Operation DuckRabbit

MathArtStream 4 ~ Kirby Urner: “Dimension" in Synergetics

Nominalist thinking involves playing "pin the tail on the donkey", where the tail is some "signifier" like the numeral 2, and the donkey is twoness itself, the essence of two, for which the numeral is but a name. We're allowed to get loquacious about what twoness truly entails. We might say "the set of all sets of two elements". If we're into Synergetics, we'll speak of axial twoness (additive, rotating) versus inside-outside twoness (multiplicative, resizing).

Take the word "infinity": to what does it point, in order to get its meaning? Those with a rich imagination may flash on several images but is that what it means to mean something, i.e. to flash on images?  Might a signifier have a use without a signified? Put another way: how might we use words as a means without needing them to have referents?

In the above talk, I will focus on the word "dimension" and suggest this word has meaning through embeddings in language games, such as our "three-tuples determine a location in space" talk. What about 4-tuples? Let's talk about quadrays.

I introduce the concept of namespaces by showing how we already had a bifurcation in the meaning of "fourth dimension" by the time we got to Minkowski versus Hilbert spaces. 

The possibility of non-Euclidean geometries had already led to forks at the fifth postulate. But might we venture away from the Euclidean lineage by going in some other direction, meaning-wise, such as by rewiring "dimension talk"? 

Enter Karl Menger and the geometry of lumps.

Enter Synergetics and yet another namespace wherein 4D is a tentpole, helping to anchor another circus tent. 

A minimum box or room, a container, with no sides or lids left off, no windows or doors left open, has four walls, not the six a hexahedron has. Ergo space is four-directional (or say dimensional). Call that austere primitive beginning "pre-frequency" and then subdivide from there, adding spin, color, energy, nuance.

So there we have at least three namespaces using "dimension" in characteristic language games by the end of the 20th Century and onward.

Plus lets not forget the fractional dimensions of discrete maths. "Infinity" rears its head in this namespace as well, like the monster in Loch Ness, even as we zoom into a Mandelbrot Set (or Mandelbulb, as the case may be).

Countering nominalism, is operationalism, or Wittgenstein's "meaning as use". The rules emerge from the playing, while meaning is more or less fleeting, depending on the half-life of the containing corpus.