I always have had a list of cool ideas people came up with it (that I fascinated as a school kid). The list is mostly mathematical, for obvious reasons. Here are the first five -

1) The notion of continuity:

In some sense, our understanding of continuity (and hence of the real world) marked the beginning of modern mathematics and physics (well, probably not physics!). This has always been at the top of my list.

2) **Invariance of c**:

The speed of light in a vacuum is a universal constant (

*c*) which is independent of the motion of the light source. Just the assumption that there was a maximum velocity might not have been that profound. But add to that the insignificance of the light source, and you have got an Einstein in your hands.

However, to be more fair, the profundity lies in the mathematics behind special relativity, the credit for which goes also to Maxwell and Poincare.

3) Gravity:

There is nothing worthwhile that I can say about this, almost everything I might want to point out has been said already. I particularly liked the biography by James Gleick, where he tries to tell us how Newton was the least Newtonian entity in a world that was rapidly becoming Newtonian.

Not withstanding A(nir)bit’s qualms, the world we live in is still Newtonian.

However, I don’t count Newton’s invention of calculus as a fundamentally new idea. Calculus had been in making for a long time, and Newton happened to be one of the few first rate minds who simply stepped in and put it all together.

At the heart of Calculus lies *Continuity* and *Differentiability*. I have already mentioned *Continuity*, and *Differentiability* is an extension/generalisation of *Continuity*.

4) Heisenberg’s uncertainty principle:

My views of him (and his principle) are biased because of my intense admiration for the man. I rationalised my initial fascination later when I struggled to understand his principle of uncertainty in its mathematical rigour without getting lost in idle philosophical speculations.

For me, he was the coolest physicist, almost failing his practicals yet getting his degree by a brilliant thesis, he epitomised the arrogance of the theorists who look down upon the practical work (I am not saying it was his personal attitude, but one couldn’t find a better idol to imitate).

5) Godel’s incompleteness theorem:

I finally shook up the philosophical rants and looked through the theorem itself only recently. I had a misconception about the way Lobachevsky went about the fifth postulate of Euclid, which A(rule) corrected sometime back.

I forget who said this – “God exists because mathematics is consistent. The Devil exists because we can’t prove it.”

The argument goes something like this: “I refuse to prove that I exist,” says God, “for proof denies faith, and without faith I am nothing.”“But,” says Man, “the Babel fish is a dead giveaway isn’t it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don’t. QED”

“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic.

I recently tried to explain Godel incompleteness to a bio geek and what I realized was that it’s nearly impossible to lay-man-ize Godel. Even the great wiki fails on this one…

I always found the idea of mathematics itself to be superb. The fact there could be a small set of axioms, and a small set of rules, and lo behold, you have the whole of mathematics. I guess this includes the idea of proof, which in itself is another awesome idea. The simplicity with which you can convince another person of truth.

@anshul Here is an incomplete proof of Godel’s incompleteness theorem. Take a well constructed square matrix, and look at the diagonal. (This is the rightly famous and aptly named Cantor’s diagonal slash, used earlier by Cantor and later by Turing).

“God exists since mathematics is consistent, and the Devil exists since we cannot prove it” – AndrÃ© Weil (commenting on Kurt GÃ¶del’s work)

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