Actually, I wanted to write about Cantor, but then I realized that I would better like to tell about a notable example of how a deep, general & fundamental mathematical concept may influence the math creation, not only by inspiring novel ideas – and the creation of new math – but also by finally achieving the exact and lasting establishment of old ones.

Even professional mathematicians are many times not enough aware of the history behind the mathematical creation. Nevertheless, the historic data are more than relevant, since in math the development of new concepts always assumes or relays on others already existing.

This knowledge turns out to be fundamental to really capture the essence of math evolution, and even more, to understand the mathematical thinking. It highlights the paths and give sense to the ideas that mathematicians follow. We could say

… omitting this information in a math lecture is like teaching

skimmed–powder–math. For whole math you need to add somestory! 😉 …

It is most amazing that so many profound & fundamental concepts have come to light inspired or requested by the study of the so called Trigonometric Series.

The question of the representation of a periodic function F(x) by a Trigonometric series – i.e.finding coefficients A0, A1, A2, A3, … and B1, B2, B3, … such that F(x) = A0 + A1 cos(x) + A2 cos(2x) + A3 cos(3x) + … … … + B1 sin(x) +B2 sin (2x) + B3 sin(3x) + … … (*) for all x in (0, 2Pi) – arises in a controversy between EULER and Daniel BERNOULLI (around 1750) about the general form of the solution of the *swinging* *string *problem. The problem – given a function, finding a formula for the coefficients – remained open more of 70 years, until FOURIER (in 1822) in his “Theorie analytique de la chaleur” also came to the question of representing a function by a trigonometric series. Thus now he achieved finding the coefficients in a general way – in terms of certain definite integrals – 🙂 … Although he did not really prove the convergence of the series 😦 …

7 years later DIRICHLET gave (or found!) conditions for the Fourier Series to converge and represent a function at *all* *points* x where it is continuous.

… By doing this, he was force to *exactly* define the concepts of *function* & *continuity*!!…

Since – believe it or not – 200 years ago they were not so clear and obvious, and his authority as a prominent mathematician made them the “common knowledge” in math they are today.

Source: An exceptional mathematical biography – George Cantor – VITAMATHEMATICA (German Ed. Birkhäuser, 1987).

Fourier Transform of a Function f

Fourier Series Square Wave – Circles Animation