Saturday, November 24, 2007

A GKC Debate about Math

Back in October of 2006, our Dr. Thursday elicited some interesting discussion when he brought up Math, GKC and an Ignatian Asylum, after which a spirited debate ensued.

"Wild Goose", citing the One-Should-Always-Have-A-Healthy-
Wikipedia, stated:
It is not quite true that “When somebody [Newton] discovered the Differential Calculus there was only one Differential Calculus he could discover.”

Leibniz also discovered Differential Calculus, in a different form, arguably, a more durable and suitable calculus:

“The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. “
To which Dr. Thursday responded:
I should not have added [Newton] to GKC's words. It was in the 1660s (or so) that both Newton and Leibniz "discovered" calculus; yes, almost simultaneously, though Newton seems to have priority.

Alas, "a quarrel arose between the followers of Newton and the followers of Leibniz, and unhappily it grew into a quarrel between the great men themselves..." [The World of Mathematics 143, 286 et seq; Purcell, Calculus 156, 278]
And, if you're still following this (for full arguments, please see the comments section of the above referenced posting), "Wild Goose" continued:
You have brought up an excellent topic. The way I see it, the main point of your post was that math and science have their own fixed rules, while poetry has its own, due to the free will of the author or the creator. (“But when Shakespeare killed Romeo he might have married him to Juliet's old nurse if he had felt inclined.”) You are saying that there is only one calculus, while there may be a virtually infinite number of plays or plots along the lines of Romeo and Juliet, limited only by the author’s imagination. But I think that would be like comparing apples and oranges.
After which "DavyMax", a new commenter as far as one can tell that sort of thing, just today responded:
To simplify things a bit. Godel proved the essential intuitive nature of mathematics. But Wild Goose, you seem to be implying that intuition and imagination are one in the same. They are in fact quite different. Without seeing a proof I may intuitively think that there are infinitely many primes or that the Reimann hypothesis is true or the Axiom of Choice. However, this is because I would think, for instance, as I do, that the Axiom of Choice is in fact the truth. Clearly, this is quite different from writing a different ending of Romeo and Juliet (or preferably some other story I rather like the ending of Romeo and Juliet). Mathematicians do not intuitively think something because that's the way it sounds nice or because they think it's "cool." It's because they think it is TRUE. That is the ultimate goal. They may and often do choose to explore an idea because it is pretty or beautiful, but not because of those goals in mind but because they know from experience that the truth most often is pretty and beautiful.

This reminds me also of the line from V for Vendetta that says something of Artists telling the truth with lies. This is true in the sense that they are trying to express some inner truth through any means neccesary and we all understand what they're doing. Mathematicians are seeking rather than expressing truth when using intuition. Whereas, artists are expressing a truth already experienced when using imagination.
I thought I'd bring the whole thing to the fore because with a post that old, it's hard for people to jump back into the conversation. But I wanted to thank DavyMax for finding us and joining in the conversation by attempting to revisited this topic, if others are interested. Anyone want to respond to DavyMax? I thought he brought up an excellent point about Truth, and the difference between intuition and imagination.


  1. Only one calculus? Ah, it depends on what one means by "the calculus," doesn't it?

    I can very easily imagine a calculus that offers all the same mechanics and functions of Newton's (or Leibniz), but uses different symbols. Instead of integration being represented by a long, skinny "S" like symbol, perhaps we use a crescent moon shape, or a shape like a star.

    Does this then constitute a different calculus? Probably not. It is merely a different symbol for the same old calculus. For it to be different, it should do something different.

    Similarly, if we rewrite Romeo and Juliet to be the story of Roman and Julie, is it a different story (a rose by another name, perhaps)? Or have we only changed the externals, the symbolic elements which point to a deeper truth? What if it takes place on the moon? Is it still Romeo and Juliet?

    To build on DavyMax's comment, profound mathematics *and* profound plays are not admired because we think they sound nice or cool. We admire Newton and Shakespere because they expressed things that are true. Perhaps we could even change some of the ending elements to R&J, without it becoming a "different" play.

  2. Dr. Thursday,

    The Wild Goose flew away after being chased out, and, as any nature lover knows, animals , and especially geese, are very careful and sensitive. Some people, especially hunters, even call them smart.

    But here are a few counterpoints:

    1) If you have been following the Wiki vs. Britannica controvery, it turns out Wiki is just as reliable, or more reliable. Chesterton would have loved Wiki - a true global democratic effort of the common man to find truth, as opposed to the snobbish effort of the ellites or "experts." I think Chesterton would agree that one should be just as careful about modern science books and science magazines. I could have just as well provided you with such "hard" references, but, for a simple reason, it was my decision to use wiki whenever I could, because it is reasonably accurate and immediately available to the blog readers.

    2) I still maintain that comparing mathematics to poetry is like comparing apples and oranges. (Curiously, I had a somewhat related debate with a friend of mine, a university professor of mathematics, about Newton's alchemistic approach to science and mathematics - I hope you don't want to go there in defense of Newton. He lost the debate.)

    3) DavyMax did not simplify things a bit by bringing in Godel and his Incompleteness theorem and his Axion of Choice. A fascinating subject, or subjects, but, I am afraid, somewhat beyond the average reader of this blog, or this blog's capability to explain the problem in a few words.

    4) Intuition in mathematics - another fascinating subject.

    One could write a book, or books on any of these topics.

    Wild Goose

  3. After thinking this over for a couple years....

    First off, in response to (2), I think you hit the nail on the head. Indeed, precisely on the head. Comparing poetry and math is like comparing apples and orange. No one will deny that they are different things, however, nonetheless, no one will deny that they are both fruits. Mathematics and Poetry are the fruits of man's labor to seek truth. Very different fruits, you are certainly right, but what Dr. Thursday was pointing out is that although different they are similar in an important way.

    As to (3), I wasn't responding to make the post easier for the average reader, the average reader didn't respond to the post in disagreement; whereas, you did. So, you see, I was responding to clarify something, and in so doing used particular examples that I thought you would understand and would be easier for you. I did not intend for these examples to be elucidating for everyone.

    Furthermore, I think that in less space than an average Dr. Thursday post, I can explain to the average reader how Godel's Incompleteness theorem and the Axiom of Choice are relevant to the conversation. They need not know their precise details or make-up in order to understand their effects. Surely, if as Aquinas argues, we can talk about God even though we can never know the essence of God, then we can talk about the ideas of mere mortals without fully understanding them.


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