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It's always appreciable to try to comprehend stuff through new ways and tools, but I fail to see the aim of the post. Isn't it a matter of properly defining the

by cmehdy 6y ago

It's always appreciable to try to comprehend stuff through new ways and tools, but I fail to see the aim of the post. Isn't it a matter of properly defining the domain of application and associating a way to visualize it? Representing the result of a function from R to R requires two orthogonal axes because the element pre-transformation is 1-dimensional and the result is also 1-dimensional. For C to R, this would require 2+1=3 orthogonal axes, so it can be visualized with a 3D representation. Likewise for R to C. From C to C that would be 4-dimensional and becomes already trickier without some effort to conceptualize it, and certainly becomes less intuitive without resorting to alternate ways to conceptualize dimensions. It quite probable that beyond that, one would really encounter decreasing returns on trying to visualize the situation because the cost of abstraction would increase in order to rely on multisensorial approaches to compensate for our inability to visually perceive much beyond 3D. It's entirely possible that the whole post flew way over my head and I absolutely did not get it though, in which case I am truly just a confused commenter.

perl4ever 6y ago
One of the things that is painful because it's out of reach for me intellectually and yet tantalizingly straightforward sounding is extending the concept of an n-dimensional space to infinite dimensions. The idea of a mathematical function of a function doesn't sound like a big deal; it sounds vaguely similar to mundane abstractions in programming. This sort of thing is beyond my ability to cope with and yet it sounds like counting 1, 2, 3, not like abracadabra... https://en.wikipedia.org/wiki/Functional_analysis https://en.wikipedia.org/wiki/Functional_analysis https://en.wikipedia.org/wiki/Hilbert_space https://en.wikipedia.org/wiki/Hilbert_space
- JadeNB 6y ago
  One of the very important things to know is that, while an infinite-dimensional, separable Hilbert space is basically "n-dimensional space, but more so", there are more general kinds of infinite-dimensional spaces (I'm thinking of Banach spaces, but you can certainly get still more general than that) that are much more general than that. The prototypical examples of these are the L^p spaces, where p stands for ∞, or a real number p ≥ 1—but p isn't the dimension, as in ℝ^n (they're all infinite dimensional, at least for a reasonable underlying metric space). Rather, it's a parameter that controls, in some sense, how close the geometry is to being governed by the Pythagorean theorem, so that only p = 2 gives (a Hilbert space, and hence) the 'usual' geometry. I think your point with functional programming is spot on. Just as one learns, to pick that bête noire, monads not by reading yet another clever re-packaging that somehow only manages to make them sounds more difficult, but rather by finding a problem for which they're relevant and getting a feel for them by using them, so too does one learn about infinite-dimensional space not by treating it as some sort of philosophical profundity, but by finding a problem for which it's the right setting, and realising that it's just a mathematical tool like any other.
- QuesnayJr 6y ago
  If you are interested in developing some intuition on Hilbert spaces, I recommend learning some Fourier analysis. Thinking about how vibrating strings conceptually have infinitely-many modes, but the higher frequency modes matter less and less, gets across a lot of the intuition. The Fourier integral and the spectrum of light is also intuitive.
nadrray 6y ago
This is an incarnation of the https://en.wikipedia.org/wiki/Functor_represented_by_a_scheme#Functor_of_points https://en.wikipedia.org/wiki/Functor_represented_by_a_schem... perspective on https://en.wikipedia.org/wiki/Scheme_(mathematics) https://en.wikipedia.org/wiki/Scheme_(mathematics), the fundamental object of study in modern algebraic geometry. This perspective could be interpreted as treating equations as recipes and number systems, (fields/rings) as ingredients. It turns out to be extremely fruitful to study the "R-points" of an equation for all sorts of rings R, even more exotic ones like p-adic numbers or finite fields. Algebraic geometry and the study of schemes is a truly beautiful field of mathematics, linking number theory to theoretical physics, and much more.

by cmehdy 6y ago

perl4ever 6y ago

One of the things that is painful because it's out of reach for me intellectually and yet tantalizingly straightforward sounding is extending the concept of an n-dimensional space to infinite dimensions. The idea of a mathematical function of a function doesn't sound like a big deal; it sounds vaguely similar to mundane abstractions in programming. This sort of thing is beyond my ability to cope with and yet it sounds like counting 1, 2, 3, not like abracadabra... https://en.wikipedia.org/wiki/Functional_analysis https://en.wikipedia.org/wiki/Functional_analysis https://en.wikipedia.org/wiki/Hilbert_space https://en.wikipedia.org/wiki/Hilbert_space

JadeNB 6y ago
One of the very important things to know is that, while an infinite-dimensional, separable Hilbert space is basically "n-dimensional space, but more so", there are more general kinds of infinite-dimensional spaces (I'm thinking of Banach spaces, but you can certainly get still more general than that) that are much more general than that. The prototypical examples of these are the L^p spaces, where p stands for ∞, or a real number p ≥ 1—but p isn't the dimension, as in ℝ^n (they're all infinite dimensional, at least for a reasonable underlying metric space). Rather, it's a parameter that controls, in some sense, how close the geometry is to being governed by the Pythagorean theorem, so that only p = 2 gives (a Hilbert space, and hence) the 'usual' geometry. I think your point with functional programming is spot on. Just as one learns, to pick that bête noire, monads not by reading yet another clever re-packaging that somehow only manages to make them sounds more difficult, but rather by finding a problem for which they're relevant and getting a feel for them by using them, so too does one learn about infinite-dimensional space not by treating it as some sort of philosophical profundity, but by finding a problem for which it's the right setting, and realising that it's just a mathematical tool like any other.
QuesnayJr 6y ago
If you are interested in developing some intuition on Hilbert spaces, I recommend learning some Fourier analysis. Thinking about how vibrating strings conceptually have infinitely-many modes, but the higher frequency modes matter less and less, gets across a lot of the intuition. The Fourier integral and the spectrum of light is also intuitive.

nadrray 6y ago

This is an incarnation of the https://en.wikipedia.org/wiki/Functor_represented_by_a_scheme#Functor_of_points https://en.wikipedia.org/wiki/Functor_represented_by_a_schem... perspective on https://en.wikipedia.org/wiki/Scheme_(mathematics) https://en.wikipedia.org/wiki/Scheme_(mathematics), the fundamental object of study in modern algebraic geometry. This perspective could be interpreted as treating equations as recipes and number systems, (fields/rings) as ingredients. It turns out to be extremely fruitful to study the "R-points" of an equation for all sorts of rings R, even more exotic ones like p-adic numbers or finite fields. Algebraic geometry and the study of schemes is a truly beautiful field of mathematics, linking number theory to theoretical physics, and much more.