> mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe?
Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the same for those orders not yet established, that is, the future of science. Nevertheless, I do believe there’s a principle which allows order, but it’s not order itself, but a foundation of order, which I believe to be Unit. Unit is not mere duality, because duality implies two, Unit would be more like Cause-Effect, wherein one is the same as the other (either Cause-Cause, or Effect-Effect, doesn’t really matter). This is also different than yin-yang, since each is discrete, and discreetness itself cannot exist prior to Unit. I like to think that all numbers are qualities of Unit, and the whole of mathematical theories are different theories of Unit, so they will be consistent every time Unit is maintained consistently, when something “follows” from what has already “followed”, following some definition of “following”, whatever it may be. It would explain the effectiveness of mathematics in the sense that the whole Universe is Unit out of self-similar Unit. The fact that all of information can be encoded in 0s and 1s is a great illustration of the power of Unit, and the fact that binary streams only makes “sense” upon interpretation is a great illustration of consciousness, which is an expression of Bias. Even if the Universe would change so all of the physical “laws” would mutate, Unit would still persist unchanged. New things would still “follow”. I haven’t come across anyone realizing that a theory of “everything” can exist but be useless, just like the concept of “everything” is useless as a particularity, and theories aim to be particularly applicable, so a theory which really applies to every thing applies to no thing. This would be the utmost conclusion of Gödel‘s incompleteness. How would a theory of absolutely everything be different than an infinitely long ruler without any subdivisions, or even with infinitely many zero-spaced subdivisions? One wouldn’t be able to measure any particular thing with such universal ruler.
Does your "principle which allows order" presume the existence of space and time, and more specifically the ordering of time? I would say that in a universe without time, "cause and effect" have little meaning.
I think I lost your train of thought when you say that numbers are qualities of Unit. Does your universe involve only "Unit", or are there other principles at play as well?
The Peano arithmetic hints at the proposition that numbers are gradations within a unified principle, in this case the principle of “succession”. If you start with nothing (zero) and recursively apply the same quality (succession) you get all integers. So we can rationally assume that all numbers are different qualifications of some primitive. As for time and space, they do not fundamentally exist and they do not configure a necessity within the universe. It takes a being able to record and internally persist events for time to appear. Space is similar, being also a referentiality. In essence, time and space are different framings of the same phenomenon. The length distance between two points is also a temporal ratio between the points. It is just two perceptions, two expediences, not grounded in singular reality. I believe “beings” are Unit juxtaposed over Unit, in the spirit of Wheeler ideas [1]. I don’t think there are fundamental necessities other than Unit. Unit is not causality, I have just used the terms for illustration. If there are other fundamental necessities other than Unit then one would need to explain how they came into being, and it would eventually recurse into Unit. From nucleosynthesis to procreation, it’s Unit all over. The way I grasp it, mathematics is a coloring of Unit, just like for seeing wind one need to sprinkle something over it. It boils down to the ascription of Parts within Wholes. Parts are mere subjections.
Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the same for those orders not yet established, that is, the future of science. Nevertheless, I do believe there’s a principle which allows order, but it’s not order itself, but a foundation of order, which I believe to be Unit. Unit is not mere duality, because duality implies two, Unit would be more like Cause-Effect, wherein one is the same as the other (either Cause-Cause, or Effect-Effect, doesn’t really matter). This is also different than yin-yang, since each is discrete, and discreetness itself cannot exist prior to Unit. I like to think that all numbers are qualities of Unit, and the whole of mathematical theories are different theories of Unit, so they will be consistent every time Unit is maintained consistently, when something “follows” from what has already “followed”, following some definition of “following”, whatever it may be. It would explain the effectiveness of mathematics in the sense that the whole Universe is Unit out of self-similar Unit. The fact that all of information can be encoded in 0s and 1s is a great illustration of the power of Unit, and the fact that binary streams only makes “sense” upon interpretation is a great illustration of consciousness, which is an expression of Bias. Even if the Universe would change so all of the physical “laws” would mutate, Unit would still persist unchanged. New things would still “follow”. I haven’t come across anyone realizing that a theory of “everything” can exist but be useless, just like the concept of “everything” is useless as a particularity, and theories aim to be particularly applicable, so a theory which really applies to every thing applies to no thing. This would be the utmost conclusion of Gödel‘s incompleteness. How would a theory of absolutely everything be different than an infinitely long ruler without any subdivisions, or even with infinitely many zero-spaced subdivisions? One wouldn’t be able to measure any particular thing with such universal ruler.