
You don't understand something until you think it's obvious. - mebassett
http://mebassett.blogspot.com/2011/06/you-dont-understand-something-until-you.html
======
lmkg
And now that it's obvious to you, it's suddenly really hard to explain to
someone else.

My friend was TAing a freshman calculus class in college while working on his
PhD. He clearly remembers that his (and everyone's) biggest problem with
calculus was limits. Hands-down, limits were the most frustrating and un-
intuitive part of the class. So he sat down to try to figure out a good,
intuitive explanation of limits that would help his students avoid the
frustration that he had.

He immediately ran into the problem that they weren't hard to understand. They
were clear and intuitive. He could not for the life of him remember why he
thought they were difficult when he was first learning them. He asked me for
help, and I had the same response. I remember clearly banging my head against
them and screaming in rage and frustration at those goddamn limits. But I look
at them now, and I can't figure out how I could fail to understand something
so obvious.

And thus, the cycle of pain continues, with limits remaining the eternal bane
of freshman calculus students.

~~~
btilly
The problem with limits is simple. We force people to learn multiple concepts
in a poor order.

1\. We introduce limits. Which are blindingly artificial, and so it is hard
for us to form a concept about why people want this.

2\. We introduce the definition of the derivative. Which instantly results in
a practical reason for limits. But typically we make the mistake of
introducing the derivative _as a function_ , which means considering the limit
of a lot of things at a lot of points.

There really needs to be an intermediate step, which is to introduce the
definition of the tangent line. And how to calculate it. Then work up how to
calculate tangent lines of various combinations of functions at a single
point. And after the person is well and truly comfortable with that, then
introduce the derivative.

~~~
Fargren
_1\. We introduce limits. Which are blindingly artificial, and so it is hard
for us to form a concept about why people want this._

I don't see limits as artificial as all. For example, Zeno's Paradox[1] is
resolved thanks to the limit of a succession. And the concept of the
infinitesimal, which was something I had thought about before I learnt about
limits, is another rather intuitive thing that is explained by limits.

Derivatives are only one of the most common uses of limits, they are
definetely not the only reason they are useful.

[1][http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and...](http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise)

~~~
psykotic
The limit concept was introduced by Cauchy but lacked a rigorous definition
until Weierstrass came along. In the absence of such a definition, Cauchy
committed some infamous errors involving nested limits, e.g. he ascribed
properties to continuous functions that actually require the stronger
assumption of uniform continuity. Take into account that Cauchy's treatment of
calculus was by far the most rigorous for its time, so he was hardly careless
in these matters. If everything was as straightforward and obvious as you
imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus
would have served any meaningful purpose.

The more you learn, the more you come to realize that the nature of the real
numbers is deeply mysterious and not to be treated lightly. That's not to say
you cannot convey many useful intuitions to students without the somewhat
abstruse formal machinery.

~~~
Fargren
_That's not to say you cannot convey many useful intuitions to students
without the somewhat abstruse formal machinery._

That's more or less what I was trying to say. There are some deeply intuitive
concepts for which limits are, at the very least, a very good aproximation. It
may occur in some cases that the actual case and the intuition that one has
about it doesn't match, but that doesn't mean that the model is arbitrary or
artificial.

------
pash
John von Neumann once told a confused young scientist, "Young man, in
mathematics you don't understand things. You just get used to them."

I don't think that's quite true, but developing familiarity with a difficult
concept is tantamount to coming to understand it, in my experience.

For me, the process seems to go like this:

1) assemble the disparate pieces of a concept, and try to wrap my head around
the novel ones;

2) work out how Piece A relates to B, and B to C, and ... ;

3) forget what the hell Piece X is;

4) repeat steps 1–4 several times;

5) convince myself that all the pieces work together as they should.

And only later, after I've worked with this collection of pieces several
times, do I stop seeing the pieces and start seeing something new, the
emergent phenomenon of the new concept. And it's not until I've forgotten the
details of how Piece X fits into Piece Y, and maybe why we need Piece Z in the
first place, that I feel like I really grok this new concept.

~~~
Lost_BiomedE
I think this way as well. I was told that I am a slow processor of
information, by a credible source. Also, that I would have a harder time
learning on the fly but would have a deeper understanding due to making more
cognitive links to each idea, and this would allow for tackling harder
problems and long projects. It has turned out to be spot on, and if I would
have to guess, I would say about a quarter of engineers that I know seem to
think this way. In the general population it seems much lower.

------
sivers
Brilliant! Here's my similar, now animated take on it:

Obvious to you. Amazing to others.

<http://www.youtube.com/watch?v=-GCm-u_vlaQ>

~~~
szany
I don't necessarily disagree, but it's worth pointing out that this might not
always be the case:

"Every mathematician worthy of the name has experienced the state of lucid
exaltation in which one thought succeeds another as if miraculously. This
feeling may last for hours at a time, even for days. Once you have experienced
it, you are eager to repeat it but unable to do it at will, unless perhaps by
dogged work." -André Weil

------
csomar
I think this has to do with the structure of the brain. When you are learning
something new, the brain is creating new routes and making new connections. It
can even creates new neurons (something known as neuro-plasticity).

My guess is that when you finally understood the subject, your brain has
completed the route for that particular problem. It's now clear how to solve
the problem for the brain, since a specific route exists. It's obvious and
very easy for the brain to do that with the route shortcut.

For you, the problem becomes obvious and simple. It's no longer complicated
and requires less brain-processing-power to solve. You think you were stupid
you didn't figure out that from the beginning, but you are not!

------
JadeNB
It would be lovely if this were true, but I think it's actually a fairly
poisonous mindset. What the author's post seems actually to be saying, with
which I heartily agree, is “obviousness is only in hindsight”; deep things
become trivial only because of the time you've put into understanding them.

On the other hand, I think the things that become obvious once you've
understood them _tend_ to be (though are not always) the things that have
already been fairly well digested by others, and so are presented to you in a
smooth, flowing way to which you just have to accustom yourself.

My experience with (mathematical) research is that understanding has a roughly
equal possibility of meaning that you find it trivial, or that you finally
understand all the (apparently) irreducibly complex difficulties. Indeed, my
feeling if anyone tells me that, say, Deligne–Lusztig representations are
obvious is that he or she hasn't probably fully understood them (disclaimer:
neither have I, not even close, which may mean that I'm just illustrating the
author's point).

I don't mean at all by this that you shouldn't go on searching for the
'obvious' simplification—that way lies great insight. (As someone said much
more elegantly, if things aren't already obvious in mathematics, we tend to
make them obvious by changing the definitions.) What I mean is that you
shouldn't drag yourself down by saying “I thought I understood that, but it's
hard, not obvious!”

------
bluekeybox
Corollary: if you are angry at something, it's because you don't understand
it. Nobody gets angry at obvious things such as water being wet.

Another corollary: understanding brings peace.

~~~
MatthewPhillips
Yes, this is something I try to remind myself whenever I find myself getting
angry. I find it helps tremendously; I quickly realize that it's something
beyond my control or something that I don't completely understand and I find
that calming. I think I've gotten better about just skipping over the part
where you worry about things that are outside of your control (not that it
doesn't still happen).

------
kylemaxwell
This is precisely why I get nervous about offering to speak at professional
groups / conferences. When I see a CFP or similar, everything cool that I do
seems obvious and I don't think it'll be useful.

But then I get into conversations with my peers and realize that some of the
stuff I do apparently _isn't_ as obvious to everyone as I'd thought.

------
bdhe
I think a better measure of understanding is how well you are able to explain
it to a layperson. As Einstein once said, "If you can't explain it to a six
year old, you don't understand it yourself."

As a researcher, who has also TAed classes, it takes a lot of effort not only
to teach a particular topic, but also simultaneously present the crux of the
idea as well as the right way to think about it and its place in the context
of other information. And learning to think about everything with these in
mind will make you understand things much much better. In particular, one of
the most challenging things is not to explain breakthrough ideas to cutting-
edge researchers but rather explaining breakthrough ideas to complete
outsiders and laypeople.

~~~
qq66
Einstein may have said that but it's just not true. How do you explain the
Higgs boson to a six-year-old? If you succeed, it's because you made an
analogy so simple that it has no meaning.

~~~
chrismsnz
Maybe that's because we don't understand it fully?

~~~
qq66
There are plenty of things we understand fully (e.g. translation lookaside
buffers) that are impossible to explain to a six-year-old.

~~~
aristus
Off the cuff, I would come up with something involving parking valets and the
pegboard for keys. It would illustrate the important bits of the mechanism and
its purpose. I don't know if you would dismiss that as meaningless analogy,
but explaining TLB (or Traveling Salesman, or functional decomposition, etc)
to kids is not impossible.

~~~
xyzzyz
There are things in mathematics that I find difficult to explain even to
someone who finished freshman calculus and linear algebra courses. How should
one explain singular homology to someone without strong background in
topology, algebra and pure math itself? Same thing with theory of sheaves, for
instance. These things are not terribly difficult themselves (but they're no
easy, I admit), but the amount of time required to explain what they're really
about is breathtaking.

~~~
aristus
I don't know anything about those so I can't say. Not everything has a facile
real-world analogy, and there may be long chains of dependent concepts to get
through first.

Here's the thing, though: if you make an honest effort to explain something
like that to a lay audience, you may fail. Or you may not. Or you'll give them
a workable, but incomplete and strictly wrong idea. Either way you yourself
will end up with a deeper understanding.

A huge part of "real" mathematics is finding isomorphisms. Teaching is more or
less finding isomorphisms between new concepts and concepts that the student
already has.

------
hrabago
This is partially why it can be frustrating to work with someone who isn't as
good as you are. Things that come easily to you doesn't for others. You
sometimes have to remind yourself that things that you consider obvious is
only obvious for you.

~~~
adamdecaf
This reminds me of a story from Feynman where he was with a math friend and
they were trying to multi-task. Yet, they ended up going about the two things
(counting for a minute while reading or speaking) in completely different
ways. (Different enough that one could only count and talk while the other
could only count and read.)

[http://www.youtube.com/watch?v=Cj4y0EUlU-Y&feature=playe...](http://www.youtube.com/watch?v=Cj4y0EUlU-Y&feature=player_detailpage#t=140s)

------
drcube
Richard Feynman said you don't really understand something unless you can
explain it to freshmen in a single lecture. Same concept, really.

The interesting consideration, to me, is how much we can get done with hardly
any understanding at all. That's human nature. We do things, then we wonder
how, and then we eventually come to understand what the hell we just did. It
seems bass-ackwards, but that's the way it goes most of the time.

------
ZackOfAllTrades
Thought: my linear algebra teach this past semester told me that some of the
greatest mistakes in mathematics came from people assuming something was
obvious without a proof. After building a huge intricate structure around the
assumption, they would watch everything fall apart as it turned out the
obvious fact wasn't true. Now, whenever I hear somebody say "it's obvious" I
always get skeptical now.

~~~
motxilo
You are just describing what motivated Bertrand Russell to write Principia
Mathematica.

------
protomyth
There is a great line from Jonathan Ive in the documentary Objectified where
he says: "A lot of what we seem to be doing is getting design out of the way.
And I think when forms develop with that sort of reason, and they’re not just
arbitrary shapes, it feels almost inevitable. It feels almost undesigned. It
feels almost like, 'well, of course it’s that way. Why would it be any other
way?'"

------
espeed
As Galileo said, "All truths are easy to understand once they are discovered;
the point is to discover them." Insights are often so simple, so obviously
true once you see them. But we are often blinded by our fixations, our hubris,
our limited frames of reference -- our narrow perspectives -- which prevent us
from seeing.

------
boredguy8
The intersection of "things I thought were obvious at one point in time" and
"things I currently think are wrong" is a rather large set. It's hard for me
to claim I "understood" those thing I now think I was wrong about.

~~~
polynomial
I was going to point out that, ironically, many people who think something is
"just obvious" completely fail to understand it at all.

So yes it cuts both ways.

------
orangecat
Not sure about this. I understand the proofs that there is no largest prime
number, and that the integers can't be put into a one-to-one correspondence
with the reals, but I wouldn't say either is "obvious".

~~~
pash
Funny you picked those examples, because I recall them both seeming quite
obvious to me when I learned them. And I'm no great mathematician.

It's, um, obviously quite hard to explain why you find something obvious, but
I think in the first case it's because you can just think about rescaling the
number line (using a bigger unit) or shifting the origin, which to me makes it
clear that bigness is completely arbitrary and implies that there can't
possibly be any biggest number in this sense; in the second case, there
clearly is no "next" number in the reals as there is in the naturals, and that
insight gets right at the heart of the concept of countability.

~~~
xyzzyz
_in the second case, there clearly is no "next" number in the reals as there
is in the naturals, and that insight gets right at the heart of the concept of
countability._

I think you missed the point. There is no "next number" concept in the
rationals either, and yet they're countable. There _is_ "next number" concept
in the first uncountable ordinal, and, as the name suggests, it is
uncountable.

~~~
pash
You're right. The "next number" test (under the natural ordering) makes the
concept of _density_ obvious, not countability.

So I think I've given credence to the notion that those who claim to find
something obvious might not fully understand it. Or at least not at four in
the morning.

(Oh, and a point that Stephen Abbott made in his wonderful intro analysis book
now springs to mind: proofs are useful also as a _check on our intuition_.)

------
yhlasx
I guess the point can be summarized like

\- When someone says that something is obvious, we cannot claim that he/she
actually understands it well.

\- But, if someone understands something well, it will seem obvious to
him/her.

------
codyguy
Yet, we might not understand the things that seem obvious. Gravity, light etc
can be "obvious" to most people, but not really understood.

------
redtwo
You don't understand something until you can explain it to your grand mother.

~~~
Resonemang
In that case I don't even understand English... ;)

------
petegrif
This is why so many people think patents are unfair - because they are so
obvious. :)

