
Confessions of a Math Idiot - blasdel
http://funcall.blogspot.com/2009/07/confessions-of-math-idiot.html
======
onedognight
Learning math on the internet is hard. Different people use different
notations and assume different backgrounds.

I encourage the poster to read and learn more about exterior algebras! Once
you've seen Stoke's theorem, the Hodge decomposition, Maxwell's equations,
etc. written using them you'll never go back and those old school t-shirts
with "and god said..."[1] will make you cringe.

I hope/expect to see them taught in basic calculus and physics, but it takes
generations for things to trickle down.

[1]
[http://images6.cafepress.com/product/96599546v11_350x350_Fro...](http://images6.cafepress.com/product/96599546v11_350x350_Front.jpg)

~~~
slackenerny
_see them taught_

I think it was done that way once, but somewhere around seventies people
dropped or moved this to advanced courses. You can't immediately solve a
transformer in differential forms like the way you can by mindless symbolic
manipulation over the standard formalism.

------
mcantor
I was chagrined to find myself in this very same position. Sadly, I have not
yet had the time to track down the resources necessary to "re-educate" myself.
Computer science has given me a new love for math that I didn't even possess
during my education, but I know of very few ways to express that enthusiasm.

------
edw519
When I first saw this, it was the beginning of the beginning:

6

28

496

8128

33550336

8589869056

137438691328

2305843008139952128

2658455991569831744654692615953842176

191561942608236107294793378084303638130997321548169216

13164036458569648337239753460458722910223472318386943117783728128

144740111546645244279463731260859884815736774914748358 89066354349131199152128

2356272345726734706578954899670990498847754785839260071014302
7597506337283178622239730365539602600561360255566462503270175
0528925780432155433824984287771524270103944969186640286445341
2803383143979023683862403317143592235664321970310172071316352
7487298747400647801939587165936401087419375649057918549492160555646976

1410537837067120690632079580860631898814867435147156678388386
7599995486774265238011410419332903769025156195056870982932716
4087724366370087116731268159313652487450652439805877296207297
4467232951666582288469268077866528701889208678794514783645693
1392206037069506473607357237869517647305526682625328488638371
5072974324463835300053138429460296575143368065570759537328128

5416252628436584741265446537439131614085649053903169578460392
0818387206994158534859198999921056719921919057390080263646159
2800138276054397462627889030573034455058270283951394752077690
4492443149486172943511312628083790493046274068171796046586734
8720992572190569465545299629919823431031092624244463547789635
4414813917198164416055867880921478866773213987566616247145517
2696430221755428178425481731961195165985555357393778892340514
6222324506715979193757372820860878214322052227584537552897476
2561793951766244263144803134469350852036575847982475360211728
8040378304860287362125931378999490033667394150374722496698402
8240806042108690077670395259231894666273615212775603535764707
9522501738583051710286030212348966478513639499289049732921451
07505979911456221519899345764984291328

Perhaps OP should have studied number theory. No number theorist has ever
complained about the quality of his teacher. It's just you and all that ever
was and all that ever will be. Now go discover.

------
thunk
See also Sussman's talk on how terrible math notation is:

[http://video.google.com/videoplay?docid=-2726904509434151616...](http://video.google.com/videoplay?docid=-2726904509434151616&hl=en)

~~~
onedognight
Sussman's critique of mathematical notation is hollow. The point of good
notation is to hide the details while conveying the high level compactly and
succinctly.

Lagrange's equations written the way he derides are a perfect example. His
need to think about what spaces the symbols live in based on their context in
the expression is typical when translating math to code. He should be talking
about about what kind of type inference engine one needs to allow the original
notation and end up with his positional notation internally.

~~~
thunk
Maybe, but a notation arrived at by an accretion of historical accident is
_not_ the way to introduce this stuff to beginners, inertia be damned.

Edit: No _ta_ tion

~~~
WilliamLP
So raise your kids in Esperanto? Or does it _really_ matter that much in the
end?

~~~
thunk
I thought someone might try that tack. Math is completely decoupled from its
representation, so you can use whatever notation you want and not lose
anything. I'm not gonna guess to what extent linguistic communication is bound
up with _its_ representation, but it's certainly more than math, meaning
meaning can be lost. So your analogy doesn't work. Also, since mathematical
notation is a tool primarily to facilitate mechanical transformations, and
only secondarily for communication, you should use whatever lets you think
biggest.

~~~
timwiseman
An excellent point as long as you do not need to actually communicate your
results to anyone. As soon as you do, it becomes precisely as bound as
language is.

------
brent
Why does he say that the arity is different for derivatives and integrals?
Isn't there an arity 1 symbolic integration? If you were numerically computing
an integral over a range, shouldn't you be evaluating the derivative at a
point? These don't have the same arity, but at least the ideas are parallel.

~~~
bkovitz
I'm a math idiot, too, but I know that he's confusing indefinite and definite
integration. Definite integration returns a number; indefinite integration
returns a function. Differentiation does undo indefinite integration.

~~~
ggchappell
> ... I know that he's confusing indefinite and definite integration.

Yes he is. The real solution is to banish the term "indefinite integral" from
the language, and substitute "antiderivative". We should also not use the
integral sign for antiderivatives, but unfortunately there is no well known
alternative. :-(

Using the same terminology and symbols for antiderivatives as for (definite)
integrals is confusing, as here. It also makes the Fundamental Theorem of
Calculus -- a very profound fact if there ever was one -- appear to be
something trivial about getting rid of limits on an integral sign. But
consider: one can compute a definite integral using an antiderivative ...
whodathunkit?

~~~
bkovitz
Isn't it weird how awful math terminology is? Even the word "derivative" is an
awful choice. The special and wonderful thing about a derivative is that it's
the rate at which another function changes, not that it's derived somehow.

------
sophacles
I have a similar, but slightly different problem. While I know that CS and
programming in general can be translated into useful math knowledge, I always
mistrust my math. I suspect that i don't understand the math correctly because
I understand it, if that makes any sense. The weird terminologies and
reputation for being way hard always erode my confidence in my understanding.

~~~
silentOpen
I've found that mostly it's just that mathematics is usually taught
_extremely_ poorly. Mathematicians and mathematics educators typically gloss
over definitions and use terrible notation. Further compounding the problem,
the mechanical aspects of mathematical computation are rarely formally defined
-- talking about elementary operators and objects of a mathematical subfield
is rare.

I've recently discovered abstract algebra and I love it. We need to find and
celebrate good explanations of mathematics both to do better work and to give
future generations the mathematical education we didn't get.

~~~
bkovitz
What do you love about abstract algebra?

I'm asking because I found myself deeply disappointed with it when I took it
in school. It seemed to me just a haphazard collection of definitions and
puzzles. Why, for example, does anyone care how many non-abelian groups of
order 18 there are? I found myself unable to retain much of anything, due to
the lack of any coherent explanation of "What is the point?"

In computers, by contrast, no one hides the point. Searching, sorting, bits
and bytes, languages that make it easier to express things without making
mistakes—there's no mystery about the purpose or importance of anything.

I wonder, if someone showed me what is attractive about abstract algebra,
maybe I would find myself just as addicted to math as to programming.

~~~
silentOpen
I love that it provides a systematic way of thinking about structures and
proving properties of said structures that I am working with in code. It's
amazingly powerful to say "type t defines a monoid with operation k: t -> t
and identity Nil" because it comes with a whole slew of properties and
theorems. It meshes very well with strongly typed functional languages that
support algebraic data types (e.g. Haskell and OCaml).

I don't much care for theoretical abstract algebra (e.g. non-abelian groups of
order 18) and I am by no means a master (or even an amateur!) but I find
abstract algebra's concepts useful and I hope to learn more!

------
jhancock
My high school math teacher refused to teach calculus. We spent year after
year on algebra, trig, geometry, and did proofs. Lots and lots of proofs. He
told me not to worry about when I went to college and studied calculus as I
would be much better at it than the students that studied calc in high school.

Smart man, he was right. The first two weeks in my first college calc class I
felt a little uneasy and understood that I was the only person in class that
had not studied this stuff in high school. After those first two weeks though,
it was not a problem and I swept through 6 calc classes without a sweat.

20 years later of course, I'm a math idiot again...use it or lose it ;)

------
rawr
I enjoyed this post a lot because I traveled the opposite path. I started out
in computer science with little interest whatsoever in math all through high
school.

Then when I got to college they introduced the notion of "program correctness"
where you tried to prove, mathematically, that your computer programs were
correct. This convinced me that computer science was simply a knock-off
approximation of mathematics and I drifted away from it.

From your derivative example above I’d say it is this inexact nature of
computer science that you especially like. Different strokes for different
folks, I suppose.

~~~
arakyd
It's not "different strokes," it's a change of perspective causing massive
confusion all 'round. Computer science isn't a knock-off approximation of
mathematics, it's a subfield of mathematics. Programming, which is what the
post is talking about, is a third thing. The derivative is the inverse of the
indefinite interval (and has the same arity), but the function he refers to
computes a definite integral which is not the same thing. Etc.

Confusion about this stuff is a sign that you didn't really understand it the
first time. Believe me, I've been (am still) there...

~~~
scott_s
Regarding where CS stands in relation to math:
<http://news.ycombinator.com/item?id=690798>

~~~
arakyd
Yeah, I tend to say "computer science" when I mean "theoretical computer
science." To me there is a clear split between the stuff that's math/logic and
the rest of it which I'm happy to lump under the catch-all of engineering (and
don't feel comfortable labeling as any sort of science).

~~~
scott_s
I'm not comfortable labeling the concepts behind operating systems, networks,
computer architecture and programming languages as either math or engineering.

