The nature of connectedness. Great starting point.
I agree: every boundary connects and separates. By saying that, however, we already assume that we can talk about separate entities. That’s interesting, because it touches real toughies:
I think a lot about fractals. (Fractals are structures which show more and more detail the further you change your measuring rod/your level of description when you look at a spatial or temporal structure. They can be perfectly self-similar but don’t have to – sometimes the parts resemble the whole in every detail, sometimes only roughly).
In the past, I have always distinguished between ideal (mathematical) fractals, such as the Mandelbrot set, and natural fractals, like a tree or bronchial tubes. The difference is that trees are self-similar to a certain extent over a limited range of nestings (branches, twigs,..) and the Mandelbrot seems to have no limit when you zoom into it. To my knowledge, nobody has yet been able to show that the Mandelbrot set is connected. It may go on scaling forever, showing more and more, finer and finer detail as we zoom in. Maybe the question of connectedness only makes sense, if we specify a level of description, a resolution, at which we look at it (or experience it). On each specific level, it makes perfect sense to us that the structure is connected. It is only because we know that we can look at a further nesting, that the question arises whether it will still be connected on the next level of resolution.
And here, I believe, Otto, your idea comes in that we need a new kind of maths which involves the observer-participant: Resolution-dependence is directly linked to the amount of internal complexity an observer-participant has developed. We are aware – either through the different levels of communication in our bodies or with the help of “observer-extensions” like microscopes – of more or less nestings within ourselves and our environment (depending on our level of complexity). So, I think, we have to specify how many levels communicate simultaneously.
To me, fractals come in two shapes.
They can be bifurcations, if we look at them in terms of the length of time. Then we see succession.
They can also be nestings (contextualizations), if we look at them in terms of the depth of time (nestings). Then we see simultaneity.
Both dimensions, the length of time (succession) and the depth of time (simultaneity) are important to human observers. The way we arrange our world in these two mutually exclusive temporal dimensions determines the way we interact/communicate. Therefore, I believe that the new kind of maths you envisage should take account of both, as they shape our individual perspectives.
The length of time does not provide any connectedness. It is the depth of time which provides the nestings – that is to say, the background against which we can contextualize and therefore arrange experiences as succession. Simultaneity is observer-participant-dependent. I believe it is our only way to generate connectedness.
An example is Husserl’s description of a tune. If we could only hear individual successive notes, we would never be able to hear a tune. In order to do this, we must embed the present notes by remembering the preceding note and anticipating the note which may follow.
For the time being, I think that connectedness can only be brought about by simultaneity (the depth of time).
Enough. This may be a good moment to stop and hand over to you, Otto …