Dear Huebler,
Fantastic ! What creativity and fertility of mind ! No, I am not offering these generous words for your line of thinking that Mathematics or order alone must be the cause of the existence [I too have been worrying about it for quite sometime], but rather for the jugglery and ingenuity of associating the elements of logic with elements of physical universe, and for the attempt to bind the unlimited scope of mathematics to the realm of observable physicalism.
Our minds are built to parse pattern and build models, arriving at a rational viewpoint that without a pattern, one cannot derive or predict anything. Therefore, we necessarily seek to understand how universe must have a rational beginning; we just cannot reconcile with the arbitrariness of nondeterministic 'process' to create the universe. Certainly, if a process does not have a degree of determinism, we cannot even call it a process.
Mathematical rationality of emergence of the universe is riddled with several difficult issues.
1. Eternity. Even the induction rule has an eternal existence. It does not matter what mathematics develops out of induction, but what ever develops is not bounded by time. Given the eternity, every notion would have played out in the past and continue to do so for ever. Only truly cyclic processes can be allowed. But then, again what fixes the periodicity of such cycles? A periodicity of infinity, or frequency of 0, could be of particular interest.
2. There does not exist uniqueness in mathematics. So, I am not sure which dynamical, or induction based mathematics should one consider as the basis for the existentiality of the universe.
3. Randomness allows unpredictable variations to occur. If randomness does not exist then following issues emerge. A static null configuration remains null for eternity. Changes are required to bring in time. Regular changes with finite elements may bring in time but it may create endless cycles. but then, if randomness is allowed then it does not remain mathematical. Of course, in our mathematical thinking, inifinity is the only savior to bring in whatever fundamental indeterminism. However, bringing elements of indeterminism via the concept of infinity and zero, would render the universe entirely non deterministic, and it would be difficult to bring in order, without an arbitrary rule.
4. The reason that we cannot reconcile with the idea of indeterminism is that once there is a breakdown of rule, we cannot employ rational thinking to resolve issues. Yet, we cannot do without indeterminism. One way to handle it is to hypothesize a limited indeterminism of certain specific 'kind' and 'limit'. For example, we may allow the most fundamental level of entities to have analog / continuous detail, which cannot be defined with finite number of parameters, yet they could change only in quanta, and have finite number of measurable states -- something like infinity modulo N. Furthermore, these entities may have constancy of relation among measurable states, individually they do not have entirely deterministic description, only relations have consistency.
5. It would be interesting to know if author believes that the process could be simulated even at smaller scale?
The most fundamental trouble that one faces is that nothing comes out of absolute null, cipher. Point to pixel posses an infinite abyss in logical construction. No matter what we do with objects of measure zero, we cannot bring about a finite dimensional object from it. The unfathomable leap in the argument is in this statement, "but as soon as the newly generated mathematical points are connected in very large dimensions and categories with a huge number of different meta-information paths, we can expect a delay of the mathematical frequency."
HoTT defines, "'a' is a point of the space A'. A bit circularity exists here. How do we first define A? We cannot, unless we consider A as a class whose members are constituted of 'a's. 'a' does not receive an independent qualifying definition from A. So, I take it as that 'a' and A have existence in mutual relation, in particular that of relation between a class and its members. Since no object can be defined with any absolute reference frame, therefore, this mutual existential relation is acceptable. You have used this consideration to get rid of physicalism.
The empty set is just an information, if we give it a status of the realism of substratum, then why only points, what cannot be imagined to have an empty set of. Information requires a substratum of its own -- e.g. registers for bits, quantum states of particles for qubits, and neurons for everything. That is to say, it is always the physical state of an entity that correlates with, or represents information, as I have worked out in my essay. And all interactions resulting in definitive consequence carry out information processing by virtue of natural causation. Without this physical substrate, information has no basis, no existence. One is able to talk about empty set, only because such a notion can be conceived of by the function of neurons.
"The impossible existence (_) leads logically to its opposite, the possible or potential existence (т--Л)". This is a pure magic of a magician working on the minds of the readers (audience). It took me so much to understand this. If the meaning, or notion, or information of 'the impossible existence' exists then of course, it leads to its own opposition. In a context of nothingness of all existence, even this information has no basis. I could grant this under the notion of 'cogito, ergo sum'. But then, we all already know, we exist. The existence is brought about by simply asking the right contradictory question, a discrete question, leads to discrete contradictory existence.
As you sought -- a logical step forward could be to seek an evolutionary path for laws or function (natural causation) of universe, such that starting from nearly no determinism, the laws acquire (evolve) greater determinism, without ever achieving absolute determinism. The absolute determinism is as much of a trouble as absolute null to begin with. Or, a hypothesized limited determinism may also work.
By the way, natural causation is non-violative, as the information processing emerges from it as I have shown. Therefore, one has to show exactly how this version of causation may come about.
Rajiv
P.S. I had to read your essay so many times, yet do not understand fully.
