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“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental myst

“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?

kevinventullo 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. 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.
joppy 8y ago
Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc. On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.
- aaaaaaaaaaab 8y ago
  But aren’t we missing out on something important by glossing over these “pathological” cases? Like how we kept ignoring nonlinear differential equations, because they aren’t that well-behaving as their linear counterparts... And then when we eventually looked into it, we found a completely new paradigm: chaos theory.
  - joppy 8y ago
    For sure we are missing something. But what hope do we have of understanding the more complicated case, before we figure out what we can know in simpler cases? The eventual goal is to move back to the more complicated cases.
    - phkahler 8y ago
      What if the approach to the simpler cases is not applicable to the harder ones. Studying the simple may shape ones thoughts in a way that prevents seeing the way to harder problems. Maybe? Sometimes new things are discovered because someone didn't know any better and tried something they were not ready for. Innovation sometimes comes from the self taught.
  - deleted 8y ago
    [deleted]
  - 8y ago
notduncansmith 8y ago
Complex numbers is one area where sine is very relevant. Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/

kevinventullo 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. 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.

joppy 8y ago

Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc. On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.

aaaaaaaaaaab 8y ago
But aren’t we missing out on something important by glossing over these “pathological” cases? Like how we kept ignoring nonlinear differential equations, because they aren’t that well-behaving as their linear counterparts... And then when we eventually looked into it, we found a completely new paradigm: chaos theory.
- joppy 8y ago
  For sure we are missing something. But what hope do we have of understanding the more complicated case, before we figure out what we can know in simpler cases? The eventual goal is to move back to the more complicated cases.
  - phkahler 8y ago
    What if the approach to the simpler cases is not applicable to the harder ones. Studying the simple may shape ones thoughts in a way that prevents seeing the way to harder problems. Maybe? Sometimes new things are discovered because someone didn't know any better and tried something they were not ready for. Innovation sometimes comes from the self taught.
- deleted 8y ago
  [deleted]
- 8y ago

notduncansmith 8y ago

Complex numbers is one area where sine is very relevant. Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/