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.
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[1] Or, rather, discovered. Although we'd better stop there, because I won't be able to cope with the ensuing philosophical back-and-forth.
Completely disagree. Math isn't manipulation of symbols. At all. Math is the study of mathematical objects, a practice often done using a formal language for the convenience and power it provides.
"0" is a formal symbol with particular formal behavior
"empty/missing/none" is a well-known physical concept
Zero is a precise, powerful mathematical object which can be represented by them both.
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This is difficult to deny. Unless you want to deny the providence of most widely recognized mathematicians throughout history, you have to accept that formal language of math is relatively new. Furthermore, it's alive and growing, inconsistant and incomplete. There is a meaningful frontier, and there you observe mathematicians are really studying something else and furiously creating the formal language to describe it.
In this light, metaphor is absolutely a useful tool in the same class as formal language for explaining and reasoning about math. You're right to point out the non-equivalence of the two, but the author's Kill Math project is in no way not math. Furthermore, I'm anecdotally a supporter of the author's belief that doing math competently requires knowing a the metaphorical side since your symbolic projects may fail or be unclear.
I'd be willing to accept that metaphor will never be as powerful as formal language, but it does discredit to the way (I'd wager) most people understand math to deny the metaphorical.
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At the heart of this trouble of definitions is Gödel's Incompletenesses. The practical effect of their discovery was the destruction of the dreams of formalists who had for years hope to discover the essential shape of the formal language from which all math would spring. With Incompleteness however, we are forced to admit that we can study, meaningfully, the behavior of mathematical objects for which the language of math cannot be used to reason about.
I have no time to craft a nuanced reply so let me just take a shortcut and concede: If you can get the student from "here are a bunch of physical concepts" to "here is a mathematical object, with interesting abstract properties that you can reason about" without introducing formalism you'll have succeeded in teaching math. Excellent.
Is this plan really going to work very often? It is easy to say that you can derive, say, the utility of "zero" without ever doing any arithmetic -- just as it is easy to say that you can be a full-fledged computer scientist without ever touching a computer -- but in practice?
It's true that the presence of a supercomputer in everyone's pocket will change this argument significantly. But simulations go only so far. They, too, are only metaphors, and if you don't know enough to tinker under their covers they are rather inflexible metaphors. Your classical mechanics simulator is not going to discover quantum mechanics for you.
In my experience learning about mathematical abstractions requires all of the above tools -- you tinker with the formalism, you ponder the physical analogies, you draw mental pictures of clouds and colors, you play with a simulator, you build some circuits in the lab, you go for a walk, you tinker with the formalism again, and six years later you finally get it.
I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In particular, it's easy to see how formal descriptions can push mathematical generalization forward far before we have a suitable concept of the mathematical object we're describing.
I think we're all (incl. the op) in some kind of agreement here about the didacticism of math. The op didn't disregard the power and utility of mathematical languages — he came from being trained pretty heavily in engineering math, at least up to playing with higher-order differential equations — but instead was, perhaps not directly, arguing for increased metaphorical/physical descriptions in taught mathematics. He's just responding to the rather eye-opening feeling one gets when one starts to realize that math is so interpretable!
I think that's a perfectly fair argument to have. I know that in my own experience, I never understood the joy of math until the day that linear algebra took an interpretation as linear space transformation.
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So we're just sort of all oscillating here in strong rebuttals of whatever interpretation of the "heart" of math the prior author champions for a while. Which is fun but unproductive.
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I like the challenge of taking someone from physical concepts and metaphors directly to a mathematical object. I think it'd be possible, and maybe even useful when someone first starts to learn real math, but certainly it's not the most efficient way to become well-read. It'd be a lot like explaining the meaning of, I dunno, Día de los Muertos without immersing someone in Mexican language and culture. A single point of contact can be forced, but you lose so much context and fluency.
I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.
Are you sure you're accurately describing Godel's Incompleteness theorem? It's been a while, but I understood it at a technical level briefly. I'd be interested in knowing more about how you think it applies here.
No, I'm not. I'm interpreting it pretty loosely. Sorry about that. Correct me if I'm wrong here, though.
More directly, what I meant to say is that since there exist true theorems which cannot be proven within any particular choice of mathematical formalism, we need to operate with tools beyond simply symbolic manipulation. That was the death knell of Hilbert's Program and solidly separated the formal specification of math from "that thing which we're studying".
I think you are wrong - even though there are true, unprovable statements, they're unusual and to the best of my knowledge, not the subject of extensive study. More like curiosities (http://en.wikipedia.org/wiki/List_of_statements_undecidable_...). Even if that were not the case, there is no alternate approach to studying them.
I'm not really sure that you mean about the "formal specification" for math vs. the "that thing we're studying". An informal (ie, not expressed in ZFC + 1rst order predicate calculus) proof of something non-trivial can go on for dozens, if not hundreds of dense pages of symbols. If I recall correctly, Whitehead's Principae Mathematica derived arithmetic from ZCF and predicate calculus, and it took the whole book.
I did a little reading to refresh myself on the subject, and this stood out as a good summary of the topic:
"In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided."
Your picture of mathematics is quite a bit more robust, so I won't deny that what you wrote here is more accurate than my "Gödel say it won't work, woe is our field!" brimstone version. So I'll explain my reasoning for invoking Gödel.
I mostly wanted to walk around the historical event I mentioned, the breaking of the Hilbert Program. At the time, it seemed that formal specification of math would provide a complete picture of what math was! Once the Program was finished then the job of mathematician would eke out into "computer" (of the abacus sort) or into other fields which interpreted the canon.
I'm not sure which death stroke was stronger, the incredible opaqueness and complexity of proof systems like ZFC or Gödel just saying what he was trying was outright impossible, but Hilbert's Program was killed before it even seriously took off, leaving the study of mathematics and the practical formalisms we use to study it pretty ad-hoc instead of grand and unified.
I'm unifying that with the fact that the way math seems to be practiced never comes from the formal language but instead first comes from imagining some kind of "mathematical object" and then taming its behavior with formalisms. You could consider them to be one and the same and argue that the difference is highly philosophical, and then this is where I'd invoke Gödel and inform you that there definitely exist things we could benefit from reasoning about that your formal language would fail to describe. This existence proof separates the classes of true things and provable things and makes their distinction more than philosophical.
Now, talking about what a "mathematical object" is gets you to the bleeding heart of the philosophy of science and epistemology. It's a tough question!
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As a final note, ZFC is ZF + Axiom of Choice... which, yes, most practicing mathematicians just accept AoC so that they can integrate or whatever. The formal world without AoC is very sparse, but nobody has any sort of idea what the arbitrary decision means. I know that there has been some significant study of ZF-C, though it's been "impractical", I don't know if anyone is willing or capable of stating that ZF-C is in any way worse than ZFC. Impractical is a Mathematicians favorite adjective, so they're just two extant formal systems which disagree quite a lot on important things but we mostly pay attention to ZFC.
roughly correct insofar as there are undecidable mathematical statements. I don't understand though, how "we can study, meaningfully" these objects that our formal systems cannot compute. These statements do exist, and we can (interestingly) express them in language and understand them (as a gut feeling), but all we can do is stare awkwardly at them, undecidedly. imvvho, we cannot claim that metaphors (or language) are more powerful than formal systems by that.
I didn't mean to imply that one is stronger than the other. Instead that if they're well-designed, then they overlap (largely). Where they don't overlap is highly interesting.
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.
An excellent example of how math poorly taught is hard to understand. Nearly all math is poorly taught, therefore nearly all math students drop out in frustration. Only a handful of students can overcome poor instruction.
seems to me that different people can grok stuff in different ways. teaching is about trying to find a way to explain something so the learner can grok it, and in your case a lot of teachers failed before one finally succeeded.
I think anyone is capable of grokking anything, just the time taken to do so is variable. people that give up on "learning" something (academia in your case) just don't want to spend that time.
p.s. I also gave up on academia for the same reasons :)
* Math is full of symbology with implied meaning. For example, theta is often used for 'angle'. How many other symbols have implied meaning like that? Granted, it forms dense, concise, precise papers. Which brings me to my second point.
* If you don't know the symbology, it's difficult to read it. I believe that people suffer reading comprehension problems if they don't know how to verbalize a symbol like 'θ'.
* Lastly, the symbols make it very difficult to google for concepts.
>More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
Math is not at all the manipulation of symbols though. That is one of many tools that is useful in doing math, but it is not the only one. The famous proof that there are an infinite number of prime numbers is normally expressed in plain English (or your language of choice) without a symbol in sight, for one example.
Math, fundamentally, is about abstract concepts, not symbols.
Sure the bucket isn't doing math, but how else would you describe it? Integration is the word we have come to use to communicate the meaning of what is happening. In other words, math isn't just about understanding. It is about communication.
Sure we have intuition. But intuition can be wrong. Not all problems are as simple to explain as zero and empty buckets. Intuition is also worthless if you cannot communicate those ideas in an unambiguous fashion.
So if the OP really develops a method for communicating the concepts described by math in a much more efficient way - I am all for it! But I find that highly unlikely when even it is admitted in the article itself that he has no idea what this will look like.
I appreciate that symbols can be an odd way to represent every day quantitative problems, but I have never had a problem with it. Sure, some people do, but this guy seems really royally pissed off at ordinary mathematics.
I have always been a proponent of having more than one way to teach a certain subject. Studies have shown that different people perceive and interact with information in very different ways. The current "one size fits all methodology" to teaching is lame, to say the least. So a new way of teaching people mathematics is welcome.
But that's not to say the old way didn't work for people like me. If this site kills math so that English majors can cope, I hope someday someone will kill poetry so that I can cope.
(Compacted for space. Original. Use tr '/' '\n' | sed -e 's/^ //' -e 's/ $//' to reëxpand, with a paragraph break being "\n\n". Sorry for the repeated initial edits and such; I had to try to work out the HN formatting rules.)
When humans try to learn symbolic math / How many of them struggle with the test! / The teacher thought of like a psychopath / Dishonoring the realm of human zest
“We must have our emotions!” students cry / “Or else we'll run around like apes, confused / Our brains are built for stories, not to scry / A world of numbers, strangled and abused.”
The teacher sighs, “They always drag their feet / Unless they're cornered, up against the wall. / To risk my job with answers incomplete! / They'll never use it later, after all.”
Then, big surprise! The math is found at fault / Tear-stained by cringing memories of school / “Dispense with all the symbols, and Exalt / Thine Intuition”—that shall be the rule.
Professors' lamentations curse the air / Hung out to dry for calling any bluff / “To shun defective math must be unfair / For surely no one understands the stuff.”
So woe to ye from near the world of forms / Who strain to show the populace your realms / They're immunized against your grand transforms / And explanation only overwhelms.
(Now, please don't take this poem at its word / Or treat it as authoritative fact / Exaggerated story and absurd / Polemic leave specifics inexact
The author's nearly made of symbols, note— / Despite the slow decay of some to blanks / So though he doesn't mean to seem to gloat / He'd rather keep his “freakish” symbols, thanks.)
What's you background? What kind of math did you learn by looking at mathematical symbols rather than natural language explanations? I have hard time envisioning how such a leaning process would work.
There are several pretty obvious problems with mathematical notation.
- Meaningless one-letter names.
- Meaning of notations is usually highly context-dependent. (Exponentiation and matrix inverse operation are denoted in exactly the same way, for example.)
- A lot depends on arbitrary conventions.
- People rarely explain how notations work syntactically. It's all ad-hoc,learn by example.
I'm sorry to be the one to have to bring this up. But not all mathematics can be represented visually. Maths breaks into a number of main branches, among them principally algebra, analysis and geometry.
Usually only the third can be represented visually, though often mathematicians develop diagrams which help internalise the complex notions involved in analysis and algebra too.
For example, in algebraic geometry, the more familiar notions of geometric objects such as curves and surfaces are replaced with purely algebraic notions, such as schemes. This is because of various categorical equivalences between geometric objects (on the geometric side) and various algebraic objects (on the algebraic side). But on the algebraic side, schemes are a very expansive generalisation of things that actually correspond to geometric (and visualisable) objects.
I once went to a teaching seminar on the use of a package called GeoGebra for the teaching of mathematics. None of us mathematicians could bring ourselves to put up our hands and ask how one might represent a complex of modules over a noetherian ring pictorially in GeoGebra. There's this fundamental misunderstanding amongst educators that symbolic mathematics is not essential to understanding maths.
This is an important insight when it comes to computer programs though. The same thing happens in computer science. You get splits between things that are geometric, symbolic and purely computational.
Often I get really annoyed at people showing off their latest concurrent programming paradigm by implementing a GUI or event loop for some graphical or network application. They forget that many things simply don't fit into that paradigm.
I equally get annoyed at computer scientists for forgetting that the number of integers is not about 10. Sometimes us mathematicians really want to do things with matrices of ten thousand by ten thousand entries.
One of the great things about math is that it lets us surpass our own intuition. The spatial intuition we evolved in the wild doesn't work well for a number of things, which is why symbols and abstraction are so useful in the first place.
We don't need mathematics to figure out how a swinging pendulum works; we can do that intuitively. (Actually, this isn't completely true, but we can at least get the general idea.) Simple problems like that are worked out in classes so that students can get used to the mathematics. In domains where our intuition fails us - e.g. quantum mechanics, high-energy physics, statistical mechanics, not to mention 11+ dimensional formulations of string theory, infinite or fractional dimensional spaces, and more esoteric theoretical mathematics and physics - we rely on mathematical symbols and abstraction to guide us, because our finely honed physical intuition is useless (and sometimes worse than useless).
I'm interested in seeing what the author does with concepts like superposition and n-dimensional spaces. Replacing them with graphs and animations is not going to cut it.
Two fairly simple improvements (to mathematics) I can think of:
1) Abandon the absolutely batty practice of representing everything with a single symbol. It's crazy. We still can't reliably represent mathematical symbols over the internet in text, we have to rely on images. I'm obviously biased, but mapping mathematical functions to actual words (ala programming) would be a big win, imo.
2) Lock up all the physicists, mathematicians, engineers, logicians, you name it, and have them agree on a single unified notation. Every symbol ought to have one meaning and everyone needs to stop stealing symbols from a different field and giving it a new meaning! I'd wager that more than half of the symbols in this list (http://en.wikipedia.org/wiki/List_of_mathematical_symbols) have several valid interpretations. No thanks!
You know, one of the reason that notations differ is because that's useful. The symbol \times might mean cross product or direct product of groups, or cartesian product. Why? Because these operations are conceptually similar --- they all involve making a multiplicatively larger thing from smaller things. Using the same symbol for different things might be polymorphism, not conflict.
1) What do you mean? It depends on field, but in most mathematical papers I read, the ideas are explained with words, not with symbols -- they only serve as helpful shortcuts. Anyway, I believe that properly used symbolism usually clarify the issue at hand, especially when word explanation is very long. For instance, what is faster to read and comprehend:
"Let V1 and V2 be a subspaces of W, such that their intersection is zero. Let f be a mapping from a direct sum of V1 and V2, such that it takes a vector, whose first component is x and second y to a difference of x and y multiplied by two."
And this was easy example, I can think of _a lot_ harder.
There is no regulating body of mathematical notation -- it can be (and usually is) created by introducing it in some paper or book by some mathematician who invented it and regards as useful. Frequently, there are more than notation introduced, but usually only one survives -- hopefully the best one. The only possibilities of encountering several different notations in use at once are either reading very old works, which is not good anyway, or the most recent ones, but I presume that people who are able to read them are also able to get over such a minor problem.
Seriously, I believe that the mathematic notation is a lot clearer, more intuitive and easier to understand than syntactic rules of many programming languages, for instance C++. Symbol overloading almost never pose a problem, since the intended meaning is usually obvious from the context. If one frequently misunderstands the intended meaning, it is a sign he does not really get the concepts involved, and the fact he is confused by notation is his smallest problem.
Even symbol overloading most often takes places only if the sign represents the same idea in all contexts. For instance, one usually uses '+' sign to represent a binary commutative operation whatever structure we all talking about, because, well, it represents similar idea. One can go even further and say that symbols like \oplus and \times in most contexts they are used in (Cartesian product of sets, direct product/sum of rings/groups/modules/vector spaces/mappings) are actually representing exactly the same idea -- namely, the notion of product/coproduct in some category.
There are a lot of different symbols in use in math. If we abandoned symbol overloading, we would need to introduce many, many new symbols, and this would create real confusion.
Symbol overloading almost never pose a problem, since the intended meaning is usually obvious from the context.
Heh. Heh. Heh.
So I would have believed until I tried to learn differential geometry. The default is to eliminate all parts of the notation that are unambiguous. Proving that they are unambiguous is left as an exercise to the reader, and the exercise is often non-trivial. Furthermore widely used constants vary by factors of 2 pi depending on who is using it.
About the relation of the "computer science" approach to symbolic expression, and the "mathematics" approach: remember that lambda-calculus was originally introduced to make functional substitution, composition, etc., completely rigorous. Part of the early/mid 20th century project to turn mathematics into a completely formal system.
Of course, I agree that any idea of a "symbol standardization committee" for mathematics is both crazy and stupid.
Maybe, but I don't think that's where the OP was going. My understanding of the OP was that instead of, say using the large S symbol to mean integral, why not use this type of notation:
integrate(start, end, function)
Of course, I can think of a few problems with this. The large 'S' symbol is understood by everybody, regardless of their language, is one advantage of symbols that comes immediately to mind.
Kind of tangentially related (no pun intended), this reminded me of an interesting article by Tom Apostol, well known calculus teacher and textbook author. Here's the opening paragraph:
Calculus is a beautiful subject with a host of
dazzling applications. As a teacher of calculus for
more than 50 years and as an author of a couple of
textbooks on the subject, I was stunned to learn that
many standard problems in calculus can be easily solved
by an innovative visual approach that makes no use of
formulas. Here’s a sample of three such problems:
Is this guy trolling? First two paragraphs sound to me similar to what some of my math untalented friends said about it in highschool. "Power to understand and predict" is not limited only to those who can manipulate abstract symbols - for some reasons it seems that such abstract manipulation is the best way to do it now.
The second paragraph is even worse. This guy have simply no idea what he is writing about. 'Assigning meaning to set of symbols' is just abstracting unnecessary details and focusing on important information in problem, 'blindly shuffling symbols according to arcane rules' - um, calling math 'arcane' is a clear indicator of person's lack of understanding. Rules are not arcane, everything has explanation (proof) and is derived from other things in logical way (and as far as we know, world acts logcailly) - some of the are axioms, which seem to be abstractios of most basic properties. Also, 'shuffling blindly' is in fact spotting patterns in things on different levels.
Sure, explainig things on more intuitive level, using e.g. graphical representations is sometimes really helpful. However sometimes intuition doesn't work, and how do you graph 3-dimensional manifold embedded in R^4? Or finite field? Also, I can't imagine of other way of doing math that would be consistent and useful other than the one we are using now.
This "guy" studied electrical engineering and computer science at Caltech and UC Berkeley, I am sure his grasp of mathematics is ok. Perhaps you should try to stop feeling offended at his proposal and try to see where he might be coming from. Also as he admits in the article, it is by no means a comprehensive set of thoughts, but rather a few hunches hinting at a bigger idea.
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.
> It's really a shame that even now, after so many centuries of teaching math to people, the effectiveness of the process is still so utterly dependent on the teacher.
Maybe it's not a shame. Maybe it's just an indication that teaching is hard, like art, science, programming, and discovering new theorems, not easy like answering phones in a call center or being a short-order cook.
I am really in love with the sentiment/insight here- that the reason people are turned off by math (even just numerals) is because it's a complex mess of symbols that they maybe don't understand.
I am all for finding a way to explain quantitative concepts in a new way. However, it will be extremely difficult to avoid falling into the trap of "reinventing the wheel" if all we're talking about is coming up with a new set of symbols.
A certain recipe serves 3, but the cook is only cooking for 2, so she needs to 2/3 all of the ingredients. The recipe calls for 3/4 cup of flour. The cook measures out 3/4 cup of flour, spreads it into a circle on the counter, takes a 1/3 piece out of the circle and puts it back into the bag. That's 2/3 of 3/4.
Much easier to eyeball 1/3 when it's laid out in a rectangle as opposed to a circle. Author credibility -1
mindless tradition
Did you just call the set of symbols evolved by mathematicians for thousands of years mindless? Credibility -2
Finally the two animated examples given are clever but not groundbreakingly clear. -3
It's a neat project but maybe you could think a little harder about defining your problem.
"Much easier to eyeball 1/3 when it's laid out in a rectangle as opposed to a circle."
Maybe for you, but certainly not for me, and I'm guessing most bakers would agree with me. Bakers are used to circles because of pies. I can eyeball a third of a circle, but I'd have trouble eyeballing a third of a rectangle that I couldn't fold.
Further, the baker often works by feel, so an exact is not needed in these circumstances.
"Did you just call the set of symbols evolved by mathematicians for thousands of years mindless?"
Perhaps a better word would have been arbitrary, but there's no fundamental reason we pick y=mx+b. Y, M, X, and B are picked arbitrarily, and we do pick them without questioning whether these are optimal for initial learning.
I grokked math as a kid, but it was precisely because I was able to make the leap that the language of math was arbitrary and substitutable while other kids were stuck not understanding the meaning.
I grokked math as a kid, but it was precisely because I was able to make the leap that the language of math was arbitrary and substitutable while other kids were stuck not understanding the meaning.
Then we need to teach them that, not a new set of symbols. Again, I think the crucial insight here which you uncovered is that people are distracted/confused by the symbology, perhaps trying to take everything too literally.
By the way, the way to "eyeball" a third is to use your two hands (rotate them so palms facing each other) to divide into sections A B and C; since we can very accurately eyeball a 50/50 split, you simply compare A to B and B to C, then adjust your hands until A=B and B=C. Bam, you have thirds. Once you get good at this you just mentally visualize invisible dividers instead of actually using your hands.
When it comes to a circle, if you are staring at pies all day then maybe you are better than average, but many studies have shown that humans are horrible at discerning angles other than 180 and 90 degrees.
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.
This is not completely true. Modern calculus (and generally, mathematical) notation is a result of many, many years of evolution. Leibniz's notation only barely resembles the modern one.
seriously? and your contribution to helping people understand maths is ...?
You've given him some arbitrary credibility rating of -3 because his examples weren't exactly how you would have done them, except that you didn't write them, he did. You contribution was to write some bitchy comment about it.
Ugh. Math is a collection of concrete symbols for simplifying different problems. It's evolved over thousands of years to help us understand difficult things. We don't make it that way just to piss people off.
His argument is that analyzing a differential equation without exploring it in phase space was like analyzing a piece of sheet music without actually hearing it.
And if music education were taught in that way-- by looking at music purely as the manipulation of concrete symbols-- I imagine some of us would be writing "kill music (as it is currently taught)" blog posts as well.
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!
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.
Talking about Math as a single subject is like talking about Sports. Chess, Boxing, Baseball, Snowboarding, Curling, and Luge may have some things in common. However, suggesting that DiffEq and Fractions are equally meaningless to most people seems just as out of place as assuming a chess grandmaster is also a champion snowboarder.
Most of the symbols are pretty recent; if you read mathematics from even 300 years ago, there's a much bigger proportion written in prose. Along the lines of, "Consider two quantities, such that the latter is at least twice the former ...".
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.
I remember trying for hours to understand Adaboost before it all clicked (it's an ingenious algorithm, by the way). The paper could certainly do with some more explanation, rather than just "this is the weight updating function, this is the evaluation step, done".
I think AI and machine learning are worse than math in that respect for some reason. Part of it might be the greater focus on 6-page conference papers, which tends to require everything to get squished.
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.
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.
Math is a language. The problem is that this language is archaic, difficult to teach, and is generally taught very poorly.
I remember learning calculus in college. The professor went up to the board, scribbled down symbols, and took us through various procedures. I was absolutely, utterly lost until my father (an engineer) told me that "a derivative is a rate of change."
At that instant, I understood everything. My professor never said this.
This taught me to approach math concepts-first, and that helped, but I've always had a problem with math. To make a long story short: I hate math for the same reason that I hate Perl. My mind recoils in horror from messy, crufty languages.
At the very least, all math lessons should begin by teaching the language and the concepts that the various symbols, arrangements, etc. refer to. Only once the language is thoroughly grasped should they proceed to methods, procedures, and problems. Right now it's like teaching Chinese literature before teaching Chinese...
When I was a child, I always found math easy. Fractions, calculus, formulas—they were always visual. Fractions were pizzas; equations were deformations of the plane; integration was tracing out a shape using a slope.
So I'm sympathetic to the author's desire for better visualization and teaching tools.
But when I reached college, I became frustrated with math. It just wasn't easy anymore, the way programming was: I could pick up a programming book, read it in a weekend, and understand it. But when I tried to read an advanced math text, I became lost after 10 pages.
Eventually, I figured out what had happened: The information density of college-level math texts is insane. Even if you're bright and talented, it may take you a day to understand a single page. And there's no substitute for working carefully, finding concrete examples, and slowly building a deep understanding.
Here's an example that involves programming. Once upon a time, I needed to understand monads, in hope of finding a better way to represent Bayesian probabilities.
I started with the monad laws, a handful of equations relating unit, map, and join. I read countless monad tutorials, and dozens of papers. I read every silly example of how monads are like containers, space suits, C++ templates, and who knows what else.
I wrote little libraries. I learned category theory. I wrote a monad tutorial. I eventually wrote a paper explaining a whole family of probability monads:
And then one day, I thought about the monad laws again. I realized, "Hey, that's it. That's all. Just unit, map, join, and a handful of equations. Anything which quacks like a monad, is a monad. How did I ever think this was complicated?"
But when I look at the monad laws today, there's this huge structure of connections in my head. All that work, just to grasp something so simple, and so easy.
So I'm all for building better visualizations, and for helping people to understand math intuitively. That's an important step along the path. But math doesn't stop at an intuitive understanding. When you really understand it, the equations will suddenly be easy, and everything will fit together.
And then you'll encounter the miracle of math: Your deep understanding will become the raw material for the next level. Counting prepares you for addition, addition prepares you for multiplication, basic arithmetic for algebra, algebra for calculus, and so on. And someday, I hope that my rudimentary understanding of category theory will prepare me to understand why adjoint functors are interesting.
I had a different problem, I found that word density increases 10 fold, but symbol density and proofs go down as you go through it.
My favorite example to use is this: Complex Variables and Applications by Brown and Churchill. This book has been in print for 70 years or something, and it's somewhere around 400 pages in the current edition I believe. My professor I did research for had a early 80s edition, and it had almost 100 less pages than the current edition. There wasn't really anything new added between the versions (chapters are only about 5-10 pages, so there's something like 65 of them) I ended up using mine for the problems and his for reading because I have ADHD, and the wordiness absolutely kills me. Symbols and relationships are much more meaningful to me than words describing them. The real nightmare with the ADHD sets in because of the break in context when you have to switch between two or three pages to find the next theorem, formula or proof.
I retook that class twice.
On the other hand, I utterly and completely rocked my Advanced Electrodynamics course, outscoring even the graduate students, in a course which even made use of the stuff we were learning in Complex Variables (as well as PDEs and all that fun stuff) Why? I had a crazy russian professor who hated all the current textbooks (I'm looking at you, Griffiths) for the same reason that I hated textbooks, too much words and not enough symbols. So he wrote his own notes to every lesson and made his own homework. He said he originally wrote those notes when he first came here, and his english was worse, so there's little or no explanation, just proofs -- math and symbols. A few of these would span two pages, and very rarely three, but there wasn't the context break you get in many college level books, just beautiful math and lots of intermediate steps. The intermediate steps, almost never provided in most textbook proofs, really help the visual learners like me and provide stepping stones for the inevitable manipulation you will perform with those equations in your homework and on tests.
I still have all his notes, I want to bind them up some day when I get a chance.
Any chance of contacting said professor and obtaining permission to share said notes online?
Then... y'kno. Doing so :)
(I realize that scanning tons of a pages and ensuring quality isn't exactly a small undertaking, but I'm sure the HN community would greatly appreciate your efforts if such a thing were possible)
I really liked Griffiths. I thought it had struck a great balance between explanations, mathematics, and examples. That book had great examples and homework questions, which I find the most useful anyway.
Griffiths' E+M book (actually, also his QM book) is a shining star to me, a perfect example of the way introductory textbooks should be written. It ends up being criticized a lot once you get to the advanced level ("Who'd you learn quantum mechanics from, Griffithslol?"), but that's fine. It's perfect, even - if you really need all the gory stuff, sure, go get a copy of Jackson's book, or some 800 page tome on perturbation theory and the rigorous mathematical underpinnings of quantum theory, if you're going into the field you should absolutely know more than you can get from an intro text. But an introductory textbook should be exactly that. Rigor is awesome once you have the intuition down, but it's often very difficult to go the other way (mainly because a rigorous understanding is difficult to achieve without having the intuition first).
I think Griffiths EM/QM texts are fine if your not studying physics. However if you are they just skim the surface of material and this is were I think a lot of the bashing comes in. Physics curriculum (presumably like all) are a constant stepping stone and when your senior level professors expect you to know Clebsch Gordon coefficients in detail and Griffiths QM didn't then some questioning about his texts start to rise. In my opinion I thought Sakurai's QM was excellent in delivering a solid foundation in quantum mechanics
I don't mind it, this professor hated it, although it was the "recommended" book for the course. He recommended the following book, but it was out of print so he couldn't teach the course with it: http://www.amazon.com/Electromagnetic-Fields-Waves-Paul-Lorr...
It's similar to Griffiths, but I feel like it has more examples and is a little more concise in those examples than Griffiths (although I did borrow Griffiths occasionally) I grabbed one on ebay for $5 or something, and it was well worth it.
I have read both of them and I strongly prefer Griffiths to Corson. Lorraine and Corson are extremely rigorous but IMO their explanation of EM is upside down :-) They first explain special relativity and derive the EM equations using it. But historically, we had the EM equations first which forced Einstein to abandon Galilean transformations and discover special relativity.
But Lorraine and Corson just start out with special relativity, without really providing any impetus for why it is necessary, which can be found in the very first paragraph of Einstein's paper on relativity. Historically, physics has been a mystery that keeps unraveling with time. However LnC destroy the mystery by telling us who the murderer was in the opening chapter :-) We don't really appreciate how relativity was discovered. I think teaching the mystery is a very important and easily overlooked part of Physics education which Griffiths seems to appreciate but LnC don't.
Sounds like a great application for digital textbooks on some future color e-ink reader. For students who prefer verbal explanation, some of the intermediate steps can be set to display: none, and the text reflows accordingly. For symbolic learners, hide unnecessary <span>s of text.
Don't know that I've ever seen a resume that covered everything from hardware engineering all the way up to some of the most highly regarded user interface designs. He has certainly covered the entire gamut of "things you can do with a computer."
Which makes him the perfect Apple employee, as they cover the space from hardware engineering through user experience better than anyone.
I also have a problem with the way mathematics is taught currently where the student is more adept at the mechanics than the philosophy of mathematics. If anybody is attempting to change that, my best wishes. I don't necessarily have a problem with the current system of notations. It is just the method of teaching that lacks.
Just started with 'What is Mathematics' by Courant and I am totally hooked on. He talks about everything from why we chose to adopt the decimal system to why pi was needed to solve certain problems.
As Hammock says, the author has to be clear about defining the problem itself. Is it mathematics as it is represented today which is lacking or the method of teaching which is lacking
> This "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots.
> using concrete representations and intuition-guided exploration.
Is it just me, or are these two lines very much at odds with each other?
Pictures aid understanding. But in the vast majority of sophisticated mathematics (that I've seen), pictures would be unable to accurately represent the real assertions.
For instance, if I want to talk about 26-dimensional discontinuous space (the space of the basic alphabet), there is no visualization that can help you grasp the totality of the matter.
I believe there are a few interpretations of what math actually is (formal reasoning, interpretation, etc), philosophy of math 'junk'. I'll leave that to a more-beered time. :-)
"those with a freakish knack for manipulating abstract symbols"
The downside of the growing public awareness of people on the autism spectrum, some of whom are geeks, has been the slow trend towards conflating intellectualism with atypical mental function.
Some people like to call right now the "victory of the geeks". I suspect that within the next 10 years, nerdy kids will start being diagnosed as having Asperger's by school counselors and the like with about as much care, caution, and accuracy as we saw with ADHD.
This post reminds me the visual explanation of pythagoras' theorem :
http://www.mathsisfun.com/pythagoras.html
Also, the math language symbols are great for communicating easily and efficiently among us, but it's not the best materials for learning. Distinguish learning (with visual) and communicating (with language symbols) activity in math should help...
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?
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!
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.
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.
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).
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 ?
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.
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.
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).
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.
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[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.