A brief note on mathematics

This short excerpt started off an essay which will never see the light of day. I hope to eventually have a suitable way to complete this idea, but until then, maybe somebody will find it good food for thought. (You may also substitute “mathematics” for the theoretical subject of your choice, with varying degrees of success.)

In mathematics, a proof is no more than a convincing argument of a statement’s validity. But as justification for a theorem, a proof alone is wanting. Why state one theorem over another?

The metric of real-world utility seems not to apply. After all, the very objects of study are, in some sense, artificial.

But the allure of mathematics–to me, at least–is how well those objects fit together: the degree to which mathematics unifies apparently different concepts is staggering. Perhaps nobody said it better than G.H. Hardy in A Mathematician’s Apology: “The ‘seriousness’ of a mathematical theorem lies in…the significance of the mathematical ideas it connects.”

If research mathematics is about connecting ideas, surely math education ought to convey that interconnectedness. On the contrary; until reaching senior-level courses, mathematics appears an amalgam of essentially disparate concepts.

Posted January 26th, 2010 in Academia, Math.

11 comments:

  1. Anonymous: July 29th, 2010 at 6:36 pm
  2. Jonah Greenthal:

    I don't agree with the last sentence, though.

    January 26th, 2010 at 9:06 am
  3. Jerry Vinokurov:

    I wonder what your thesis is here. I would agree that mathematics classes do a poor job of connecting seemingly disparate areas of math, though.

    January 26th, 2010 at 10:07 am
  4. Carlo Angiuli:

    I don't have a concrete thesis at the moment, except perhaps "mathematics classes should be structured differently."

    January 26th, 2010 at 10:14 am
  5. Sarah Brodsky:

    I believe math is a language for describing the world around us, giving us a means of taking the ideas and concepts within our own mind and transferring them to another's in a way which best attempts to preserve the initial content. Proofs describe, create, and discover such content. The education system definitely does not illuminate the true nature of mathematics until much later in the game; which is extremely unfortunate. I think if more "advanced" mathematics were taught early on then perhaps more non-mathematicians, in particular, would look at the subject in a different, more useful light.good morning.

    January 26th, 2010 at 11:02 am
  6. John Brown:

    I think that the standard math curriculum (up until the point where you might be planning to do this thing for a living) is structured like a progression. People often have this false idea of going "higher" in math, as if it's this really tall ladder that you ascend until you fall off. "How far did you get in math?" is a question that betrays this false premise. It's not just a matter of teaching things as disconnected. They're also pigeonholed into a strict hierarchy.

    January 26th, 2010 at 3:18 pm
  7. Brian Slattery:

    Have you read Lockhart's Lament? http://www.maa.org/devlin/LockhartsLament.pdfMight be of interest

    January 26th, 2010 at 6:34 pm
  8. Carlo Angiuli:

    Sarah: That's a good way to put it. I definitely agree that "advanced" topics should be taught early on; it's a shame that people don't really see the beauty of it all until at least halfway through an undergraduate degree.John: In some sense, definitions and theorems do naturally fall into a partial order under the "prerequisite" comparator. (Assuming you choose one canonical version of each. For instance, you can't really take analysis unless you're familiar with basic calculus.) But the connectedness of these ideas is highly understated, especially pre-college.Brian: Yes, I have. It's a bit overstated, but has some very good points.

    January 26th, 2010 at 7:36 pm
  9. Joe Pacold:

    If I end up teaching algebra in high school (or at the 100/200-level in college) I am going to work in elementary number theory somehow. It's elegant, accessible, and historically important, but somehow never shows up in the standard classes.

    January 27th, 2010 at 1:59 am
  10. Carlo Angiuli:

    Lots of elegant, accessible things slip through the cracks like that. Another one is Cantor's diagonal argument, which doesn't seem to come up nearly early enough.

    January 27th, 2010 at 2:06 am
  11. Luke Pacold:

    I think I saw it in "Images of Infinity" when I was much younger, but didn't really understand it until last semester in M380, c/o Prof. Dadok.

    January 31st, 2010 at 11:39 pm

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