"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
-- Albert Einstein
You seem to be trying to, somewhat independently of the way it is done in science and mathematics typically, arrive at distinct concepts of accurate, precise, and predictive. At least how I'm using them, these terms can each describe a theory.
Precise - well defined and repeatable, with little or no error.
Accurate - descriptive of reality, as far as we know.
Predictive - we can extend this theory to describe new results accurately.
A theory can be any of these things, on different domains, to different degrees. A mathematical theory is precise*, but that need not make it accurate or predictive. It's of course a dangerous mistake to conflate these properties, or even to extend them beyond the domains on which they apply, when using a theory. Which is why these form part of the foundation of scientific method.
*There's a caveat here, but that would be an entire different discussion.
I have not read Kuhn's work, but I have read some Ptolemy, and if I recall correctly he is pretty careful not to claim that the circles in his astronomy are present in some mechanical sense. (Copernicus, on the other hand, literally claims that the planets are moved by giant transparent spheres centered around the sun!)
In his discussion of his hypothesis that the planets' motions are simple, Ptolemy emphasizes that what seems simple to us may be complex to the gods, and vice versa. (This seems to me to be very similar to the distinction between concepts ...
Please see this previous comment of mine.
The point here is that it "1+1=2" should not be taken as a statement about physical reality, unless and until we have agreed (explicitly!) on a specific model of the world -- that is, a specific physical interpretation of those mathematical terms. If that model later turns out not to correspond to reality, that's what we say; we don't say that the mathematics was incorrect.
Thus, examples of things not to say:
"Relativity disproves Euclidean geometry."
"Quantum mechanics disproves classica
As the other commenters have indicated, I think that your distinction is really just the distinction between physics and mathematics.
I agree that mathematical assertions have different meanings in different contexts, though. Here's my attempt at a definition of mathematics:
Mathematics is the study of very precise concepts, especially of how they behave under very precise operations.
I prefer to say that mathematics is about concepts, not terms. There seems to me to be a gap between, on the one hand, having a precise concept in one's mind and, on the other...
Grahams point s straightforward if expressed as 'pure maths is the study of terms with precise meannigs'.
Which begs the question of whether it really matters if the conceptualisation is wrong, as long as the numbers are right? Isn’t instrumental correctness all that really matters?
I'm not in the business of telling people what values to have, but if you are a physcalist, you are comited to more than instrumental.
The fact that predictiveness has almost nothing to do with accuracy, in the sense of correspondence is one of the outstanding problems with physicalism
Relativity teaches us that "the earth goes around the sun" and "the sun goes around the earth, and the other planets move in complicated curves" are both right. So to say, "Those positions [calculated by epicycles] were right but they had it conceptualised all wrong," makes no sense.
Hence, when you say the epicycles are wrong, all you can mean that they are more complicated and harder to work with. This is a radical redefinition of the word wrong.
So, basically, I disagree completely with your conclusion. You can't say that a representation gives the right answers, but lies.
Isn’t instrumental correctness all that really matters? We might think so, but this is not true. How would Pluto’s existence been predicted under an epicycles conceptualisation? How would we have thought about space travel under such a conceptualisation?
Your counterexamples don't seem apposite to me. Out of sample predictive ability strikes me as an instrumental good.
To add to what others have already commented...
It is theoretically possible to accurately describe the motions of celestial bodies using epicycles, though one might need infinite epicycles, and epicycles would themselves need to be on epicycles. If you think there's something wrong with the math, it won't be in its inability to describe the motion of celestial bodies. Rather, feasibility, simplicity, usefulness, and other such concerns will likely be factors in it.
While 'accurate' and 'precise' are used as synonyms in ordinary language, please never use ...
Well I agree that I can think just with verbal signs, so long as the verbal sentences or symbolic statements mean something to me (could potentially pay rent*) or the symbols are eventually converted into some other representation that means something to me.
I can think with the infinity symbol, which doesn't mean anything to me (unless it means what I first said above: in short, "way big enough"), and then later convert the result back into symbols that do mean something to me. So I'm fine with using infinity in math, as long as it's just a formalism (a symbol) like that.
But here is one reason why I want to object to the "realist" interpretation of infinity via this argument that it's just a formalism and has no physical or experiential interpretation, besides "way big enough": The Christian god, for example, is supposed to be infinite this and infinite that. This isn't intended - AFAIK - as a formalism nor as an approximation ("way powerful enough"), but as an actual statement. Once you realize this really isn't communicating anything, theological noncognitivism is a snap: the entity in question is shown to be a mere symbol, if anything. (Or, to be completely fair, God could just be a really powerful, really smart dude.) I know there are other major problems with theology, but this approach seems cleanest.
*ETA: This needs an example. Say I have a verbal belief or get trusted verbal data, like a close friend says in a serious and urgent voice, "(You'd better) duck!" The sentence means something to me directly: it means I'll be better off taking a certain action. That pays rent because I don't get hit in the head by a snowball or something. To make it into thinking in words (just transforming sentences around using my knowledge of English grammar), my friend might have been a prankster and told me something of the form, "If not A, then not B. If C, then B. If A, then you'd better duck. By the way, C." Then I'd have to do the semantic transforms to derive the conclusion: "(I'd better) duck!" which means something to me.
To know reality we employ physics. Physics employs calculus. Calculus employs limits. Limits employ infinite sequences. Does that pay enough rent?
[edit: sorry, the formatting of links and italics in this is all screwy. I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]
In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham: