5 ms·

Reforming the Mathematical Language of Physics (2002) [pdf]

41 points by MaysonL 11y ago

jacobolus 11y ago
Previous submissions: https://news.ycombinator.com/item?id=3284160 https://news.ycombinator.com/item?id=3284160 https://news.ycombinator.com/item?id=9202543 https://news.ycombinator.com/item?id=9202543 Related conversation currently on HN: https://news.ycombinator.com/item?id=9746051 https://news.ycombinator.com/item?id=9746051
MichaelCrawford 11y ago
Kuro5hin's trane is a classical linguist (ie. Ancient Greek) who knows enough about physics and math to shoot his own foot off. For example he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty. He's a real smart guy as well as one of my very best friends but it's hard to debate someone who got honors in his Linguistics Masters degree.
- jacobolus 11y ago
  > he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty It’s very oversimplified to say that “mathematics does not permit division by zero”. Mathematics (at least in the modern sense, descending from Euclid) is just the logical extension of whatever set of axioms you want to set up. It’s entirely possible to operate on a concept of numbers which allows division by zero. For example using the extended complex numbers (the complex plane plus a point at infinity). See https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere Anyhow, what does your friend have to do with Hestenes’s paper?
  - DougMerritt 11y ago
    > what does your friend have to do with Hestenes’s paper? The comment is clearly totally irrelevant, so I did some googling, and it appears that "trane" is a bot created on Kuro5hin by MichaelCrawford, and that references to "Kuro5hin's trane" are some sort of inside joke. Which in turn makes it look like the non sequitur above was posted by that bot, even if other posts from that account were by human. In any case it's just adding noise to the conversation.
    - MichaelCrawford 11y ago
      His website is http://www.subbot.org/ http://www.subbot.org/ His real name is Robert Mitchell. He has a BA in Classical Greek as well as a Master's of some sort from U of Chicago. He worked for several years as a Java coder but for reasons I don't fully understand, decided that paying work is somehow evil and so flatly refuses even to apply for a job. In part that's because he is very shy and so cannot tolerate office politics, and in part - again for reasons I don't understand - he has a passionate hatred for women and so cannot tolerate being in their presence. He's heavily into jazz and swing music; he likes to call himself "trane" because that was John Coltrane's nickname.
jessriedel 11y ago
> I have noted that perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus! It's hard to tell what the author is proposing, but from the surrounding pages it sure sounds like he's talking about teaching vectors as geometric objects with various manipulation rules and avoiding coordinate descriptions. This is actually the sort of angle that may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system. But for physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction. In my teaching experience as a graduate student, students adopt the geometric picture of a vector as soon as they are able to, because it's vastly easier and more pleasant to work with. (That's why the abstraction was originally developed.) The students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.
- jblow 11y ago
  I disagree completely. I am not a physicist but I make video games which has the same kind of constant grounding in applicability. (We are always dealing with running physical systems, it's just that they are simulated.) The week I learned to treat vectors as abstract objects, rather than arrays of coordinates, I experienced a drastic phase shift in my ability to program geometric operations effectively and clearly. The coordinates are still there, of course, but you have a lot more power over them. The book "Linear Algebra Done Right" is all about this, and I absolutely recommend reading it if you haven't.
  - 11y ago
sullyj3 11y ago
Not being a mathematician, I have no idea whether Geometric Algebra is in fact superior to the systems in general use currently, but it sounds exciting.

41 points by MaysonL 11y ago

Previous submissions: https://news.ycombinator.com/item?id=3284160 https://news.ycombinator.com/item?id=3284160 https://news.ycombinator.com/item?id=9202543 https://news.ycombinator.com/item?id=9202543 Related conversation currently on HN: https://news.ycombinator.com/item?id=9746051 https://news.ycombinator.com/item?id=9746051

MichaelCrawford 11y ago

Kuro5hin's trane is a classical linguist (ie. Ancient Greek) who knows enough about physics and math to shoot his own foot off. For example he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty. He's a real smart guy as well as one of my very best friends but it's hard to debate someone who got honors in his Linguistics Masters degree.

jacobolus 11y ago
> he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty It’s very oversimplified to say that “mathematics does not permit division by zero”. Mathematics (at least in the modern sense, descending from Euclid) is just the logical extension of whatever set of axioms you want to set up. It’s entirely possible to operate on a concept of numbers which allows division by zero. For example using the extended complex numbers (the complex plane plus a point at infinity). See https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere Anyhow, what does your friend have to do with Hestenes’s paper?
- DougMerritt 11y ago
  > what does your friend have to do with Hestenes’s paper? The comment is clearly totally irrelevant, so I did some googling, and it appears that "trane" is a bot created on Kuro5hin by MichaelCrawford, and that references to "Kuro5hin's trane" are some sort of inside joke. Which in turn makes it look like the non sequitur above was posted by that bot, even if other posts from that account were by human. In any case it's just adding noise to the conversation.
  - MichaelCrawford 11y ago
    His website is http://www.subbot.org/ http://www.subbot.org/ His real name is Robert Mitchell. He has a BA in Classical Greek as well as a Master's of some sort from U of Chicago. He worked for several years as a Java coder but for reasons I don't fully understand, decided that paying work is somehow evil and so flatly refuses even to apply for a job. In part that's because he is very shy and so cannot tolerate office politics, and in part - again for reasons I don't understand - he has a passionate hatred for women and so cannot tolerate being in their presence. He's heavily into jazz and swing music; he likes to call himself "trane" because that was John Coltrane's nickname.

jessriedel 11y ago

> I have noted that perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus! It's hard to tell what the author is proposing, but from the surrounding pages it sure sounds like he's talking about teaching vectors as geometric objects with various manipulation rules and avoiding coordinate descriptions. This is actually the sort of angle that may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system. But for physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction. In my teaching experience as a graduate student, students adopt the geometric picture of a vector as soon as they are able to, because it's vastly easier and more pleasant to work with. (That's why the abstraction was originally developed.) The students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.

jblow 11y ago
I disagree completely. I am not a physicist but I make video games which has the same kind of constant grounding in applicability. (We are always dealing with running physical systems, it's just that they are simulated.) The week I learned to treat vectors as abstract objects, rather than arrays of coordinates, I experienced a drastic phase shift in my ability to program geometric operations effectively and clearly. The coordinates are still there, of course, but you have a lot more power over them. The book "Linear Algebra Done Right" is all about this, and I absolutely recommend reading it if you haven't.
- 11y ago

sullyj3 11y ago

Not being a mathematician, I have no idea whether Geometric Algebra is in fact superior to the systems in general use currently, but it sounds exciting.