> So, we can only agree or disagree, and there is no reason to believe your definition of 'definition'?
I'm not really sure what you are asking?!
I am telling you what I think the generally accepted definition is. That's an empirical claim, and it's up to you whether you take my word for it, check other sources, or do an investigation yourself, just as with every other empirical claim?
> How might we empirically determine whether such agreements exist without having agreement on what the term 'agree' means?
By not using the term 'agree' in the empirical investigation?! Though I don't quite see what the point of that limitation is? If there is agreement on what 'agree' means, as seems to be common among users of the English language, then why would you want to avoid the term?
> How do you determine that there are actual unresolved references, and that it is not just a deficiency in your definitions?
I am not quite sure what you mean by "deficiency in the definitions"?
If you mean that there are terms in the statement that I have no definition for, then I try to obtain a definition for those terms. So, if I can ask whoever made the statement, I might ask them for the definition they were using. If that isn't an (easy) option, I might look it up in dictionaries. Or I might try to derive it from the context in which the statement was made (in particular when I do know definitions, but it's multiple mutually-exclusive ones).
If you mean that my definitions are not the ones that the person making the statement meant ... well, then I get the wrong idea of what they were saying?
But mind you that when I said "there are unresolved references remaining", I don't mean "unresolved references to definitions". When I evaluate "Polar bears are green", then "polar bear", for example, refers to a concept that refers to an observable entity. So, I can resolve "polar bear" to that concept using the definition of "polar bear". But at the end there remains the reference to the observable thing. That's what I am talking about.
> Why is that your definition of "true"?
Because that is the generally accepted definition, and as such it is useful to have a definition that matches what other people understand when you use the term. Also, it's probably the generally accepted definition because those two concepts are useful, even though having them both use the same label maybe is unfortunate.
> Also what are your definitions of "consistent" and "derived"?
In this particular context:
Consistent: Not leading to contradictions.
Derived: Arrived at by applying transformations/identities specified by the axioms. You could also say proven using the axioms.
> It seems that you agree with calling 'consistent with reality' a 'real truth' and 'derived from axioms' a 'formal truth'.
Seemed like a useful short hand, so I went with it, sure.
> Now you've stated that "... every 'formal truth' is a 'real truth' ...". Wouldn't that in some way negate the most of the emphasis provided by the term 'formal truth'?
No, that is just a confusion of layers of abstraction. Within the framework of mathematics, noone empirically investigates whether "1 + 9 = 10", because that would be a category error, simply by definition. Within mathematics, 1 + 9 = 10 is defined to be formally true (unless explicitly stated that standard axioms don't apply), but noone is making a claim about real truth, which is why investigating that question empirically is a category error: Empirical investigation applies to claims of real truth, not to claims of formal truth, just as proofs apply only to formal truths, but not to real truths.
But that doesn't mean that the claim that "'1 + 9 = 10' is a formal truth in mathematics" isn't an empirical claim. For every supposed formal truth in mathematics, you can investigate empirically whether that is indeed the case, namely by investigating what the consensus on definitions among mathematicians is.
> A number is 'a more abstract word'??
Seems like a pretty succinct description? Though, as I also wrote, it probably needs to be usable for measuring or counting, otherwise lots of mathematical objects would fall under that definition that are not usually considered numbers.
> Does that mean that the math isn't a model of reality though?
Yes, of course, that's exactly what it means!
Mind you though that that doesn't mean that some parts of mathematics can't be used to model some aspects of reality. But that is a matter of empirical investigation.
Does Newton's law of gravity predict the movement of masses that don't collide? That is an empirical question of whether a particular mathematical structure is useful for modeling an aspect of reality. As it turned out, it is quite useful indeed. But as it later also turned out, it's not actually a (perfect) model of reality, and the theory of relativity gives more accurate results, even though it uses a quite different mathematical structure. But then, there is no reason to think that that is a perfect model of reality either.
> Does that establish that the content of a thought doesn't really exist though?
I'm not sure what you are trying to say here? What do you mean by "really exist"? Are you saying that someone who is hallucinating doesn't have those ideas that we usually call "hallucinations"?
> Are you making a distinction between a representation and the thing being represented?
Well, yes?!
> It seems that your definition of 'representation' also includes the thing being represented, but that is a definition you deny later on when you state "For one, there is reason to think that things that are not (just) representations do exist, like, say, a rock (unless you count everything as a representation of itself, then I guess that's true by definition, but there is still a distinction between self-representation and "other-representation")".
I'm not sure where you see a contradiction here?! I am just pointing out that no matter how you prefer to generally define 'representation' (and different definitions are useful in different contexts), you can distinguish between things representing themselves and things representing other things, i.e., you can (conceptually) distinguish a rock from a picture of a rock or from the word "rock".
> Again this could be another case of confusing abstraction and the underlying representation, and if that is the case, what is 'abstraction', what does it mean for something to be an abstraction, and do particular abstractions have physical representation?
I suppose it is.
When you say "movie X is funny", it is completely irrelevant to the meaning of that statement how that movie is physically represented. The thing that you refer to as "the movie" here is a concept that is not concerned with material aspects. You can use pretty much any object (as long as it has sufficient degrees of freedom) to encode that movie into it, and that statement would still have the same meaning.
So, movies are not material in the sense that any statement that you make about a movie retains its meaning no matter what representation you choose for the movie/the statement applies to all representations of it that there are, have been, or will be. Really, as far as the abstraction is concerned, things would still work perfectly fine with a movie that had no physical representation--if that were a thing somehow.
As for what an abstraction is, I think the summary at the top of the wikipedia article is a good one:
https://en.wikipedia.org/wiki/Abstraction
In the specific case, the material representation has no subjectively valued purposes to the concept of a movie.
As to whether particular abstractions have physical representation: Well, given that our brains constantly operate using abstractions (all of language is nothing but abstractions), I would say that obviously yes?!
Mind you, none of this is about "empirical properties of movies" or anything of that sort. The claim isn't "if you take a DVD and put it under the microscope, you will find that it is immaterial" or anything like that. It's simply a matter of what abstractions are concerned with by definition, and the definition of "movie" simply is not concerned with physical representation.
> Now, if you're not saying "that all representations are physical", and are saying "that no non-physical representation has ever been demonstrated", wouldn't that, at the very least, leave the question open to axioms being claims about reality?
No. What claims are about is a matter of intention, and mathematicians don't intend to make statements about reality when they declare axioms, thus they are not claims about reality. It's simply a matter of definition: Axioms are defined to be unproved statements that are assumed to be (formally) true for the purpose of making derivations from them.
When I say "Joe is a baker", that doesn't leave the question open whether I am talking about John, and when a mathematician says "I will assume that 1 is a natural number", that doesn't leave the question open whether it's an assumption.
That doesn't mean that John isn't a baker, nor that the structure that can be derived from the Peano axioms can't be used to model aspects of reality. But that's not what either of those claims is about.
> Why are you equating the "symbolic or conceptual representation" of '1 < 2' to the "evaluation rules" of '1 < 2'?
I am not. I am saying that there is no such thing as the meaning of a statement in a formal language absent the rules of that formal language. It's like asking "is a pun funny regardless which language it is in?" It's a category error. The pun is a property of the language, or possibly of a family of related languages: If you remove the language from it, there is no pun left. "A pun in English, but absent English" has no referent.