The us-eless geeks
Originally published June 2005
This reminds me of the Asimov book on numbers. That we classify or attribute a certain nature to numbers in mathematics and then assign words to describe them, has little to do with what those words mean out of the mathematical context.
He narrates an interesting anecdote. A mathematics teacher was teaching an arts class once. One of the more cocky students, complained that the class was fairly useless, as mathematics in itself was flawed; that it dealt with “unreal” concepts like “imaginary numbers”. The teacher smiled. (The last sentence, is surprisingly frequent in such anecdotes!) He then asked the student to come forth and give him half a piece of chalk. The student promptly broke a fresh piece of chalk into two and displayed proudly the result of his efforts. The teacher then told him that what he held in his hand was a piece of chalk, not half a piece of chalk. The student realised that while imaginary numbers are explicitly named as such, they are no more or no less imaginary than fractions and no more or less real than “natural” numbers.
Of course as with the case of the argument(s) between Golu and Viral, I imagine that this has little to do with what kind of a person you are and more with what you look at mathematics as.
If you look at math as a means of measuring reality, then you are bound to be the ones who check the units on their answers and bound them in a box to emphasize the truth and accuracy of their efforts. When Pythagoras came up with his theorem, he hardly had the tools of abstraction that we consider so elementary. In fact if I am not mistaken, the proofs he gave hardly used numbers as concepts of measurement. He used tiles to prove his theorem. You can’t specify a triangle as a set of numbers, you need units. Until then numbers are just that. Add the dose of Euclidean Geometric axioms. Add units and references and then you get a reference to real (or as Viral very weirdly puts it — “tangible”) triangles. They are not in themselves the representatives of the triangle and have little to do with the essence of Pythagoras’ work.
The other kind is of course the kind that absorbs mathematics in abstraction. It is true that Pythagorean triplets consist of natural numbers; not because that has anything to do with the nature of Pythgaoras theorem, or that triangles can have only positive lengths for their edges; but because they are defined thus. Pascal’s triangle is what it is because it is defined thus. And there is reason why it is not extended to integers (or more correctly to any set larger that the natural number set — including whole numbers), because it leads to trivial solutions of the kind that Golu pointed out. In fact he doesn’t even need negative integers. A more interesting question, that if the Pythagorean triplets are an ordered set, does a given natural number, in a given position, generate a unique triplet, if at all it does generate a triplet in that position.
Enough.
I can’t imagine that I am actually posting a reply to someone’s else posting, and it got so out of hand that it had to be an entry in my blog.
And then I wonder how all these bums who had no time, when they were around here, to either run the entertaining and sprawling bogs that they now maintain; or to discuss/argue academic/geek issues, and how suddenly the US of A seems to grant them the will and flimsy time to engage in all of that.
But most importantly, I need to figure out the procedure of Trademarking things, to prevent further abuse of my identity or appearance!