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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 do

What we have here is a failure to define our terms. Math is the manipulation of abstract symbols according to abstract rules. If you don't like symbols, you don't like math. If you are illiterate in symbols you are illiterate in math. The word is often used imprecisely, though. Because so many real-world problems can be translated into math, there is a temptation to equate "math" with "any problem that can be expressed in math". Thus you frequently see poetic statements like "my cat is great at solving differential equations", or "music is math because it's all about harmonic series and Fourier analysis". But these things aren't literally true. You can put a bucket under a flowing faucet, and it will collect all the water, but that isn't really integration. The bucket isn't doing math. And being ignorant of math isn't the same as being stupid. As the OP points out, you can get a lot of quantitative reasoning done without using math. A classic Fun Fact About Math is that it took thousands of years to invent the number zero. And it's true. But that doesn't mean that the ancient Egyptians used to waste hours staring into newly-emptied buckets and baskets in stunned amazement, murmuring "what on earth is that" to themselves in Coptic. People understood what "having no objects" meant long before there was a symbol for "the number of objects in an empty basket". It was the highly abstract symbol "zero", and the highly abstract operations involving zero, that had to be invented. [1] It's worthwhile to recognize that interpreting the real world in terms of abstract symbols, and vice versa, is a terribly difficult skill that requires lots of practice. (In my case, I was well through grad school before many bits of physics clicked.) And it's worthwhile to recognize that you can often do without math: You can reason quantitatively without it. Birds do it! Bees do it! But don't pretend that you're doing math unless you are actually doing math. The abstractions are the math. --- [1] Or, rather, discovered. Although we'd better stop there, because I won't be able to cope with the ensuing philosophical back-and-forth.

StavrosK 15y ago
You know, I'm glad that you're able to easily manipulate symbols, but when I look at them, my eyes glaze over. It took years for me to learn linear algebra because not a single teacher ever told me "a matrix is also mapping from one space to another one", until Massi Pontil of UCL. At that point, all the linear algebra I couldn't figure out for the life of me all those years finally made sense. And it was the same for most of my classmates. After that, whenever I saw xY, I thought "the vector x is being moved into a new space", and all the equations made sense to me. You could explain what an SVM is with equations to me all day, but it's only when you say "you're trying to get the plane to separate your data by a margin as wide as possible" do I actually get it, and then all the math becomes easy. Different people have different ways of manipulating the abstract symbols, and for me it's to equate them to something I already have experience in. Then I can get the solutions intuitively, rather than pore over pages and pages of equations. In the end, I quit academia precisely because I couldn't manipulate symbols, and thus my way of learning wasn't compatible with everyone's way of teaching. Maybe I can come up with something better if someone explains things to me in terms I can understand.
- etherealG 15y ago
  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 :)
  - StavrosK 15y ago
    It's not about not wanting to spend the time. How, for example, would I know about linear algebra intuitively if nobody ever put it in these terms?
  - stevep98 15y ago
    Here's my comments: * 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.
- warmfuzzykitten 15y ago
  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.
nightski 15y ago
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.
tel 15y ago
Completely disagree. Math isn't manipulation of symbols. At all. Math is the study of mathematical objects, a practice often done using a formal language for the convenience and power it provides. "0" is a formal symbol with particular formal behavior "empty/missing/none" is a well-known physical concept Zero is a precise, powerful mathematical object which can be represented by them both. --- This is difficult to deny. Unless you want to deny the providence of most widely recognized mathematicians throughout history, you have to accept that formal language of math is relatively new. Furthermore, it's alive and growing, inconsistant and incomplete. There is a meaningful frontier, and there you observe mathematicians are really studying something else and furiously creating the formal language to describe it. In this light, metaphor is absolutely a useful tool in the same class as formal language for explaining and reasoning about math. You're right to point out the non-equivalence of the two, but the author's Kill Math project is in no way not math. Furthermore, I'm anecdotally a supporter of the author's belief that doing math competently requires knowing a the metaphorical side since your symbolic projects may fail or be unclear. I'd be willing to accept that metaphor will never be as powerful as formal language, but it does discredit to the way (I'd wager) most people understand math to deny the metaphorical. --- At the heart of this trouble of definitions is Gödel's Incompletenesses. The practical effect of their discovery was the destruction of the dreams of formalists who had for years hope to discover the essential shape of the formal language from which all math would spring. With Incompleteness however, we are forced to admit that we can study, meaningfully, the behavior of mathematical objects for which the language of math cannot be used to reason about. Then we extend that language, of course.
- mechanical_fish 15y ago
  I like this response. 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.
timwiseman 15y ago
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.
bobbyi 15y ago
> Math is the manipulation of abstract symbols according to abstract rules. That's the definition of a calculus, not the entirety of math. http://en.wikipedia.org/wiki/Calculus http://en.wikipedia.org/wiki/Calculus : >More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.

StavrosK 15y ago

You know, I'm glad that you're able to easily manipulate symbols, but when I look at them, my eyes glaze over. It took years for me to learn linear algebra because not a single teacher ever told me "a matrix is also mapping from one space to another one", until Massi Pontil of UCL. At that point, all the linear algebra I couldn't figure out for the life of me all those years finally made sense. And it was the same for most of my classmates. After that, whenever I saw xY, I thought "the vector x is being moved into a new space", and all the equations made sense to me. You could explain what an SVM is with equations to me all day, but it's only when you say "you're trying to get the plane to separate your data by a margin as wide as possible" do I actually get it, and then all the math becomes easy. Different people have different ways of manipulating the abstract symbols, and for me it's to equate them to something I already have experience in. Then I can get the solutions intuitively, rather than pore over pages and pages of equations. In the end, I quit academia precisely because I couldn't manipulate symbols, and thus my way of learning wasn't compatible with everyone's way of teaching. Maybe I can come up with something better if someone explains things to me in terms I can understand.

etherealG 15y ago
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 :)
- StavrosK 15y ago
  It's not about not wanting to spend the time. How, for example, would I know about linear algebra intuitively if nobody ever put it in these terms?
- stevep98 15y ago
  Here's my comments: * 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.
warmfuzzykitten 15y ago
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.

nightski 15y ago

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.

tel 15y ago

Completely disagree. Math isn't manipulation of symbols. At all. Math is the study of mathematical objects, a practice often done using a formal language for the convenience and power it provides. "0" is a formal symbol with particular formal behavior "empty/missing/none" is a well-known physical concept Zero is a precise, powerful mathematical object which can be represented by them both. --- This is difficult to deny. Unless you want to deny the providence of most widely recognized mathematicians throughout history, you have to accept that formal language of math is relatively new. Furthermore, it's alive and growing, inconsistant and incomplete. There is a meaningful frontier, and there you observe mathematicians are really studying something else and furiously creating the formal language to describe it. In this light, metaphor is absolutely a useful tool in the same class as formal language for explaining and reasoning about math. You're right to point out the non-equivalence of the two, but the author's Kill Math project is in no way not math. Furthermore, I'm anecdotally a supporter of the author's belief that doing math competently requires knowing a the metaphorical side since your symbolic projects may fail or be unclear. I'd be willing to accept that metaphor will never be as powerful as formal language, but it does discredit to the way (I'd wager) most people understand math to deny the metaphorical. --- At the heart of this trouble of definitions is Gödel's Incompletenesses. The practical effect of their discovery was the destruction of the dreams of formalists who had for years hope to discover the essential shape of the formal language from which all math would spring. With Incompleteness however, we are forced to admit that we can study, meaningfully, the behavior of mathematical objects for which the language of math cannot be used to reason about. Then we extend that language, of course.

mechanical_fish 15y ago
I like this response. 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.

timwiseman 15y ago

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.

bobbyi 15y ago

> Math is the manipulation of abstract symbols according to abstract rules. That's the definition of a calculus, not the entirety of math. http://en.wikipedia.org/wiki/Calculus http://en.wikipedia.org/wiki/Calculus : >More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.