Algebra in Wonderland
nytimes.com
nytimes.com
"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?
In a sense, yes. A parabola is simply an ellipse with one of the foci at infinity.
the author of this article should stick to what she knows...english literature, not algebra.
In the affine real plane, yes. In the projective real plane, all (non-degenerate) conics have the same fundamental group.
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.
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.
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.
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.)
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-...
[2] http://plagiarist.com/poetry/1760/
[3] http://books.google.com/books?id=lZDCequ4E4kC&pg=PA170#v...
I was told an anecdote by someone in a course by Sherry Turkle: he said that one day there were many absences due to a Grateful Dead concert and that she couldn't comprehend this. Unfortunately I can't remember the details of her incomprehension, but at least as of the early 80s I didn't know anyone who knew her who thought she had a clue about what she was studying.
Perhaps I should say that there was nothing in the culture (as construed above) of those in "the humanities" that was out of bounds for us, whereas e.g. they'd never get jokes like the ones about an Abelian Grape, or better, a spherical cow ("Let us solve Schrodinger's equation for a cow. First assume...").
For every lit-crit type who'll never understand what quaternions are, I'll find you a math geek with a shelf full of Star Trek novels, who'll never understand what real literature is.
(I once tried, and failed, to convince a mailing list chock full of some of the most brilliant programmers I knew that there's more to fiction than science fiction).
The humanities and the STEM fields (I didn't know this abbreviation) are not exactly balanced in this way; there is something to what you're saying; but the something that's there is not anything like a "superset" relationship, not even close.
Once you restrict to people who both understand what quaternions are, and prefer Kafka to Zelazny, you're talking about a superset of both sides, not a STEM subgroup that also assimilated knowledge from the humanities, but remained essentially STEM. I don't think it works that way.
E.g. the equivalence you draw in your second paragraph is not relevant to my point, which is more like "while there are math geeks who understand what real literature is, there are essentially no lit-crits who will ever understand what quaternions are".
(And Star Trek novels might be a bad choice, seeing as how Patrick Stewart introduced a lot of Star Trek fans to Shakespeare. :-)
Let me take this further by discussing people who I'll call "bridges". In my subgroup of "lit-geeks" there was one seriously into philosophy. and she got some of us to appreciate some of it. Similarly, I was the one who dragged people to the first showing of the Ring Cycle (although pretty quickly that became a larger group thing as someone else e.g. dug up a fantastic Wagner scholar, it's a real shame he died as he was finished his commentary on Die Walküre.)
Despite your lack of success with the members of that mailing list, what I'm suggesting is that it's in theory possible to make that bridge and that it's not so with lit-crits and quaternions (well, not without years of math study they've essentially shown they won't or can't do).
However, being that either scenario is astonishingly rare and unlikely, I just don't see that as a good argument for asserting any kind of superiority of STEM culture over humanities culture. And when, astonishingly rarely, that does happen, I suspect the human in question doesn't see themselves as primarily STEM or primarily humanities, but just, you know, curious about stuff. Knowledge-hungry, culture-hungry, and screw the tags.
Isn't that what you're supposed to do when "analyzing" literature?
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.
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.
http://www.amazon.com/Annotated-Alice-Definitive-Lewis-Carro...