"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?
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.
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.
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.
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.
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.
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.)
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.
I think I was unclear WRT to your last point. I'm saying that "we", the STEM types, live in a popular and to a certain degree also a higher culture that is a superset of the culture that those in "the humanities" live in. It's a very broad generalization, perhaps colored by my being a part of a sub-group that would go to the ballet, Shakespeare productions, Ring Cycles (and each of us had at least one poet and painter we really liked, etc. etc.) ... but then again those who didn't like attending "operas [that] last three or four days" nonetheless had at least a minute appreciation, e.g. we weren't denigrated for our particular interests.
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.
I agree that "superset" is too crude a term for what I'm trying to say, but as you acknowledge, there is something to what I'm saying.
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).
I agree that it's much more likely for a math geek to develop serious knowledge of lit-crit than it is for a lit-crit type to develop serious knowledge of math. That's the something that there is to what you're saying.
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.
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.
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.
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.
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.
"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?