9 ms·

Algebra in Wonderland

28 points by robg 17y ago

nazgulnarsil 17y ago
19th century writing in general makes a mockery of today's level of education. I believe this is due, in part, to the more limited availability of print at the time. most of the people printing things in any sort of quantity that ensured it would survive to our times was generally pretty worldly. not that i'm arguing that such times were better. the diaspora of artistic expression fueled by cheaper means has been a boon for humanity.
- Groxx 17y ago
  I'd argue that the limited availability meant that only the extreme few who could afford to print did so, and when they did they kept things to what their elitist groups would find interesting. Meanings hidden within meanings not because they were any more intelligent than others, simply because they needed to do so to fit in with the elitist habits / aesthetics. You see the same thing in art prior to the modern art shift. Art was so elitist that it wasn't focused on being artistic any more, it was instead a conglomeration of meanings that could be examined and critiqued by the social elite. All social elite groups do this, it raises them above others who don't know the language and gives them a feeling of superiority.
anatoly 17y ago
I read this argument before, in a paper by the author I think. I thought it was nonsense then and this article doesn't change my opinion about it. The purported explanations are really unconvincing, the analogies very strained.
- hga 17y ago
  Agreed, the author just doesn't supply any strong evidence for her thesis. Would it be impolite to point out that she's an "English Major"? Then again, her A Levels might have given her a far better grounding in math than is the norm in the US.
  - tsally 17y ago
    Would it be impolite to point out that she's an "English Major"? You already know the answer to this question. Also note that she's not an "English Major" in the sense you imply, she's getting a PhD in English Lit from Oxford. If indeed her arguments are incorrect, you should be able to frame a refutation on ideas alone instead of resorting to an ad hominem argument. Ironically, this is a basic skill even "English Majors" possess.
    - hga 17y ago
      Impolite does not mean irrelevant. What I'm trying to point out is that there is no sign in what little we know of her biography or in this article that she knows math at this level, the modern math she thinks Carroll is skewering. Let's put it this way: how certain is anyone here that she understands how quaternions are non-commutative and how that was such a leap from all previous algebra? Such a leap that it took Hamilton years before he made it, such that it is commemorated on the bridge where he first wrote (carved) a solution? Something I impolitely point out is that all too often I've seen people from "the humanities" make incorrect claims about "us" without any understanding of our work or our culture, which they don't realize is a superset of their's. (And I'm annoyed about all the attention they get in doing this, but I suppose that's obvious.)
      - tsally 17y ago
        I'll join you in being annoying that she got published in the Times and I'll agree there are blatant inaccuracies in the article, but there are more effective arguments than insulting the author or the humanities at large. There's plenty to disagree with in the article without resorting to that. As for mathematics being a superset of the humanities, that is just patently false. Here are a few counter examples [1,2,3]. I always find it interesting when fellow engineers/scientists argue for the superiority of their discipline when compared to the humanities. Personally I find the hard sciences awe inspiring enough to stand by themselves. [1] http://upload.wikimedia.org/wikipedia/commons/e/e1/Brueghel-tower-of-babel.jpg http://upload.wikimedia.org/wikipedia/commons/e/e1/Brueghel-... [2] http://plagiarist.com/poetry/1760/ http://plagiarist.com/poetry/1760/ [3] http://books.google.com/books?id=lZDCequ4E4kC&pg=PA170#v=onepage&q=&f=false http://books.google.com/books?id=lZDCequ4E4kC&pg=PA170#v...
- jey 17y ago
  > The purported explanations are really unconvincing, the analogies very strained. Isn't that what you're supposed to do when "analyzing" literature?
  - anatoly 17y ago
    No. If you read the genuinely good critics, that's not so at all. They open your eyes to great things you haven't noticed, but have actually been there.
georgecmu 17y ago
Her exposition of math concepts is ridiculous. "De Morgan had proposed a more modern approach to algebra, which held that any procedure was valid as long as it followed an internal logic. This allowed for results like the square root of a negative number" Imaginary numbers were introduced by Descartes in 17th century and were widely accepted by the middle of 18th (think Euler: e^(i \pi) + 1 = 0). "This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties — a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin)." Huh? You can bend or stretch a circle into a parabola?
- cperciva 17y ago
  Huh? You can bend or stretch a circle into a parabola? In a sense, yes. A parabola is simply an ellipse with one of the foci at infinity.
  - dmvaldman 17y ago
    you cannot stretch a parabola into a circle because a circle is a closed loop, while a parabola is not. any stretching would necessarily have to tear the circle. a topologist would say they have different fundamental groups. the author of this article should stick to what she knows...english literature, not algebra.
    - cperciva 17y ago
      a topologist would say they have different fundamental groups In the affine real plane, yes. In the projective real plane, all (non-degenerate) conics have the same fundamental group.
      - gambling8nt 17y ago
        To elaborate on cperciva's (correct) point: On a projective plane a parabola contains a single point at infinity, connected (both figuratively and topologically) to the two open ends. A hyperbola contains two distinct points at infinity each connected to one end of each of the two usual components.
eitally 17y ago
I think everyone who posted here would be well served reading Martin Gardner's The Annotated Alice. Gardner, who was the puzzle editor for Scientific American for decades, is also a Lewis Carroll scholar, and his annotated versions of Alice in Wonderland and Through the Looking Glass are tremendous. http://www.amazon.com/Annotated-Alice-Definitive-Lewis-Carroll/dp/0393048470 http://www.amazon.com/Annotated-Alice-Definitive-Lewis-Carro...

28 points by robg 17y ago

19th century writing in general makes a mockery of today's level of education. I believe this is due, in part, to the more limited availability of print at the time. most of the people printing things in any sort of quantity that ensured it would survive to our times was generally pretty worldly. not that i'm arguing that such times were better. the diaspora of artistic expression fueled by cheaper means has been a boon for humanity.

Groxx 17y ago
I'd argue that the limited availability meant that only the extreme few who could afford to print did so, and when they did they kept things to what their elitist groups would find interesting. Meanings hidden within meanings not because they were any more intelligent than others, simply because they needed to do so to fit in with the elitist habits / aesthetics. You see the same thing in art prior to the modern art shift. Art was so elitist that it wasn't focused on being artistic any more, it was instead a conglomeration of meanings that could be examined and critiqued by the social elite. All social elite groups do this, it raises them above others who don't know the language and gives them a feeling of superiority.

anatoly 17y ago

I read this argument before, in a paper by the author I think. I thought it was nonsense then and this article doesn't change my opinion about it. The purported explanations are really unconvincing, the analogies very strained.

hga 17y ago
Agreed, the author just doesn't supply any strong evidence for her thesis. Would it be impolite to point out that she's an "English Major"? Then again, her A Levels might have given her a far better grounding in math than is the norm in the US.
- tsally 17y ago
  Would it be impolite to point out that she's an "English Major"? You already know the answer to this question. Also note that she's not an "English Major" in the sense you imply, she's getting a PhD in English Lit from Oxford. If indeed her arguments are incorrect, you should be able to frame a refutation on ideas alone instead of resorting to an ad hominem argument. Ironically, this is a basic skill even "English Majors" possess.
  - hga 17y ago
    Impolite does not mean irrelevant. What I'm trying to point out is that there is no sign in what little we know of her biography or in this article that she knows math at this level, the modern math she thinks Carroll is skewering. Let's put it this way: how certain is anyone here that she understands how quaternions are non-commutative and how that was such a leap from all previous algebra? Such a leap that it took Hamilton years before he made it, such that it is commemorated on the bridge where he first wrote (carved) a solution? Something I impolitely point out is that all too often I've seen people from "the humanities" make incorrect claims about "us" without any understanding of our work or our culture, which they don't realize is a superset of their's. (And I'm annoyed about all the attention they get in doing this, but I suppose that's obvious.)
    - tsally 17y ago
      I'll join you in being annoying that she got published in the Times and I'll agree there are blatant inaccuracies in the article, but there are more effective arguments than insulting the author or the humanities at large. There's plenty to disagree with in the article without resorting to that. As for mathematics being a superset of the humanities, that is just patently false. Here are a few counter examples [1,2,3]. I always find it interesting when fellow engineers/scientists argue for the superiority of their discipline when compared to the humanities. Personally I find the hard sciences awe inspiring enough to stand by themselves. [1] http://upload.wikimedia.org/wikipedia/commons/e/e1/Brueghel-tower-of-babel.jpg http://upload.wikimedia.org/wikipedia/commons/e/e1/Brueghel-... [2] http://plagiarist.com/poetry/1760/ http://plagiarist.com/poetry/1760/ [3] http://books.google.com/books?id=lZDCequ4E4kC&pg=PA170#v=onepage&q=&f=false http://books.google.com/books?id=lZDCequ4E4kC&pg=PA170#v...
jey 17y ago
> The purported explanations are really unconvincing, the analogies very strained. Isn't that what you're supposed to do when "analyzing" literature?
- anatoly 17y ago
  No. If you read the genuinely good critics, that's not so at all. They open your eyes to great things you haven't noticed, but have actually been there.

georgecmu 17y ago

Her exposition of math concepts is ridiculous. "De Morgan had proposed a more modern approach to algebra, which held that any procedure was valid as long as it followed an internal logic. This allowed for results like the square root of a negative number" Imaginary numbers were introduced by Descartes in 17th century and were widely accepted by the middle of 18th (think Euler: e^(i \pi) + 1 = 0). "This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties — a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin)." Huh? You can bend or stretch a circle into a parabola?

cperciva 17y ago
Huh? You can bend or stretch a circle into a parabola? In a sense, yes. A parabola is simply an ellipse with one of the foci at infinity.
- dmvaldman 17y ago
  you cannot stretch a parabola into a circle because a circle is a closed loop, while a parabola is not. any stretching would necessarily have to tear the circle. a topologist would say they have different fundamental groups. the author of this article should stick to what she knows...english literature, not algebra.
  - cperciva 17y ago
    a topologist would say they have different fundamental groups In the affine real plane, yes. In the projective real plane, all (non-degenerate) conics have the same fundamental group.
    - gambling8nt 17y ago
      To elaborate on cperciva's (correct) point: On a projective plane a parabola contains a single point at infinity, connected (both figuratively and topologically) to the two open ends. A hyperbola contains two distinct points at infinity each connected to one end of each of the two usual components.

eitally 17y ago

I think everyone who posted here would be well served reading Martin Gardner's The Annotated Alice. Gardner, who was the puzzle editor for Scientific American for decades, is also a Lewis Carroll scholar, and his annotated versions of Alice in Wonderland and Through the Looking Glass are tremendous. http://www.amazon.com/Annotated-Alice-Definitive-Lewis-Carroll/dp/0393048470 http://www.amazon.com/Annotated-Alice-Definitive-Lewis-Carro...