> Given how long it can take to make responses, for us to start dipping into doing two part posts when we aren't making much progress is likely a waist time for the both of us [...]
Sure, I felt a bit like doing that as well, especially when I couldn't submit my previous response in one piece ;-)
> What does it mean for something to be incompatible with science?
Well, I guess that's a question that can be answered in a million ways, depending on the context, but the relevant aspect to this discussion probably is this:
To be incompatible with science means to be making factual (as in real/non-formal) claims based on methodology that is not demonstrated to be reasonably reliable for determining the truth of such a claim.
> What determines that not being testable is "about as incompatible with science as you can get"?
The definition of what science/"the scientific method" is?
> I'm asking whether I am following your statements correctly. You, to paraphrase, seemed to have stated that we can only agree or disagree on definitions, having no reason to believe any definition is true because all definitions are arbitrary "by definition". Does this sum up your position correctly?
No, and that is for the simple reason that "having no reason to believe any definition is true" is a category error, and hence the question doesn't even make sense.
Let's say I define that X = 5. Is that definition true? Does it even make sense to ask whether me defining X as 5 is true?
The act of assigning meanings to labels is not in the category of things that have a truth value. You might as well be asking whether pouring juice into a glass is true. Or whether singing a song is true. When someone says "I will use the word 'apple' to refer to large, grey animals with trunks" (i.e., they give you the definition they are using), and you ask "but is that definition true?", that is simply a non-sequitur. It's as sensible as asking in response to "I will use 'X' to refer to 5" with "but is that definition true?".
And just to make sure this doesn't cause confusion: That does not mean that asking for the truth value of "'apple' as a word for large, grey animals with trunks is commonly accepted in the English language" is a category error. Whether something is a commonly accepted definition for a word in a widely used language is in the category of things that have a truth value, more specifically a "real truth" value, and obviously in this particular case that truth value is false.
> How is it really a succinct description if "lots of mathematical objects would fall under that definition that are not usually considered numbers"?
Well, then it's not. Combined with what I added after that, it still is, so what's the point of harping on this? I gave you a definition, you think half of that definition is not good enough ... well, how about you use the full definition, then? Especially when I obviously kindof agree that the first part is pretty ambiguous, which presumably is why I added the second part?
> What do you intend to mean by adding the modifier "conceptually" to "distinguish"? I probably agree, but I want to make sure I'm understanding you correctly.
I am simply trying to avoid the pitfall of hard solipsism. After all, every rock you see is mediated through a picture of that rock on your retina. If there actually is a retina, that is. So, can you really distinguish a rock from a picture of a rock?
I think none of that is really all that relevant to the question at hand, and it's kindof enough that you can conceptualize the difference, which is why I added the "(conceptually)", to avoid that possible distraction--but if you agree that we can distinguish a rock from a picture of a rock, then you might as well ignore it.
> I would add that you can distinguish a rock from a picture of a rock, from the word "rock", and from the concept of a rock.
Yeah, sure.
> Can or can you not do the same with mathematics, and why or why not?
Well, obviously not the same, as there is no such thing as "a picture of a mathematics"?! But obviously, we can, say, distinguish mathematics from the word "mathematics". So, I guess I am not really sure what the question is?
> So, in that sense, would you agree that the non-physical exists in reality?
I agree that instances of abstractions that are not concerned with physical representation do exist in reality. That does not mean that I agree that instances of those same abstractions exist without physical representation, for lack of evidence of that being a thing.
> How do you know that mathematicians don't intend to make statements about reality when they declare axioms?
Because that is how "axiom" is defined (by mathematicians)?
> Even if they don't intend to make claims about reality, couldn't mathematicians intend to do so?
You mean whether a hypothetical mathematician could be intending to make a claim about reality when declaring it an "axiom"?
Well, yes, in the sense that anyone can obviously redefine words however they want? It's the same sense in which I could be talking about a mammal when I say 'apple'. The point is, that still doesn't make elephants a fruit, so the question is kindof pointless. Even if you somehow manage to make someone apply the label 'axiom' to a claim about reality, that helps you nothing with trying to show that all of the not-a-claim-about-reality axioms are claims about reality.
> Yes, asking them to substantiate their claim does not mean that you are taking the opposite view point. Having not taking the affirmative or negative position means that you are taking the middle ground...
Well, if that is how you use the term, then that's how you use the term, I guess, but it seems to me that it's more appropriate for someone taking a position that is meaningfully on a scale between two other positions--which is very much distinct from not coming to a conclusion.
> That being said, asking someone to substantiate their claim does mean that you are claiming that their claim is unsubstantiated, meaning that you are not taking the middle ground regarding the evidence they do or don't have to back up their claim.
Sure. Though it's just as much just an expression of how convincing you find someone's arguments as it is some claim of an objective measure of substantiation.
> Your objection, that my claim is unsubstantiated, is an unsubstantiated claim. (This claim hear is unsubstantiated because I have given no evidence to support it. Evidence for or against it might be found in previous replies)
Well, sure, you can read it that way. But ultimately, in a discussion between two, it's about how convincing your arguments are to the other party, so it's kindof besides the point?
> Sure, but be careful to substantiate your rejections.
Yeah, I think I am. That is, I explain why your arguments are not convincing (unless they are ...).
> The assumption it self is unjustified, yes, but why am I unjustified in assuming it?
My point was that you are neither justified nor unjustified (in the sense of being (in)sufficiently justified), because that's a category error. To make an assumption means to declare that you will treat whatever you are saying you assume as true (for the purposes of a particular statement/discussion/argument/whatever), regardless whether there is justification for it or not. Assumptions are not in the category of things that have a level of sufficient justification. If you give a justification for a statement, that that by definition makes it not an assumption. "Justified assumption" is a contradiction in terms like "wet dryness".
> Arguments can't be made without assumptions, but the assumptions of arguments should be obvious.
And they should be something the other party agrees with, if you goal is to convince. If you assume something to be true that the other party doesn't (yet) think is true, any arguments that you build on that assumption will be unconvincing.
Which also means that if the other party challenges your assumption, you then should justify it (making it no longer an assumption).
> I think that "'1 < 2' has a referent" is an obvious assumption in light of the Peano axioms (in part since we both think that the Peano axioms exist (ignoring what we think regarding the nature of the Peano axioms and their derivations)).
Well, but that is what this is all about? If the Peano axioms are a purely artificial construct of humans, then "'1 < 2' absent humans" has no referent, and your hypothetical scenario that you were asking about was roughly the equivalent of that, arguably.
Also, it doesn't really make too much sense to respond to an objection to an assumption with "it should be obvious". I mean, you might have thought that, and that's why you made the assumption, and that's all fine. But the moment the other party tells you they don't agree with your assumption, the equivalent of "but I think you should agree" is not really going to help anyone.
> If you think that is not obvious, why?
Why I think that it is not obvious that '1 < 2' absent an evaluator aware of the Peano axioms (and more) has a referent? Because as far as I can see that is a statement in a formal system, which has no meaning without that formal system.
> If you think arguments shouldn't make assumptions at all, then tell me why and give me an example argument with no assumptions.
I am not saying that at all. It just doesn't make sense to make "justified assumptions", because that's a contradiction in terms.
> So if the "copies" are material, are the "originals" also material?
How would I know? I mean, I have only ever experienced material (in the farthest sense) copies and originals, so I see no reason to think that non-material ones are a thing. But who knows, maybe you have the evidence for non-material copies of movies?!