Sandeep's Blog

March 22, 2010

A reflection on my last post

Filed under: Uncategorized — Sandeep Thilakan @ 7:18 pm

After a little thought on my last post, I felt that I had gone way too far in time with my experiences. Infact, the very practice of learning a concept and sometimes even an entire subject without knowledge of its foundations has been living with us since childhood. Now, these examples would be the most natural justifications for my statements in the last post.

The following lines would be the storyline of each one of us from kindergarten to graduation unless you belong to the hierarchy of the great mathematician Gauss who happened to have discovered the formula for the sum of numbers in an arithmetic progression right when he was in first grade.

Primary School Arithmetic

1,2,3… are called numbers. 1+1=2, 2+1=3 and so on.” Hey this is natural and straight forward. Not even the slightest mention that this entire system of numbers and operations like addition and multiplication are built on a beautiful set of axioms by Peano. But even a hint about those axioms by the teacher at that time would have lead to chaos. Just think about the millions of mathematical operations you have done ever since without giving a thought about its foundations.

High school Geometry

Lines, angles, triangles, circles, congruence, and what not. I always admired these ojects and enjoyed manipulating with them. But the realisation that the system in which I am manipulating is one form of Geometry called Euclidean Geometry and that it can be developed from a set of five seemingly trivial axioms occured quite recently. And further, there are something called non Euclidean geometries where all our high school theorems break down, and that the world we live in is in fact also one among them. Just imagine your high school teacher introducing geometry with Euclid’s axioms and further generalizing geometry to higher dimensions and saying that its dynamic i.e your geometry may be different from my geometry! I am sure these concepts would simply fly past innocent high schoolers.

Electricity and Magnetism

Ya, this is exactly how it is taught in secondary school. Even I had a course in my sophomore year titled Electric and Magnetic fields. We read Coulumb’s law for electrostatics, Biot-Savart’s law for magnetism, Faraday’s law for electromagnetic induction, and so on . Again in this case, we have a deeper unity between the two phenomena and its just the reference frame of the observer that decides whether the observed force feels like electric or magnetic. Here in fact, we later come across, in my view, the most elegant unification in physics: the Maxwell’s equations.

I have no intentions of challenging the curriculum designers for hiding the most interesting and deep results from us in school. Somehow I too feel that our brains accept the current plan quite well. And being an engineer, I can always argue that in the real world, even Peano would never think about his axioms when asked ” How much is 1+1?”

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