Robert Tragesser's peculiar riff on the math-poetry correspondence in the Weierstrass-Kovalevskaia correspondence warrants, I feel, a correspondingly peculiar response.
Where Tragesser sees "a sign that Kovalevskaja was clueless about the nature of poetic thought," I see a sign that Tragesser is clueless about Kovalevskaia. (Please don't mistake my irony for insult; I'm only following Tragesser's lead in making bold, provocative statements. Forgive me, Robert, and kindly reply in kind!) He seems to be unaware that Kovalevskaia had keen interest in literature, if little talent for it. (He also seems to be unaware that Kovalevskaia was a woman.) As Ann Hibner Koblitz puts it in her biography of Kovalevskaia, "A Convergence of Lives" (from which I've taken my transliteration of her name),
"Sofia Kovalevskaia wrote a number of plays, novellas, poems, essays and sketches, many of which were unfinished at the time of her death in 1891. With the exception of her Memories of Childhood, however, most of her works are of marginal literary interest at the present time. Sofia's style is uneven, her characterizations are inconsistent, and her narrative point of view too naive and sentimental to appeal to modern tastes."
Of course, Koblitz's assessment can be taken as proving Tragesser's point: Kovalevskaia's literary weaknesses show she didn't really know anything about poetry. But maybe not. Creative limitations don't automatically correlate with misunderstanding, as plenty of English majors, art historians, and others will protest (perhaps too much). Conversely, it's entirely possible to create great works of art without thinking too deeply about the nature of what you're doing.
I suppose it all depends on what's meant by understanding the nature of poetic thought. Tragesser criticizes Kovalevskaia for wandering off into generalities about poetry when she decries the popular misimpression of mathematics as arid; he would rather have her trope the word "arid" into something florid and sublime. But consider who Kovalevskaia was writing to. Alfred Ross, who introduced the passage in the present discussion, didn't stipulate a source (shame on him!), but one can be found in Koblitz: Kovalevskaia was writing to "the belle lettrist A.S. Shabelskaia (Montvid) in the autumn of 1890." I'm not exactly sure what a belle lettrist is, either, but I think we can assume that Shabel- and Kovalevskaia shared common views on the value of literature. I don't think Sofia was trying to convert her correspondent over to an appreciation of mathematics; I think she was just trying to clarify her own stance. She may not elaborate her views to Tragesser's taste, but that doesn't make her clueless. Keep in mind this was a private correspondence, not a treatise. (But what do I know about nineteenth-century letter writers' views on the purpose of their prose? All I can say is that I'll be deeply embarrassed if anyone ever collects the platitudes and trivialities I've used over the years as bland seasoning in letters to friends and family.)
As for the math-poetry nexus itself, I'm inclined to agree with Tragesser (if I understand him correctly) that the quote is all too often offered up without regard for its implications. The assertion that a mathematician can't be complete without a component of poetry can't be taken seriously as a statement of fact, but only as an expression of attitude. In this sense, it falls roughly into the same category as the familiar phrase, "Real men don't eat quiche." Each claim simply serves to convey a synoptic sense of what's meant by "complete mathematician" or "real man." There are (I conjecture) a great many mathematicians with no facility for poetry or much interest in it; likewise there are undoubtedly many manly men who'll dive into a plate of quiche, especially if the wife's away and it's the only frozen dinner in the fridge.
The manly prohibition on quiche is, of course, primarily played for laughs (though who knows what effect it's had on the grocery-shopping habits of bachelors). The mathematician-poet combo is more often advanced in a serious vein (though, again, who knows what chuckles it elicits among the limerickmeisters). The Weierstrass quote is typically trotted out to establish that mathematicians have an appreciation for the finer things of life (namely poetry) and hopefully thereby jar the clueless view of math as uncreative---though perhaps we should do with "uncreative" what Tragesser suggests for "arid," and turn it with a tropical trick (or trompeur trope).
Finally, when it comes to context, I'm of the clan that doesn't care. More precisely, I often don't care. I'm always looking for good, juicy quotes that succinctly summarize a point, even if they're ripped entirely out of context. One of my favorites is from a poem by Rita Dove, a poet-laureate of the United States:
He is weary of analysis, the small predictable truths.
Dove is, of course, not referring to mathematical analysis (though others of her poems do deal with mathematical themes), but the lines are nonetheless delightful when read as unintended. Where context does become important is if your use of the quote depends on the original, intended meaning---if, for example, you're engaged in historical study of Weierstrass's view of mathematics, or if you're appealing to the source of a quote as an authority to support your own interpretation. (It would be a blunder to cite Dove in a debate on the relative merits of, say calculus and abstract algebra.)
The Weierstrass quote certainly gains some of its force from the (mathematical) stature of its author, as do the words of Kovalevskaia, but it also stands on its own, just as Weierstrass's mathematics does. However, the reach of his authority is, I daresay, rather limited. It's comforting to mathematicians to hear the math-poetry comparison from one of their legends, but it's not clear Weierstrass's words carry much weight outside their own sphere. How many poets or poetasters know or care who Weierstrass was or what he did? The effort to establish his credentials in order to impress people with the wisdom of his words may be too taxing and ultimately counterproductive (especially if you're dealing with people still smarting from the epsilon-delta rigors of an old calculus professor).
It can also be dangerous, as we learn from Hans Lausch, whose posting shows the Weierstrass quote was written cheek by jowl with some anti-semitic remarks. Mathematicians arguing for a humanistic view of mathematics can wind up hoist by their own petard if they're not careful. (Of course the humanities have their own mine fields to dance through. Vide Ezra Pound.) That's maybe one more reason for knowing the context of a quote.
To wrap things up, it's worth quoting the next paragraph from Kovalevskaia's letter to her belle-lettrist buddy, after her allusion to Weierstrass. The passage appears in Koblitz (pg. 231):
"As for me, I have never been able to decide toward which I have the greater inclination: mathematics or literature. As soon as my mind becomes tired of purely abstract speculations, it immediately begins to turn toward observation of life, toward the urge to retell what I see. And conversely, at other times everything in life suddenly seems unimportant and uninteresting, and only the eternal, immutable laws of science attract me. It is very possible that I would have done more in either one of these fields if I had devoted myself exclusively to it, but at the same time I simply cannot abandon either one of them."