Isomorphismes

Categorial decomposition of Galilean spacetime.

Sean Carroll tells us that it was Galileo who first si rese conto che motion can be separated into:

• motion in the x direction — ẋ or x′[t]
• motion in the y direction — ẏ or y′[t]
• motion in the z direction — ż or z′[t]

and, importantly, that physical laws should be the same for all the 360° × 360° orthonormal choices of (x,y,z). It was Galileo’s idea that you can draw axes, that forces can be decomposed onto those axes, and that forces along one axis behave independently of each other.

For example if you kick a football, it goes forward x′[t], chips up y′[t], and bends left z′[t]. If you kicked it off a cliff, it would retain its exact same forward x'[t] speed even after it dropped y<0 below the plane of the cliff at an ever increasing speed. (NB: That’s not actually true, which is why we say “in a vacuum”.)

The traditional way to talk about a path γ is talking in tuples:

• First, you have some points
• Then, you have a 3-basis.
• Then, you have an interval.
• If you want to talk about kicking the ball, you would probably call the ball a point, say “there is” a vector space tangent to the ball, and your single kick of the ball constitutes a single force-vector applied (instantaneously) to the point, I mean ball. “Then” — by which I mean “at higher values of t∈interval” — the ball “is” chipped up in the air, “then” back on the ground.
• The path γ is any member of the product (pairing) of 3-basis with interval.

path γ ∈ time × space*

* space in the geographer’s sense; the casual, not mathematical, sense of the word space. Lawvere calls mathematical space a “universe” … like the theoretical universe that the theory lives in

All of this “you have” — it’s a violation of E′. The “false subject” in English sentences that start with “There are” is repeated over, and over, and over again in mathematics (hence the invention of the symbol ∃).

Now cometh F William Lawvere, 3 centuries later, with a conceptual breakthrough.

path γ : time → space

The categoryists use labelled dots and labelled arrows to sketch concepts. So in pictures 2 and 3 you can see projection arrows splitting 3-space into a 2-plane (ground) and a 1-line (air). (Arrows sometimes seem backwards in category theory. Galileo projects 3D onto 1D + 2D, so something like “coprojection” would be the natural piecing together of independent sub-motions to get the full picture.)

And the Galileo example is just meant to be a shared thing we can all discuss. But this same thought-pattern — categorial decomposition — I can use on non-chalkboard things from my life as well. Gottman-style 2-eqn relationship dynamics; speculating about some economics in the news; love triangles; the deeper you plant this seed, the more places you see it.

(Source: amazon.com)