> If you agree with this definition, then do you also think that there is "completely" NO REASON to assume that the Peano axioms are true? If so, is this because the Peano axioms are merely assumptions, and not themselves proven?
Well, yes, that is what axioms are, by definition?
(And mind you, we are talking about pure mathematics here--there absolutely are good reasons for generally accepting certain axioms while rejecting others because of the applicability of the mathematics that can be derived from them, but that is where you enter the realm of the testable.)
> How would someone rejecting axioms, make statements consistent with those axioms untrue?
Well, how is truth of mathematical statements determined? How would you determine that "1 + 9 = 10" is mathematically true without the Peano axioms (and without new axioms that replace them, obviously)?
> I'll agree that the truth of any mathematical statement is dependent upon the framework under which you view it, but that doesn't stop the statement from being true or false under those frameworks regardless of which frameworks a person chooses to use. Those frameworks don't depend on human minds, so even before mathematical axioms were formulated by humans "1 + 9 = 10" was still true under the Peano axioms, and that will still be true after humans have died off. That which is true is "in accordance with fact or reality".
That sounds like one big category error?
Mathematics is pretty much like a language. You might as well be saying that "The sky is blue" was a meaningful sentence before English was invented. It seems like you are constantly crossing the border between pure mathematics (analogous to the grammar and vocabulary of a language) and its application (analogous to the semantics of a language). The sky was very much blue before English was a thing, but "The sky is blue" was not a meaningful sentence. In the same way, those things that we describe using mathematics had the same structure before mathematics was invented. When there was one rock in some place, and 9 rocks fell from the sky next to it, there were then 10 rocks in that place. But that is obviously a testable claim (about (physical) reality), not an untestable proof (derived from axioms).
If I were to try and boil down your argument, I think it would be this:
We use mathematics to describe physical reality. Physical reality with the patterns that we describe using mathematics existed before we existed. Therefore, mathematics existed before we existed.
And analogously:
We use English to describe physical reality. Physical reality with the patterns that we describe using English existed before we existed. Therefore, English existed before we existed.
Yes, the patterns existed. But the patterns are a perfectly testable thing. So, you can't use the testable patterns to demonstrate the existence of untestable abstract ideas.
> A) If potential states of the universe do not exist, then the universe can't change states
Could you define what you mean by the word "exist"?
> C) If potential states of the universe are physical, then potential states of the universe would be observable hear and now.
I can not observe the contents of your screen here and now. Does that mean that your screen is not physical?
> E) Potential states of the universe are not actual and potential at the same time.
So, the universe does not exist? (You defined "potential" as "possible", so if a state can not be actual and potential at the same time, all actual states are impossible, hence there is no state for the universe to be in, thus the universe can not exist, right?)
(And yeah, I suspect you mean something else, but it's kinda difficult to point out the flaws in an argument if you don't quite know what the argument is ...)
Overall, I suspect this is all one big exercise in false equivocation, but we'll first have to nail down some definitions and details before I can put my finger on it.
> Saying I intentionally avoided making a claim is disingenuous, you may have merely misinterpreted what I was trying to say.
Well, yeah, let me rephrase: You probably didn't have the intention of expressing something that is obviously identifiable as a claim (because it's a pretty strange way to formulate a sentence that is intended to express a claim)?
> How might you respond to a rephrasing such as "It is immoral to <X>"?
Assuming I didn't agree, I would probably ask you for the evidence to support that claim.
If you were to object that asking for evidence for moral claims is a category error (which I suspect is where you are going with this), I would ask you to define "morality".
> "God is a trinity of three divine persons each fully possessing one divine nature."
Does this god have any observable effects in physical reality?
Well, yes, that is what axioms are, by definition?
(And mind you, we are talking about pure mathematics here--there absolutely are good reasons for generally accepting certain axioms while rejecting others because of the applicability of the mathematics that can be derived from them, but that is where you enter the realm of the testable.)
> How would someone rejecting axioms, make statements consistent with those axioms untrue?
Well, how is truth of mathematical statements determined? How would you determine that "1 + 9 = 10" is mathematically true without the Peano axioms (and without new axioms that replace them, obviously)?
> I'll agree that the truth of any mathematical statement is dependent upon the framework under which you view it, but that doesn't stop the statement from being true or false under those frameworks regardless of which frameworks a person chooses to use. Those frameworks don't depend on human minds, so even before mathematical axioms were formulated by humans "1 + 9 = 10" was still true under the Peano axioms, and that will still be true after humans have died off. That which is true is "in accordance with fact or reality".
That sounds like one big category error?
Mathematics is pretty much like a language. You might as well be saying that "The sky is blue" was a meaningful sentence before English was invented. It seems like you are constantly crossing the border between pure mathematics (analogous to the grammar and vocabulary of a language) and its application (analogous to the semantics of a language). The sky was very much blue before English was a thing, but "The sky is blue" was not a meaningful sentence. In the same way, those things that we describe using mathematics had the same structure before mathematics was invented. When there was one rock in some place, and 9 rocks fell from the sky next to it, there were then 10 rocks in that place. But that is obviously a testable claim (about (physical) reality), not an untestable proof (derived from axioms).
If I were to try and boil down your argument, I think it would be this:
We use mathematics to describe physical reality. Physical reality with the patterns that we describe using mathematics existed before we existed. Therefore, mathematics existed before we existed.
And analogously:
We use English to describe physical reality. Physical reality with the patterns that we describe using English existed before we existed. Therefore, English existed before we existed.
Yes, the patterns existed. But the patterns are a perfectly testable thing. So, you can't use the testable patterns to demonstrate the existence of untestable abstract ideas.
> A) If potential states of the universe do not exist, then the universe can't change states
Could you define what you mean by the word "exist"?
> C) If potential states of the universe are physical, then potential states of the universe would be observable hear and now.
I can not observe the contents of your screen here and now. Does that mean that your screen is not physical?
> E) Potential states of the universe are not actual and potential at the same time.
So, the universe does not exist? (You defined "potential" as "possible", so if a state can not be actual and potential at the same time, all actual states are impossible, hence there is no state for the universe to be in, thus the universe can not exist, right?)
(And yeah, I suspect you mean something else, but it's kinda difficult to point out the flaws in an argument if you don't quite know what the argument is ...)
Overall, I suspect this is all one big exercise in false equivocation, but we'll first have to nail down some definitions and details before I can put my finger on it.
> Saying I intentionally avoided making a claim is disingenuous, you may have merely misinterpreted what I was trying to say.
Well, yeah, let me rephrase: You probably didn't have the intention of expressing something that is obviously identifiable as a claim (because it's a pretty strange way to formulate a sentence that is intended to express a claim)?
> How might you respond to a rephrasing such as "It is immoral to <X>"?
Assuming I didn't agree, I would probably ask you for the evidence to support that claim.
If you were to object that asking for evidence for moral claims is a category error (which I suspect is where you are going with this), I would ask you to define "morality".
> "God is a trinity of three divine persons each fully possessing one divine nature."
Does this god have any observable effects in physical reality?