The Long Assault on Intuition

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To anyone who has been paying attention, it has to be clear that what we call intuition has been under a steady attack.

Let's take law enforcement, for example.  It used to be that a policeman could accost anyone who looked suspicious, and subject him or her to an on-the-spot questioning.  Over the years, as the police force grew, and money grew tight, and for various reasons employment in the police force became attractive to people of questionable talent, it became possible to correlate the tendency of a policeman to think someone suspicious with the ethnicity of the suspect.  Meanwhile, of course, many more dollars were stolen by white middle-class males, I'm ashamed to say, than by any minority group, so the harassment of minorities was given a name: profiling.  What had happened was that the intuition of a law officer had become replaced by an objectionable rule of thumb: "Immigrants are suspicious, just because," and of course that would not be allowed to continue.

Mathematics is an area where intuition is very important and useful.  When I was a lad, my classmates and I were taught how to solve mathematical problems using at least 50% intuition.  But, as mathematics was seen to be important, teaching mathematics was an area that saw an influx of people who needed to teach more than they wanted to teach.  Furthermore, mathematicians became classified into categories such as pure mathematician and applied mathematician.  All mathematicians solved problems, of course, but while applied problems came from outside mathematics, pure problems came from withing mathematics: how could you find a such-and-such that did such-and-such?  I remember when I discovered the fascination of pure mathematical problems.  There was none of this dirtying your hands with principles of chemistry or physics; you looked at a problem just in order to perfect your technique.  Unfortunately, though, as pure mathematicians began to edge out the applied mathematicians from the calculus-teaching cadre, they systematically replaced the intuition with theorems, and mechanical rules of thumb.  Textbooks began to fill up with theorems (which were readily converted into rules of thumb), and proofs began to replace intuition.

Pure mathematicians had a ready defense of the gradual replacement of intuition with theorems: intuition is fallible, theorems are (if correctly proved) never wrong.  So, in theory, a class that survived the more theoretically oriented calculus course could do a few things perfectly.  They might not understand, in intuitive terms, the heuristic behind the logic, but they know the facts.

In the mid Eighties, the applied mathematicians rebelled.  They labeled the proof-based calculus approach as sterile and unintuitive, and pleaded to be allowed to teach calculus intuitively, so that their students would understand why --in intuitive ways; we know that logically everything is supposed to work as advertized-- why the methods make sense.  This was partially successful.  The compromise was that the subject should be presented (1) logically, (2) intuitively, (3) geometrically, (4) numerically, and (5) using technology, e.g. computers.  This was a tall order, but everyone bought into it, for a while.  But, over the years, instructors taught calculus any way they wanted to.  Quietly, the computer lab periods that used to accompany calculus were dropped, (hey, I did it, too) and calculus became highly theoretical again.  Now, the emphasis is on learning calculus via YouTube.  Online calculus teaching enthusiasts emphasize the intuitive approach; they can show lots of graphics, and the classroom instructor can only do that through a lot of work with software and PowerPoint; some instructors do it, others do not.

Along with the opening up of mathematics teaching opportunities, I also observed the power of students to challenge grades.  Why have I lost three points here?  Oh, I just thought your work was not up to standard.  But in exactly what way?  You see where you did B instead of A?  You do know that that's the wrong thing to do, don't you?  Oh, I see.  But why three points, and not two, for instance?

It became convenient --nay, necessary-- to invent rules for everything, so that everything was done mechanically.  There were rules about when a state trooper could pull over a car.  Rules about how many points you could deduct, because, of course, deducting points had to be defensible.  To every occupation rules were required.  On the one hand, this eliminated wayward and fanciful ways of doing things.  On the other hand, the rules empowered no-talent hacks to do the work of people with intuition and insight.  But, as we observe around us today, it is convenient to hire no-talent hacks  in preference to talented people because:
(1) No-talent hacks are predictable, and obey orders.
(2) No-talent hacks can be made to be consistent.
(3) No-talent hacks are not temperamental.
(4) No-talent hacks are (usually) happy with lower pay.  They cost less.
(5) No-talent hacks do not think for themselves.
(6) No-talent hacks don't usually ask troublesome questions, because they're really not interested in what they're doing.  They just do it for the nickel.

Consider music.  For centuries, music was an art, which is to say, you did it by ear.  But then, some clever fellow decided that he would actually make rules by which music could be written, enabling, famously, any idiot to be able to write correct harmony, or what have you.  And, many idiots did.  Mozart railed against inspiredly bad correct music.  Bach lost his temper against an idiot bassoonist, and drew his sword on him.  (Actually, we don't really know how the bassoonist had offended Bach.)

Around the time of Leonardo Da Vinci, and Albrecht Durer, artists were beginning to discover mathematical and geometrical rules for drawing representative art.  But it appears that artists were among the first to abandon the rules, because too many people were drawing according to the rules, and in a field crowded with paint-to-rule hacks, artists felt the need to jettison rules in favor of individualism.  Similar things happened in literature.  Musicians held onto rules longer than most others, and every time they tossed out one category of rules, they seemed to adopt an alternative set.  Because of the peculiar problems that a musician faces, there must be rules.  Communication through music is fragile, and a composer must be "commercial" to at least some degree, or he or she has no livelihood.

I think we must make room for intuition in today's world, even at the cost of alienating the critics.  On one hand, having rules makes the work of critics easier, because they merely have to assess to what extent the rules have been satisfied.  On the other hand, having no rules makes it easy for critics, too, because they can simply claim that a work has "no soul", for instance.  Whether you do have or don't have rules, there are winners and losers.  But if someone has an intuitive approach or skill about something, it seems to me that it is a capability that we must, in most cases, encourage.  It would be a terrible world if everyone dealt with everything according to rules alone.

Because of various missteps, I find myself teaching all my classes without the support of printed textbooks.  This means that I'm not at the mercy of the numerous rules that typical mathematics texts are full of, and can encourage my students to proceed using their intuition a lot of the time.  But, on the other hand, there are going to be numerous complaints that I did not provide rules for the students.  While I think that developing their intuition is the greatest service I can render my students, some of them think that this is all very well for students who have some intuition to start with, but what about the rest of them?  How are they going to go out into the schools and teach, if I don't supply them with all the rules they need?  Yes; it seems that teaching today proceeds on the wheels of rules, rather than intuition and understanding.  Schools, colleges and universities are full of people wanting to be taught without appeal to intuition.  Wanting to be taught things that they have absolutely no interest in, too.

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