Off-topic, but who uses an IE document icon as a favicon? got freaked out here in the office thinking IE managed to crawl back anywhere I can mistakenly open it somehow.
The site was made using Word.
How to Read Mathematics
41–44 of 44 posts
Re: How to Read Mathematics
#42Earlier quoted context omitted.
>You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math. My point is that these goals are nearly one in the same when you get to the high level. Many mathematical relationships are very complex, usually not the step-by-step procedures that is common in code. Thus being able to hold the entire relationship in your head at once is crucial. Single letter…
Sorry, if all you have left is the (very typical) argument of "if you think that way you just don't understand it", then I'll consider this discussion over. I have no interest in hearing endless appeals to tradition. I'm sure people made similar arguments for GOTO back in the day. Good thing we moved on.
I would like to think my argument is far more nuanced than you're giving it credit. This isn't an appeal to tradition; I'm making an argument that explains why traditionally math has stayed with the "cryptic" notation rather than descriptive variable names.
You're completely ignoring the working memory argument I'm making. A large part of math is pattern matching: recognizing a common relationship between variables (a common pattern or "theme") and investigating that relationship further. This ability is critical to mathematical ability. Visual compactness is crucial here. Using descriptive names will completely bog down your visual system with reading words rather than identifying patterns.
Pattern matching is a critical skill when one becomes an expert in any field; in math its of utmost importance. Math is optimized for reading by other experts who have those same visual patterns committed to memory. This is the optimal approach for the work's target audience. Yes, it makes advanced math largely inaccessible to outsiders, but that's a part of the trade off.
Re: How to Read Mathematics
#43Earlier quoted context omitted.
Sorry, if all you have left is the (very typical) argument of "if you think that way you just don't understand it", then I'll consider this discussion over. I have no interest in hearing endless appeals to tradition. I'm sure people made similar arguments for GOTO back in the day. Good thing we moved on.
I would like to think my argument is far more nuanced than you're giving it credit. This isn't an appeal to tradition; I'm making an argument that explains why traditionally math has stayed with the "cryptic" notation rather than descriptive variable names. You're completely ignoring the working memory argument I'm making. A large part of math is pattern matching: recognizing a common relationship between variables (…
>You're completely ignoring the working memory argument I'm making.
I'm not ignoring it. I said I don't buy it. I generally don't believe that you need visual compactness to to point of being cryptic to make use of pattern matching. I'm not saying this is necessarily easy. Neither is choosing good variable names in programming, especially at high abstraction levels. But that's no excuse for not doing it.
The other thing is that mathematicians will finally have to accept that they aren't the only ones who need math. It's not, for the most part, a "field for experts" (to paraphrase you) which only those who are willing to become experts at it are allowed to understand. It's a beautiful science with lots of appliance in almost every other field. But it's inaccessible as hell in large parts due to mathematicians with your attitude.
I strongly believe that math can be made more accessible and more "user-friendly" for people who aren't experts at it. It will be hard to do that without sacrificing its tremendous power and flexibility (which I am totally against in software, as well). But it must (and will) eventually be done.
Take note, by the way, that the accessibility problem is less and less important the more specialized the math you are doing becomes. I don't really care if some hyper-abstract field of math that a few dozen people in the world are even able to grasp the basics of is arcane and cryptic. That's something only experts will care for and if they are fine with their niche field being overly cryptic, so be it. What I'm concerned about is math in general.
I firmly agree with the position that math should not be taught as an appliance, because that is missing the point. However, it's downright fatuous to ignore the fact that math has appliances, and that therefore, a lot of people will have to learn a subset of it. To excuse bad practices which have survived by virtue of no one questioning them with nothing but "it's optimized for edge cases (experts)" is a cop-out. It ignores the problem.
I also find it funny that I already met with such vehement opposition for just proposing meaningful variable and function names. That was just an example. There are a lot more problems to be solved. Stuff like "this is trivial" or "left as an exercise" as a way to avoid writing out cumbersome, but nonetheless important (for non-experts) parts, the general terseness of mathematical texts, including the parts in plain English. I could probably find more.
Let me conclude with this: Math as a whole has accessibility problems, just like a lot of CS and computer stuff. Flat out ignoring them or even asserting "it must be this way" is incredibly ignorant. And we need dialogue to do this - not condescending "there is no problem" rebuttals.
Re: How to Read Mathematics
#44Earlier quoted context omitted.
I would like to think my argument is far more nuanced than you're giving it credit. This isn't an appeal to tradition; I'm making an argument that explains why traditionally math has stayed with the "cryptic" notation rather than descriptive variable names. You're completely ignoring the working memory argument I'm making. A large part of math is pattern matching: recognizing a common relationship between variables (…
>You're completely ignoring the working memory argument I'm making. I'm not ignoring it. I said I don't buy it. I generally don't believe that you need visual compactness to to point of being cryptic to make use of pattern matching. I'm not saying this is necessarily easy. Neither is choosing good variable names in programming, especially at high abstraction levels. But that's no excuse for not doing it. The other th…
You make a lot of points that I agree with. I just don't think your solution is the right one. Math, when it comes to writing proofs and peer review, should absolutely be optimized for other experts. Whatever makes communicating ideas precisely among themselves is what's important here. I don't see the benefit of changing this process so those of us as outsiders can take a peek.
But as you said, there is a very large part of math that deals with applying these concepts and communicating them to non-mathematicians. The way we go about this is definitely in need of a do-over. Finding more intuitive ways to communicate these ideas is critical a critical part of this (possibly including more descriptive variable names). But, I think there will always be an inherent chasm between the math experts and those who are using the results they discover. An example is the difference between the calculus track of courses that every science major takes, and say, abstract algebra. The math that mathematicians do and the math that the rest of us learn will always be vastly different. It's just the nature of the beast.