facts, which means they struggle with application of the facts. The latest example from a few weeks ago in a junior level stats course with calc 2 prerequisite (which many people take 2 or 3 times here apparently) was when apparently no one but me recognized (or bothered to speak up after 20 seconds) that (1+1/x)^x limited to infinity is the definition of e, which we then immediately used with (1 - const/x)^x = e^(-const). (Which is immediately related to the continual compound interest formula that changes (1+r/n)^nt to e^rt as n->infty which everyone should have seen multiple times in algebra 2 / pre calc when learning about logarithms! The two most common examples are money and radioactive decay.)
Sure that's a memorized fact for me, it's not necessarily intuitive apart from the 1/x being related to logs which are related to e, but come on. In my calc 3 course that I didn't get to waive out of, where it was just me and two others, I was the only one who passed, and when the teacher would try to make the class more interactive by asking us to complete-the-step instead of just a pure lecture the other two were seemingly incapable at remembering even the most basic derivative/integral pairs like D[sin(x)]=cos(x). A lot of 'education' is students cramming for and regurgitating on a test where the teacher hopes (often in vain) that the cramming resulted in long-term memorization. I do this too, cram and forget, but maybe I just do it less frequently or for less important subjects or my brain's wired to memorize things more easily than average (but I disfavor hypotheses that suppose unexplained gaps in human ability as largely static). My high school teacher's pre-calc and calc courses had frequent pure memorization tests such that there wasn't a need to cram elsewhere because that iterative cramming was enough to get long-term memorization (with cache warmups every so often; I did have to breeze through a calculus book in preparation for a waiver exam last year).
A person can only hold so many things in their active, conscious memory at once (some people think it's only 7 plus or minus 3 but that seems like too weird a definition of 'thing' that results in so little; human brains are not CPUs with only registers r0 to r6), so when you start looking at a math proof with x's and y's and r's and greek letters and other uninterpreted single-character symbols all over the place, it's incredibly easy to get lost. If you start in the middle, if your brain hasn't memorized that "x means this, y means this" for any value of "this", you have to do a conscious lookup and dereference the symbol to some meaning even if the actual math itself is clear. My control systems book uses C and R for output/input (I already forgot which is which) but it's so embedded in my brain that Y is typically output and X is input that I use those instead, as does the teacher. I agree with Dijkstra that practice at manipulating uninterpreted symbols makes it a bit easier, but there are quickly diminishing returns and ever since I started programming and saw how PHP uses the $var syntax to denote variables I've been thinking "Why hasn't this caught on in math? It makes so many things much clearer!" But it's not so much the "$" in $var (or @var and other sigils in Perl) but the "var". Saving your brain a dereference step is pretty useful.
Single-character symbols (and multiple character symbols) that not only hide tons of meaning through a dereference, which is naturally how language works, that also suggest a meaning themselves, are stupid and promote natural diseased human thinking. My poster-child here is global warming. Every winter you'll hear the same comments on "When's global warming going to hit here?" The poor choice of global warming as a variable that points to a bigger theory has cost it heavily in PR because humans look at only the phrase instead of what it points to, it's still so bad that people's every day experiences, which they use to form a subconscious prior probability in global warming as a theory, contain "it's really hot all the time and everywhere I've been!" and so they look at "global warming" and dismiss any evidence for/against it based purely on the name and how their experience seems to contradict the name. It's like a computer thinking that a pointer address that happens to correspond to 0xADD means it should invoke addition when it should figure out what 0xADD points to instead which could be anything.
Another use of long names is with Bayes' Theorem in probability. It is only through memorization of the uninterpreted symbols A,B,C themselves that I remember prob(A | B, C) = prob(A | C) * prob(B | A, C) / prob(B | C) But it never made intuitive sense to me and I never bothered memorizing until it was expressed as prob(hypothesis | data, bg_info) = prob(hypothesis | bg_info) * prob(data | hypothesis, bg_info) / prob(data | bg_info). (Sometimes data is replaced by model.) The notion that it relates reason with reality, that it encapsulates the process of learning and scientific reasoning, elevates the equation to the status of a tool instead of something you cram for and regurgitate on a test. An immediately useful application for the tool is using Naive Bayes to filter spam.