"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 ... (read more)

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

Or: "Physics is not Math"

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... (read more)

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.

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 ... (read more)