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HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, mean

by ivan_ah 13y ago

HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, meaning the concepts can be understood intuitively, e.g., the connections to physics. The trick is to covertly throw in some formal proofs to prepare the reader for the more abstract stuff. Going directly into proofs might be too much of a jump for some people, though I agree that, ultimately, this is what mathematics is about. As for the foreign notation, I wrote an appendix which explains how to read things: http://mcgillweb.alwaysdata.net/notation_appendix.pdf http://mcgillweb.alwaysdata.net/notation_appendix.pdf

wolfgke 13y ago
> HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, meaning the concepts can be understood intuitively, e.g., the connections to physics. IMHO there's little mathematics that can't be applied to the real world rather directly. I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons. Especially if you consider connections to physics as intuitive understandability - for example in string theory or quantum field theory you'll find lots of highly complicated mathematics.
- xyzzyz 13y ago
  I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons. It must have been applied mathematician, for I cannot even imagine pure mathematician uttering anything like this. I mean, seriously, I know quite a lot of modern pure mathematics, and for most of it I cannot picture even far-fetched connection with anything existing in real world, not to mention an actual application in solving some problem that wasn't created only for this application. Most of the time, for almost every field of mathematics, the way it works is that for an enormous amount of knowledge, and enormous amount of research happening and results and papers being published, only very, very small amount will actually get applied any time soon (soon as in next 200 years). For some fields, like calculus, or probability, or partial differential equations, amount of applicable stuff is larger (mostly it's just old, one or two centuries old stuff anyway), and for other fields, like say homological algebra, or algebraic topology, or descriptive set theory, the applicable stuff is almost nonexistent - I'll buy a beer to anyone who'll point me to an application of descriptive set theory to any real life problem. Almost all of the mathematics existing is purely abstract, and not applicable to real world, and I find it hard to even imagine anyone trying to argue otherwise. One can argue that what now is considered abstract and not applicable can become very useful in real life problems in future, and indeed, it happens quite often, but I think it will not be much, since we're applying to real life problems only a small fraction of 200 years old mathematics today anyway.
  - wolfgke 13y ago
    Unluckily I know little about homological algebra, algebraic topology and descriptive set theory. But I surely have already seen papers where methods from algebraic topology are applied for pattern detection or finding whether some high-dimensional data are homeomorphic. Also simplicial complexes (that are used in homological algebra and algebraic topology) can also come from triangulating high-dimensional point sets (for example from big data). I could well imagine that methods from homological algebra or algebraic topology could give us some insight into some properties of this data.
  - j2kun 13y ago
    I don't think it's fair to say some idea is not applicable because you don't know of an application. Number theory was like that before the advent of computers, and I'm a firm believer that there are thoughts that cannot be thunk until the right framework comes along to allow it. That being said, a lot of these pure subjects do have applications. For example: algebraic geometry (the crown jewel of pure mathematics, my colleagues would have be believe) has applications to tons of industrial problems in the form of solving systems of polynomial equations (see homotopy continuation). Algebraic geometry has also been applied to robot motion planning, etc. Descriptive set theory is applied in functional analysis and in ergodic theory, which in turn is applied to statistical physics. Not to mention that descriptive set theory is the pure-logic equivalent of computational complexity theory, and that there is potential to connect the two fields and resolve some big open problems (though it's doubtful that P vs NP will be resolved this way). And almost all of modern physics is based on more or less modern mathematics: tensor analysis and other flavors of linear algebra, lie theory, etc. Algebraic topology is starting to find some traction in the subfield of persistent homology, which aims to study high-dimensional data sets in the context of homological algebra. I even gave a talk earlier this year on the concrete attempts people have made to apply persistent homology to real-world problems [1]. It's still an extremely young field, but shows some promise. I'll give you that mathematics is extremely abstract, because I believe it. But to say that it's not applicable and being applied wherever possible is a bit naive. And to say that it's only 200 year old mathematics is to ignore the most applicable fields which did not exist even a hundred years ago: combinatorial optimization, mathematical computer science, and modern statistics and probability theory. [1] http://jeremykun.com/2013/04/27/persistent-homology-talk-at-uic-slides/ http://jeremykun.com/2013/04/27/persistent-homology-talk-at-...

by ivan_ah 13y ago

wolfgke 13y ago

> HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, meaning the concepts can be understood intuitively, e.g., the connections to physics. IMHO there's little mathematics that can't be applied to the real world rather directly. I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons. Especially if you consider connections to physics as intuitive understandability - for example in string theory or quantum field theory you'll find lots of highly complicated mathematics.

xyzzyz 13y ago
I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons. It must have been applied mathematician, for I cannot even imagine pure mathematician uttering anything like this. I mean, seriously, I know quite a lot of modern pure mathematics, and for most of it I cannot picture even far-fetched connection with anything existing in real world, not to mention an actual application in solving some problem that wasn't created only for this application. Most of the time, for almost every field of mathematics, the way it works is that for an enormous amount of knowledge, and enormous amount of research happening and results and papers being published, only very, very small amount will actually get applied any time soon (soon as in next 200 years). For some fields, like calculus, or probability, or partial differential equations, amount of applicable stuff is larger (mostly it's just old, one or two centuries old stuff anyway), and for other fields, like say homological algebra, or algebraic topology, or descriptive set theory, the applicable stuff is almost nonexistent - I'll buy a beer to anyone who'll point me to an application of descriptive set theory to any real life problem. Almost all of the mathematics existing is purely abstract, and not applicable to real world, and I find it hard to even imagine anyone trying to argue otherwise. One can argue that what now is considered abstract and not applicable can become very useful in real life problems in future, and indeed, it happens quite often, but I think it will not be much, since we're applying to real life problems only a small fraction of 200 years old mathematics today anyway.
- wolfgke 13y ago
  Unluckily I know little about homological algebra, algebraic topology and descriptive set theory. But I surely have already seen papers where methods from algebraic topology are applied for pattern detection or finding whether some high-dimensional data are homeomorphic. Also simplicial complexes (that are used in homological algebra and algebraic topology) can also come from triangulating high-dimensional point sets (for example from big data). I could well imagine that methods from homological algebra or algebraic topology could give us some insight into some properties of this data.
- j2kun 13y ago
  I don't think it's fair to say some idea is not applicable because you don't know of an application. Number theory was like that before the advent of computers, and I'm a firm believer that there are thoughts that cannot be thunk until the right framework comes along to allow it. That being said, a lot of these pure subjects do have applications. For example: algebraic geometry (the crown jewel of pure mathematics, my colleagues would have be believe) has applications to tons of industrial problems in the form of solving systems of polynomial equations (see homotopy continuation). Algebraic geometry has also been applied to robot motion planning, etc. Descriptive set theory is applied in functional analysis and in ergodic theory, which in turn is applied to statistical physics. Not to mention that descriptive set theory is the pure-logic equivalent of computational complexity theory, and that there is potential to connect the two fields and resolve some big open problems (though it's doubtful that P vs NP will be resolved this way). And almost all of modern physics is based on more or less modern mathematics: tensor analysis and other flavors of linear algebra, lie theory, etc. Algebraic topology is starting to find some traction in the subfield of persistent homology, which aims to study high-dimensional data sets in the context of homological algebra. I even gave a talk earlier this year on the concrete attempts people have made to apply persistent homology to real-world problems [1]. It's still an extremely young field, but shows some promise. I'll give you that mathematics is extremely abstract, because I believe it. But to say that it's not applicable and being applied wherever possible is a bit naive. And to say that it's only 200 year old mathematics is to ignore the most applicable fields which did not exist even a hundred years ago: combinatorial optimization, mathematical computer science, and modern statistics and probability theory. [1] http://jeremykun.com/2013/04/27/persistent-homology-talk-at-uic-slides/ http://jeremykun.com/2013/04/27/persistent-homology-talk-at-...