On intuition and learning, or, why art matters
Art loves chains
This little blurb is just a reflection on what I feel to be a
underappreciated and mistunderstood facet of learning any subject.
admit it might be just peculiar to me but I've had such great and good
success with it I thought I'd share. I've used this approach over
decades of teaching martial arts and Mathematics at the university
level. As with all things understand it if
you choose and let me know if it jibes with your experience.
Three important definitions
What is art?
There are a few definitions in my dictionary. These are
The last one is what I think is the more important for what I'm writing
However, the first with its inclusion of aesthetic value is important
What I am trying to get here is the idea that one has skill that is
acquired and an aesthetic sensibility
for what is right and not.
- High quality of conception or execution, as found in works of
beauty; aesthetic value
- Skill in doing or performing that is attained by study, practice,
Intuition and what it's not
Hand in hand with art as aesthetic-based skill goes intuition, which my
The more common use is that one 'just knows' something. We see this in
the popular mindset all
the time whereby someone has a hunch with no supporting facts or prior
knowledge. This borders
often on the supernatural. Since I managed to avoid being gifted in
this way, I must plod
on with reality as best I can. Crucial here is that intuition can
be tutored, in other
words you may work with something enough to be able to hop over much
This is one of the foundations of skill.
- The act or faculty of knowing or sensing without the use of
rational processes; immediate cognition.
- A sense of something not evident or deducible; an impression.
Talent, IQ and all that
I have to chat about talent, because attitudes about talent usually get
way of acquiring skill. Moreover, IQ often surfaces too in strangely
ways. Here is what my dictionary says about talent:
These are the definitions, but I don't buy into the idea of talent at
all. I should warn you that I started out as a musician and was
something of a child prodigy, so I am basing this on my experience. The
key here again is that talent is something that
is seen as being natural or innate. In truth, nobody I know of who is
talented in music gets by without having practiced a tremendous amount.
thinking is that music is just there and one has musical ability, much
being born with a pair of wings. Nobody has to teach a bird to fly.
Many consider being called talented a great compliment. It is this
attitude towards innate abilities that causes much grief and suffering
because it confuses the complimented that hard work is not important
and excuses the complimenter from having to try.
Sure I've seen people who start some field of study or activity and
at it almost immediately. In those cases there always one of two things
on. Either they have had some similar training elswhere so that there
a cross-training effect, or they are effective self-learners.
- A marked innate ability, as for artistic accomplishment.
- Natural endowment or ability of a superior quality.
really is not much more than liking something enough to do it
repeatedly and therefore
get better at it. I have to admit bitterness here, since I grew up in a
system which felt that talent was innate, so that there were no
programs for me to develop. I was lucky since I was quite hostile to
this attitude – I knew that I
was practicing 4 to 6 hours a day and resented like Hell having that
swept under the
rug – and I made it a point of
tracking down instructors at the local university to teach me those
things I wanted
to know. I started taking college-level instruction in the eighth grade
theory and composition. I just want you to understand that what I write
here is from
the perspective of someone who was universally accorded that status of
So what about IQ? How does this fit in? Well, it doesn't, at least
the way most folks want it to. This is usually the pseudo-scientific
manifestation of talent. It was not always so. The first person to
posit an intelligence
quotient (IQ) was Alfred Binet at the end of the 19th century. I
strongly reading Stephen J. Gould's book "The Mismeasure of Man"
for a more thorough and probing discussion.
Binet's original idea – and it is a good one – is that people who we
consider impaired have a difficult time with mundane tasks. No mention
"gifted" people or geniuses was really there. As such, the first test
not to measure intelligence per se, but inability to do basic tasks. It
really is a good measure of this. The idea caught one and got quite a
of its own and became the darling of the racists back in the early part
the twentieth century, where the distinctions were made to coincide
(so a mongoloid was someone who was supposedly as stupid as a Mongol.
didn't they almost conquer the world once, and we were all saved
from a nasty drubbing because they changed their minds?).
American Science didn't really exist until after WW II (I don't know
everyone thinks this is classified material, but the US was about the
place anyone with brains would go up until the late 1930's), and Gould
rightly points out
that the dismally low levels of Science pre WW I is at least one reason
such bad research
got a foothold. Gould certainly does have his opinions, but he argues
quite well and the book is well worth a read.
A nice parallel (in Gould's book, as I recall, although I don't have it
front of me for reference), is measuring water in a bucket. If we
the water repeatedly we should get the same result or there is
either seriously wrong with the bucket or our measuring device. But,
a person the exact same IQ test everyday for a year would not do that.
You'd see a rise in their supposedly immutable intelligence and by the
they'd be right up there with Einstein.
Soapbox: I am a Mathematician by
profession and bristle when people try to introduce numbers into a
discipline. Most of the time they rig it so it comes out the way they
and we are none the wiser for their efforts, so it is incumbent on a
researcher to demonstrate that they really can introduce Math
Gould nicely shows how IQ researchers masturbated with Statistics and
academically blind because of it.
In my experience, those people whom I find are the most intelligent all
share a common
attribute: They are gifted at teaching themselves. I think a better
measure of intelligence, therefore, would be measuring how well
people learn from
others and how well they can self-instruct. To the best of my
is no such field of research for doing this.
All together now
Let's start with my bias: Art is the only proper human undertaking
The quote at the top of this page from Madame Boulanger is very
important. She was one of
the most noted teachers of music in the first half of the twentieth
century and managed to list among her pupils Stravinsky, Copeland, Piazzolla and
many others. Her comment was
aimed squarely at those people who think that music just sort of
happens all on its
own. Her point is that it is only by completely and utterly controlling
it as a medium
that we can make it do what we want. My music teacher, Anthony
Newman was a pupil of hers as well.
And so it goes with all fields of endeavor. Our great claim to fame as
human beings is
that we are thinking and learning machines. We don't sprout fur when it
gets cold, we think about fluid dynamics and thermodynamics then invent
Jacuzzi. If the sabre-toothed tiger threatens you, invent the uzi. My
point here is
that the way people actually function is by acquiring skills and
This builds the intuition needed. Then there is an aesthetic sense of
right and wrong that goes hand in hand with this.
The way that prople operate on a day to day basis is this way. Consider
how the Sciences
are done. There is no such thing as sitting down and ploughing straight
through a serious
research problem from start to finish. That only occurs in very
carefully contrived textbook
A rant. Mathematicians often harp on 'proofs' of
do not explain that to the laity in any meaningful way. When one lives
with a hard
problem for an extended time, gradually convincing oneself of the
wrongness constitutes the actual proof. However, since books could be
that, the convention has evolved that very carefully controlled prose
mannered stylism, called technical prose, is used to
give a condensation of this. Including a comment
that "I'd just finished a huge meal and during my constitutional I
got to thinking about the way that the main course was sliced which in
that I might want to take whatever the current bane of my existence was
slice it up similarly, leading to the result that..." would be forcibly
struck by any
editor worth his salt. The condensed version is the proof you get
to read in a book. Students always read a proof and flatly state they
have no idea
of how to write one and they are correct. Mathematicians should put
where their mouths are and offer courses in how to do proofs and then
up, rather than just sort of picking it up from the ether and
for their stupidity.
In reality, one looks for patterns and these shifting patterns
associations. Jacques Hadamard was a famous Mathematician and wrote a
book called "Psychology of Invention in the Mathematical Field." There
awful lot of cognitive theories of learning and I've yet to find many
I really think of as credible. Hadamard's thesis is still rather
Essentially, he chatted up his friends and found that the way he and
they operated was along the following lines.
Human thinking is not hierarchical = arranged with some
scheme, like a table of contents or an index, but heterarchical
= arranged as a network,
like the World Wide Web.
There are many ways of linking or interconnecting bits of information
and the actual way that
the network is organized varies by person. A good example of this is
can memorize extremely long numbers at a glance (such as the first few
digits of pi). One person who was particularly
good at this said that he did this by having a story he remembered that
from one digit to another. There was nothing even slightly mathematical
about his process. In this case the first part of Hadamard's thesis has
insinuating the facts into the network and building in the sorts of
to this information that allows for recall. Intuition is then built and
then one may let the patterns and associations interact. It is
human brains are excellent pattern recognition machines and the musings
that run full tilt.
- Live with the problem, familiarizing yourself with it for days or
- Go do something else.
- Muse on the problem and the answer will likely come to you.
As Nietzsche pointed out, thinking is not a faculty such as vision.
We have full
control of vision and if there were an instance when we couldn't see
some shape, even
briefly, we'd hot foot it to the doctor. On the other hand, this is
exactly what happens
in thinking. How often do we know some fact but can't quite seem to
recall it at the
instant we need it? Or there is some bit of logic that escapes us for
There are far too many variables that affect thinking such as being
tired, hungry, grumpy, etc. Consequently part of education is
the discipline to counteract this, at least a bit.
Yet another bias is on the role of a good education. Just
willy-nilly learning facts
as is in vogue in much education is the worst way to learn. Part of
learning a subject
is doing so in a way that the links are usable and make sense for the
topic. This is
what the role of a good teacher is. Hate to sound like an old fogey,
but wisdom is
the crucial element for a good education and the reductionist attempt
all branches of learning into facts simply makes it easy to crank out
which is really what the US higher education system has perfected.
Facts are easily
quantifiable on tests but wisdom is not, therefore facts are what is
Another bias of mine is that a good classical education is crucial, not
because you will have to quote Shakespeare to get a job, but because it
you with the best of the best in human thinking. Human beings are
raised while homo
sapiens are born.
A good model for all learning is human language. It is, I think, the
model since humans do learn languages as a matter of course from birth,
I am not a linguist (nor a psychologist).
One does not laboriously form sentences
but takes several years of familiarization and pattern internalization
the thought comes and the sentence is there (and perhaps even it is out
out mouths sooner than we would like...) This is the way, I have come
all human activities function, be it Music, the Sciences, martial arts
or any other discipline.
Of course, some activities, such as Music, do require a certain modicum
memory, but that is not what I'm talking about.
Before I hear a yelp that Mathematics is logical I am not disputing
that. I just maintain
Mathematicians aren't. We don't require that Botanists photosynthesize,
do we? In the
case of Mathematics, intuition has gotten us into trouble in the past
with things that
were supposedly obvious. Now we have an enormously sophisticated model
Logic, to helps us not shoot ourselves in the foot. "It is by intuition
we discover and
Logic we prove" is a quote I heard in graduate school that has stuck
with me through the years.
So here is the moment you've been waiting for. No matter what field
you decide to study,
aim for art in it. That is to say, practice it until it is second
nature, tutoring your new intuition, but be mindful
of the style since that will be the easiest way to know right from
wrong. Seek the wisdom
for the subject since it is there that you will understand it in a
profound way. I am a notoriously
quick study and what I have outlined here is just what I do. There is
old samurai maxim: "He who masters one art shows it in all he does". Of
art here meant war art, such as swordsmanship, but the point is well
taken. The Japanese have this as a cultural concept. The idea was that
one pursued the Way of a given
discipline and strove for mastery, refinement and elegance.