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Food for Thought

Philosophy of Mathematics

I’m reading Scale, by Geoffrey West, and enjoying it. One of its heroes is Lewis Fry Richardson, a Quaker, banned from academic jobs because of his pacifism in a jingoist economy.

He threw himself into studying war, from a somewhat physics-based angle. If you’ve checked out Scale, you’ll recognize the Santa Fe Institute approach, which has been seminal.

Richardson takes us to Mandelbrot, and the question “How Long is the Coast of Britain?” which we now know depends on what resolution we choose.

The phenomenon of scale is not hard to introduce and has to do with linear, areal and volumetric growth rates having different exponents. But then metabolic rates do not increase in direct proportion to volume.

Following Mandelbrot, and others, we enter the namespace of “fractional dimensions”. Another meaning of “four dimensional” emerges, in the way spaghetti, packed closely together, almost fills space.

Likewise a two-dimensional object can almost fill volume. The “spaghetti ball” pushes up against a next higher power.

These higher powers might be separated off from the dimension scale if we like. Separating dimension from power is not commonly done, but I know of at least one namespace that takes this route.

Here’s a hint:

1033.611 Dimension begins at four. Four-dimensionality is primitive and exclusively within the primitive systems’ relative topological abundances and relative interangular proportionment. Four-dimensionality is eternal, generalized, sizeless, unfrequenced.

1033.612 If the system is frequenced, it is at minimum linearly five-dimensional, surfacewise six-dimensional, and volumetrically seven-dimensional. Size is special case, temporal, terminal, and more than four-dimensional.

What’s that all about? What namespace is this?

That’s R. Buckminster Fuller, talking up his shape versus size distinction. The different behaviors we observe in physical phenomena, based on scale, take us into the energetic realm (P for physical), versus the Platonic realm (M for metaphysical). U = MP (U for Universe). In the Platonic world, we have shape without any specific energy involvement, which is what keeps such considerations generic (generalized).

Fuller considered the conceptual yet timeless world of Platonic forms to be four dimensional in the sense that a tetrahedron is four directional. The XYZ “jack” of six spokes, floating in space, is not his anchoring image. Think “caltrop” instead.

Let me back up and ask a question. Did Kant, in his Critique of Pure Reason, ever commit to space being three dimensional? Let me know in the comments if you’re knowledgeable in that area.

I know Cantor had issues with the “dimension” concept of his day. His thinking was another influence in the emergence of fractional dimensions (fractals).

D’Arcy Thompson expresses some skepticism, in a letter to Alfred North Whitehead, about whether “three dimensions” are the right number, for ordinary spatial experience. I mention that in my slides.

If you’ve never questioned said dogma, that of space being 3D, then I recommend trying it. I have some workshop exercises to help a person see another way.

We talk about “aspect shifts” like the Necker Cube, or Duck-Rabbit, also themes in Wittgenstein. Lets see if we can “pop the cube” and see the world a little differently.

Keep digging through my writings here on Medium if you want to explore these concepts more deeply. I’ll give you a starting place, but you might find others, and that’s quite OK.

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