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I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They'r

I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. Going back to the 1800's, in the theory of Riemann surfaces (one-dimensional complex manifolds), the only meromorphic (complex differentiable) functions are ratios of polynomials! And all closed projective complex manifolds are algebraic varieties. See https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...

wolfgke 8y ago
> I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. This is rather a tautology since polynomials are exactly defined this way. I can imagine that if we built a (fictional) number system rather on properties of the Fourier transformation, sines and cosines would be very natural operations that would have a deep generelization when sufficiently abstracted.
- sdenton4 8y ago
  The Fourier transform is just a change of basis, if you squint at it properly. So at least some of the time (eg, when you're not mixing frequency and time domains like a crazy person) you can convert to frequency space and the functions become much simpler, and usual polynomial operations are relevant again.