Cf: Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9 http://inquiryintoinquiry.com/2020/07/19/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-9/
Re: FB | Systems Sciences https://www.facebook.com/groups/2391509563/permalink/10158218873839564/ Re: Kenneth Lloyd https://www.facebook.com/groups/2391509563/permalink/10158218873839564/?comment_id=10158219473119564 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 … Regards, Jon
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