### Ere Riemann

One of the best books I ever picked up second hand [1] begins with the very old problem of the vibrating string, as studied by Euler and contemporaries. In the author's historical opinion, the young Lagrange took on board both (i) Euler's preference for introducing the concept of non-differentiable functions and (ii) Euler's confusion over, and lack of enthusiasm for, expressing general solutions in terms of differentiable periodic functions. In other words, Lagrange was happy to work with generalised functions, but unhappy about Euler's Leibnizian infinitesimals. This prompted Lagrange to reinterpret calculus using Taylor series, in his own words "independent of all metaphysics". Nobody can know exactly what he meant by this, but perhaps it is a reference to the philosophy of Leibniz.

These events must strike a chord with anyone educated in physics in the 20th century, heavily influenced by the thinking of Lagrange. Of course, modern physicists also have Fourier analysis, but this was a later development. In fact, Fourier's first paper on the heat equation (1807) was never published because of Lagrange's objections to it.

[1] I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press (1970)

These events must strike a chord with anyone educated in physics in the 20th century, heavily influenced by the thinking of Lagrange. Of course, modern physicists also have Fourier analysis, but this was a later development. In fact, Fourier's first paper on the heat equation (1807) was never published because of Lagrange's objections to it.

[1] I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press (1970)

## 2 Comments:

I see Leibniz with his monads as a visionary of mathematics the depth of whose ideas have not been understood at all by pragmatic physicists who fail to realize that "infinite" and "infinitesimal" are topological and hence relative concepts and therefore think that these notions are unpractical and useless.

Infinitesimals are much more than a practical tool to develop formulas. They have their own fascinating number theory following from the notion of infinite prime. The identification of point of real axis as an infinitely structured object (due to existence of infinite number of real units expressible as ratio of infinite integers) object with complex number theoretic anatomy allows a possible realization for Leibniz's monadism. I would call it algebraic holography. Or number theoretic Brahman=Atman if you wish.

Just a note about the mathematician you cite, Dr Ivor Grattan-Guinness. Grattan-Guinness was a close friend of the cross-talk electonics engineer Ivor Catt (until the latter got divorced and Grattan-Guinness took sides with his wife). I believe from memory that this began when Catt, Catt's wife and Grattan-Guinness tried to co-author a book about mathematician Oliver Heaviside and his work, especially the censored out stuff. (That book was never published, surprise-surprise!)

Catt wrote in his article "The deeper hidden message in Maxwell’s Equations", Electronics & Wireless World, December 1985:

"Dr. Ivor Grattan-Guinness once pointed out to me that the decline, or ossification, of science into ‘maturity’ was a necessary result of the introduction of universal education in the midt-19th century, because it caused the growth of a powerful group with a vested interest in knowledge, the professional teachers."

The theory here is that a lot of progress due to amateurs like Faraday (who started without formal education in science by cleaning test tubes in a lab) was due to eccentric styles. Paul Feyerabend's view in "Against Method" is that whatever method works is science.

The mainstream view now of course is exactly the opposite: that method defines science.

Nowadays you are a proper scientist if you use mainstream methods, and deemed a real fraud if you don't.

"Grattan-Guinness said that the introduction of universal education, in around 1850, which instituted the new class of knowledge professionals, meant that in the end knowledge would be frozen. We have been feeding through this process, and finally progress comes to a halt."

- http://www.ivorcatt.com/3600.htm

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