July 2nd, 2014

This is my exercise in shoveling out the blogpile…


  1. Jeff Hess says:

    As I grow older, I find increasing comfort in maths. For me there is a beauty in equations that I have often found in an elegant bit of code (an elegance that I think has been lost in a world of computers with memories and processing speeds measured in insane quantities.

    I think that is why this comment in the What is it like to understand advanced mathematics? piece spoke to me:

    Mathematicians will often spend days figuring out why a result follows easily from some very deep and general pattern that is already well-understood, rather than from a string of calculations. Indeed, you tend to choose problems motivated by how likely it is that there will be some “clean” insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities.

    In A Mathematician’s Apology, the most poetic book I know on what it is “like” to be a mathematician], G.H. Hardy wrote:

    “In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

    Others that made me think include:

    No. 1—Mathematicians don’t really care about “the answer” to any particular question; even the most sought-after theorems, like Fermat’s Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it.

    Two professions that I’ve considered over the years were becoming a Lawyer or a Talmudic Scholar because both have a certain cerebral cleanness about them. I think, if I had had the proper education in the beginning, I might have considered Mathematics as well.

    This topic, and thread, is a rabbit hole if I ever saw one.


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