10 ms·

Kill Math

227 points by pw 15y ago

hotdox 15y ago
After killing math, this site kill my browser.
- filobloomz 15y ago
  Same here hotdox. It killed my machine. I had to reboot.
hessenwolf 15y ago
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.
- michael_dorfman 15y ago
  But math is more than that. 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
  - hammock 15y ago
    I doubt anyone here learned math (at least up through calculus) without concrete examples. They were called word problems and lab assignments.
    - generalk 15y ago
      Word problems are still abstract manipulation of symbols, you just have to hunt around in the phrasing for the right operands or operations, which arguably makes it harder.
    - fhars 15y ago
      But the jury is still out on the question of whether concrete examples are usefull in maths education: http://researchnews.osu.edu/archive/mathed.htm http://researchnews.osu.edu/archive/mathed.htm
      - hessenwolf 15y ago
        Weak study. They made some students learn something using concrete examples only, and then expected them to be able to abstract. That's just dumb. I learn the maths, and apply to various situations in my head to make sure I understand it.
    - hessenwolf 15y ago
      I'll be honest; I absolute love graph theory. If I had a million lives and financial independence, I would spend several of them solving graph theory problems. I don't really give a shit what the nodes and edges are. Computer network? Rigid structure? Conditional independence of personality traits? I don't care!
- _delirium 15y ago
  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 ...".
  - StefanKarpinski 15y ago
    The shift to more symbols and more and better notation has also drastically increased mathematical productivity and rigor. In fact, most of the notation emerged together with the axiomatization of the foundations of mathematics in the early 20th century. If we were to follow this guy's ideas it would cause even more of a class divide between those who can understand the "magical symbols" and those who can't. Doing what he suggests would mean that the non-cognoscenti wouldn't even have access to the understanding of simple algebraic equations.
  - eru 15y ago
    And you will understand, why we switched from prose to symbols. But that doesn't mean we should go overboard with the symbols.
    - hessenwolf 15y ago
      http://upload.wikimedia.org/wikipedia/commons/5/56/Actuarial_notation.svg http://upload.wikimedia.org/wikipedia/commons/5/56/Actuarial... Overboard?
      - eru 15y ago
        Depends on how you embed it in text.
    - StavrosK 15y ago
      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>"
- simpleTruth 15y ago
  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.
hammock 15y ago
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.
- cubicle67 15y ago
  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.
  - true_religion 15y ago
    You can't ask us to "kill" something that's foundational to modern society without having a brilliant replacement. A mere incremental change isn't worth the hardship of giving up the status quo.
- hassy 15y ago
  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.
  - 15y ago
jgrodziski 15y ago
This post reminds me the visual explanation of pythagoras' theorem : http://www.mathsisfun.com/pythagoras.html 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...
deleted 15y ago
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Jacobi 15y ago
Math is not a collection of symbols. It's language + reasoning.
- Jacobi 15y ago
  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 ?
karamazov 15y ago
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 15y ago
  I agree with you, not to mention the beauty of some very abstract mathematical concepts.
gfodor 15y ago
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 15y ago
  I'll second that. I was thrilled to see him pop up again in my RSS reader last month, after years of radio silence.
cubicle67 15y ago
I'm just really curious to know what Brett was involved with at Apple [Edit: wow! http://worrydream.com/cv/bret_victor_resume.pdf http://worrydream.com/cv/bret_victor_resume.pdf]
- jimbokun 15y ago
  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 15y ago
    Magic Ink is my single favorite statement on software design: http://worrydream.com/#!/MagicInk http://worrydream.com/#!/MagicInk
jxcole 15y ago
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.
- mechanical_fish 15y ago
  If this site kills math so that English majors can cope, I hope someday someone will kill poetry so that I can cope. Modded up for this line alone. [Not that there's anything wrong with the rest of the lines. ;) ]
- romaniv 15y ago
  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.
- premchai21 15y ago
  (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.)
hassy 15y ago
Highly recommended reading by the same author: http://worrydream.com/#!/MagicInk 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 15y ago
  There's some discussion on that here http://news.ycombinator.com/item?id=600799 http://news.ycombinator.com/item?id=600799 (from about 2 years ago)
palish 15y ago
This is awesome: http://worrydream.com/SimulationAsAPracticalTool/ http://worrydream.com/SimulationAsAPracticalTool/
nithyad 15y ago
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
rednum 15y ago
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 15y ago
  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 15y ago
    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.
mechanical_fish 15y ago
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.
- StavrosK 15y ago
  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.
pavel_lishin 15y ago
> 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?
ltnately 15y ago
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.
Ixiaus 15y ago
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.
adbge 15y ago
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 http://en.wikipedia.org/wiki/List_of_mathematical_symbols) have several valid interpretations. No thanks!
- pavpanchekha 15y ago
  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.
- xyzzyz 15y ago
  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 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.
wbhart 15y ago
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.
smogzer 15y ago
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 15y ago
  what python code should people write for non-constructive results?
  - deleted 15y ago
    [deleted]
- xyzzyz 15y ago
  I wish software came in beautiful ponies instead of ugly source code, so that I can just get right on it and ride where the sun rises. Sadly, ponies are not the best (or even appropriate) way to represent software, just as Python code is not useful in describing math.
ekidd 15y ago
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/probability-monads.pdf 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 15y ago
noonespecial 15y ago
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.
T_S_ 15y ago
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 15y ago
  > 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_ 15y ago
    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.
- pbiggar 15y ago
  The goal isn't to speak French, it's to communicate with French people.
api 15y ago
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...
tzs 15y ago
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 http://eands.caltech.edu/articles/Apostol%20Feature.pdf
pnathan 15y ago
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 15y ago
"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.
headbiznatch 15y ago
Always happy to read what is on Bret Victor's mind. Chris Knight, Mitch Taylor? Morons. Bret is the shit.
ignifero 15y ago
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 15y ago
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.
comex 15y ago
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 15y ago
  Computers have not reduced the convenience of writing symbols on PAPER or on a BLACKBOARD, the places where math is actually done.
perlgeek 15y ago
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.
waterhouse 15y ago
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).

Kill Math

227 points by pw 15y ago

hotdox 15y ago

After killing math, this site kill my browser.

filobloomz 15y ago
Same here hotdox. It killed my machine. I had to reboot.

hessenwolf 15y ago

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.

michael_dorfman 15y ago
But math is more than that. 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
- hammock 15y ago
  I doubt anyone here learned math (at least up through calculus) without concrete examples. They were called word problems and lab assignments.
  - generalk 15y ago
    Word problems are still abstract manipulation of symbols, you just have to hunt around in the phrasing for the right operands or operations, which arguably makes it harder.
  - fhars 15y ago
    But the jury is still out on the question of whether concrete examples are usefull in maths education: http://researchnews.osu.edu/archive/mathed.htm http://researchnews.osu.edu/archive/mathed.htm
    - hessenwolf 15y ago
      Weak study. They made some students learn something using concrete examples only, and then expected them to be able to abstract. That's just dumb. I learn the maths, and apply to various situations in my head to make sure I understand it.
  - hessenwolf 15y ago
    I'll be honest; I absolute love graph theory. If I had a million lives and financial independence, I would spend several of them solving graph theory problems. I don't really give a shit what the nodes and edges are. Computer network? Rigid structure? Conditional independence of personality traits? I don't care!
_delirium 15y ago
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 ...".
- StefanKarpinski 15y ago
  The shift to more symbols and more and better notation has also drastically increased mathematical productivity and rigor. In fact, most of the notation emerged together with the axiomatization of the foundations of mathematics in the early 20th century. If we were to follow this guy's ideas it would cause even more of a class divide between those who can understand the "magical symbols" and those who can't. Doing what he suggests would mean that the non-cognoscenti wouldn't even have access to the understanding of simple algebraic equations.
- eru 15y ago
  And you will understand, why we switched from prose to symbols. But that doesn't mean we should go overboard with the symbols.
  - hessenwolf 15y ago
    http://upload.wikimedia.org/wikipedia/commons/5/56/Actuarial_notation.svg http://upload.wikimedia.org/wikipedia/commons/5/56/Actuarial... Overboard?
    - eru 15y ago
      Depends on how you embed it in text.
  - StavrosK 15y ago
    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>"
simpleTruth 15y ago
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.

hammock 15y ago

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.

cubicle67 15y ago
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.
- true_religion 15y ago
  You can't ask us to "kill" something that's foundational to modern society without having a brilliant replacement. A mere incremental change isn't worth the hardship of giving up the status quo.
hassy 15y ago
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.
- 15y ago

jgrodziski 15y ago

This post reminds me the visual explanation of pythagoras' theorem : http://www.mathsisfun.com/pythagoras.html 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...

deleted 15y ago

[deleted]

Jacobi 15y ago

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

Jacobi 15y ago
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 ?

karamazov 15y ago

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 15y ago
I agree with you, not to mention the beauty of some very abstract mathematical concepts.

gfodor 15y ago

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 15y ago
I'll second that. I was thrilled to see him pop up again in my RSS reader last month, after years of radio silence.

cubicle67 15y ago

I'm just really curious to know what Brett was involved with at Apple [Edit: wow! http://worrydream.com/cv/bret_victor_resume.pdf http://worrydream.com/cv/bret_victor_resume.pdf]

jimbokun 15y ago
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 15y ago
  Magic Ink is my single favorite statement on software design: http://worrydream.com/#!/MagicInk http://worrydream.com/#!/MagicInk

jxcole 15y ago

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.

mechanical_fish 15y ago
If this site kills math so that English majors can cope, I hope someday someone will kill poetry so that I can cope. Modded up for this line alone. [Not that there's anything wrong with the rest of the lines. ;) ]
romaniv 15y ago
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.
premchai21 15y ago
(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.)

hassy 15y ago

Highly recommended reading by the same author: http://worrydream.com/#!/MagicInk 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 15y ago
There's some discussion on that here http://news.ycombinator.com/item?id=600799 http://news.ycombinator.com/item?id=600799 (from about 2 years ago)

palish 15y ago

This is awesome: http://worrydream.com/SimulationAsAPracticalTool/ http://worrydream.com/SimulationAsAPracticalTool/

nithyad 15y ago

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

rednum 15y ago

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 15y ago
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 15y ago
  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.

mechanical_fish 15y ago

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.

StavrosK 15y ago
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.

pavel_lishin 15y ago

> 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?

ltnately 15y ago

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.

Ixiaus 15y ago

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.

adbge 15y ago

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 http://en.wikipedia.org/wiki/List_of_mathematical_symbols) have several valid interpretations. No thanks!

pavpanchekha 15y ago
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.
xyzzyz 15y ago
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 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.

wbhart 15y ago

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.

smogzer 15y ago

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 15y ago
what python code should people write for non-constructive results?
- deleted 15y ago
  [deleted]
xyzzyz 15y ago
I wish software came in beautiful ponies instead of ugly source code, so that I can just get right on it and ride where the sun rises. Sadly, ponies are not the best (or even appropriate) way to represent software, just as Python code is not useful in describing math.

ekidd 15y ago

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/probability-monads.pdf 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 15y ago

noonespecial 15y ago

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.

T_S_ 15y ago

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 15y ago
> 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_ 15y ago
  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.
pbiggar 15y ago
The goal isn't to speak French, it's to communicate with French people.

api 15y ago

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...

tzs 15y ago

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 http://eands.caltech.edu/articles/Apostol%20Feature.pdf

pnathan 15y ago

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 15y ago

"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.

headbiznatch 15y ago

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

ignifero 15y ago

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 15y ago

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.

comex 15y ago

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 15y ago
Computers have not reduced the convenience of writing symbols on PAPER or on a BLACKBOARD, the places where math is actually done.

perlgeek 15y ago

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.

waterhouse 15y ago

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).