- #1

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1 - Maxwells electromagnetics. In his original work there were 2 parts. Part 1 was a touchy feel thought experiment where he imagined an ether existed and proceeded to describe the workings of this ether by means of cogs and wheels. An intuitive approach. In part 2 he put skin and bones on it with a mathematical description of what an imaginary ether would be. Finally after another 30 years Heavside re wrote Maxwells equations in the modern form we have today. My gripe is that we now learn Electromagnetic without the intuition that Maxwell used. Its like someone building a skyscraper then taking down all the scaffolding and being left left with this beatiful edifice but no comprehension of how it was built

2 - Euler Lagrange equation in calculus of variations. Eulers derivation was a localised derivation which was intuitively simple and gives a great and easy intuitive way of looking at it. Lagrange derivation and the one accepted and taught now is less intuitive but more analytic and precise. But I bet my house that Lagrange used the same intuitive approach as a first attempt at a derivation before he did his more analytic derivation. Again creating academic skyscrapers and hiding the dirty scaffolding.

3 - Engineering students learning Laplace transforms. I was taught formulas and processes and when I looked at it it appeared that an astounding genius must have thought it up, 'approximate a function using a family of decreasing and increasing rotating phasers , spirals. But it turns out inow order to understand it properly you need to think of Fourier and laplace transforms in the context of vector spaces and metrics within these spaces , linear algebra. Then all becomes apparent. Again a case of building skyscrapers then throwing away the inconvenient and ghastly looking scaffolding.

So in summary why can we not learn from intuition when of value 1,2 and real maths when of value 3.