math – update

blogging & searching for true math …

Math Story & Creation III

3 Comments

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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Author: Math - Update

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

3 thoughts on “Math Story & Creation III

  1. 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 ^_^

    Liked by 1 person

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