Cf: Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9
Re: FB | Systems Sciences
Re: Kenneth Lloyd
Dear Kenneth, All ...
Mulling over recent discussions put me in a pensive frame of mind
and my thoughts led me back to my first encounter with category theory.
I came across the term while reading and I didn't fully understand it.
But I distinctly remember a short time later catching up with my math TA —
it was on the path by the tennis courts behind Spartan Stadium — and asking
him about it.
The instruction I received that day was roughly along the following lines.
“Actually . . . we’re already doing a little category theory, without
quite calling it that. Think about the different types of spaces we’ve
been discussing in class, the real line R, the various dimensions of
real-value spaces, R^n, R^m, and so on, along with the various types
of mappings between those spaces. There are mappings from the real
line R into an n-dimensional space R^n — we think of those as curves,
paths, or trajectories. There are mappings from the plane R^2 to
values in R — we picture those as potential surfaces over the plane.
More generally, there are mappings from an n-dimensional space R^n to
values in R — we think of those as scalar fields over R^n — say, the
temperature at each point of an n-dimensional volume. There are
mappings from R^n to R^n and mappings from R^n to R^m where n and m
are different, all of which we call transformations or vector fields,
depending on the use we have in mind.”
All that was pretty familiar to me, though I had to admire the panoramic
sweep of his survey, so my mind’s eye naturally supplied all the arrows
for the maps he rolled out. A curve γ through an n-dimensional space
would be typed as a function γ : R → R^n, where the functional domain R
would ordinarily be regarded as a time dimension. A mapping α from the
plane to a real value would be typed as a function α : R^2 → R, where
we might be thinking of α(x, y) as the altitude of a topographic map
above each point (x, y) of the plane. A scalar field β defined on an
n-dimensional space would be typed as a function β : R^n → R, where
β(x_1, …, x_n) is something like the pressure, the temperature, or the
value of some other dependent variable at each point (x_1, …, x_n) of the
n-dimensional volume. And rounding out the story, if only the basement
and ground floor of a towering abstraction still under construction, we
come to the general case of a mapping f from an n-dimensional space to
an m-dimensional space, typed as a function f : R^n → R^m.
To be continued …