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.
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.
You know, I'm glad that you're able to easily manipulate symbols, but when I look at them, my eyes glaze over. It took years for me to learn linear algebra because not a single teacher ever told me "a matrix is also mapping from one space to another one", until Massi Pontil of UCL.
At that point, all the linear algebra I couldn't figure out for the life of me all those years finally made sense. And it was the same for most of my classmates. After that, whenever I saw xY, I thought "the vector x is being moved into a new space", and all the equations made sense to me.
You could explain what an SVM is with equations to me all day, but it's only when you say "you're trying to get the plane to separate your data by a margin as wide as possible" do I actually get it, and then all the math becomes easy.
Different people have different ways of manipulating the abstract symbols, and for me it's to equate them to something I already have experience in. Then I can get the solutions intuitively, rather than pore over pages and pages of equations.
In the end, I quit academia precisely because I couldn't manipulate symbols, and thus my way of learning wasn't compatible with everyone's way of teaching. Maybe I can come up with something better if someone explains things to me in terms I can understand.
>More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
Math is not at all the manipulation of symbols though. That is one of many tools that is useful in doing math, but it is not the only one. The famous proof that there are an infinite number of prime numbers is normally expressed in plain English (or your language of choice) without a symbol in sight, for one example.
Math, fundamentally, is about abstract concepts, not symbols.
Sure the bucket isn't doing math, but how else would you describe it? Integration is the word we have come to use to communicate the meaning of what is happening. In other words, math isn't just about understanding. It is about communication.
Sure we have intuition. But intuition can be wrong. Not all problems are as simple to explain as zero and empty buckets. Intuition is also worthless if you cannot communicate those ideas in an unambiguous fashion.
So if the OP really develops a method for communicating the concepts described by math in a much more efficient way - I am all for it! But I find that highly unlikely when even it is admitted in the article itself that he has no idea what this will look like.
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.
(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.)
What's you background? What kind of math did you learn by looking at mathematical symbols rather than natural language explanations? I have hard time envisioning how such a leaning process would work.
There are several pretty obvious problems with mathematical notation.
- Meaningless one-letter names.
- Meaning of notations is usually highly context-dependent. (Exponentiation and matrix inverse operation are denoted in exactly the same way, for example.)
- A lot depends on arbitrary conventions.
- People rarely explain how notations work syntactically. It's all ad-hoc,learn by example.
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.
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.
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!
You know, one of the reason that notations differ is because that's useful. The symbol \times might mean cross product or direct product of groups, or cartesian product. Why? Because these operations are conceptually similar --- they all involve making a multiplicatively larger thing from smaller things. Using the same symbol for different things might be polymorphism, not conflict.
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:
"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.
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:
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.
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.
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.
"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.
Most mathematical symbols haven't been in use for longer than a couple of hundred of years. Leibniz's now standard calculus notation was introduced near the end of 17th century.
seriously? and your contribution to helping people understand maths is ...?
You've given him some arbitrary credibility rating of -3 because his examples weren't exactly how you would have done them, except that you didn't write them, he did. You contribution was to write some bitchy comment about it.
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.
His argument is that analyzing a differential equation without exploring it in phase space was like analyzing a piece of sheet music without actually hearing it.
And if music education were taught in that way-- by looking at music purely as the manipulation of concrete symbols-- I imagine some of us would be writing "kill music (as it is currently taught)" blog posts as well.
Talking about Math as a single subject is like talking about Sports. Chess, Boxing, Baseball, Snowboarding, Curling, and Luge may have some things in common. However, suggesting that DiffEq and Fractions are equally meaningless to most people seems just as out of place as assuming a chess grandmaster is also a champion snowboarder.
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 ...".
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.
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...
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:
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.
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.
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.
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
> 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?
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. :-)
"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.
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...
[+] [-] mechanical_fish|15 years ago|reply
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|15 years ago|reply
"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.
[+] [-] StavrosK|15 years ago|reply
At that point, all the linear algebra I couldn't figure out for the life of me all those years finally made sense. And it was the same for most of my classmates. After that, whenever I saw xY, I thought "the vector x is being moved into a new space", and all the equations made sense to me.
You could explain what an SVM is with equations to me all day, but it's only when you say "you're trying to get the plane to separate your data by a margin as wide as possible" do I actually get it, and then all the math becomes easy.
Different people have different ways of manipulating the abstract symbols, and for me it's to equate them to something I already have experience in. Then I can get the solutions intuitively, rather than pore over pages and pages of equations.
In the end, I quit academia precisely because I couldn't manipulate symbols, and thus my way of learning wasn't compatible with everyone's way of teaching. Maybe I can come up with something better if someone explains things to me in terms I can understand.
[+] [-] bobbyi|15 years ago|reply
That's the definition of a calculus, not the entirety of math.
http://en.wikipedia.org/wiki/Calculus :
>More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
[+] [-] timwiseman|15 years ago|reply
Math, fundamentally, is about abstract concepts, not symbols.
[+] [-] nightski|15 years ago|reply
Sure we have intuition. But intuition can be wrong. Not all problems are as simple to explain as zero and empty buckets. Intuition is also worthless if you cannot communicate those ideas in an unambiguous fashion.
So if the OP really develops a method for communicating the concepts described by math in a much more efficient way - I am all for it! But I find that highly unlikely when even it is admitted in the article itself that he has no idea what this will look like.
[+] [-] jxcole|15 years ago|reply
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|15 years ago|reply
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.)
[+] [-] mechanical_fish|15 years ago|reply
Modded up for this line alone.
[Not that there's anything wrong with the rest of the lines. ;) ]
[+] [-] romaniv|15 years ago|reply
There are several pretty obvious problems with mathematical notation. - Meaningless one-letter names. - Meaning of notations is usually highly context-dependent. (Exponentiation and matrix inverse operation are denoted in exactly the same way, for example.) - A lot depends on arbitrary conventions. - People rarely explain how notations work syntactically. It's all ad-hoc,learn by example.
[+] [-] wbhart|15 years ago|reply
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|15 years ago|reply
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|15 years ago|reply
[+] [-] adbge|15 years ago|reply
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!
[+] [-] pavpanchekha|15 years ago|reply
[+] [-] xyzzyz|15 years ago|reply
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.
[+] [-] tzs|15 years ago|reply
[+] [-] rednum|15 years ago|reply
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|15 years ago|reply
[+] [-] hammock|15 years ago|reply
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|15 years ago|reply
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.
[+] [-] hassy|15 years ago|reply
[+] [-] cubicle67|15 years ago|reply
You've given him some arbitrary credibility rating of -3 because his examples weren't exactly how you would have done them, except that you didn't write them, he did. You contribution was to write some bitchy comment about it.
[+] [-] hessenwolf|15 years ago|reply
[+] [-] michael_dorfman|15 years ago|reply
His argument is that analyzing a differential equation without exploring it in phase space was like analyzing a piece of sheet music without actually hearing it.
And if music education were taught in that way-- by looking at music purely as the manipulation of concrete symbols-- I imagine some of us would be writing "kill music (as it is currently taught)" blog posts as well.
EDIT: minor grammar fix
[+] [-] simpleTruth|15 years ago|reply
[+] [-] _delirium|15 years ago|reply
[+] [-] T_S_|15 years ago|reply
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|15 years ago|reply
That's basically the approach Rosetta Stone takes, and it works fairly well.
[+] [-] pbiggar|15 years ago|reply
[+] [-] api|15 years ago|reply
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|15 years ago|reply
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...
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|15 years ago|reply
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.
[+] [-] hassy|15 years ago|reply
(To pique your curiosity: read it to understand why Hipmunk is awesome - there's much more to it than just that of course)
[+] [-] cubicle67|15 years ago|reply
[+] [-] cubicle67|15 years ago|reply
[Edit: wow! http://worrydream.com/cv/bret_victor_resume.pdf]
[+] [-] jimbokun|15 years ago|reply
Which makes him the perfect Apple employee, as they cover the space from hardware engineering through user experience better than anyone.
[+] [-] nithyad|15 years ago|reply
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|15 years ago|reply
> using concrete representations and intuition-guided exploration.
Is it just me, or are these two lines very much at odds with each other?
[+] [-] palish|15 years ago|reply
[+] [-] pnathan|15 years ago|reply
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|15 years ago|reply
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|15 years ago|reply