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A more recent example I've come across are n-dimensional spaces. I realized that most people have problems to make sense of the concept behind that because they

A more recent example I've come across are n-dimensional spaces. I realized that most people have problems to make sense of the concept behind that because they've been stuck in thinking mathematically. I think of points in an n-dimensional space as objects holding n different types of information, which is a practical approach, and has lit a candle for some other people I explained it to this way as well. For example, images are often stored as 4 dimensional structures, with the dimensions representing height, width, depth and color value. Every point in that structure is a pixel of the image. You can easily expand this to any number of dimensions, and it still makes sense - you're just adding information (eg, add a fifth dimension for time, and you've got yourself a sequence of images - a simple video). My beef with the way math is often approached is that on the one hand, it's asserted that math is the universal solution to all problems - including practical ones - but on the other hand, more often than not math is removed so far from reality that it's nearly impossible to make any practical use of it.

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."
barumrho 15y ago
I don't think any mathematician will assert that math is a practical subject or the universal solution. In math, we just define n-dimensional Euclidean space, but no implication is made about the physical world.
jl6 15y ago
That's totally the standard way that all mathematicians think about n-dimensional spaces. I've never heard a mathematician claim math is the universal solution to all problems... sounds like you had a bad high school experience. Do you need a math-hug? :)
- slowpoke 15y ago
  >sounds like you had a bad high school experience. Not exactly high school. I'm a university student in Germany. But yes, it's pretty much a bad experience. I've seen quite a lot of other students which are otherwise bright persons struggle with math simply because they aren't good at learning to interpret arcane sequences of symbols which pretty much represent nothing relevant to reality. Math the way it is taught here is essentially useless for 95% of all CS students.
  - eru 15y ago
    I studied math and CS in Germany. The mathematicians were usually adept at CS, but the computer scientist didn't--with some exceptions of course--even get what math was about at all. They should probably stop trying to dumb down the math for computer scientist. Math is hard, and pretending otherwise, won't save one from reality.
  - kbolino 15y ago
    I think you are obsessing about minutiae. Perhaps mathematical instruction is very different in Germany than in the United States, or perhaps our brains are simply incompatible, but I've never found symbology to significantly affect my understanding. Symbols are just convenient and useful abbreviations. In my experience, they facilitate both speed and precision in mathematical work, far and above what is possible in a natural language like English. There is no particular reason why certain concepts are associated with certain symbols apart from convention. Any decent mathematician should be able to operate with any set of symbols, although not necessarily with equal proficiency. I won't deny that many mathematicians have a hard time expressing themselves in natural-language speech and writing. Perhaps for them the symbols have become a crutch. But I still think you have erected a mental block around an ultimately inconsequential aspect of mathematics, which is needlessly impeding your understanding. That having been said, I think there are definitely bad ways to approach the use of symbols in instruction. Concepts and their related symbols should be introduced incrementally, and you should always have ample opportunity to practice with the ones you have just learned before having to learn new ones. I think that a well rounded mathematics curriculum should include a course with a significant focus on understanding and manipulating common mathematical and logical techniques, with mathematical language being a significant if not necessary part thereof.

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."

barumrho 15y ago

I don't think any mathematician will assert that math is a practical subject or the universal solution. In math, we just define n-dimensional Euclidean space, but no implication is made about the physical world.

jl6 15y ago

That's totally the standard way that all mathematicians think about n-dimensional spaces. I've never heard a mathematician claim math is the universal solution to all problems... sounds like you had a bad high school experience. Do you need a math-hug? :)

slowpoke 15y ago
>sounds like you had a bad high school experience. Not exactly high school. I'm a university student in Germany. But yes, it's pretty much a bad experience. I've seen quite a lot of other students which are otherwise bright persons struggle with math simply because they aren't good at learning to interpret arcane sequences of symbols which pretty much represent nothing relevant to reality. Math the way it is taught here is essentially useless for 95% of all CS students.
- eru 15y ago
  I studied math and CS in Germany. The mathematicians were usually adept at CS, but the computer scientist didn't--with some exceptions of course--even get what math was about at all. They should probably stop trying to dumb down the math for computer scientist. Math is hard, and pretending otherwise, won't save one from reality.
- kbolino 15y ago
  I think you are obsessing about minutiae. Perhaps mathematical instruction is very different in Germany than in the United States, or perhaps our brains are simply incompatible, but I've never found symbology to significantly affect my understanding. Symbols are just convenient and useful abbreviations. In my experience, they facilitate both speed and precision in mathematical work, far and above what is possible in a natural language like English. There is no particular reason why certain concepts are associated with certain symbols apart from convention. Any decent mathematician should be able to operate with any set of symbols, although not necessarily with equal proficiency. I won't deny that many mathematicians have a hard time expressing themselves in natural-language speech and writing. Perhaps for them the symbols have become a crutch. But I still think you have erected a mental block around an ultimately inconsequential aspect of mathematics, which is needlessly impeding your understanding. That having been said, I think there are definitely bad ways to approach the use of symbols in instruction. Concepts and their related symbols should be introduced incrementally, and you should always have ample opportunity to practice with the ones you have just learned before having to learn new ones. I think that a well rounded mathematics curriculum should include a course with a significant focus on understanding and manipulating common mathematical and logical techniques, with mathematical language being a significant if not necessary part thereof.