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Math is first and foremost a language, whose symbols are generally well-defined, although they differ between contexts in some cases. In many of those differing

Math is first and foremost a language, whose symbols are generally well-defined, although they differ between contexts in some cases. In many of those differing contexts, the symbols are re-used to give an intuitive sense. For example, re-using "x" for cross product. Reducing math to words means English math is different from Chinese math. If I, a native English speaker, were to look at such math, I would have to more or less take it on faith that the translation I'm reading is accurate. And god forbid I'm trying to read a translation of a work done jointly by (say) a Romanian and a Chinese. The symbols generally remove ambiguity. Consider the English word "bi-weekly." It's so muddied that it is almost useless. It means either twice a week or once every two weeks. It means, literally, both multiplication and division; and context is rarely helpful for this particular word. You can certainly use the symbols poorly and ambiguously, as many viral "Solve this math problem!" memes exemplify. But used properly they're generally clear, concise, and well-defined. I do, however, have one nitpick about symbols, and that's with the use of the ellipsis in math. You'll often see things like {1, 2, ..., n} which is generally intended to mean (say) the natural numbers up and including to n. But does it? Couldn't it also represent powers of 2? Or the set of fibonacci numbers? And since it is a set, order doesn't matter. Almost anything at all could be in that set. We can only be sure it's not something like "all the odd numbers" because 2 is in the set. Frustratingly, it's not even necessary in most cases, since we have set-builder notation and other tools.

horsawlarway 7y ago
Sure, but this is also exactly why latin was used for as long as it was - It was a language that the educated elites from many different and diverse countries used as a command ground. I'm not saying that math shouldn't be using symbols, I'm saying it's near intentionally hostile to those first picking up the subject. If the goal of math education is to provide the groundwork of understanding to the general populace... Cater to the general populace. The vast majority of those folks are not going to be discussing complex math across several languages. They're going to be using it for accounting, taxes, construction, cooking, etc. They don't need to memorize an entire language to do that effectively, and we shouldn't be wasting their time in school trying to make them. Those that choose to specialize are welcome to. In my opinion that doesn't justify the use of specialized language in basic education settings.
- cgriswald 7y ago
  Can you give an example of which symbols you object to and what you anticipate as a replacement? The most difficult things I can think of that someone doing accounting or cooking might have to do is compound interest or converting units; so we're talking about ()+-x÷ and exponents. I'm guessing you have something else in mind or I'm not getting it. I do have criticism with symbols insofar as math is taught which roughly fall in line with "A Mathematician's Lament" by Paul Lockhart[0]. Loosely, that much math is taught as symbol manipulation rather than... actual math. But that's not a problem with the symbols themselves so much as with their use to obfuscate what's actually being done mathematically. You might be interested in the book Burn Math Class by Jason Wilkes [1]. I don't really recall what his arguments against symbols were. He ends up inventing his own notation as he goes along (which is what ultimately turned me off the book about halfway through -- it just became an exercise in translating notation for me). [0] - https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... [1] - https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics-ebook/dp/B017QL8UHU/ https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics-... Edit: Fixed list of links.
  - horsawlarway 7y ago
    I think you're vastly over estimating the capability of many students (particularly young students without previous exposure). It's easy to fall into a trap where you forget what it's like to be completely new to a subject. It's particularly hard when you tend to surround yourself in educated communities where this "in-knowledge" becomes assumed and standard (eg hacker news). Take just less than ( < ) and greater than ( > ). Think about how many stupid rhymes or memorization techniques you see in classrooms to help learners memorize which is which. Here are three separate sites, with an entire page dedicated to helping students remember which is which: https://math.wonderhowto.com/how-to/remember-greater-than-less-than-symbols-302956/ https://math.wonderhowto.com/how-to/remember-greater-than-le... https://numberock.com/lessons/comparing-numbers-to-100/ https://numberock.com/lessons/comparing-numbers-to-100/ https://myhomeworkdone.com/blog/greater-than-less-than-sign/ https://myhomeworkdone.com/blog/greater-than-less-than-sign/ One of them includes a whole damned song for the purpose. All to avoid writing out smaller/bigger. Like any language, once you learn it's hard to remember all the places you struggled. Why make "change" Δ Why make "square root of -1" i In how many classes do we see rote memorization of the quadratic formula, with no context around why you should even bother to learn it? (I've seen quite a few). Now, not all of those are really the fault of the language (math), but using the language for each of those problems facilitates lazy teaching, and it changes the goal from "understand how math relates to the world" to "memorize this language construct". One is much more helpful than the other.
viraptor 7y ago
> Couldn't it also represent powers of 2? You'd use a different notation then. `{1, 2, 4, ..., n^2}` or `{2^0, 2^1, ..., n}` or something similar which indicates what you mean. There's nothing wrong with elipsis, if you spend a second thinking about what you're writing. If someone wanted to be obnoxious and confusing then they could say: Haha, `n` is not a symbol either; I'm using base-50 and it's digit 23(10). So you can't really stop bad / intentionally misleading communication.
- cgriswald 7y ago
  Like I said, it’s a nitpick. I can know what a person means most of the time, but it’s imprecise and technically the ellipsis can be almost anything depending on the explicit values/symbols. In the cases where the pattern is clear it’s clear. Where it’s not, using proper set builder notation in the first place would have made it clear. I’ve also seen it used in contexts other than sets, including for summations. In those cases often just having the summations in proper form can make consequences and manipulations plainer and easier.

horsawlarway 7y ago

Sure, but this is also exactly why latin was used for as long as it was - It was a language that the educated elites from many different and diverse countries used as a command ground. I'm not saying that math shouldn't be using symbols, I'm saying it's near intentionally hostile to those first picking up the subject. If the goal of math education is to provide the groundwork of understanding to the general populace... Cater to the general populace. The vast majority of those folks are not going to be discussing complex math across several languages. They're going to be using it for accounting, taxes, construction, cooking, etc. They don't need to memorize an entire language to do that effectively, and we shouldn't be wasting their time in school trying to make them. Those that choose to specialize are welcome to. In my opinion that doesn't justify the use of specialized language in basic education settings.

cgriswald 7y ago
Can you give an example of which symbols you object to and what you anticipate as a replacement? The most difficult things I can think of that someone doing accounting or cooking might have to do is compound interest or converting units; so we're talking about ()+-x÷ and exponents. I'm guessing you have something else in mind or I'm not getting it. I do have criticism with symbols insofar as math is taught which roughly fall in line with "A Mathematician's Lament" by Paul Lockhart[0]. Loosely, that much math is taught as symbol manipulation rather than... actual math. But that's not a problem with the symbols themselves so much as with their use to obfuscate what's actually being done mathematically. You might be interested in the book Burn Math Class by Jason Wilkes [1]. I don't really recall what his arguments against symbols were. He ends up inventing his own notation as he goes along (which is what ultimately turned me off the book about halfway through -- it just became an exercise in translating notation for me). [0] - https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... [1] - https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics-ebook/dp/B017QL8UHU/ https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics-... Edit: Fixed list of links.
- horsawlarway 7y ago
  I think you're vastly over estimating the capability of many students (particularly young students without previous exposure). It's easy to fall into a trap where you forget what it's like to be completely new to a subject. It's particularly hard when you tend to surround yourself in educated communities where this "in-knowledge" becomes assumed and standard (eg hacker news). Take just less than ( < ) and greater than ( > ). Think about how many stupid rhymes or memorization techniques you see in classrooms to help learners memorize which is which. Here are three separate sites, with an entire page dedicated to helping students remember which is which: https://math.wonderhowto.com/how-to/remember-greater-than-less-than-symbols-302956/ https://math.wonderhowto.com/how-to/remember-greater-than-le... https://numberock.com/lessons/comparing-numbers-to-100/ https://numberock.com/lessons/comparing-numbers-to-100/ https://myhomeworkdone.com/blog/greater-than-less-than-sign/ https://myhomeworkdone.com/blog/greater-than-less-than-sign/ One of them includes a whole damned song for the purpose. All to avoid writing out smaller/bigger. Like any language, once you learn it's hard to remember all the places you struggled. Why make "change" Δ Why make "square root of -1" i In how many classes do we see rote memorization of the quadratic formula, with no context around why you should even bother to learn it? (I've seen quite a few). Now, not all of those are really the fault of the language (math), but using the language for each of those problems facilitates lazy teaching, and it changes the goal from "understand how math relates to the world" to "memorize this language construct". One is much more helpful than the other.

viraptor 7y ago

> Couldn't it also represent powers of 2? You'd use a different notation then. `{1, 2, 4, ..., n^2}` or `{2^0, 2^1, ..., n}` or something similar which indicates what you mean. There's nothing wrong with elipsis, if you spend a second thinking about what you're writing. If someone wanted to be obnoxious and confusing then they could say: Haha, `n` is not a symbol either; I'm using base-50 and it's digit 23(10). So you can't really stop bad / intentionally misleading communication.

cgriswald 7y ago
Like I said, it’s a nitpick. I can know what a person means most of the time, but it’s imprecise and technically the ellipsis can be almost anything depending on the explicit values/symbols. In the cases where the pattern is clear it’s clear. Where it’s not, using proper set builder notation in the first place would have made it clear. I’ve also seen it used in contexts other than sets, including for summations. In those cases often just having the summations in proper form can make consequences and manipulations plainer and easier.