
Kill Math - pw
http://worrydream.com/KillMath/
======
mechanical_fish
What we have here is a failure to define our terms.

Math _is_ the manipulation of abstract symbols according to abstract rules. If
you don't like symbols, you don't like math. If you are illiterate in symbols
you are illiterate in math.

The word is often used imprecisely, though. Because so many real-world
problems can be translated into math, there is a temptation to equate "math"
with "any problem that can be expressed in math". Thus you frequently see
poetic statements like "my cat is great at solving differential equations", or
"music is math because it's all about harmonic series and Fourier analysis".
But these things aren't literally true. You can put a bucket under a flowing
faucet, and it will collect all the water, but that isn't really integration.
The bucket isn't doing math.

And being ignorant of math isn't the same as being stupid. As the OP points
out, you can get a lot of quantitative reasoning done without using math. A
classic Fun Fact About Math is that it took thousands of years to invent the
number zero. And it's true. But that doesn't mean that the ancient Egyptians
used to waste hours staring into newly-emptied buckets and baskets in stunned
amazement, murmuring "what on earth is that" to themselves in Coptic. People
understood what "having no objects" meant long before there was a _symbol_ for
"the number of objects in an empty basket". It was the _highly abstract
symbol_ "zero", and the highly abstract operations involving zero, that had to
be invented. [1]

It's worthwhile to recognize that interpreting the real world in terms of
abstract symbols, and vice versa, is a terribly difficult skill that requires
lots of practice. (In my case, I was well through grad school before many bits
of physics clicked.) And it's worthwhile to recognize that you can often do
without math: You can reason quantitatively without it. Birds do it! Bees do
it! But don't pretend that you're doing math unless you are actually doing
math. The abstractions _are_ the math.

\---

[1] Or, rather, discovered. Although we'd better stop there, because I won't
be able to cope with the ensuing philosophical back-and-forth.

~~~
tel
Completely disagree. Math _isn't_ manipulation of symbols. At all. Math is the
study of mathematical objects, a practice often done using a formal language
for the convenience and power it provides.

"0" is a formal symbol with particular formal behavior

"empty/missing/none" is a well-known physical concept

Zero is a precise, powerful mathematical object which can be represented by
them both.

\---

This is difficult to deny. Unless you want to deny the providence of most
widely recognized mathematicians throughout history, you have to accept that
formal language of math is relatively new. Furthermore, it's alive and
growing, inconsistant and incomplete. There is a meaningful frontier, and
there you observe mathematicians are really studying something else and
furiously creating the formal language to describe it.

In this light, metaphor is absolutely a useful tool in the same class as
formal language for explaining and reasoning about math. You're right to point
out the non-equivalence of the two, but the author's Kill Math project is in
no way not math. Furthermore, I'm anecdotally a supporter of the author's
belief that doing math competently requires knowing a the metaphorical side
since your symbolic projects may fail or be unclear.

I'd be willing to accept that metaphor will never be as powerful as formal
language, but it does discredit to the way (I'd wager) most people understand
math to deny the metaphorical.

\---

At the heart of this trouble of definitions is Gödel's Incompletenesses. The
practical effect of their discovery was the destruction of the dreams of
formalists who had for years hope to discover the essential shape of the
formal language from which all math would spring. With Incompleteness however,
we are forced to admit that we can study, meaningfully, the behavior of
mathematical objects for which the language of math cannot be used to reason
about.

Then we extend that language, of course.

~~~
3am
Are you sure you're accurately describing Godel's Incompleteness theorem? It's
been a while, but I understood it at a technical level briefly. I'd be
interested in knowing more about how you think it applies here.

~~~
tel
No, I'm not. I'm interpreting it pretty loosely. Sorry about that. Correct me
if I'm wrong here, though.

More directly, what I meant to say is that since there exist true theorems
which cannot be proven within any particular choice of mathematical formalism,
we need to operate with tools beyond simply symbolic manipulation. That was
the death knell of Hilbert's Program and solidly separated the formal
specification of math from "that thing which we're studying".

~~~
3am
I think you are wrong - even though there are true, unprovable statements,
they're unusual and to the best of my knowledge, not the subject of extensive
study. More like curiosities
([http://en.wikipedia.org/wiki/List_of_statements_undecidable_...](http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC)).
Even if that were not the case, there is no alternate approach to studying
them.

I'm not really sure that you mean about the "formal specification" for math
vs. the "that thing we're studying". An informal (ie, not expressed in ZFC +
1rst order predicate calculus) proof of something non-trivial can go on for
dozens, if not hundreds of dense pages of symbols. If I recall correctly,
Whitehead's Principae Mathematica derived arithmetic from ZCF and predicate
calculus, and it took the whole book.

I did a little reading to refresh myself on the subject, and this stood out as
a good summary of the topic:

"In a sense, the crisis has not been resolved, but faded away: most
mathematicians either do not work from axiomatic systems, or if they do, do
not doubt the consistency of ZFC, generally their preferred axiomatic system.
In most of mathematics as it is practiced, the various logical paradoxes never
played a role anyway, and in those branches in which they do (such as logic
and category theory), they may be avoided."

([http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foun...](http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis))

~~~
tel
Your picture of mathematics is quite a bit more robust, so I won't deny that
what you wrote here is more accurate than my "Gödel say it won't work, woe is
our field!" brimstone version. So I'll explain my reasoning for invoking
Gödel.

I mostly wanted to walk around the historical event I mentioned, the breaking
of the Hilbert Program. At the time, it seemed that formal specification of
math would provide a complete picture of what math was! Once the Program was
finished then the job of mathematician would eke out into "computer" (of the
abacus sort) or into other fields which interpreted the canon.

I'm not sure which death stroke was stronger, the incredible opaqueness and
complexity of proof systems like ZFC or Gödel just saying what he was trying
was outright impossible, but Hilbert's Program was killed before it even
seriously took off, leaving the study of mathematics and the practical
formalisms we use to study it pretty ad-hoc instead of grand and unified.

I'm unifying that with the fact that the way math seems to be practiced never
comes from the formal language but instead first comes from imagining some
kind of "mathematical object" and then taming its behavior with formalisms.
You could consider them to be one and the same and argue that the difference
is highly philosophical, and then this is where I'd invoke Gödel and inform
you that there definitely exist things we could benefit from reasoning about
that your formal language would fail to describe. This existence proof
separates the classes of true things and provable things and makes their
distinction more than philosophical.

Now, talking about what a "mathematical object" is gets you to the bleeding
heart of the philosophy of science and epistemology. It's a tough question!

\---

As a final note, ZFC is ZF + Axiom of Choice... which, yes, most practicing
mathematicians just accept AoC so that they can integrate or whatever. The
formal world without AoC is very sparse, _but_ nobody has any sort of idea
what the arbitrary decision means. I know that there has been some significant
study of ZF-C, though it's been "impractical", I don't know if anyone is
willing or capable of stating that ZF-C is in any way worse than ZFC.
Impractical is a Mathematicians favorite adjective, so they're just two extant
formal systems which disagree quite a lot on important things but we mostly
pay attention to ZFC.

------
jxcole
I appreciate that symbols can be an odd way to represent every day
quantitative problems, but I have never had a problem with it. Sure, some
people do, but this guy seems really royally pissed off at ordinary
mathematics.

I have always been a proponent of having more than one way to teach a certain
subject. Studies have shown that different people perceive and interact with
information in very different ways. The current "one size fits all
methodology" to teaching is lame, to say the least. So a new way of teaching
people mathematics is welcome.

But that's not to say the old way didn't work for people like me. If this site
kills math so that English majors can cope, I hope someday someone will kill
poetry so that I can cope.

~~~
premchai21
(Compacted for space. Original. Use tr '/' '\n' | sed -e 's/^ //' -e 's/ $//'
to reëxpand, with a paragraph break being "\n\n". Sorry for the repeated
initial edits and such; I had to try to work out the HN formatting rules.)

When humans try to learn symbolic math / How many of them struggle with the
test! / The teacher thought of like a psychopath / Dishonoring the realm of
human zest

“We must have our emotions!” students cry / “Or else we'll run around like
apes, confused / Our brains are built for stories, not to scry / A world of
numbers, strangled and abused.”

The teacher sighs, “They always drag their feet / Unless they're cornered, up
against the wall. / To risk _my_ job with answers incomplete! / They'll never
use it later, after all.”

Then, big surprise! The math is found at fault / Tear-stained by cringing
memories of school / “Dispense with all the symbols, and Exalt / Thine
Intuition”—that shall be the rule.

Professors' lamentations curse the air / Hung out to dry for calling any bluff
/ “To shun defective math must be unfair / For surely no one understands the
stuff.”

So woe to ye from near the world of forms / Who strain to show the populace
your realms / They're immunized against your grand transforms / And
explanation only overwhelms.

(Now, please don't take this poem at its word / Or treat it as authoritative
fact / Exaggerated story and absurd / Polemic leave specifics inexact

The author's nearly _made_ of symbols, note— / Despite the slow decay of some
to blanks / So though he doesn't mean to seem to gloat / He'd rather keep his
“freakish” symbols, thanks.)

------
wbhart
I'm sorry to be the one to have to bring this up. But not all mathematics can
be represented visually. Maths breaks into a number of main branches, among
them principally algebra, analysis and geometry.

Usually only the third can be represented visually, though often
mathematicians develop diagrams which help internalise the complex notions
involved in analysis and algebra too.

For example, in algebraic geometry, the more familiar notions of geometric
objects such as curves and surfaces are replaced with purely algebraic
notions, such as schemes. This is because of various categorical equivalences
between geometric objects (on the geometric side) and various algebraic
objects (on the algebraic side). But on the algebraic side, schemes are a very
expansive generalisation of things that actually correspond to geometric (and
visualisable) objects.

I once went to a teaching seminar on the use of a package called GeoGebra for
the teaching of mathematics. None of us mathematicians could bring ourselves
to put up our hands and ask how one might represent a complex of modules over
a noetherian ring pictorially in GeoGebra. There's this fundamental
misunderstanding amongst educators that symbolic mathematics is not essential
to understanding maths.

This is an important insight when it comes to computer programs though. The
same thing happens in computer science. You get splits between things that are
geometric, symbolic and purely computational.

Often I get really annoyed at people showing off their latest concurrent
programming paradigm by implementing a GUI or event loop for some graphical or
network application. They forget that many things simply don't fit into that
paradigm.

I equally get annoyed at computer scientists for forgetting that the number of
integers is not about 10. Sometimes us mathematicians really want to do things
with matrices of ten thousand by ten thousand entries.

------
karamazov
One of the great things about math is that it lets us surpass our own
intuition. The spatial intuition we evolved in the wild doesn't work well for
a number of things, which is why symbols and abstraction are so useful in the
first place.

We don't need mathematics to figure out how a swinging pendulum works; we can
do that intuitively. (Actually, this isn't completely true, but we can at
least get the general idea.) Simple problems like that are worked out in
classes so that students can get used to the mathematics. In domains where our
intuition fails us - e.g. quantum mechanics, high-energy physics, statistical
mechanics, not to mention 11+ dimensional formulations of string theory,
infinite or fractional dimensional spaces, and more esoteric theoretical
mathematics and physics - we rely on mathematical symbols and abstraction to
guide us, because our finely honed physical intuition is useless (and
sometimes worse than useless).

I'm interested in seeing what the author does with concepts like superposition
and n-dimensional spaces. Replacing them with graphs and animations is not
going to cut it.

~~~
Jacobi
I agree with you, not to mention the beauty of some very abstract mathematical
concepts.

------
adbge
Two fairly simple improvements (to mathematics) I can think of:

1) Abandon the absolutely batty practice of representing everything with a
single symbol. It's crazy. We still can't reliably represent mathematical
symbols over the internet in text, we have to rely on images. I'm obviously
biased, but mapping mathematical functions to _actual words_ (ala programming)
would be a big win, imo.

2) Lock up all the physicists, mathematicians, engineers, logicians, you name
it, and have them agree on a single unified notation. Every symbol ought to
have one meaning and everyone needs to stop stealing symbols from a different
field and giving it a new meaning! I'd wager that more than half of the
symbols in this list
(<http://en.wikipedia.org/wiki/List_of_mathematical_symbols>) have several
valid interpretations. No thanks!

~~~
xyzzyz
1) What do you mean? It depends on field, but in most mathematical papers I
read, the ideas are explained with words, not with symbols -- they only serve
as helpful shortcuts. Anyway, I believe that properly used symbolism usually
clarify the issue at hand, especially when word explanation is very long. For
instance, what is faster to read and comprehend:

<http://mathbin.net/62276>

or

"Let V1 and V2 be a subspaces of W, such that their intersection is zero. Let
f be a mapping from a direct sum of V1 and V2, such that it takes a vector,
whose first component is x and second y to a difference of x and y multiplied
by two."

And this was easy example, I can think of _a lot_ harder.

There is no regulating body of mathematical notation -- it can be (and usually
is) created by introducing it in some paper or book by some mathematician who
invented it and regards as useful. Frequently, there are more than notation
introduced, but usually only one survives -- hopefully the best one. The only
possibilities of encountering several different notations in use at once are
either reading very old works, which is not good anyway, or the most recent
ones, but I presume that people who are able to read them are also able to get
over such a minor problem.

Seriously, I believe that the mathematic notation is a lot clearer, more
intuitive and easier to understand than syntactic rules of many programming
languages, for instance C++. Symbol overloading almost never pose a problem,
since the intended meaning is usually obvious from the context. If one
frequently misunderstands the intended meaning, it is a sign he does not
really get the concepts involved, and the fact he is confused by notation is
his smallest problem.

Even symbol overloading most often takes places only if the sign represents
the same idea in all contexts. For instance, one usually uses '+' sign to
represent a binary commutative operation whatever structure we all talking
about, because, well, it represents similar idea. One can go even further and
say that symbols like \oplus and \times in most contexts they are used in
(Cartesian product of sets, direct product/sum of rings/groups/modules/vector
spaces/mappings) are actually representing _exactly_ the same idea -- namely,
the notion of product/coproduct in some category.

There are a lot of different symbols in use in math. If we abandoned symbol
overloading, we would need to introduce _many, many_ new symbols, and this
would create real confusion.

~~~
btilly
_Symbol overloading almost never pose a problem, since the intended meaning is
usually obvious from the context._

Heh. Heh. Heh.

So I would have believed until I tried to learn differential geometry. The
default is to eliminate all parts of the notation that are unambiguous.
Proving that they are unambiguous is left as an exercise to the reader, and
the exercise is often non-trivial. Furthermore widely used constants vary by
factors of 2 pi depending on who is using it.

------
tzs
Kind of tangentially related (no pun intended), this reminded me of an
interesting article by Tom Apostol, well known calculus teacher and textbook
author. Here's the opening paragraph:

    
    
        Calculus is a beautiful subject with a host of
        dazzling applications. As a teacher of calculus for
        more than 50 years and as an author of a couple of
        textbooks on the subject, I was stunned to learn that
        many standard problems in calculus can be easily solved
        by an innovative visual approach that makes no use of
        formulas. Here’s a sample of three such problems:
    

Here's the article, in PDF format:
<http://eands.caltech.edu/articles/Apostol%20Feature.pdf>

------
rednum
Is this guy trolling? First two paragraphs sound to me similar to what some of
my math untalented friends said about it in highschool. "Power to understand
and predict" is not limited only to those who can manipulate abstract symbols
- for some reasons it seems that such abstract manipulation is the best way to
do it now.

The second paragraph is even worse. This guy have simply no idea what he is
writing about. 'Assigning meaning to set of symbols' is just abstracting
unnecessary details and focusing on important information in problem, 'blindly
shuffling symbols according to arcane rules' - um, calling math 'arcane' is a
clear indicator of person's lack of understanding. Rules are not arcane,
everything has explanation (proof) and is derived from other things in logical
way (and as far as we know, world acts logcailly) - some of the are axioms,
which seem to be abstractios of most basic properties. Also, 'shuffling
blindly' is in fact spotting patterns in things on different levels.

Sure, explainig things on more intuitive level, using e.g. graphical
representations is sometimes really helpful. However sometimes intuition
doesn't work, and how do you graph 3-dimensional manifold embedded in R^4? Or
finite field? Also, I can't imagine of other way of doing math that would be
consistent and useful other than the one we are using now.

~~~
hassy
This "guy" studied electrical engineering and computer science at Caltech and
UC Berkeley, I am sure his grasp of mathematics is ok. Perhaps you should try
to stop feeling offended at his proposal and try to see where he might be
coming from. Also as he admits in the article, it is by no means a
comprehensive set of thoughts, but rather a few hunches hinting at a bigger
idea.

~~~
bermanoid
_This "guy" studied electrical engineering and computer science at Caltech and
UC Berkeley_

I'm actually quite surprised by that. Those are both excellent schools with
top notch math departments, and the example that he mentioned in the article
(about not really grokking the second order ODE and what it meant re: the
phase space plot, etc.) indicates that he took a _seriously_ badly taught
class. I'm kind of surprised you could get through a diff eq's course at
either school without having such basic stuff taught to you...makes me wonder
who the teacher was, maybe it was pawned off on a grad student?

Edit: whoops...just looked back at the article, turns out he _didn't_ take
that diff eq course at Caltech or Berkeley, it was at a local college. That
explains a lot. I'm not going to say that there are no good teachers at
mediocre schools, nor that there are no bad teachers at the good ones, but on
average there's a huge discrepancy in the quality of the classes.

I had similar experiences, where I took classes at a local college while in
high school, and thought I was stupid or something when I didn't "get" them,
only to find that when I took them again at a better school they were, in
fact, very easy topics.

I blame the textbook writers in part: for one example, Serge Lang's math books
are _extremely_ difficult as a rule, and leave a lot of the scaffolding out.
Scaffolding which, when he taught classes himself, he always filled in to make
for an amazingly smooth and effortless learning experience, but which lesser
teachers would never think to talk about (perhaps because they don't
understand it themselves, or at least don't understand how important it is to
explain). It's really a shame that even now, after so many centuries of
teaching math to people, the effectiveness of the process is still so utterly
dependent on the teacher. Hopefully things like the Khan Academy will begin to
rectify these problems.

~~~
kragen
> It's really a shame that even now, after so many centuries of teaching math
> to people, the effectiveness of the process is still so utterly dependent on
> the teacher.

Maybe it's not a shame. Maybe it's just an indication that teaching is hard,
like art, science, programming, and discovering new theorems, not easy like
answering phones in a call center or being a short-order cook.

------
hammock
I am really in love with the sentiment/insight here- that the reason people
are turned off by math (even just numerals) is because it's a complex mess of
symbols that they maybe don't understand.

I am all for finding a way to explain quantitative concepts in a new way.
However, it will be extremely difficult to avoid falling into the trap of
"reinventing the wheel" if all we're talking about is coming up with a new set
of symbols.

 _A certain recipe serves 3, but the cook is only cooking for 2, so she needs
to 2/3 all of the ingredients. The recipe calls for 3/4 cup of flour. The cook
measures out 3/4 cup of flour, spreads it into a circle on the counter, takes
a 1/3 piece out of the circle and puts it back into the bag. That's 2/3 of
3/4._

Much easier to eyeball 1/3 when it's laid out in a rectangle as opposed to a
circle. Author credibility -1

 _mindless tradition_

Did you just call the set of symbols evolved by mathematicians for thousands
of years _mindless?_ Credibility -2

Finally the two animated examples given are clever but not groundbreakingly
clear. -3

It's a neat project but maybe you could think a little harder about defining
your problem.

~~~
ebiester
"Much easier to eyeball 1/3 when it's laid out in a rectangle as opposed to a
circle."

Maybe for you, but certainly not for me, and I'm guessing most bakers would
agree with me. Bakers are used to circles because of pies. I can eyeball a
third of a circle, but I'd have trouble eyeballing a third of a rectangle that
I couldn't fold.

Further, the baker often works by feel, so an exact is not needed in these
circumstances.

"Did you just call the set of symbols evolved by mathematicians for thousands
of years mindless?"

Perhaps a better word would have been arbitrary, but there's no fundamental
reason we pick y=mx+b. Y, M, X, and B are picked arbitrarily, and we _do_ pick
them without questioning whether these are optimal for initial learning.

I grokked math as a kid, but it was precisely because I was able to make the
leap that the language of math was arbitrary and substitutable while other
kids were stuck not understanding the meaning.

~~~
hammock
_I grokked math as a kid, but it was precisely because I was able to make the
leap that the language of math was arbitrary and substitutable while other
kids were stuck not understanding the meaning._

Then we need to teach them that, not a new set of symbols. Again, I think the
crucial insight here which you uncovered is that people are
distracted/confused by the symbology, perhaps trying to take everything too
literally.

By the way, the way to "eyeball" a third is to use your two hands (rotate them
so palms facing each other) to divide into sections A B and C; since we can
very accurately eyeball a 50/50 split, you simply compare A to B and B to C,
then adjust your hands until A=B and B=C. Bam, you have thirds. Once you get
good at this you just mentally visualize invisible dividers instead of
actually using your hands.

When it comes to a circle, if you are staring at pies all day then maybe you
are better than average, but many studies have shown that humans are horrible
at discerning angles other than 180 and 90 degrees.

------
hessenwolf
Ugh. Math is a collection of concrete symbols for simplifying different
problems. It's evolved over thousands of years to help us understand difficult
things. We don't make it that way just to piss people off.

~~~
_delirium
Most of the symbols are pretty recent; if you read mathematics from even 300
years ago, there's a much bigger proportion written in prose. Along the lines
of, "Consider two quantities, such that the latter is at least twice the
former ...".

~~~
eru
And you will understand, why we switched from prose to symbols. But that
doesn't mean we should go overboard with the symbols.

~~~
StavrosK
Exactly. Why do devs shout "comment your code" but academic papers have no
explanation next to the equations?

"Next, calculate the fitness of the algorithm and add it to the pool if it is
better than the worst of the last generation: <math here>"

~~~
xyzzyz
_Why do devs shout "comment your code" but academic papers have no explanation
next to the equations?_

That's exactly the opposite of my impression. Most papers are full of text,
and symbols are not the main feature. For instance:

<http://ttic.uchicago.edu/~yury/papers/kuratowski.pdf> Graph theory, the
symbolism is next to nonexistent.

<http://math.berkeley.edu/~aboocher/math/tietze.pdf> Topology, still symbolism
does not take much space.

<http://www.jstor.org/stable/1989708> Classical and very highly technical, yet
the ratio of text to symbolism is still in favour.

~~~
StavrosK
I remember trying for hours to understand Adaboost before it all clicked (it's
an ingenious algorithm, by the way). The paper could certainly do with some
more explanation, rather than just "this is the weight updating function, this
is the evaluation step, done".

~~~
_delirium
I think AI and machine learning are worse than math in that respect for some
reason. Part of it might be the greater focus on 6-page conference papers,
which tends to require everything to get squished.

------
T_S_
Sure. Speak French without learning French. Or maybe learn French by reading a
picture book. I sympathize with the author because there is a great deal of
room for improvement in the _language_ of math, which may let us model the
world on a computer much more easily.

One example: Using Robinson infinitesimals allows you very easily to write
code for forward mode exact differentiation (not symbolic, not approximate).
But justifying these simplifications is hard. The question is how much math
can be simplified by analogous means without wrecking the foundations.

Another example: Many really useful systems have to deal with uncertainty. I
have yet to see a system that allows programmers to easily build such models.
I have seen some nice ideas probability monads, Bayesian networks, etc. But
how many non-specialists are prepared to use such tools? Happy to have HNers
prove me wrong on this one.

~~~
tzs
> Sure. Speak French without learning French. Or maybe learn French by reading
> a picture book.

That's basically the approach Rosetta Stone takes, and it works fairly well.

~~~
T_S_
Rosetta Stone for math? I'd like to see it.

I have a friend who was able to buy a laxative in Italy without knowing a word
of Italian. It's a funny story. I doubt the same approach would work with
math.

------
api
Math is a language. The problem is that this language is archaic, difficult to
teach, and is generally taught very poorly.

I remember learning calculus in college. The professor went up to the board,
scribbled down symbols, and took us through various procedures. I was
absolutely, utterly lost until my father (an engineer) told me that "a
derivative is a rate of change."

At that instant, I understood everything. My professor never said this.

This taught me to approach math concepts-first, and that helped, but I've
always had a problem with math. To make a long story short: I hate math for
the same reason that I hate Perl. My mind recoils in horror from messy, crufty
languages.

At the very least, all math lessons should begin by teaching the __language
__and the concepts that the various symbols, arrangements, etc. refer to. Only
once the language is thoroughly grasped should they proceed to methods,
procedures, and problems. Right now it's like teaching Chinese literature
before teaching Chinese...

------
ekidd
When I was a child, I always found math easy. Fractions, calculus,
formulas—they were always _visual._ Fractions were pizzas; equations were
deformations of the plane; integration was tracing out a shape using a slope.

So I'm sympathetic to the author's desire for better visualization and
teaching tools.

But when I reached college, I became frustrated with math. It just wasn't
_easy_ anymore, the way programming was: I could pick up a programming book,
read it in a weekend, and understand it. But when I tried to read an advanced
math text, I became lost after 10 pages.

Eventually, I figured out what had happened: The information density of
college-level math texts is _insane_. Even if you're bright and talented, it
may take you a day to understand a single page. And there's no substitute for
working carefully, finding concrete examples, and slowly building a deep
understanding.

Here's an example that involves programming. Once upon a time, I needed to
understand monads, in hope of finding a better way to represent Bayesian
probabilities.

I started with the monad laws, a handful of equations relating _unit, map,_
and _join_. I read countless monad tutorials, and dozens of papers. I read
every silly example of how monads are like containers, space suits, C++
templates, and who knows what else.

I wrote little libraries. I learned category theory. I _wrote_ a monad
tutorial. I eventually wrote a paper explaining a whole family of probability
monads:

[http://www.randomhacks.net/darcs/probability-
monads/probabil...](http://www.randomhacks.net/darcs/probability-
monads/probability-monads.pdf)

And then one day, I thought about the monad laws again. I realized, "Hey,
that's it. That's _all_. Just _unit, map, join,_ and a handful of equations.
Anything which quacks like a monad, _is_ a monad. How did I ever think this
was complicated?"

But when I look at the monad laws today, there's this huge structure of
connections in my head. All that work, just to grasp something so simple, and
so easy.

So I'm all for building better visualizations, and for helping people to
understand math intuitively. That's an important step along the path. But math
doesn't stop at an intuitive understanding. When you really understand it, the
equations will suddenly be easy, and everything will fit together.

And then you'll encounter the miracle of math: Your deep understanding will
become the raw material for the next level. Counting prepares you for
addition, addition prepares you for multiplication, basic arithmetic for
algebra, algebra for calculus, and so on. And someday, I hope that my
rudimentary understanding of category theory will prepare me to understand why
adjoint functors are interesting.

~~~
juiceandjuice
I had a different problem, I found that word density increases 10 fold, but
symbol density and proofs go down as you go through it.

My favorite example to use is this: Complex Variables and Applications by
Brown and Churchill. This book has been in print for 70 years or something,
and it's somewhere around 400 pages in the current edition I believe. My
professor I did research for had a early 80s edition, and it had almost 100
less pages than the current edition. There wasn't really anything new added
between the versions (chapters are only about 5-10 pages, so there's something
like 65 of them) I ended up using mine for the problems and his for reading
because I have ADHD, and the wordiness absolutely kills me. Symbols and
relationships are _much_ more meaningful to me than words describing them. The
real nightmare with the ADHD sets in because of the break in context when you
have to switch between two or three pages to find the next theorem, formula or
proof.

I retook that class twice.

On the other hand, I utterly and completely rocked my Advanced Electrodynamics
course, outscoring even the graduate students, in a course which even made use
of the stuff we were learning in Complex Variables (as well as PDEs and all
that fun stuff) Why? I had a crazy russian professor who hated all the current
textbooks (I'm looking at you, Griffiths) for the same reason that I hated
textbooks, too much words and not enough symbols. So he wrote his own notes to
every lesson and made his own homework. He said he originally wrote those
notes when he first came here, and his english was worse, so there's little or
no explanation, just proofs -- math and symbols. A few of these would span two
pages, and very rarely three, but there wasn't the context break you get in
many college level books, just beautiful math and lots of intermediate steps.
The intermediate steps, almost never provided in most textbook proofs, really
help the visual learners like me and provide stepping stones for the
inevitable manipulation you will perform with those equations in your homework
and on tests.

I still have all his notes, I want to bind them up some day when I get a
chance.

~~~
joshhart
I really liked Griffiths. I thought it had struck a great balance between
explanations, mathematics, and examples. That book had great examples and
homework questions, which I find the most useful anyway.

~~~
bermanoid
Griffiths' E+M book (actually, also his QM book) is a shining star to me, a
perfect example of the way introductory textbooks should be written. It ends
up being criticized a lot once you get to the advanced level ("Who'd you learn
quantum mechanics from, _Griffiths_ lol?"), but that's _fine_. It's perfect,
even - if you really need all the gory stuff, sure, go get a copy of Jackson's
book, or some 800 page tome on perturbation theory and the rigorous
mathematical underpinnings of quantum theory, if you're going into the field
you should absolutely know more than you can get from an intro text. But an
introductory textbook should be exactly that. Rigor is awesome once you have
the intuition down, but it's often very difficult to go the other way (mainly
because a rigorous understanding is difficult to achieve without having the
intuition first).

~~~
toponium
I think Griffiths EM/QM texts are fine if your not studying physics. However
if you are they just skim the surface of material and this is were I think a
lot of the bashing comes in. Physics curriculum (presumably like all) are a
constant stepping stone and when your senior level professors expect you to
know Clebsch Gordon coefficients in detail and Griffiths QM didn't then some
questioning about his texts start to rise. In my opinion I thought Sakurai's
QM was excellent in delivering a solid foundation in quantum mechanics

------
hassy
Highly recommended reading by the same author:
<http://worrydream.com/#!/MagicInk>

(To pique your curiosity: read it to understand why Hipmunk is awesome -
there's much more to it than just that of course)

~~~
cubicle67
There's some discussion on that here
<http://news.ycombinator.com/item?id=600799> (from about 2 years ago)

------
cubicle67
I'm just really curious to know what Brett was involved with at Apple

[Edit: wow! <http://worrydream.com/cv/bret_victor_resume.pdf>]

~~~
jimbokun
Don't know that I've ever seen a resume that covered everything from hardware
engineering all the way up to some of the most highly regarded user interface
designs. He has certainly covered the entire gamut of "things you can do with
a computer."

Which makes him the perfect Apple employee, as they cover the space from
hardware engineering through user experience better than anyone.

~~~
spenrose
Magic Ink is my single favorite statement on software design:
<http://worrydream.com/#!/MagicInk>

------
nithyad
I also have a problem with the way mathematics is taught currently where the
student is more adept at the mechanics than the philosophy of mathematics. If
anybody is attempting to change that, my best wishes. I don't necessarily have
a problem with the current system of notations. It is just the method of
teaching that lacks.

Just started with 'What is Mathematics' by Courant and I am totally hooked on.
He talks about everything from why we chose to adopt the decimal system to why
pi was needed to solve certain problems.

As Hammock says, the author has to be clear about defining the problem itself.
Is it mathematics as it is represented today which is lacking or the method of
teaching which is lacking

------
pavel_lishin
> This "Math" consists of assigning meaning to a set of symbols, blindly
> shuffling around these symbols according to arcane rules, and then
> interpreting a meaning from the shuffled result. The process is not unlike
> casting lots.

> using concrete representations and intuition-guided exploration.

Is it just me, or are these two lines very much at odds with each other?

------
palish
This is awesome: <http://worrydream.com/SimulationAsAPracticalTool/>

------
pnathan
Pictures aid understanding. But in the vast majority of sophisticated
mathematics (that I've seen), pictures would be unable to accurately represent
the real assertions.

For instance, if I want to talk about 26-dimensional discontinuous space (the
space of the basic alphabet), there is no visualization that can help you
grasp the totality of the matter.

I believe there are a few interpretations of what math actually is (formal
reasoning, interpretation, etc), philosophy of math 'junk'. I'll leave that to
a more-beered time. :-)

------
Semiapies
" _those with a freakish knack for manipulating abstract symbols_ "

The downside of the growing public awareness of people on the autism spectrum,
some of whom are geeks, has been the slow trend towards conflating
intellectualism with atypical mental function.

Some people like to call right now the "victory of the geeks". I suspect that
within the next 10 years, nerdy kids will start being diagnosed as having
Asperger's by school counselors and the like with about as much care, caution,
and accuracy as we saw with ADHD.

------
jgrodziski
This post reminds me the visual explanation of pythagoras' theorem :
<http://www.mathsisfun.com/pythagoras.html> Also, the math language symbols
are great for communicating easily and efficiently among us, but it's not the
best materials for learning. Distinguish learning (with visual) and
communicating (with language symbols) activity in math should help...

------
comex
Can we start by killing all the Greek letters and replacing them with
Mathematica-esque notation, now that computers have dramatically reduced the
convenience of writing symbols rather than words?

~~~
Muzza
Computers have not reduced the convenience of writing symbols on PAPER or on a
BLACKBOARD, the places where math is actually done.

------
gfodor
I've gone back to Bret's site for years since its so amazing. Imagine my
thrill to see he is finally out of the shadows of Apple and is sharing his
ideas with the world again. Awesome!

~~~
jashkenas
I'll second that. I was _thrilled_ to see him pop up again in my RSS reader
last month, after years of radio silence.

------
smogzer
I just wish that scientific publications came in Python code instead/besides
of "in formula (a) the variables ... represent ..." . Just get me the code and
I will just plug into the general framework for that (robotics, chemistry,
politics, finance) and see if it works as stated in the article.

And ! code cannot be misinterpreted, either it runs or it does not.

So dear mathematicians please do the evolution and cast your ideas as python
code.

~~~
boryas
what python code should people write for non-constructive results?

------
Ixiaus
For quite a long time I hated Math and was horrible at it - I remember being
up until 2AM in the morning with my father during 7th and 8th grade trying to
do my math homework, taking tests and feeling confident about all of it then
receiving my grade that said I failed. After I dropped out of high school and
did some self-discovery (involved India, a commune, and a few other journeys)
I began a self-study curriculum (created by me). I didn't bring math into the
curriculum until about three years ago, I've been fond of logic and the
methods of logic for quite some time and ended up finding "The Foundations of
Arithmetic" by Gottlieb Frege - a treatise attempting to provide a purely
logical framework for number (he influenced set theory) and the operations on
number; but, reading it really fired me up, I was absorbed and interested,
Frege showed me a completely different side if mathematics that _no one_ in my
education had cared to show me or even hint at.

With that new found interest in Mathematics I set about to find a book that I
could use to teach myself. I won't say how I came across it, but I ended up
coming across "Practical Mathematics" by C.I Palmer - the 1919 publication,
very old. The book is intended for working adults in mechanical trades; the
problems in the book were often given using real-world problems in the
technical trades. The author also used _much_ more technical language! It was
refreshing! Even my high school math textbooks felt like the authors were
trying to teach kids and regarded their audience as nothing more than
insufferable immatures. This book has been immeasurably valuable to me, I now
feel less "darkness" and confusion when I see a math problem, I'm finally
seeing the utility in my everyday life of the things I'm learning (my gf had a
wedge table and was selling it on Craigslist and needed to know the length of
the arc, for example). I actually _know_ how to add and subtract fractions,
it's no longer a mystical act of numbers disappearing here and showing up
there.

The greatest thing about that book? Certain operations that most schools only
_ever_ taught me as a mechanical process were taught to me and explained to me
with the underlying fundamentals in "Practical Mathematics". In 1919 there
were no calculators, it had to all be done by hand, even square roots, and you
couldn't really survive without knowing _why_ or _how_ certain mechanical
operations are used the way they are in Math.

I'm not sure if college or higher mathematics gets into that stuff, but, it
was a revelation for me and I'm well on my way through the Algebras and Trig
now - I didn't even make it out of pre-algebra in the traditional educational
system. I can also see the beauty of mathematics too, something I never
thought I would understand about mathematicians when I used to hate the
subject.

There will never be any "killing" of math. But our educational system has a
long way to go.

~~~
zerosanity
Free on Google Books: <http://books.google.com/books?id=EAmgAAAAMAAJ>

~~~
2mur
Thanks for this!

------
waterhouse
Technical tangent: I opened this in a tab, intending to read it later, then
came back and noticed Firefox was using 100% CPU. I thought "oh god what's
happening" and tried closing other tabs to determine the cause. Eventually I
discovered that a) this website was the culprit, and b) this is most likely
because, a few pages down, there's some kind of (processor-intensive)
simulation happening.

Lessons I draw from this:

\- In designing a website that outsiders visit, you should probably follow
this rule: If your page must have a majorly resource-intensive object on it,
it should either be visible and obvious at the top of the page, or have a
"Start" or "Play" button and not run until the user presses it.

\- Firefox should get some (easily accessible) way to see resource usage
broken down by tab. Chrome does this; it's probably made easy by the fact that
Chrome makes a separate OS process for each tab (or group of tabs). I wouldn't
recommend switching to that model, because it would be work and because I
wouldn't like it (I don't like how it clutters up the global process list in
Activity Monitor), but it would be nice if you could see CPU and memory usage
broken down by tab.

\- (Optional) Browsers shouldn't run purely graphical animations if they're
not visible (like if they're several pages down). This might not be perfectly
achievable--e.g. if animations started when you scrolled down to them, then
one animation slightly higher than another might start earlier when the
designer wanted them to be synchronized. Maybe you would instead have
background things keep time without actually rendering anything. This might
not work for cases where the nth frame depends on the conditions of the n-1th
frame, and so you'd have to run the whole simulation in the background anyway
--though maybe leaving out the "draw" part would make a big difference, I
don't know. But if it's a series of static frames, like a GIF, then I think
this would work--when the GIF is offscreen, the browser would just keep
incrementing a frame number (mod the number of frames in the GIF).

------
Jacobi
Math is not a collection of symbols. It's language + reasoning.

~~~
Jacobi
Also math provides us with a framework to work on things that we could not
visualize or even imagine ... how about visualizing an n-dimensional
riemannian manifold ?

------
noonespecial
You can do "language" without all of those pesky symbols as well but it makes
it hella hard to tell other people 100 years down the line what you were on
about.

------
ltnately
I think it misses a key point of math which is to take an idea from a concrete
example and expand it to demonstrate something in general.

------
headbiznatch
Always happy to read what is on Bret Victor's mind. Chris Knight, Mitch
Taylor? Morons. Bret is the shit.

------
perlgeek
I like the idea, but the headline is totally off. The author wants to make
math available with less formalism and less reliance on symbols, not kill math
in any way.

------
hotdox
After killing math, this site kill my browser.

~~~
filobloomz
Same here hotdox. It killed my machine. I had to reboot.

------
ignifero
Math is about abstraction. It's OK to describe elementary maths and calculus
with water-filling bottles, but it ends there, because the higher order
abstract structures of mathematics have no real-world equivalent. If we start
teaching people with Flash simulations, we 'll have to forget about the
Standard model, relativity, electromagnetism, Symmetry, string theory etc.
Sure, visual representations make some subjects more accessible, but they can
also be terribly misleading and limiting. (Just imagine how much bias our
3-dimensional world experience imposes to our visual thinking). If anything,
abstract (written) math allows us to escape the confines of perceived reality.

Math and language are 2 faces of the same ability we have to model the world
in arbitrary symbols. Just like math, language has symbols and syntax, that
have meaning to us regardless of representation (spoken, written or digital).

------
NY_USA_Hacker
He doesn't understand math at all, not at K-12, college, graduate school,
research, applications, etc. at all.

E.g., his claim about symbol manipulation is total nonsense.

His direction is a waste of time.

If he wants to improve materials for learning math, then fine, but he should
first learn some math.

He should start with the books and papers of P. Halmos, one of the best
writers of math ever.

Note: My Ph.D. dissertation research was on the math of stochastic optimal
control.

