Dear Noson,
i now read your essay in detail. It is written in clear language, simple to understand and the lines of reasoning can be traced very easily. Good work.
You contrast order with disorder, structure with chaos. You seem to have a rather pessimistic view on things like goals and intentions. But nonetheless, you argue your case very well and stringently. Let me annotate some thoughts i had during the reading.
Firstly, if considering the contest's questions, i think one has to presuppose as an axiom that all that exists does all things the right way. If we drop that assumption, we end up at nihilism. So i presuppose that the universe and its possible causes and all the rest is a consistent whole.
Your take on mathematical number systems is fascinating, innovative and thought provoking to me. The progression in the dimensionalities of the mentioned number systems is clearly an expression of logic, it is a kind of algorithm and it has therefore structure. In contrast to this - as you annotated at the end of your essay - one may arrive with such a progression at a level of description of 'all there is' which seems to say that the universe, viewed objectively, is devoid of structure. The interesting question is (and i formulated similar questions at the essay page of Cristinel Stoica) how order and disorder, randomness and information are intertwined.
If the mentioned progression of the dimensionalities of number systems indeed leads to the conclusion that the universe is devoid of structure, this would pose serious questions. For example, does it make sense right from the start to extrapolate a mathematical algorithm like the one for building up ever higher dimensional number systems? Isn't this somewhat similar to Cantors cardinalities, building up ever higher infinities out of a simple algorithm? Surely, your progression should not be infinite but terminate at a certain level where no axioms are left. Although i am not a mathematician and certainly not an expert on octonions and so forth, i ask myself how can such a progression of number systems be able to drop one axiom after the other, until there are no axioms left? But taking it as given, what you arrive at is simply a tautology, namely a 'number system' (although without any axioms) saying that the universe and all the rest is just what it is (without specifying it further). Obviously this is plainly true, indeed the universe and all the rest is just what it is. One does not need a single axiom to conclude this!
The problem only arises when one wants to specify the whole thing due to a certain category. I assume it to be true that our universe has many phenomena which do not have goals and intentions. But this does neither imply that the universe's main characteristic must be termed as 'chaos', nor does it mean that the universe's existence and the many 'mindless' phenomena in it are senseless from a higher point of view. Maybe mindless phenomena serve a higher purpose; surely, this purpose then had to be determined via the construction and rules of the universe and also surely not by some physical mechanisms, but by an entity which has goals and intentions. I see no contradiction that randomness, chaos and disorder cannot support goals and intentions. This may be a hard to swallow statement, but i will explain it further.
As you know, a perfectly random sequence of 0's and 1's follows a certain mathematical law. The digits 0 and 1 should be evenly distributed over the whole pattern. Random in this case means that the occurrence of either of the two digits does not depend on the value of the preceding or following values of such digits. Every event should be totally independent of each other event. Have we catched 'chaos' and 'randomness' with this? In no way. We only catched the extreme case at one end of the continuum of order. 'Randomness' as a nihilistic ontological fact should have other features, i suspect. It should be not catched up with any mathematical description. For example, imagine that you are a kind of Boltzmann brain, but without all the physics needed for it. Just imagine you are aware of something. This something does not reveal any correlations. At one time you see a flash, then you see the flasch forming to a vague kind of bubbeling-up of some melted cheese-like thing, you see all sorts of visual impressions and they do not make any sense to you. 'Randomness' defined as this would be just like a nightmare.
Now, let us elevate the mathematical concept of randomness to perhaps meet what you intended to say in your essay's last paragraphs. Maybe the observed structure of our universe is a lucky fluke within a chain of random events (like the 0's and 1's, randomly encoding some kind of theory of everything). Would this be a convincing scenario to explain the order in our universe and its - assumed - dichotomy to the observed chaos in our universe? I would say no, because it does not answer where the randomness comes from, how and on what existencial features it operates on and why it can be mathematically explored to the point where it produces conscious beings which indeed then mathematically have explored it.
The whole point for me here is to say that behind the concepts of randomness and order, there must be a common fundamental origin of all of this. Alternatively one must conclude that reality is somewhat irrational. Because i am not the kind of Boltzmann brain described above, i conlude that behind the interplay between randomness and order, there is some deeper origin of all there is. As you envisioned with the progressions of number systems, mathematics seems to be able to transcend itself - in the sense that its very limits show us how the universe cannot be. It cannot be infinitely dividable, it cannot be overall deterministic and at the same time be overall consistent, it cannot be overall chaotic and nihilistic in a Boltzmann-brain sense and it cannot capture the whole of existence. All this cries out for an explanation that is metaphysical. It is no wonder that the progression of number systems tends to converge towards a non-axiomatic description of reality. Because mathematics cannot capture all there is due to the nature of it. Its nature is not overall compressible. But this does not mean that for explaining the nature of reality otherwise than with mathematics, one wouldn't need some axioms. My axioms are that there are realms beyond space and time from which our universe originated. It is limited in time, duration and space. At the outer borders (at the microscale) of this universe, quantum mechanics rules and spontaneous collapses occur. These collapses may not have an explanation in terms of physical causality, but i am convinced they have an explanation in terms of purpose and intention. For me, it is no problem to think of a creator that has the power to give some order to his creations to behave spontaneously (although here again, this spontanity is restricted by the rules of QM, also given by this creator). Surely, these considerations are all axioms, choosen by me, they do not arise necessarily from what i have written. But i see no alternatives between a creator and nihilism, the latter in the sense that reality is absurd and logics cannot conclude something logical from the fact that logics has its limits in deducing the ultimate layer of reality. But it surely can *induce* these layers on the basis that logics has realized its own deductive limits. Therefore i tend to say that logics, as an expression of a formal system, is as incomplete in the sense of Gödel's results as every other formal system subject to Gödels results. The main result for me is that the limits of deducability do not necessarily mean that the universe is absurd. It is open for the possibility being not so. This openness is installed even within the very structure of logics, since logics can discriminate between a possibility and a necessity.
I would be happy if you could write what you think about this all.
Best wishes
Stefan Weckbach
