Archive for the ‘Mathematics’ Category
Two interesting papers from PNAS:
Take a look!
Histories make men wise; poets witty; the mathematics subtle; natural philosophy deep; moral grave; logic and rhetoric able to contend.
He went on to recommend the study of mathematics to make the mind “less wandering”:
Nay, there is no stond or impediment in the wit but may be wrought out by fit studies; like as diseases of the body may have appropriate exercises. Bowling is good for the stone and reins; shooting for the lungs and breast; gentle walking for the stomach; riding for the head; and the like. So if a man’s wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again.
At the AMS Graduate Student blog, Luke Wolcott writes about the effect mathematics on his emails (which reminded me of the above sentences of Bacon):
Doing math has trained me to communicate concisely, tersely even. As I became more and more socialized into my math department, my email correspondences became shorter and denser. At some point, friends in other departments (e.g. Gender Studies, Communications) started to comment on the Robot Luke that sent them emails, and I started to wonder if I should intentionally increase verbosity.
The post of Luke also reminded me of Sheila Dhar and her music teacher Pran Nath; Pran Nath believed that if you have to sing Hindustani, you should stop speaking English!
Today I heard Prof. M S Raghunathan on Mathematics: art that would rather be science? (pdf): his thesis was that mathematicians develop mathematics driven by their fascination with its beauty than usefulness; however, they tend to align themselves with scientists than artists. However, I found it curious that Hardy was not mentioned (I might be wrong about this since I came a bit late to the talk; for all I know he might have started with Hardy; but there was no reference to Hardy after I entered the hall — which was, at worst, after the first five minutes).
Prof. Raghunathan’s talk set me thinking about a couple of things that I find fascinating about mathematics: (a) Why do many people find mathematics hard (because, it is easy to make mistakes and hard to cover them up — I probably heard this first in Terence Tao’s blog) (b) Why do people have tendency to use too much of mathematics, unnecessarily (ostensibly to make a piece of work more respectable!).
On the whole, it was an enjoyable talk; and, I found some of his answers to questions (Are Indians more mathematically talented? Why do not we have a good programme to identify and nurture mathematical talent) quite sharp and funny!
Today, I heard Prof. L Mahadevan on Applied mathematics as metaphor (pdf); Prof. Mahadevan talked about curling guts, fluttering flags and swarming toothbrushes! Needless to say it was all fascinating and fun! Most of the talk was based on papers which you can download from Prof. Mahadevan’s page.
A Friday morning must-read is here; and, here is a teaser:
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.
Don’t miss it; there is more where the above quote came from, and all of it is equally interesting/inspiring!
Horn’s point is that any organized attempt to look deeply at something risks being self-defeating: you can end up disappearing down all manner of silly dead ends, and understanding less than you would with a more-is-more approach.This absolutely rings true to me. For reasons which today elude me, I decided when I was doing my A-levels in England to do what they call “double maths” — essentially taking two mathematics exams (Maths and Further Maths), in the same two years you’d normally spend studying for just one. As a result, we had a highly accelerated mathematics curriculum, and there was no time to circle back and make sure the class had understood something before moving on to the next thing. It was all rather sink-or-swim.
And at any given point in time, I was sinking — along, I think, with most of the rest of my class. I was pretty fuzzy about what we’d been taught in previous weeks, and I was very unlikely to understand what the teacher was trying to say at any given time. Maths class, for me, was a combination of panic and incomprehension, combined with a desperate attempt to bluff my way through as much as I could. (Needless to say, if you’re reduced to trying to bluff, mathematics is not the best subject to choose.)
Yet somehow my classmates and I all did very well, at the end of the two years, when it came time to taking the actual exams. As I recall, nearly everybody taking double maths wound up getting an A in their Maths A-level, and most of us got an A or a B in Further Maths as well. Somehow we had managed to gain a pretty good grasp of the subject by dint of sheer velocity: the mechanism, I think, was that a desperate attempt to understand a new concept had the effect of making earlier ideas drop into place. And that the best way of mastering the Maths curriculum was not so much to study it directly, but rather to try to study the Further Maths curriculum: even getting halfway there would bring you pretty much up to speed on the stuff that went before.
Something similar, I think, happens with blogging. Bloggers tend to be foxes, rather than hedgehogs; it’s pretty clear that Athreya is an archetypal hedgehog and has a deep-seated mistrust of foxes. We skip around a lot of different things, and much of the time we don’t really understand them. But somehow the accumulated effect of all that skipping around is to make connections and develop understandings which hedgehogs often lack. What’s more, we live, as Athreya admits, in a highly complex world — one which there are serious limits to what economics can do on its own.
A good post!
A teacher of mathematics, who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz, will then become a relict like the one nowadays who does not know the difference between an open and a closed set.
Do read the entire essay.
A piece in EPW Vela Velupillai writes (Unfortunately, EPW does not allow one to give the exact URL to the piece):
In this paper I attempt to make a case for anarchy in research against the current practice of picking winners in universities at advanced levels of education and research. By considering a paradigmatic example of freedom in speculative intellectual activities leading to unintended consequences of enormous benefit to mankind, I try to substantiate a case for this. The example I consider is the way issues in the foundations of mathematics paved the way for what came to be known as the it revolution. It is a counter-factual narrative and may – hopefully, will – provide an antidote to the current orthodoxy’s regimented non-vision of “picking winners”, ex ante, without any historical substantiation.
While you are at it, a piece on Periyar’s views on science from the same issue might also be of interest.
Update: These lines from Velupillai’s piece are too tempting to be not quoted:
…I continued to learn from [Goodwin], both in the substance of economic theory …. and in a more subtle way that I do not know how to describe except as a matter of intellectual style. The unspoken language was that if a thing is worth doing it is worth doing playfully. Do not misunderstand me: ‘playful’ does not mean ‘frivolous’ or ‘unserious’. It means, rather, that one should follow a trail the way a puppy does, sniffing the ground, wagging one’s tail, and barking a lot, because it smells interesting and it would be fun to see where it goes.