10 ms·

Kill Math

199 points by noonespecial 15y ago

tikhonj 15y ago
This is an interesting idea, and I think his final analogy summed it up perfectly: symbolic math is like a command line. But I reject his assumption that a command line is a bad interface. And that mirrors my thoughts on this post: for the layperson, a GUI may indeed be better than a command line; for a professional (a programmer) the reverse is true. A command line lets you combine and recombine different programs and easily do many things its creators never dreamed of. Symbolic manipulation and algebra are just like that! Let's imagine you know how to differentiate viscerally; you understand what a tangent line looks like and how to plot that and you care not for silly equations like d/dx x^3 = 3x^2. Naturally, you understand basic arithmetic in the same way and not as mechanical manipulations on symbols. You are perfectly well equipped to deal with real problems, and perhaps find it easier than shuffling symbols around. But you would never come up with a way to get the derivative of, say, an algebraic data type[1][2]. (For the curious, this is how you can define a zipper on a type.) [1]: http://blog.lab49.com/archives/3011 http://blog.lab49.com/archives/3011 [2]: http://strictlypositive.org/diff.pdf http://strictlypositive.org/diff.pdf And that's the problem really: mechanical manipulation of symbols lets you divorce the mathematical idea from the underlying "reality". It lets you generalize patterns with complete disregard for what they mean. And, somehow, this produces indubitably useful, nontrivial results. That's the real magic of math, and that's the magic that you don't get in your high school courses. (I think linear algebra is the first subject like this, but I'm just learning it now.)
- LesZedCB 15y ago
  Also, I think a point that could be made that if you were to teach a 'layperson' how to use a command line, given they were willing to learn it -and not pull the "I'm too dumb, this is for you computer people" card- I think they would quickly discover that there are few things they do daily that they could do quicker.
  - chimeracoder 15y ago
    The problem is that our education systems, by and large, include effectively no education about computers whatsoever. (Classes with touch-typing lessons/word-processing skills/etc. don't count in my mind.) As a result, most people have no concept of how a computer really works. With just the tiniest bit of this knowledge, the command-line would seem much less arcane, and the 'I'm too dumb, this is for you computer people' argument wouldn't really apply. It's shocking how computing is quite literally the technological advance of the millennium, both for industry and for individuals, and yet we somehow think it's acceptable to leave it up to people to learn about it all on their own, and even then only if they care to.
    - LesZedCB 15y ago
      yes. Also, people are so thrown by complexity, that they assume they aren't capable of taking steps toward figuring something out. a CLI really is NOT that hard. They all follow a very simple pattern: [command] [parameters] $ ls just a one word command. shows you directory contents. easy. not complicated looking. Lets make it harder. $ ls -a an option. list contents of directory. but ALL of them. still easy. building from the ground up is easy. For some reason, people still are scared off. Why? Why are people afraid to learn?
      - duopixel 15y ago
        It's easy to sight of just how much things you must know just to use the command line. When I read you comment I wondered what other options were available, so I tried... $ls -h This is what I've seen with most command line tools, but it doesn't work. So then I tried... $ls --help The output is > ls: illegal option -- - > usage: ls [-ABCFGHLOPRSTUWabcdefghiklmnopqrstuwx1] [file ...] What? I had to google it. Ok so... $man ls Oh, cool, a list of the options, finaly. Now, let's see what ls -a does: > -a Include directory entries whose names begin with a dot (.). If I were looking for invisible files I'd have absolutely no idea this is the option I need to use. Now, I completely understand that the power of the command line relies on its consistency, I'm confident that—at some point—I'm going to be able to grasp it's elegance. But, for now, I feel like a foreigner in a new country who just knows a few phrases and struggles greatly to understand what is being said to me. Saying that the command line is easy is like saying that English is easy.
- platz 15y ago
  I think Bret may be targeting laypeople from what I've seen, so he will always come down on the side of enabling growth for larger numbers of people, as opposed to an 'elite' few (comparatively).
gmichnikov 15y ago
This is interesting, I especially like the Scrubbing Calculator. While I am not sure how it would handle math above Algebra 1, I think this could be great for building intuition in Pre-Algebra and Algebra 1, which are the most important years of math education, in my opinion. I tutor some 11-14 year-olds who are studying these sorts of things, and it's incredible how many of them can look at something like 100/0.08 and have absolutely NO idea what neighborhood the answer should be in. (For example, they might accidentally multiply, get 8, and never think twice that the answer couldn't possibly be 8 because it has to be above 100.) Something like this might be really helpful in building intuition about the relationships between numbers.
LesZedCB 15y ago
Something that I think is important in math is the practical and theoretical going hand-in-hand. I'm still an undergrad student, but I have really enjoyed my math education so far. I have always appreciated both the beauty of the abstraction of math, as well as the beauty of applying that abstraction to solve very real problems. That being said, I think something might be lost when you take away the abstraction of math. I think a lot of math -and math abstraction especially- is difficult because of the way variables are understood. Since the beginning of algebra one, we learn that we can have an unknown, say 'x', and we can solve for it trivially by manipulating an equation to set 'x' equal to some value. But people always seem to get caught up in the naming idea. I remember in high school math classes, many of my peers would get so hung up as soon as the 'x' was turned into a 'y'. I felt like it was hopeless trying to explain that 'x' and 'y' are just names for something. This same problem percolates into every bit of math. After enough time, functions in math become first class citizens as well. That throws even more people. The exact same problem is seen in the programming world when people move from functions as an idea of a subroutine to being treated like a first class citizen. And lets not even mention Haskell or another functional language, with currying and stuff like that. I guess that's the problem that I see in math; the idea of an abstraction is difficult. However, I'm not totally convinced that making complex math more concrete is the best way to do that. At some point, having linked numbers like in interface builder looses its usefulness. So much of math is proofs. How can you do proofs when you are constantly instantiating concepts, rather than dealing with an abstraction. At some point, "there exists" needs to become "for all," which I think might be difficult in this type of mathematical environment. Again, concrete and abstraction really go best hand-in-hand, in my opinion.
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forrestthewoods 15y ago
I'm not sure how I feel about this. Sometimes complicated things are just complicated. I'm not sure making pretty pictures to make things appear more simple than they are is necessarily a good thing. Abstract thinking is difficult. Breaking the problems down such that it no longer requires abstract thought somewhat defeats the purpose.
- JumpCrisscross 15y ago
  Reminds me of the thesis behind the Learn Python/C/etc. The Hard Way series. By abstracting away the abstraction and complexity we may be turning out people who are very comfortable with the specific task but, due to a lack of familiarity with the primitives underlying what they're doing, are incapable at generalising it to even slightly different problems. There is another set of "abstract symbols" we have a "freakish knack" for manipulating: the alphabet. Literacy was once, too, deemed beyond the reach of commoners.
- icandoitbetter 15y ago
  Don't delude yourself with this Protestant hard-work-is-necessary mentality. Better representations are possible and they can save us a lot of work. It's easier to perform multiplication using Hindu-Arab numerals than it is with Roman numerals. It's easier to program with Python than Assembly. Difficulty is a property of the representation, not of some underlying 'abstract thing' that is being represented.
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  - Someone 15y ago
    Correction: Difficulty is not necessarily a property of a problem, but can also be caused by a poor choice of representation. I agree that modern technology may help provide better representations, but in the case of most of abstract mathematics, I am not convinced it currently does, or even that it can. There is some truth in the statement "a picture lies more than a thousand words"
  - 15y ago
noonespecial 15y ago
FTA: "By comparison, consider literacy. The ability to receive thoughts from a person who is not at the same place or time is a similarly great power. The dramatic social consequences of the rise of literacy are well known." I'm thinking it might be a bit on the difficult side to express the relationships written equations make easy if all you've got is an animated graph full of dials or a "scrubbing calculator". Written math is wonderfully dense and meaningful and allows the transmission of a very particular kind of knowledge with the simplest of mediums.
- firefoxman1 15y ago
  That's a good point. Sort of like when I first discovered regular expressions, I thought "Wow, I can express a whole sentence worth of very specific commands in a few characters." The same goes for Math. I don't really like Math, but I certainly won't deny its power and ability to be expressed in such a compact form.
mythz 15y ago
I pretty much read everything Bret Victor produces these days given he is responsible for one of the most insightful presentations I've ever seen: http://vimeo.com/36579366 http://vimeo.com/36579366
- ypcx 15y ago
  Thanks, I did not realize it was the same guy, I still have that video in watch-later.
- tricolon 15y ago
  Thank you for this link. I have a feeling it will change my life.
- Tycho 15y ago
  The best talk I've ever seen.
Jach 15y ago
I had some initial thoughts when I first saw this, I don't think they've changed much since. It'll be interesting to see where it ends up, for sure. I wrote again about this page after reading the somewhat recent Dijkstra lecture about the radical novelty that is programming (and other topics). Here's a slightly modified copy/paste, I'll warn that it kind of wanders after "Things Other People Have Said" so you're invited to stop reading at that point. While I sympathize with the opening, because many neat things have been made/discovered without the person having any formal math knowledge like what the "U"-looking symbol in an equation stands for: >"The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols." I heavily disagree with the conclusion: >"They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly. >And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface." I think the notion that they're unusable by the vast majority of people because of something fundamental about people is false. At some point in time reading and writing were not done by the vast majority of people, then they came into existence. Even as recent as 500 years ago, the vast majority of the population neither read nor wrote. Along came public education and that proportion flip flopped insanely fast such that the vast majority are capable of reading and writing (regardless of how good they are at it). Reading and writing were just as radical novelties as computing, just because something is a radical novelty doesn't mean most humans can't be proficient at it eventually. I think we should teach everyone to use the command line. Well, not Windows' CMD.EXE, but bash preferably in a gnome-terminal. Here is a short quote expressing why I think this is a good idea: >Linux supports the notion of a command line or a shell for the same reason that only children read books with only pictures in them. Language, be it English or something else, is the only tool flexible enough to accomplish a sufficiently broad range of tasks. -- Bill Garrett I think we should teach everyone how to interact with the computer at the most general way--which means programming. Which means commanding the computer. Describing the inverse square law in terms of pictures and intuition isn't going to make it any more of a tool than some other method, people are still going to think of it as something one is taught. The only way to make it seem like a tool is to use it as a tool, this means programming for some purpose. Maybe a physics simulation. And the beauty of tools is why the software world has exploded with utility despite Dijkstra's depression at programmers' inability to program extremely well. The beauty of tools is that they can be used without understanding the tool, just what the tool is useful for. The "Things Other People Have Said" at the end is more interesting than the essay. I wonder what Dijkstra would think of it. My two best guesses are "This is just a continuation of infantizing everything" and "We did kill math, with programming." I think a lot of people's difficulties with symbolic manipulation are due to the symbols not having immediately interpreted meaning. Dijkstra seems to recommend fixing this by drilling symbol manipulation of the predicate calculus with uninterpreted symbols like "black" and "white". My own approach I have been using more and more is to just use longer variable names in my math, whether written or typed. It really seems like this simple step can be a tremendous aid in understanding what's going on. Over the past couple of years I've realized just how important sheer memorization can be as I see almost everyone around me struggle with basic calculus facts, which means they struggle with application of the facts. The latest example from a few weeks ago in a junior level stats course with calc 2 prerequisite (which many people take 2 or 3 times here apparently) was when apparently no one but me recognized (or bothered to speak up after 20 seconds) that (1+1/x)^x limited to infinity is the definition of e, which we then immediately used with (1 - const/x)^x = e^(-const). (Which is immediately related to the continual compound interest formula that changes (1+r/n)^nt to e^rt as n->infty which everyone should have seen multiple times in algebra 2 / pre calc when learning about logarithms! The two most common examples are money and radioactive decay.) Sure that's a memorized fact for me, it's not necessarily intuitive apart from the 1/x being related to logs which are related to e, but come on. In my calc 3 course that I didn't get to waive out of, where it was just me and two others, I was the only one who passed, and when the teacher would try to make the class more interactive by asking us to complete-the-step instead of just a pure lecture the other two were seemingly incapable at remembering even the most basic derivative/integral pairs like D[sin(x)]=cos(x). A lot of 'education' is students cramming for and regurgitating on a test where the teacher hopes (often in vain) that the cramming resulted in long-term memorization. I do this too, cram and forget, but maybe I just do it less frequently or for less important subjects or my brain's wired to memorize things more easily than average (but I disfavor hypotheses that suppose unexplained gaps in human ability as largely static). My high school teacher's pre-calc and calc courses had frequent pure memorization tests such that there wasn't a need to cram elsewhere because that iterative cramming was enough to get long-term memorization (with cache warmups every so often; I did have to breeze through a calculus book in preparation for a waiver exam last year). A person can only hold so many things in their active, conscious memory at once (some people think it's only 7 plus or minus 3 but that seems like too weird a definition of 'thing' that results in so little; human brains are not CPUs with only registers r0 to r6), so when you start looking at a math proof with x's and y's and r's and greek letters and other uninterpreted single-character symbols all over the place, it's incredibly easy to get lost. If you start in the middle, if your brain hasn't memorized that "x means this, y means this" for any value of "this", you have to do a conscious lookup and dereference the symbol to some meaning even if the actual math itself is clear. My control systems book uses C and R for output/input (I already forgot which is which) but it's so embedded in my brain that Y is typically output and X is input that I use those instead, as does the teacher. I agree with Dijkstra that practice at manipulating uninterpreted symbols makes it a bit easier, but there are quickly diminishing returns and ever since I started programming and saw how PHP uses the $var syntax to denote variables I've been thinking "Why hasn't this caught on in math? It makes so many things much clearer!" But it's not so much the "$" in $var (or @var and other sigils in Perl) but the "var". Saving your brain a dereference step is pretty useful. Single-character symbols (and multiple character symbols) that not only hide tons of meaning through a dereference, which is naturally how language works, that also suggest a meaning themselves, are stupid and promote natural diseased human thinking. My poster-child here is global warming. Every winter you'll hear the same comments on "When's global warming going to hit here?" The poor choice of global warming as a variable that points to a bigger theory has cost it heavily in PR because humans look at only the phrase instead of what it points to, it's still so bad that people's every day experiences, which they use to form a subconscious prior probability in global warming as a theory, contain "it's really hot all the time and everywhere I've been!" and so they look at "global warming" and dismiss any evidence for/against it based purely on the name and how their experience seems to contradict the name. It's like a computer thinking that a pointer address that happens to correspond to 0xADD means it should invoke addition when it should figure out what 0xADD points to instead which could be anything. Another use of long names is with Bayes' Theorem in probability. It is only through memorization of the uninterpreted symbols A,B,C themselves that I remember prob(A | B, C) = prob(A | C) * prob(B | A, C) / prob(B | C) But it never made intuitive sense to me and I never bothered memorizing until it was expressed as prob(hypothesis | data, bg_info) = prob(hypothesis | bg_info) * prob(data | hypothesis, bg_info) / prob(data | bg_info). (Sometimes data is replaced by model.) The notion that it relates reason with reality, that it encapsulates the process of learning and scientific reasoning, elevates the equation to the status of a tool instead of something you cram for and regurgitate on a test. An immediately useful application for the tool is using Naive Bayes to filter spam.
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brian_campbell 15y ago
This article isn't necessarily about making ALL math less abstract / challenging. That's arguably impossible. It's about making math more accessible to the masses. Most college educated Americans have never learned ANYTHING about differential equations, linear algebra, or discrete math... That is a serious shame and you could argue impacts our national productivity/potential. Most people simply will not try to learn higher level math out of fear of failure, challenge, or whatever. Making SOME higher level math more intuitive is a great objective. I would much rather live in a world where more Americans ultimately understand a higher level of math (regardless if the path to get there was a little less challenging) than leaving math education in it's current state. Analogy: How many average American's tried to use computers with just command prompts? I would argue that it was the creation of an intuitive interface ("physical" folders where you store documents) that really started mass adoption of computers throughout the world. Is this less challenging/abstract than command prompts- yes. Is the world vastly more productive because of it? Absolutely.
nekojima 15y ago
I had thought the movement to kill math ended long ago with this video: http://www.youtube.com/watch?v=MfgX0fyNeLc http://www.youtube.com/watch?v=MfgX0fyNeLc
- JadeNB 15y ago
  No, it was done in by 2: http://dtecomic.com/?n=2 http://dtecomic.com/?n=2.
aspensmonster 15y ago
One question: How were those demonstrations programmed? I have a feeling they were developed by "manipulating abstract symbols." Math doesn't need a new interface, any more than a POSIX compliant shell does. Math is a beautiful and expressive language. Its strength is in communicating difficult, abstract concepts in a language that is manipulable. This manipulation is of the utmost importance in understanding the relationships between different concepts and even in developing new ones. Removing the single strongest aspect --the "abstract symbols"-- of mathematics is to neuter it.
- maxerickson 15y ago
  I think part of his goal is to help people that do not yet understand the abstraction have a way to manipulate a system involving it. Then play can lead to understanding, in a way that is unlikely for 2x+1=5.
nwatson 15y ago
It will be hard to come up with an accessible general framework to express general problem situations without resorting to either (a) an explosion of templates for solving the N-most-common kinds of problems (the browsing through which will be worse than learning the math and the programming); (b) a set of elaborate but limiting special-purpose programs (much like iPhone apps that might help, for example, a metal welder to choose stock feed rate, gas mix/flow, and electric current depending on metal type and thickness to be joined; or mortgage calculators). Worse yet, it doesn't take much for interesting problems to get into the land of over- or under-constraint. In an under-constrained problem there are a multitude of solutions forming their own K-dimensional space, and then one likely should apply some secondary optimization criteria to determine the best solution. In over-constrained problems there is no exact solution, but again one needs to impose some optimization to get something "close enough" to the desired criteria, according to some norm. Explaining the need for optimization, letting the user explore the solution space, and having them express these optimization criteria will be tough. A Google search for "geometric constraint solver" leads to this paper: http://www.cs.purdue.edu/homes/cmh/distribution/papers/Constraints/CAD95.pdf http://www.cs.purdue.edu/homes/cmh/distribution/papers/Const... -- what appears to be a lot of work just in helping people/machines solve geometric problems (though surely more widely applicable when taken in the abstract). Coming up with even more general "hands-on" explorers/solvers will be a lot of work.
ypcx 15y ago
Tell me if I'm wrong, but I have this theory that the whole Math could be expressed on top of a single basic operation, which is addition. For example 2 * 3 is (2 + 2 + 2). Or sin(7) is (7 * (1 + -0.1666666664*(7*7) + 0.0083333315*(7*7)*(7*7) + -0.0001984090*(7*7)*(7*7)*(7*7) + 0.0000027526*(7*7)*(7*7)*(7*7)*(7*7) + -0.0000000239*(7*7)*(7*7)*(7*7)*(7*7)*(7*7))) (the multiply operator left in for brevity, but can be easily replaced by addition) If my assumption is true, then Math is really just an arcane and archaic form of a computer-like language. And at the same time, it's possible that the operations meant by Math's various symbols, however arcane, can be understood much better (especially with the use of modern tech, animations, etc.) than we understand them from classic texts, but those who learnt them haven't spent enough effort to create more visual explanations of it, based on which others, and also themselves, could understand those operations better, and then be able to build on top of them and even evolve Math further. (I know that I don't know what I don't know, but I have tendency to assume that I know a lot more than I really know, thus I don't explore the known deeper/further.)
- thristian 15y ago
  It sounds like you're thinking of (or working towards) Peano Arithmetic¹, which (to oversimplify) basically starts with "0", "1", "=" and "+", and works from there. Whether that's "enough" rather depends on what you're trying to achieve, though. There's subgenres of mathematics that don't necessarily involve numbers at all, like geometry and topology. It would be difficult to reduce them to addition. ¹: http://en.wikipedia.org/wiki/Peano_axioms http://en.wikipedia.org/wiki/Peano_axioms
- yequalsx 15y ago
  Addition is a function and so functions would be a more basic concept. But functions are sets and sets are a more basic concept. Thus we should study sets. Everything comes from sets. Replace addition with sets and your first sentence would be closer to the truth. I strongly disagree with your last paragraph though.
  - derleth 15y ago
    And you can take sets as something built on top of categories (specifically, topoi), so category theory is more foundational than set theory. http://plato.stanford.edu/entries/category-theory/ http://plato.stanford.edu/entries/category-theory/ http://sakharov.net/foundation.html http://sakharov.net/foundation.html http://en.wikipedia.org/wiki/Topos http://en.wikipedia.org/wiki/Topos
Gravityloss 15y ago
Two things: 1. Needless complexity 2. Needed complexity 1.: Think how programming in assembler requires you to mentally keep track of registers, and how higher level programming languages did away with that by enabling variable names. The resulting program is still mostly the same thing underneath. You can get the same logical results but the former is much harder. There is a question here though: why is it easier for most humans to program in C rather than in assembler? Sure, there's more housekeeping in assembler. But I think it's mostly because there's one less layer of representation. a=5; b=10; c=a+b; Vs "load 5 into register A1", "load 10 into register A2", "calculate sum from registers A1 and A2 and store it to register A3". So here in "less layered math", instead of saying: Let alpha represent the angle, a the closer cathete and b the further cathete and c the hypotenuse, then sin(alpha) = b/c You instead say: sin is the relation of the further cathete to the hypotenuse. Again, a large portion of people can get the former way of explaining sin, perhaps when they see it all at once on the blackboard or book and go back and forth... (what was b again?) but the latter way of explaining does away with the whole exercise of variables. (Note that I picked a subject of convention, not something that can be deduced from more basic principles, I think those again can be taught somewhat differently.) The problem is, many math teacher love variables and think how beautiful all that symphony of x:s and y:s is. But for the average school student math is just one subject among others. They just want the essentials with the least amount of extra crap and layers on top. Lots of caveats here. I think people need to learn basic algebra, but it makes me sad that so few people can't use it for problems they are trying to solve. I think these would make interesting psychology research subjects. It's a very important field. 2.: Yet you can't go beyond some point in simplification.
- Geee 15y ago
  Have you seen this Bret's talk about his principle http://vimeo.com/36579366 http://vimeo.com/36579366 ? It's not really about simplification of things. The principle is basically "to see what's going on", or "to have immediate connection with what you are doing". Basically when you design something, you have to try to think or simulate in your head what's going on. Bret wants to remove that barrier so you don't have to think. There's some great examples in the video where the principle is applied to graphics, game design, electronics, programming etc.
bwarp 15y ago
Oh FFS. Everything has a barrier to entry. Just work with it until you get over it. People want instant gratification these days. It takes time to understand stuff. The Internet (and crappy TV about popstars and shit) has lead to people having completely unrealistic expectations of everything.
- derleth 15y ago
  You can use precisely the same arguments to denigrate demotic script versus hieroglyphs, and say that anyone who thinks writing complex pictures is too difficult a way to express themselves is just whining. I'm not taking a position on the argument expressed in the article; I'm merely noting that your argument is so infinitely applicable (even recursively applicable, back every step of every path of technological and notational improvement) that it is effectively useless as a way to determine the best path to follow.
- derleth 15y ago
  I see more people agree with you than with me, despite the fact upvotes aren't meant to be used that way. HN can be trying at times, can't it?
  - bwarp 15y ago
    Indeed - but that is the nature of the Internet. There should always be a counterpoint to a good view.
- JadeNB 15y ago
  > Oh FFS. Everything has a barrier to entry. Just work with it until you get over it. The problem is that most people's response is "I'd rather give up on it than get over it", so we wind up with a math-illiterate society. If we can make it easier to be math literate, then we encourage a better world. (Your response, which is quite correct, reminds me of http://www.marco.org/2012/02/25/right-vs-pragmatic. http://www.marco.org/2012/02/25/right-vs-pragmatic.)
migiale 15y ago
The idea of "geometrization" of mathematical education as opposed to "algebraization" has been discussed widely in the recent decade at least, maybe more. In Russian school of mathematics, Vladimir Arnold (http://en.wikipedia.org/wiki/Vladimir_Arnold http://en.wikipedia.org/wiki/Vladimir_Arnold), was a very strong proponent of the idea. You can read his talk of the subject here http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html .
flashingleds 15y ago
Quite aside from any of the main points being made here, the guy is dead on about Strogatz's book. I had the same reaction to that book as I did to Lipovača's Learn You A Haskell - it's a great accessible resource to learn something new and interesting (if perhaps not immediately useful). Would appeal to the HN crowd I imagine.
ilaksh 15y ago
He's identified a real problem and is going in the right direction. I don't think we need to go quite as far though to make an improvement. A simple observation that I find amazing that more people don't see, is this: Mathematics is obfuscated code that doesn't compile or run. One letter identifiers and odd symbols rather than useful textual descriptions. No way to drill down into the meaning of the symbols. With code you can take a higher-level function and look at the code for the functions it uses, then look at the implementation of the language, then look at the assembly language, etc. No such thing in mathematics.
shadowmint 15y ago
I can't help but think this is completely missing the point. TED has had people cover this a number of times, but the problem with math isn't that the the _syntax is too hard to parse_ its that the _problems in math class are stupid_. "The bucket is depth X, diameter D and full of water. When you open the tap, how long will it take to drain if the water drains out at rate Z?" I've never had to apply solving a problem like this, in my entire life (and if I were in a job that I _did_ have to, the complexity of a real life situation would mean I would still have to learn domain specific tools to solve the problem (eg. how big is the air intake? Does that limit the rate of flow? etc)). Tangible problems in the real world require mathematical models (often probabilistic models) to solve them. How do you take a real world problem, break it down into bits, and then use the mathematical tools available to solve them? By creating a model, and then guessing what the rules that govern that model are, then comparing the model to reality, and refining the rules; and when you can't figure out the rules, thats when its time to whip out the text book and say, well, guess what, someone has had that problem before and this is how they solved it~ Teaching kids how to create models and reach out into the mathematical library available to them when they need it would be vastly more helpful than trying to creating more abstract alternative ways of understanding obscure math concepts that will never to relevant to them. I've never been more frustrated than I was the other night when I was at a party and a boiler maker (who incidentally earns 3x what I do. damn mining boom) was telling me about all the cool math he's learnt since he started his job. It's all geometry and rate of flow differentials and he said "why did I have to learn matrices at school? total waste of time. they should have been teaching us useful things"
- TeMPOraL 15y ago
  > "why did I have to learn matrices at school? total waste of time.(...)" Unless you're doing game programming as a hobby, or encounter problems that can be modeled as a set of equations, which then can be easily solved using few matrix tricks, giving you lots of fun and $$$. But I do agree with your sentiment; even though math can be fun on its own, it's easier (and probably better) to get people interested in it by showing how to use it to solve complex problem. After all, math is just formalized, applied rational thinking.
- im3w1l 15y ago
mbq 15y ago
This is so wrong... The point of math is that ideas can be generalized and re-used in previously unexpected places -- your ideas remind me Middle Ages where people were learning a mnemonic poems to remember "law of proportion", treating it like a precious magic gizmo and were perfectly sure they wouldn't be able to recreate it once forgotten.
- Riesling 15y ago
  Have you read the article? One of his arguments is that math, due to its interface, is not about ideas but about abstract symbols and "symbol-pushing tricks". For example many people know how to use the chain rule when they derive a function, without even having the slightest grasp of what is going on. How is this any different from "learning a mnemonic poem to remember the law of proportion"?
pfortuny 15y ago
OK so it takes years to learn Japanese but it is the only way to truly enjoy haikus... The la guage of modern maths is what has enabled the physics at the LHC. You cannot get the latter without the former, hard as you may try.
Tycho 15y ago
I get the sense that there's two levels of understanding for most mathematics: a superficial level where you just have to manipulate symbols and be comfortable applying rules, and a deeper level where you grasp the underlying relationships and some fundamental 'why' of the problem. I used to think that if you were bad at the first, then the second would be an even tougher challenge. But I'm starting to suspect that actually people who are good at the first often just don't care about the second. And in fact it might be the very need for things to 'make sense' to you on a deeper level that is causing discomfort and difficulty at the superficial level. Some people lack curiosity about the deeper relationships, and perversely this might actually help them get along at the superficial level. For instance in finance, why is Macaulay duration(1) approximately equal to effective duration? One measures time in years, the other measures sensitivity in percentage form. How the heck do they come out at the same number approximately? For some reason i can't find any literature that's in a rush to explain this relationship... even though the two measures share the name 'duration' so presumably there's some intuitive understanding to be reached. (1) when the yield is expressed continuously compounding, at least
DennisP 15y ago
His "interactive exploration of a dynamical system" looks like some very sweet software. It's not an entirely new concept though. Modeling tools like Vensim and Stella do something very similar, modeling differential equations with "stock and flow" diagrams. Modeling problems with tools like these is a pretty active field, with applications in economics, ecology, and engineering. There are a fair number of books about it. Vensim has a free "personal" version, if anyone wants to play around with this stuff. It's definitely not as elegant as Bret Victor's demo, but might be better for more complicated systems (depending of course on how much Bret left out of the demo).
merraksh 15y ago
Previous discussion: http://news.ycombinator.com/item?id=2532271 http://news.ycombinator.com/item?id=2532271
super_mario 15y ago
Someone doesn't know the formal definition of computation. This is all there is to it really.
jpdoctor 15y ago
> When most people speak of Math, what they have in mind is more its mechanism than its essence. 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.* Wow, sounds like someone doesn't really understand math.
- deadsy 15y ago
  Brett Victor is very accomplished. I'm pretty sure he understands math. Personally speaking if Brett Victor had a different opinion about something than me than I'd pay careful attention, because I'm probably missing something.
  - rhizome 15y ago
    jpdoctor is a bomb-thrower, your words fall on deaf ears.
- lukev 15y ago
  That's precisely his point. The "UI" for math obfuscates what it's actually about. Math is full of purity and simplicity, but the notation and algorithms we start out teaching kids are anything but.
dubya 15y ago
For the more mathematically inclined, there's a really nice little book, "The Computer as Crucible" by Borwein and Devlin, about experimental math. It makes a really convincing case, that I think couldn't have been made 15 years ago, that computers really have something essential to contribute to mathematics now. I think Borwein has written a number of articles about the same, and Devlin had a column in the AMS Notices about computer math.
DanBC 15y ago
I am hopeless at math. This is something that I am ashamed of. But there are people who seem to be proud of their ignorance; they're shocked if you haven't read any Shakespeare but happily admit they can't do percentages. Will those people be helped by the author's approach? I don't know, but I don't think so. These people will see a number and throw their hands up, saying "Oh maths! I can't do sudoku, how do you expect me to do this!". This attitude is not quite as prevalent in the UK as it used to be, but it's still there. See, for example, the number of esoteric arts programmes on BBC compared to the number of advanced science programmes. (I'm not aware of any science programming that would be beyond an enthusiastic 14 year old. I do know of hours of arts programming that is unashamedly elitist. Elitist is fine, but it'd be great for some balance.) What is needed is better maths education. (I finished school many years ago; maybe things are different now.) Math is not blindly mashing numbers and symbols and hoping for the best. Maths includes a large element of careful thinking, exploring the problem, listing the known information, listing what you want to find. New techniques for using math would help reduce the gender inequality in math results too. Put the normal "more research needed" caveats around this, but: There's some suggestion that girls use inefficient techniques and "just struggle through", they manage to get correct results and so they don't get extra help. Boys tend to just stop when it gets too laborious, and thus they get taught new better techniques. (http://news.bbc.co.uk/1/hi/education/4587466.stm http://news.bbc.co.uk/1/hi/education/4587466.stm)
moxiemk1 15y ago
I'm worried that in presenting math with visible, tangible intuition dominant will do more to silo the knowledge inside the ivory tower. N-variable calculus isn't something that you can draw a picture of. Lots of (very practical) Linear Algebra is derived from the pure "Linear Transformation" interpretation rather than the "Matrix" one. Stochastic...anything (seems to be all the rage these days) is not made with helpful pictures. If we train people to do math in a non-abstract way, they won't be as easily able to grapple with the real problems, which are only approachable in the abstract.
- enjalot 15y ago
  I disagree, at least that pictures can't help with higher dimensional math. Well, regular pictures may not help so much but interaction really can. A lot of my understanding of higher level math depends on my programming ability to "touch" some of the abstract concepts. When learning something new you need to be able to associate the new concept with things you already understand, and visuals can give you an intuitive sense of how things are behaving (especially in time) I don't think the visuals could necessarily take the place of training to use the abstract symbols or lines of code any time soon, but I think they are a really necessary step in the right direction to get more people to approach math.
justinbaker 15y ago
This is the kind of shit we don't need. I'm a Computer Science & Mathematics double major. I'm also bipolar/ADHD, and this post makes extreme assumptions about learning. Not everyone learns the same way. I absolutely cannot stand to sit in class and listen to a professor babble on incessantly and draw pictures of concepts and irrelevant ideas simply to relate it for the majority to learn. I learn by doing. By code. By numbers. By command line. It has done quite well so far..for me at least. Maybe it hasn't for you..then change that and get busy learning instead of filling HN with awful blog posts.
iso8859-1 15y ago
I really like the Soulver program that he mentions on the page for the "Scrubbing Calculator". I found a similar Windows/.NET program called OpalCalc: http://www.skytopia.com/software/opalcalc/ http://www.skytopia.com/software/opalcalc/ Time to do a web edition? Surely it already exists. Somebody help me out :P
Tycho 15y ago
Are there any good apps for the iPad that let you manipulate variables and equations in a tactile manner? I have Ovium which is a calculator where you put your numbers in bubbles and then connect them however you want with different operators. I'd like like something where you could use the value of a slider to change the value of a variable.
FreshCode 15y ago
Fix your scrolling; I think your message will benefit from increased usability.

199 points by noonespecial 15y ago

This is an interesting idea, and I think his final analogy summed it up perfectly: symbolic math is like a command line. But I reject his assumption that a command line is a bad interface. And that mirrors my thoughts on this post: for the layperson, a GUI may indeed be better than a command line; for a professional (a programmer) the reverse is true. A command line lets you combine and recombine different programs and easily do many things its creators never dreamed of. Symbolic manipulation and algebra are just like that! Let's imagine you know how to differentiate viscerally; you understand what a tangent line looks like and how to plot that and you care not for silly equations like d/dx x^3 = 3x^2. Naturally, you understand basic arithmetic in the same way and not as mechanical manipulations on symbols. You are perfectly well equipped to deal with real problems, and perhaps find it easier than shuffling symbols around. But you would never come up with a way to get the derivative of, say, an algebraic data type[1][2]. (For the curious, this is how you can define a zipper on a type.) [1]: http://blog.lab49.com/archives/3011 http://blog.lab49.com/archives/3011 [2]: http://strictlypositive.org/diff.pdf http://strictlypositive.org/diff.pdf And that's the problem really: mechanical manipulation of symbols lets you divorce the mathematical idea from the underlying "reality". It lets you generalize patterns with complete disregard for what they mean. And, somehow, this produces indubitably useful, nontrivial results. That's the real magic of math, and that's the magic that you don't get in your high school courses. (I think linear algebra is the first subject like this, but I'm just learning it now.)

LesZedCB 15y ago
Also, I think a point that could be made that if you were to teach a 'layperson' how to use a command line, given they were willing to learn it -and not pull the "I'm too dumb, this is for you computer people" card- I think they would quickly discover that there are few things they do daily that they could do quicker.
- chimeracoder 15y ago
  The problem is that our education systems, by and large, include effectively no education about computers whatsoever. (Classes with touch-typing lessons/word-processing skills/etc. don't count in my mind.) As a result, most people have no concept of how a computer really works. With just the tiniest bit of this knowledge, the command-line would seem much less arcane, and the 'I'm too dumb, this is for you computer people' argument wouldn't really apply. It's shocking how computing is quite literally the technological advance of the millennium, both for industry and for individuals, and yet we somehow think it's acceptable to leave it up to people to learn about it all on their own, and even then only if they care to.
  - LesZedCB 15y ago
    yes. Also, people are so thrown by complexity, that they assume they aren't capable of taking steps toward figuring something out. a CLI really is NOT that hard. They all follow a very simple pattern: [command] [parameters] $ ls just a one word command. shows you directory contents. easy. not complicated looking. Lets make it harder. $ ls -a an option. list contents of directory. but ALL of them. still easy. building from the ground up is easy. For some reason, people still are scared off. Why? Why are people afraid to learn?
    - duopixel 15y ago
      It's easy to sight of just how much things you must know just to use the command line. When I read you comment I wondered what other options were available, so I tried... $ls -h This is what I've seen with most command line tools, but it doesn't work. So then I tried... $ls --help The output is > ls: illegal option -- - > usage: ls [-ABCFGHLOPRSTUWabcdefghiklmnopqrstuwx1] [file ...] What? I had to google it. Ok so... $man ls Oh, cool, a list of the options, finaly. Now, let's see what ls -a does: > -a Include directory entries whose names begin with a dot (.). If I were looking for invisible files I'd have absolutely no idea this is the option I need to use. Now, I completely understand that the power of the command line relies on its consistency, I'm confident that—at some point—I'm going to be able to grasp it's elegance. But, for now, I feel like a foreigner in a new country who just knows a few phrases and struggles greatly to understand what is being said to me. Saying that the command line is easy is like saying that English is easy.
platz 15y ago
I think Bret may be targeting laypeople from what I've seen, so he will always come down on the side of enabling growth for larger numbers of people, as opposed to an 'elite' few (comparatively).

gmichnikov 15y ago

This is interesting, I especially like the Scrubbing Calculator. While I am not sure how it would handle math above Algebra 1, I think this could be great for building intuition in Pre-Algebra and Algebra 1, which are the most important years of math education, in my opinion. I tutor some 11-14 year-olds who are studying these sorts of things, and it's incredible how many of them can look at something like 100/0.08 and have absolutely NO idea what neighborhood the answer should be in. (For example, they might accidentally multiply, get 8, and never think twice that the answer couldn't possibly be 8 because it has to be above 100.) Something like this might be really helpful in building intuition about the relationships between numbers.

LesZedCB 15y ago

Something that I think is important in math is the practical and theoretical going hand-in-hand. I'm still an undergrad student, but I have really enjoyed my math education so far. I have always appreciated both the beauty of the abstraction of math, as well as the beauty of applying that abstraction to solve very real problems. That being said, I think something might be lost when you take away the abstraction of math. I think a lot of math -and math abstraction especially- is difficult because of the way variables are understood. Since the beginning of algebra one, we learn that we can have an unknown, say 'x', and we can solve for it trivially by manipulating an equation to set 'x' equal to some value. But people always seem to get caught up in the naming idea. I remember in high school math classes, many of my peers would get so hung up as soon as the 'x' was turned into a 'y'. I felt like it was hopeless trying to explain that 'x' and 'y' are just names for something. This same problem percolates into every bit of math. After enough time, functions in math become first class citizens as well. That throws even more people. The exact same problem is seen in the programming world when people move from functions as an idea of a subroutine to being treated like a first class citizen. And lets not even mention Haskell or another functional language, with currying and stuff like that. I guess that's the problem that I see in math; the idea of an abstraction is difficult. However, I'm not totally convinced that making complex math more concrete is the best way to do that. At some point, having linked numbers like in interface builder looses its usefulness. So much of math is proofs. How can you do proofs when you are constantly instantiating concepts, rather than dealing with an abstraction. At some point, "there exists" needs to become "for all," which I think might be difficult in this type of mathematical environment. Again, concrete and abstraction really go best hand-in-hand, in my opinion.

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forrestthewoods 15y ago

I'm not sure how I feel about this. Sometimes complicated things are just complicated. I'm not sure making pretty pictures to make things appear more simple than they are is necessarily a good thing. Abstract thinking is difficult. Breaking the problems down such that it no longer requires abstract thought somewhat defeats the purpose.

JumpCrisscross 15y ago
Reminds me of the thesis behind the Learn Python/C/etc. The Hard Way series. By abstracting away the abstraction and complexity we may be turning out people who are very comfortable with the specific task but, due to a lack of familiarity with the primitives underlying what they're doing, are incapable at generalising it to even slightly different problems. There is another set of "abstract symbols" we have a "freakish knack" for manipulating: the alphabet. Literacy was once, too, deemed beyond the reach of commoners.
icandoitbetter 15y ago
Don't delude yourself with this Protestant hard-work-is-necessary mentality. Better representations are possible and they can save us a lot of work. It's easier to perform multiplication using Hindu-Arab numerals than it is with Roman numerals. It's easier to program with Python than Assembly. Difficulty is a property of the representation, not of some underlying 'abstract thing' that is being represented.
- deleted 15y ago
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- Someone 15y ago
  Correction: Difficulty is not necessarily a property of a problem, but can also be caused by a poor choice of representation. I agree that modern technology may help provide better representations, but in the case of most of abstract mathematics, I am not convinced it currently does, or even that it can. There is some truth in the statement "a picture lies more than a thousand words"
- 15y ago

noonespecial 15y ago

FTA: "By comparison, consider literacy. The ability to receive thoughts from a person who is not at the same place or time is a similarly great power. The dramatic social consequences of the rise of literacy are well known." I'm thinking it might be a bit on the difficult side to express the relationships written equations make easy if all you've got is an animated graph full of dials or a "scrubbing calculator". Written math is wonderfully dense and meaningful and allows the transmission of a very particular kind of knowledge with the simplest of mediums.

firefoxman1 15y ago
That's a good point. Sort of like when I first discovered regular expressions, I thought "Wow, I can express a whole sentence worth of very specific commands in a few characters." The same goes for Math. I don't really like Math, but I certainly won't deny its power and ability to be expressed in such a compact form.

mythz 15y ago

I pretty much read everything Bret Victor produces these days given he is responsible for one of the most insightful presentations I've ever seen: http://vimeo.com/36579366 http://vimeo.com/36579366

ypcx 15y ago
Thanks, I did not realize it was the same guy, I still have that video in watch-later.
tricolon 15y ago
Thank you for this link. I have a feeling it will change my life.
Tycho 15y ago
The best talk I've ever seen.

Jach 15y ago

I had some initial thoughts when I first saw this, I don't think they've changed much since. It'll be interesting to see where it ends up, for sure. I wrote again about this page after reading the somewhat recent Dijkstra lecture about the radical novelty that is programming (and other topics). Here's a slightly modified copy/paste, I'll warn that it kind of wanders after "Things Other People Have Said" so you're invited to stop reading at that point. While I sympathize with the opening, because many neat things have been made/discovered without the person having any formal math knowledge like what the "U"-looking symbol in an equation stands for: >"The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols." I heavily disagree with the conclusion: >"They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly. >And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface." I think the notion that they're unusable by the vast majority of people because of something fundamental about people is false. At some point in time reading and writing were not done by the vast majority of people, then they came into existence. Even as recent as 500 years ago, the vast majority of the population neither read nor wrote. Along came public education and that proportion flip flopped insanely fast such that the vast majority are capable of reading and writing (regardless of how good they are at it). Reading and writing were just as radical novelties as computing, just because something is a radical novelty doesn't mean most humans can't be proficient at it eventually. I think we should teach everyone to use the command line. Well, not Windows' CMD.EXE, but bash preferably in a gnome-terminal. Here is a short quote expressing why I think this is a good idea: >Linux supports the notion of a command line or a shell for the same reason that only children read books with only pictures in them. Language, be it English or something else, is the only tool flexible enough to accomplish a sufficiently broad range of tasks. -- Bill Garrett I think we should teach everyone how to interact with the computer at the most general way--which means programming. Which means commanding the computer. Describing the inverse square law in terms of pictures and intuition isn't going to make it any more of a tool than some other method, people are still going to think of it as something one is taught. The only way to make it seem like a tool is to use it as a tool, this means programming for some purpose. Maybe a physics simulation. And the beauty of tools is why the software world has exploded with utility despite Dijkstra's depression at programmers' inability to program extremely well. The beauty of tools is that they can be used without understanding the tool, just what the tool is useful for. The "Things Other People Have Said" at the end is more interesting than the essay. I wonder what Dijkstra would think of it. My two best guesses are "This is just a continuation of infantizing everything" and "We did kill math, with programming." I think a lot of people's difficulties with symbolic manipulation are due to the symbols not having immediately interpreted meaning. Dijkstra seems to recommend fixing this by drilling symbol manipulation of the predicate calculus with uninterpreted symbols like "black" and "white". My own approach I have been using more and more is to just use longer variable names in my math, whether written or typed. It really seems like this simple step can be a tremendous aid in understanding what's going on. Over the past couple of years I've realized just how important sheer memorization can be as I see almost everyone around me struggle with basic calculus facts, which means they struggle with application of the facts. The latest example from a few weeks ago in a junior level stats course with calc 2 prerequisite (which many people take 2 or 3 times here apparently) was when apparently no one but me recognized (or bothered to speak up after 20 seconds) that (1+1/x)^x limited to infinity is the definition of e, which we then immediately used with (1 - const/x)^x = e^(-const). (Which is immediately related to the continual compound interest formula that changes (1+r/n)^nt to e^rt as n->infty which everyone should have seen multiple times in algebra 2 / pre calc when learning about logarithms! The two most common examples are money and radioactive decay.) Sure that's a memorized fact for me, it's not necessarily intuitive apart from the 1/x being related to logs which are related to e, but come on. In my calc 3 course that I didn't get to waive out of, where it was just me and two others, I was the only one who passed, and when the teacher would try to make the class more interactive by asking us to complete-the-step instead of just a pure lecture the other two were seemingly incapable at remembering even the most basic derivative/integral pairs like D[sin(x)]=cos(x). A lot of 'education' is students cramming for and regurgitating on a test where the teacher hopes (often in vain) that the cramming resulted in long-term memorization. I do this too, cram and forget, but maybe I just do it less frequently or for less important subjects or my brain's wired to memorize things more easily than average (but I disfavor hypotheses that suppose unexplained gaps in human ability as largely static). My high school teacher's pre-calc and calc courses had frequent pure memorization tests such that there wasn't a need to cram elsewhere because that iterative cramming was enough to get long-term memorization (with cache warmups every so often; I did have to breeze through a calculus book in preparation for a waiver exam last year). A person can only hold so many things in their active, conscious memory at once (some people think it's only 7 plus or minus 3 but that seems like too weird a definition of 'thing' that results in so little; human brains are not CPUs with only registers r0 to r6), so when you start looking at a math proof with x's and y's and r's and greek letters and other uninterpreted single-character symbols all over the place, it's incredibly easy to get lost. If you start in the middle, if your brain hasn't memorized that "x means this, y means this" for any value of "this", you have to do a conscious lookup and dereference the symbol to some meaning even if the actual math itself is clear. My control systems book uses C and R for output/input (I already forgot which is which) but it's so embedded in my brain that Y is typically output and X is input that I use those instead, as does the teacher. I agree with Dijkstra that practice at manipulating uninterpreted symbols makes it a bit easier, but there are quickly diminishing returns and ever since I started programming and saw how PHP uses the $var syntax to denote variables I've been thinking "Why hasn't this caught on in math? It makes so many things much clearer!" But it's not so much the "$" in $var (or @var and other sigils in Perl) but the "var". Saving your brain a dereference step is pretty useful. Single-character symbols (and multiple character symbols) that not only hide tons of meaning through a dereference, which is naturally how language works, that also suggest a meaning themselves, are stupid and promote natural diseased human thinking. My poster-child here is global warming. Every winter you'll hear the same comments on "When's global warming going to hit here?" The poor choice of global warming as a variable that points to a bigger theory has cost it heavily in PR because humans look at only the phrase instead of what it points to, it's still so bad that people's every day experiences, which they use to form a subconscious prior probability in global warming as a theory, contain "it's really hot all the time and everywhere I've been!" and so they look at "global warming" and dismiss any evidence for/against it based purely on the name and how their experience seems to contradict the name. It's like a computer thinking that a pointer address that happens to correspond to 0xADD means it should invoke addition when it should figure out what 0xADD points to instead which could be anything. Another use of long names is with Bayes' Theorem in probability. It is only through memorization of the uninterpreted symbols A,B,C themselves that I remember prob(A | B, C) = prob(A | C) * prob(B | A, C) / prob(B | C) But it never made intuitive sense to me and I never bothered memorizing until it was expressed as prob(hypothesis | data, bg_info) = prob(hypothesis | bg_info) * prob(data | hypothesis, bg_info) / prob(data | bg_info). (Sometimes data is replaced by model.) The notion that it relates reason with reality, that it encapsulates the process of learning and scientific reasoning, elevates the equation to the status of a tool instead of something you cram for and regurgitate on a test. An immediately useful application for the tool is using Naive Bayes to filter spam.

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brian_campbell 15y ago

This article isn't necessarily about making ALL math less abstract / challenging. That's arguably impossible. It's about making math more accessible to the masses. Most college educated Americans have never learned ANYTHING about differential equations, linear algebra, or discrete math... That is a serious shame and you could argue impacts our national productivity/potential. Most people simply will not try to learn higher level math out of fear of failure, challenge, or whatever. Making SOME higher level math more intuitive is a great objective. I would much rather live in a world where more Americans ultimately understand a higher level of math (regardless if the path to get there was a little less challenging) than leaving math education in it's current state. Analogy: How many average American's tried to use computers with just command prompts? I would argue that it was the creation of an intuitive interface ("physical" folders where you store documents) that really started mass adoption of computers throughout the world. Is this less challenging/abstract than command prompts- yes. Is the world vastly more productive because of it? Absolutely.

nekojima 15y ago

I had thought the movement to kill math ended long ago with this video: http://www.youtube.com/watch?v=MfgX0fyNeLc http://www.youtube.com/watch?v=MfgX0fyNeLc

JadeNB 15y ago
No, it was done in by 2: http://dtecomic.com/?n=2 http://dtecomic.com/?n=2.

aspensmonster 15y ago

One question: How were those demonstrations programmed? I have a feeling they were developed by "manipulating abstract symbols." Math doesn't need a new interface, any more than a POSIX compliant shell does. Math is a beautiful and expressive language. Its strength is in communicating difficult, abstract concepts in a language that is manipulable. This manipulation is of the utmost importance in understanding the relationships between different concepts and even in developing new ones. Removing the single strongest aspect --the "abstract symbols"-- of mathematics is to neuter it.

maxerickson 15y ago
I think part of his goal is to help people that do not yet understand the abstraction have a way to manipulate a system involving it. Then play can lead to understanding, in a way that is unlikely for 2x+1=5.

nwatson 15y ago

It will be hard to come up with an accessible general framework to express general problem situations without resorting to either (a) an explosion of templates for solving the N-most-common kinds of problems (the browsing through which will be worse than learning the math and the programming); (b) a set of elaborate but limiting special-purpose programs (much like iPhone apps that might help, for example, a metal welder to choose stock feed rate, gas mix/flow, and electric current depending on metal type and thickness to be joined; or mortgage calculators). Worse yet, it doesn't take much for interesting problems to get into the land of over- or under-constraint. In an under-constrained problem there are a multitude of solutions forming their own K-dimensional space, and then one likely should apply some secondary optimization criteria to determine the best solution. In over-constrained problems there is no exact solution, but again one needs to impose some optimization to get something "close enough" to the desired criteria, according to some norm. Explaining the need for optimization, letting the user explore the solution space, and having them express these optimization criteria will be tough. A Google search for "geometric constraint solver" leads to this paper: http://www.cs.purdue.edu/homes/cmh/distribution/papers/Constraints/CAD95.pdf http://www.cs.purdue.edu/homes/cmh/distribution/papers/Const... -- what appears to be a lot of work just in helping people/machines solve geometric problems (though surely more widely applicable when taken in the abstract). Coming up with even more general "hands-on" explorers/solvers will be a lot of work.

ypcx 15y ago

Tell me if I'm wrong, but I have this theory that the whole Math could be expressed on top of a single basic operation, which is addition. For example 2 * 3 is (2 + 2 + 2). Or sin(7) is (7 * (1 + -0.1666666664*(7*7) + 0.0083333315*(7*7)*(7*7) + -0.0001984090*(7*7)*(7*7)*(7*7) + 0.0000027526*(7*7)*(7*7)*(7*7)*(7*7) + -0.0000000239*(7*7)*(7*7)*(7*7)*(7*7)*(7*7))) (the multiply operator left in for brevity, but can be easily replaced by addition) If my assumption is true, then Math is really just an arcane and archaic form of a computer-like language. And at the same time, it's possible that the operations meant by Math's various symbols, however arcane, can be understood much better (especially with the use of modern tech, animations, etc.) than we understand them from classic texts, but those who learnt them haven't spent enough effort to create more visual explanations of it, based on which others, and also themselves, could understand those operations better, and then be able to build on top of them and even evolve Math further. (I know that I don't know what I don't know, but I have tendency to assume that I know a lot more than I really know, thus I don't explore the known deeper/further.)

thristian 15y ago
It sounds like you're thinking of (or working towards) Peano Arithmetic¹, which (to oversimplify) basically starts with "0", "1", "=" and "+", and works from there. Whether that's "enough" rather depends on what you're trying to achieve, though. There's subgenres of mathematics that don't necessarily involve numbers at all, like geometry and topology. It would be difficult to reduce them to addition. ¹: http://en.wikipedia.org/wiki/Peano_axioms http://en.wikipedia.org/wiki/Peano_axioms
yequalsx 15y ago
Addition is a function and so functions would be a more basic concept. But functions are sets and sets are a more basic concept. Thus we should study sets. Everything comes from sets. Replace addition with sets and your first sentence would be closer to the truth. I strongly disagree with your last paragraph though.
- derleth 15y ago
  And you can take sets as something built on top of categories (specifically, topoi), so category theory is more foundational than set theory. http://plato.stanford.edu/entries/category-theory/ http://plato.stanford.edu/entries/category-theory/ http://sakharov.net/foundation.html http://sakharov.net/foundation.html http://en.wikipedia.org/wiki/Topos http://en.wikipedia.org/wiki/Topos

Gravityloss 15y ago

Two things: 1. Needless complexity 2. Needed complexity 1.: Think how programming in assembler requires you to mentally keep track of registers, and how higher level programming languages did away with that by enabling variable names. The resulting program is still mostly the same thing underneath. You can get the same logical results but the former is much harder. There is a question here though: why is it easier for most humans to program in C rather than in assembler? Sure, there's more housekeeping in assembler. But I think it's mostly because there's one less layer of representation. a=5; b=10; c=a+b; Vs "load 5 into register A1", "load 10 into register A2", "calculate sum from registers A1 and A2 and store it to register A3". So here in "less layered math", instead of saying: Let alpha represent the angle, a the closer cathete and b the further cathete and c the hypotenuse, then sin(alpha) = b/c You instead say: sin is the relation of the further cathete to the hypotenuse. Again, a large portion of people can get the former way of explaining sin, perhaps when they see it all at once on the blackboard or book and go back and forth... (what was b again?) but the latter way of explaining does away with the whole exercise of variables. (Note that I picked a subject of convention, not something that can be deduced from more basic principles, I think those again can be taught somewhat differently.) The problem is, many math teacher love variables and think how beautiful all that symphony of x:s and y:s is. But for the average school student math is just one subject among others. They just want the essentials with the least amount of extra crap and layers on top. Lots of caveats here. I think people need to learn basic algebra, but it makes me sad that so few people can't use it for problems they are trying to solve. I think these would make interesting psychology research subjects. It's a very important field. 2.: Yet you can't go beyond some point in simplification.

Geee 15y ago
Have you seen this Bret's talk about his principle http://vimeo.com/36579366 http://vimeo.com/36579366 ? It's not really about simplification of things. The principle is basically "to see what's going on", or "to have immediate connection with what you are doing". Basically when you design something, you have to try to think or simulate in your head what's going on. Bret wants to remove that barrier so you don't have to think. There's some great examples in the video where the principle is applied to graphics, game design, electronics, programming etc.

bwarp 15y ago

Oh FFS. Everything has a barrier to entry. Just work with it until you get over it. People want instant gratification these days. It takes time to understand stuff. The Internet (and crappy TV about popstars and shit) has lead to people having completely unrealistic expectations of everything.

derleth 15y ago
You can use precisely the same arguments to denigrate demotic script versus hieroglyphs, and say that anyone who thinks writing complex pictures is too difficult a way to express themselves is just whining. I'm not taking a position on the argument expressed in the article; I'm merely noting that your argument is so infinitely applicable (even recursively applicable, back every step of every path of technological and notational improvement) that it is effectively useless as a way to determine the best path to follow.
derleth 15y ago
I see more people agree with you than with me, despite the fact upvotes aren't meant to be used that way. HN can be trying at times, can't it?
- bwarp 15y ago
  Indeed - but that is the nature of the Internet. There should always be a counterpoint to a good view.
JadeNB 15y ago
> Oh FFS. Everything has a barrier to entry. Just work with it until you get over it. The problem is that most people's response is "I'd rather give up on it than get over it", so we wind up with a math-illiterate society. If we can make it easier to be math literate, then we encourage a better world. (Your response, which is quite correct, reminds me of http://www.marco.org/2012/02/25/right-vs-pragmatic. http://www.marco.org/2012/02/25/right-vs-pragmatic.)

migiale 15y ago

The idea of "geometrization" of mathematical education as opposed to "algebraization" has been discussed widely in the recent decade at least, maybe more. In Russian school of mathematics, Vladimir Arnold (http://en.wikipedia.org/wiki/Vladimir_Arnold http://en.wikipedia.org/wiki/Vladimir_Arnold), was a very strong proponent of the idea. You can read his talk of the subject here http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html .

flashingleds 15y ago

Quite aside from any of the main points being made here, the guy is dead on about Strogatz's book. I had the same reaction to that book as I did to Lipovača's Learn You A Haskell - it's a great accessible resource to learn something new and interesting (if perhaps not immediately useful). Would appeal to the HN crowd I imagine.

ilaksh 15y ago

He's identified a real problem and is going in the right direction. I don't think we need to go quite as far though to make an improvement. A simple observation that I find amazing that more people don't see, is this: Mathematics is obfuscated code that doesn't compile or run. One letter identifiers and odd symbols rather than useful textual descriptions. No way to drill down into the meaning of the symbols. With code you can take a higher-level function and look at the code for the functions it uses, then look at the implementation of the language, then look at the assembly language, etc. No such thing in mathematics.

shadowmint 15y ago

I can't help but think this is completely missing the point. TED has had people cover this a number of times, but the problem with math isn't that the the _syntax is too hard to parse_ its that the _problems in math class are stupid_. "The bucket is depth X, diameter D and full of water. When you open the tap, how long will it take to drain if the water drains out at rate Z?" I've never had to apply solving a problem like this, in my entire life (and if I were in a job that I _did_ have to, the complexity of a real life situation would mean I would still have to learn domain specific tools to solve the problem (eg. how big is the air intake? Does that limit the rate of flow? etc)). Tangible problems in the real world require mathematical models (often probabilistic models) to solve them. How do you take a real world problem, break it down into bits, and then use the mathematical tools available to solve them? By creating a model, and then guessing what the rules that govern that model are, then comparing the model to reality, and refining the rules; and when you can't figure out the rules, thats when its time to whip out the text book and say, well, guess what, someone has had that problem before and this is how they solved it~ Teaching kids how to create models and reach out into the mathematical library available to them when they need it would be vastly more helpful than trying to creating more abstract alternative ways of understanding obscure math concepts that will never to relevant to them. I've never been more frustrated than I was the other night when I was at a party and a boiler maker (who incidentally earns 3x what I do. damn mining boom) was telling me about all the cool math he's learnt since he started his job. It's all geometry and rate of flow differentials and he said "why did I have to learn matrices at school? total waste of time. they should have been teaching us useful things"

TeMPOraL 15y ago
> "why did I have to learn matrices at school? total waste of time.(...)" Unless you're doing game programming as a hobby, or encounter problems that can be modeled as a set of equations, which then can be easily solved using few matrix tricks, giving you lots of fun and $$$. But I do agree with your sentiment; even though math can be fun on its own, it's easier (and probably better) to get people interested in it by showing how to use it to solve complex problem. After all, math is just formalized, applied rational thinking.
im3w1l 15y ago

mbq 15y ago

This is so wrong... The point of math is that ideas can be generalized and re-used in previously unexpected places -- your ideas remind me Middle Ages where people were learning a mnemonic poems to remember "law of proportion", treating it like a precious magic gizmo and were perfectly sure they wouldn't be able to recreate it once forgotten.

Riesling 15y ago
Have you read the article? One of his arguments is that math, due to its interface, is not about ideas but about abstract symbols and "symbol-pushing tricks". For example many people know how to use the chain rule when they derive a function, without even having the slightest grasp of what is going on. How is this any different from "learning a mnemonic poem to remember the law of proportion"?

pfortuny 15y ago

OK so it takes years to learn Japanese but it is the only way to truly enjoy haikus... The la guage of modern maths is what has enabled the physics at the LHC. You cannot get the latter without the former, hard as you may try.

Tycho 15y ago

I get the sense that there's two levels of understanding for most mathematics: a superficial level where you just have to manipulate symbols and be comfortable applying rules, and a deeper level where you grasp the underlying relationships and some fundamental 'why' of the problem. I used to think that if you were bad at the first, then the second would be an even tougher challenge. But I'm starting to suspect that actually people who are good at the first often just don't care about the second. And in fact it might be the very need for things to 'make sense' to you on a deeper level that is causing discomfort and difficulty at the superficial level. Some people lack curiosity about the deeper relationships, and perversely this might actually help them get along at the superficial level. For instance in finance, why is Macaulay duration(1) approximately equal to effective duration? One measures time in years, the other measures sensitivity in percentage form. How the heck do they come out at the same number approximately? For some reason i can't find any literature that's in a rush to explain this relationship... even though the two measures share the name 'duration' so presumably there's some intuitive understanding to be reached. (1) when the yield is expressed continuously compounding, at least

DennisP 15y ago

His "interactive exploration of a dynamical system" looks like some very sweet software. It's not an entirely new concept though. Modeling tools like Vensim and Stella do something very similar, modeling differential equations with "stock and flow" diagrams. Modeling problems with tools like these is a pretty active field, with applications in economics, ecology, and engineering. There are a fair number of books about it. Vensim has a free "personal" version, if anyone wants to play around with this stuff. It's definitely not as elegant as Bret Victor's demo, but might be better for more complicated systems (depending of course on how much Bret left out of the demo).

merraksh 15y ago

Previous discussion: http://news.ycombinator.com/item?id=2532271 http://news.ycombinator.com/item?id=2532271

super_mario 15y ago

Someone doesn't know the formal definition of computation. This is all there is to it really.

jpdoctor 15y ago

> When most people speak of Math, what they have in mind is more its mechanism than its essence. 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.* Wow, sounds like someone doesn't really understand math.

deadsy 15y ago
Brett Victor is very accomplished. I'm pretty sure he understands math. Personally speaking if Brett Victor had a different opinion about something than me than I'd pay careful attention, because I'm probably missing something.
- rhizome 15y ago
  jpdoctor is a bomb-thrower, your words fall on deaf ears.
lukev 15y ago
That's precisely his point. The "UI" for math obfuscates what it's actually about. Math is full of purity and simplicity, but the notation and algorithms we start out teaching kids are anything but.

dubya 15y ago

For the more mathematically inclined, there's a really nice little book, "The Computer as Crucible" by Borwein and Devlin, about experimental math. It makes a really convincing case, that I think couldn't have been made 15 years ago, that computers really have something essential to contribute to mathematics now. I think Borwein has written a number of articles about the same, and Devlin had a column in the AMS Notices about computer math.

DanBC 15y ago

I am hopeless at math. This is something that I am ashamed of. But there are people who seem to be proud of their ignorance; they're shocked if you haven't read any Shakespeare but happily admit they can't do percentages. Will those people be helped by the author's approach? I don't know, but I don't think so. These people will see a number and throw their hands up, saying "Oh maths! I can't do sudoku, how do you expect me to do this!". This attitude is not quite as prevalent in the UK as it used to be, but it's still there. See, for example, the number of esoteric arts programmes on BBC compared to the number of advanced science programmes. (I'm not aware of any science programming that would be beyond an enthusiastic 14 year old. I do know of hours of arts programming that is unashamedly elitist. Elitist is fine, but it'd be great for some balance.) What is needed is better maths education. (I finished school many years ago; maybe things are different now.) Math is not blindly mashing numbers and symbols and hoping for the best. Maths includes a large element of careful thinking, exploring the problem, listing the known information, listing what you want to find. New techniques for using math would help reduce the gender inequality in math results too. Put the normal "more research needed" caveats around this, but: There's some suggestion that girls use inefficient techniques and "just struggle through", they manage to get correct results and so they don't get extra help. Boys tend to just stop when it gets too laborious, and thus they get taught new better techniques. (http://news.bbc.co.uk/1/hi/education/4587466.stm http://news.bbc.co.uk/1/hi/education/4587466.stm)

moxiemk1 15y ago

I'm worried that in presenting math with visible, tangible intuition dominant will do more to silo the knowledge inside the ivory tower. N-variable calculus isn't something that you can draw a picture of. Lots of (very practical) Linear Algebra is derived from the pure "Linear Transformation" interpretation rather than the "Matrix" one. Stochastic...anything (seems to be all the rage these days) is not made with helpful pictures. If we train people to do math in a non-abstract way, they won't be as easily able to grapple with the real problems, which are only approachable in the abstract.

enjalot 15y ago
I disagree, at least that pictures can't help with higher dimensional math. Well, regular pictures may not help so much but interaction really can. A lot of my understanding of higher level math depends on my programming ability to "touch" some of the abstract concepts. When learning something new you need to be able to associate the new concept with things you already understand, and visuals can give you an intuitive sense of how things are behaving (especially in time) I don't think the visuals could necessarily take the place of training to use the abstract symbols or lines of code any time soon, but I think they are a really necessary step in the right direction to get more people to approach math.

justinbaker 15y ago

This is the kind of shit we don't need. I'm a Computer Science & Mathematics double major. I'm also bipolar/ADHD, and this post makes extreme assumptions about learning. Not everyone learns the same way. I absolutely cannot stand to sit in class and listen to a professor babble on incessantly and draw pictures of concepts and irrelevant ideas simply to relate it for the majority to learn. I learn by doing. By code. By numbers. By command line. It has done quite well so far..for me at least. Maybe it hasn't for you..then change that and get busy learning instead of filling HN with awful blog posts.

iso8859-1 15y ago

I really like the Soulver program that he mentions on the page for the "Scrubbing Calculator". I found a similar Windows/.NET program called OpalCalc: http://www.skytopia.com/software/opalcalc/ http://www.skytopia.com/software/opalcalc/ Time to do a web edition? Surely it already exists. Somebody help me out :P

Tycho 15y ago

Are there any good apps for the iPad that let you manipulate variables and equations in a tactile manner? I have Ovium which is a calculator where you put your numbers in bubbles and then connect them however you want with different operators. I'd like like something where you could use the value of a slider to change the value of a variable.

FreshCode 15y ago

Fix your scrolling; I think your message will benefit from increased usability.