As we wrote in Part I and Part II thanks to the study of the* Fourier Series, * concepts such as those of *Function, Continuity or Integral* were for the first time *exactly* and *generally* given to the math world!

But even more, it turns out that also CANTOR drew from questions about these series the decisive stimulation for his impressing life’s work!

In a note to a paper of 1870 Cantor proved that the* uniqueness* of the Fourier representation remains true even if one admits a *finite* number of* exceptional points.*

Cantor’s lucky idea was then to wonder, if it was possible to admit an *infinite *number of exceptional points. And if yes, what kind of *sets of points* would they be?… Would they be *dense* everywhere?… Or nowhere *dense*?… Just two years later, he published a paper with far reaching results about set theory and the uniqueness theorem of the Fourier representation!!…

By doing this, he first realized that the *theory of real numbers* did not have the the *exact* arithmetic form that was needed, so he wrote – in 4 printed pages – Cantor’s construction of the real numbers!!…

This work alone would have assure him a place in the history of mathematics, however he also formulated then the basis of his Set Theory!!… An amazing collection of new mathematical ideas that – since then – have become part of every piece of math.

… This seems to be the way good math happens and I think it is very valuable to reflect about it 😉 … Math is amazing, but its *story* may be unexpectedly amazing …

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

Fourier Transform of a Function f

Fourier Series Square Wave – Circles Animation

Square Wave – Fourier Arrows

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Updating Math In Our Mind & Heart!!...

December 11, 2015 at 16:38

I like the way to made the connections and the idea of all 3 posts for Math&Creation. I didn’t have time to relate all these things like this and it gave me a lot of information about the inside story. Thank you ^_^

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December 12, 2015 at 22:40

I’m very glad you liked it! 🙂 … I was wondering how to give all this information in a nice & easy way, and I think I found a good solution 😉 … Slowly finding my style …

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December 12, 2015 at 22:45

It is quite interesting ^_^ You will slowly find your style and I think it will be really nice ^_^

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