Tuesday, July 31, 2018

Topology for artists - workshop sketch 2

Topology for artists - workshop sketch 2
One of my little dreams is to make and give a workshop about topology for artists.
It might be called: What you didn't know about spaces. Topology for artists.
These are preliminary notes. Previous ones are here: YouTube Topology, Workshop sketch 1.

I started reading popular and introductory texts about topology two years ago. Since then I've become more sensitive to spatial metaphors. That's why I think a topology workshop might be interesting for artists. See for example the concept of a mapping between two complicated spaces.
For example:
When Mary speaks to Peter, she has a certain meaning in mind that she intends to convey: say, that the plumber she just called is on his way. To convey this meaning, she utters certain words: say, ‘He will arrive in a minute.’ What is the relation between Mary’s intended meaning and the linguistic meaning of her utterance?
A simple (indeed simplistic) view is that for every intended meaning there is a sentence with an identical linguistic meaning, so that conveying a meaning is just a matter of encoding it into a matching verbal form, which the hearer decodes back into the corresponding linguistic meaning.
But, at least in practice, this is not what happens. There are always components of a speaker’s meaning which her words do not encode. ... Indeed, the idea that for most, if not all, possible meanings that a speaker might intend to convey, there is a sentence in a natural language which has that exact meaning as its linguistic meaning is quite implausible.
… While it might seem that a speaker’s meaning should in principle be fully encodable, attempts to achieve such a full encoding in practice leave an unencoded, and perhaps unencodable, residue.
Here we see a mapping (a function, a relation) from the space of intended meanings to the space of sentences, (verbal utterances,  linguistic meanings). Then the nature of this mapping is discussed.

In topology we might ask how the two spaces look like and what their properties are.

We also might ask whether the mapping is homeomorphic. This will depend upon the nature of the mapping but also on the properties of the two spaces.
Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.
I believe in the idea of the workshop. But at this moment I'm still in this stage. Bear with me and help me if you can:
Some interesting links for future use:

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