Jason (jcreed) wrote,

Sometimes when I'm looking for random mathematical things, I run across Mark Jason-Dominus's blog, since he posts a lot about, well, random mathematical things. His name's familiar to me from my past life as a perl hacker. I can't recall exactly what he did to be so famous, but he was up there in the pantheon of Important Well-Known People in the community. I have half a mind to send him an email about this entry explaining that yes, "contravariant" and "covariant" do have everything to do with category theory, but I'm afraid I would then be compelled to argue against his notion that "subclass" and "superclass" is bad terminology because really "up" ought to mean "more specialized" and "down" ought to mean "less specialized". My opinion is of course biased by actually talking about these matters in terms of subtypes, but my opinion is not that there is one best "orientation" in terms of up-down metaphors, since you can just as easily pull the usual Yoneda/Galois-ish move back and forth all day 'till your face turns blue.

And ugh, coincidentally my annoyance at trying to understand what is really going on with metaphorical associations like this is not being helped by the fact that I've been reading Johnson and Lakoff's "Philosophy in the Flesh" lately. You'd think it would help, since it's exactly their subject matter, but god help me if I can go two pages without disagreeing with them in twelve different ways.

Anyway! The actual random mathematical thing I found in this case was a nice little discussion about the "dot notation" for precedence that one sometimes comes across in very old (i.e. early 20th-century) logic texts, which Peano apparently invented (along with an indimidating lot of other notations, it turns out) and was popularized by Whitehead, Russell, Quine et al., although it's largely died out by now. More info can be found here. It's certainly almost impossible for me to read, but it seems like it might have some advantages relative to matching parentheses in that it's sometimes easier to find the "matching" element when you have a distinct shape to search for. Maybe having parentheses with tiny numbers hovering above or below them would be a pareto improvement over both...
Tags: math, web
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