(non) Archimedean Dreams

In the past, I’ve ventured to consider the hypothesis (ansatze) that a noetic realm, a rough analogue of the Platonic realm of Ideals, has a real existence, in a parallel universe of sorts to our own. Part of this ansatze is that these two universes are not completely disconnected, and that the human intellectual machinery glimpses this realm and it is through this mechanism that our brain’s machinery accomplishes the semiotic scaffold and bridges gap between pattern and synapse to thought, meaning and intention.

What sorts of features might we imagine a noetic realm to have?

  • Would it have any notion of time evolution? I suggest while objects in the noetic realm were suggested by Plato to be “eternal” that this might be wrong. That in a noetic realm, in which objects or “points of space” are conceptual monadic points, there can be some sort of movement or time evolution. An analogy to how that might work might be the flip-flop. A flip-flop is a metastable two state simple electric circuit and without going into the technical details the analogy I’m bringing to this table is that noetic ideas can also demonstrate similar features. Consider some of the simple logic paradoxes, like the statement about the veracity of Cretans made by a Cretan with the postulate that Cretans always lie. The points is there exist logical complexes (ideas) that are metastable like the flip/flop, i.e., paradoxes. In a noetic realm, this might be seen as motion. With motion … time evolution.  Godel’s incompleteness theorem suggests that all logical axiomatic systems (noetic complexes) have un-proveable statements, not all as simple as the always dishonest Cretans. Might this be seen as the same as a statement that all noetic complexes exhibit “movement” when looked at carefully enough?
  • If we take points in a noetic space to be monadic concepts, more complicated ideas will be structures or linkages in this space … which themselves are also a point in that space. This suggests that if we were to give a metric to this space it might be non-Archimedean or ultrametric space. Tree structures are ultrametric if the distance between elements of a tree is counted by how many generations one traverses up the tree before a common element is found. It seems natural that concepts also have a distance relationship that is akin to a tree.
  • Consider for a moment, that this realm has a physics. A series of natural laws that given time evolution describe a dynamical relationship between objects and motion in this space. Imagine too then, that it contains life … and further intelligent life. It is hard to imagine what existence, perception and other notions which are clearly definable in our universe might be like in a realm such as the one I dimly describe above. Maths (a UK term for mathematics that I find attractive which is my excuse for using it) is a concept that is often argued is a purely noetic art. That it doesn’t depend on science or perception but is purely an intellectual (noetic) exercise. Concepts like integer, line and point from which we derive maths it is argued are universal. If we met technologically advanced aliens … we would be able to communicate because we would share a common mathematical technology. In a prior post, I argued that is not necessarily the case, that our mathematical concepts are aligned with our commonly held perceptions of the universe. A creature dwelling in the noetic universe might perceptions sufficiently distinct from our own to render this assumption false (and my argument in that prior post valid).
  • In maths, a common arithmetic simplification which yields a natural ultrametric space are based on p-adic numbers. One of the ideas lurking in the p-adic analytic realm is the Adele ring, which is a infinite vector of p-adic fields with a point at infinity added … which is naturally seen as the real numbers. Might an adelic ring analogy be seen linking noetic reality (realities) to ours in which is the archimedean point at infinity?

And if you think that time evolution or changes in connections is impossible. Consider what you know of the number 2 and other simple counter numbers and from them the integers. Then read this … (or for more fun … get this book: Surreal Numbers with combines the numbers noted in that prior wiki link with an entertaining story about those numbers has narrative parallels with Genesis 1).

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