Math: Eternal? If so Who Created It?

Over the years I've gone back and forth on this, but for a while now I've settled more on a combination of 2 and 3 mostly as a result of reading about Godel's theorem and Quantum Theory. The former tying into the whole area of model theory that shows the ambiguity in what any mathematical statement refers to "ontically" and the latter in recent years looking more and more like it points to a non-mathematically modellable layer of reality.

Model Theory can basically show you that a single mathematical statement can have several models with completely different properties and a single formal statement can have multiple realisations.So let's say the statement "The Reals are not countable". This is a provable statement in the ZFC formalisation of mathematics and basically refers to the fact that there isn't a bijection from the Natural numbers to the Reals.However there are several universes of sets that satisfy the ZFC axioms, these are models of the ZFC axioms. In some models of ZFC the statement "Reals are uncountable" is true because in that model the set that obeys the axioms of the Reals genuinely have a higher cardinality than the set that obeys the axioms of the Naturals. In other models the set filling the role of the Reals actually happens to be of the same cardinality as the set that is the Naturals but a bijection between them doesn't exist.So even a simple statement like "The Reals are uncountable", which seems to say something concrete about the Real numbers, is ambiguous because it's not totally fixed what "Reals" or "Uncountable" refer to. It's a purely formal/linguistic statement in ZFC.Edited by Son Goku, : Slight changes

It does. That doesn't change the fact that it might be countable though and thus have the same cardinality from the perspective of a more powerful model.Basically there are models of the reals no larger than the standard model of the Naturals.So even when the Reals have the cardinality of the power set of the Naturals, it can be because they are genuinely larger than the Naturals or they're the same size but the construction of the Power Set is restricted in some way.

I mentioned this in my last post, by the construction of the power set being limited within the model they may be of the same size. The lack of appropriate bijections would make them have differing cardinalities within the model, despite being of the same cardinality when viewed externally.Also one must distinguish a model of one set theory (i.e. different models of ZFC) from different kinds of set theory (i.e. ZFC, Tarski-Grothendieck, Kripke-Platek). The former are different realisations of the same syntactic statements, the others have different syntactic statements.Edited by Son Goku, : No reason given.

The specifics of the case Sarah Bellum asked about aren't too important, those only really have a technical exposition since they're a technical question.The basic point is that what a mathematical proof refers to is a bit ambiguous and mathematics can throw up problems similar to language like contradiction from over self-referencing. Which leads people to think that it is in fact a language not some externally existing "thing" fully independent of humanity.For example there are axioms that both the Whole Numbers (Naturals being their technical name) and the Real Numbers obey. These axioms are things like:
For any two numbers a,b if a,b > 0 then a + b > 0 and a.b > 0If you then try to prove the statement:
There is a number between 0 and 1
Using only those axioms you can't prove it true or false. That's because two different models (i.e. two different systems obeying them) exist. In one of them (Reals) the statement is true, in the other (Naturals) it is false. So from the axioms themselves you can't prove it true or false, because the axioms can only deal with statements common to both.Work after Gdel's incompleteness theorem showed that this is always true, no matter how precise you are with your axioms there will always be multiple models and so there will be statements you cannot prove. Also statements can be true in the different models for different reasons.Due to all these models it's ambiguous what a mathematical statement means. And this combined with paradoxes that result if you allow too much self-reference make many people think mathematics is just a language and a creation of human thought.In addition to this, for myself, the fact that quantum theory suggests a layer of reality that cannot be described with mathematics (i.e. is non-algorithmic or non-mechanical) makes me inclined even more strongly toward mathematics being a human creation.Edited by Son Goku, : No reason given.Edited by Son Goku, : No reason given.

I meant to reply to this.I wouldn't say the complexity of the Monster Group comes from the group axioms. It is a very complicated mathematical object that is a group, however the complexity doesn't arise in any sense from the group axioms."Group" is a type of object like "Vehicle". You could have a very simply designed vehicle like a bob sleigh and an incredibly complex one like a plane, but the complexity of a plane doesn't arise from the basic properties that make it a vehicle.

You get the generic properties of all groups not specific blueprints. You cannot derive the structure of the Monster Group from the Group axioms hence its complexity does not arise from them.

This gets very complex, but at its most "basic" it's the set of symmetries of the Griess algebra. This is how it was first constructed. None of its particular structure follows directly from the Group axioms.

I'm not certain what "the reason for the structure" means.However all the complexity of the Monster group is related to it being the set of symmetries of the Griess algebra. It does not come from the Group axioms.It is literally impossible to derive the details of the Monster Group from the group axioms because they are generic conditions for all groups and hence say nothing about the specific structure of any group.The rest of your post appears to be an analogy for Platonism. Whether one is a Platonist and thinks the Monster Group "already existed" in some sort of eternal sense is not relevant to this point. Even if you are a Platonist you still cannot derive the structure of the Monster group from the group axioms.