5 ms·

The way you think of it is precisely the mathematical way of thinking about n-dimensions. Mathematically, an n-dimensional space is R^n, that is, the cross pro

by mreid 15y ago

The way you think of it is precisely the mathematical way of thinking about n-dimensions. Mathematically, an n-dimensional space is R^n, that is, the cross product of the real numbers (R) with itself n times. Each copy of R lets you specify a number independently of all the others. Mathematics is just careful abstraction. If you are abstracting away certain aspects of your problem and carefully making deductions about what is left, you are doing mathematics. As you point out, this can be taken too far but any tool can be used poorly in the hands of a novice.

slowpoke 15y ago
>Mathematically, an n-dimensional space is R^n, that is, the cross product of the real numbers (R) with itself n times. Each copy of R lets you specify a number independently of all the others. See, that's exactly what I'm talking about. This gibberish makes sense to no one except a mathematician. The explanation I've given - while maybe the same in essence - conveys the meaning in a comprehensible way that's not completely removed from reality. My point is that mathematicians often do neither realize that they are talking gibberish that makes no sense to anyone but themselves nor do they accept that math without proper context is devoid of any meaning (outside of math, that is). NOTE: I would like to clear up that I'm not attacking math itself. I neither think math for math's sake is bad, nor do I think math is useless. I respect math as a self-contained field. The problems I'm pointing out are all happening where there is application within other fields - such as CS.
- oblique63 15y ago
  Thank you! I've had this same feeling for a long time. It strikes me as odd that probably a lot of the same people that would find it unacceptable to write cryptic C code with one-letter variable names, find typical math notation/jargon to be completely fine and legible. If code is so important that we go out of our way to make it comprehensible to future maintainers, why don't we have the same feelings about math? (which I would argue is much more important for people to comprehend, seeing as how it's the core foundation of all of science) Personally, I'm really into physics, so I've always really wanted to like math so that I could delve deeper into the subject more easily. It's not that I'm terribly bad at it, but most math texts are so obscenely terse and cryptic, that it makes you wonder whether the authors are actually even trying to teach people about what they're talking about...
- hackerblues 15y ago
  I certainly agree that an intuitive understanding of a concept can be helpful as a guide and emotionally satisfying. However, I don't think that it is at all sufficient and it is precisely "mathematical gibberish" which resolves the problem. An isolated concept is worthless. It is only when you are about to apply it by reasoning with it that becomes valuable. The problem with intuitive explanations is that they don't nail down enough details to allow a person to reason with them. "I think of points in an n-dimensional space as objects holding n different types of information" Imagine walking into a room and drawing a straight line on the floor. What is the dimension of that line? Answer = One dimensional. Proof: We can describe each point by one type of information. Point = (Distance of that point from the start of the line.) Answer = Two dimensional. Proof. We can describe each point by two types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall.) Answer = Three dimensional. Proof: We can describe each point by three types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall, Distance of that point from the roof.) Answer = Four dimensional.... So by the intuitive explanation we can make this single line any dimension that we want. This isn't just a problem within mathematics. For a simple programming example: Question: How does a computer program work? Intuitive Answer: You give the computer a list of instructions for it to carry out. Result: The guy opens up notepad and types in "Make a computer game where I walk around shooting Zombies."
  - slowpoke 15y ago
    >So by the intuitive explanation we can make this single line any dimension that we want. And if this makes sense in the given context - sure, why not? You seem to be missing my point. What I'm saying is that it's useless to have totally generalized abstractions (outside of pure math) since more often than not, they are so far removed from reality that most people can no longer make any connection to use cases. My entire argument is that there's a heavy communication failure between mathematicians and scientists of every other field where math is used as a tool. Sure, it's convenient for a mathematicians to be able to use shorthand gibberish to talk to other mathematicians. It doesn't justify pushing this jargon on other fields. Besides which, convenience is no excuse for making something hard to understand. Sure it's convenient to name all your variables in a program a, b, c etc but you're going to get lynched by any programmer that tries to read your code later, including yourself. When it comes to a point where gibberish becomes the only way to explain mathematical abstractions, then you should step back and ask yourself "where the hell did this go wrong?". >This isn't just a problem within mathematics. For a simple programming example:[...] I believe in giving simple explanations and expanding them whenever there's a loophole that needs to be fixed. You don't make programs that cover every single niche use case, either (if you don't have to, that is). That's the problem with math - the generalizations, while useful in math itself, are sheer overkill in many situations outside of math. Lastly, sorry taking this out of order, but: >An isolated concept is worthless. So is a generalized abstraction without any context. I'm saying that the sweet spot is somewhere in between for most people to understand and apply concepts, and that it's better to generalize upwards from reality and actual use cases instead of starting utterly removed from reality and trying to apply the generalizations downwards.
- mreid 15y ago
  I was writing it mathematically as I thought you would appreciate how your intuitive grasp of what n-dimensions is described mathematically. The terminology (gibberish) I used is nothing more than a convenient but very precise shorthand for communicating abstractions. The reason these particular abstractions are given special names by mathematicians is that they occur over and over again. There is a lot of conceptual leverage to be gained if you are able to see the same patterns in seemingly unrelated problems. As Poincaré put it, "Mathematics is the art of giving the same name to different things". I'd argue it's worth knowing some of those names.

by mreid 15y ago

slowpoke 15y ago

>Mathematically, an n-dimensional space is R^n, that is, the cross product of the real numbers (R) with itself n times. Each copy of R lets you specify a number independently of all the others. See, that's exactly what I'm talking about. This gibberish makes sense to no one except a mathematician. The explanation I've given - while maybe the same in essence - conveys the meaning in a comprehensible way that's not completely removed from reality. My point is that mathematicians often do neither realize that they are talking gibberish that makes no sense to anyone but themselves nor do they accept that math without proper context is devoid of any meaning (outside of math, that is). NOTE: I would like to clear up that I'm not attacking math itself. I neither think math for math's sake is bad, nor do I think math is useless. I respect math as a self-contained field. The problems I'm pointing out are all happening where there is application within other fields - such as CS.

oblique63 15y ago
Thank you! I've had this same feeling for a long time. It strikes me as odd that probably a lot of the same people that would find it unacceptable to write cryptic C code with one-letter variable names, find typical math notation/jargon to be completely fine and legible. If code is so important that we go out of our way to make it comprehensible to future maintainers, why don't we have the same feelings about math? (which I would argue is much more important for people to comprehend, seeing as how it's the core foundation of all of science) Personally, I'm really into physics, so I've always really wanted to like math so that I could delve deeper into the subject more easily. It's not that I'm terribly bad at it, but most math texts are so obscenely terse and cryptic, that it makes you wonder whether the authors are actually even trying to teach people about what they're talking about...
hackerblues 15y ago
I certainly agree that an intuitive understanding of a concept can be helpful as a guide and emotionally satisfying. However, I don't think that it is at all sufficient and it is precisely "mathematical gibberish" which resolves the problem. An isolated concept is worthless. It is only when you are about to apply it by reasoning with it that becomes valuable. The problem with intuitive explanations is that they don't nail down enough details to allow a person to reason with them. "I think of points in an n-dimensional space as objects holding n different types of information" Imagine walking into a room and drawing a straight line on the floor. What is the dimension of that line? Answer = One dimensional. Proof: We can describe each point by one type of information. Point = (Distance of that point from the start of the line.) Answer = Two dimensional. Proof. We can describe each point by two types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall.) Answer = Three dimensional. Proof: We can describe each point by three types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall, Distance of that point from the roof.) Answer = Four dimensional.... So by the intuitive explanation we can make this single line any dimension that we want. This isn't just a problem within mathematics. For a simple programming example: Question: How does a computer program work? Intuitive Answer: You give the computer a list of instructions for it to carry out. Result: The guy opens up notepad and types in "Make a computer game where I walk around shooting Zombies."
- slowpoke 15y ago
  >So by the intuitive explanation we can make this single line any dimension that we want. And if this makes sense in the given context - sure, why not? You seem to be missing my point. What I'm saying is that it's useless to have totally generalized abstractions (outside of pure math) since more often than not, they are so far removed from reality that most people can no longer make any connection to use cases. My entire argument is that there's a heavy communication failure between mathematicians and scientists of every other field where math is used as a tool. Sure, it's convenient for a mathematicians to be able to use shorthand gibberish to talk to other mathematicians. It doesn't justify pushing this jargon on other fields. Besides which, convenience is no excuse for making something hard to understand. Sure it's convenient to name all your variables in a program a, b, c etc but you're going to get lynched by any programmer that tries to read your code later, including yourself. When it comes to a point where gibberish becomes the only way to explain mathematical abstractions, then you should step back and ask yourself "where the hell did this go wrong?". >This isn't just a problem within mathematics. For a simple programming example:[...] I believe in giving simple explanations and expanding them whenever there's a loophole that needs to be fixed. You don't make programs that cover every single niche use case, either (if you don't have to, that is). That's the problem with math - the generalizations, while useful in math itself, are sheer overkill in many situations outside of math. Lastly, sorry taking this out of order, but: >An isolated concept is worthless. So is a generalized abstraction without any context. I'm saying that the sweet spot is somewhere in between for most people to understand and apply concepts, and that it's better to generalize upwards from reality and actual use cases instead of starting utterly removed from reality and trying to apply the generalizations downwards.
mreid 15y ago
I was writing it mathematically as I thought you would appreciate how your intuitive grasp of what n-dimensions is described mathematically. The terminology (gibberish) I used is nothing more than a convenient but very precise shorthand for communicating abstractions. The reason these particular abstractions are given special names by mathematicians is that they occur over and over again. There is a lot of conceptual leverage to be gained if you are able to see the same patterns in seemingly unrelated problems. As Poincaré put it, "Mathematics is the art of giving the same name to different things". I'd argue it's worth knowing some of those names.