The two meanings of mathematical terms

"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 that are verbally simple and concepts that are mathematically simple, which EY and others have referenced repeatedly here and at OB.) And while the device of the Equant* is fairly simple mathematically, it would raise so many mechanical complications that Copernicus rejected it, not because it's inaccurate, but because he considered it too mechanically complicated.

Ptolemy also tended to demonstrate equivalence between two ways of accounting for observations, which again suggests that he was not trying to describe the mechanics of the planets' motion, but rather only the mathematics of their motion.

I am not familiar with later Geocentric astronomy, and it may for all I know be the case that later thinkers thought that the epicycles had their own existence and moved the planets mechanically, but the history of astronomy is a little more complex than the popular account of the Copernican revolution would suggest.

If anything, Copernicus's insight was to permit the mathematics to inform his physical judgment rather than the other way around. Ptolemy rejected the heliocentric hypotheses intentionally and explicitly on common-sense grounds. Copernicus permitted the famous Ptolemaic coincidences (e.g. the fact that all the planets' epicycles tend to follow the mean motion of the sun) to suggest the sun as a more simple and natural center.

*The Equant is a device for describing planets whose uneven motion relative to the earth cannot be adequately accounted for by eccentricity and epicycles. In essence, Ptolemy assigns three centers to the planet's motion. One is the earth (the point of observation), another is the center of the circle the planet describes in space (the eccenter), and the third is the point with respect to which the angular motion of the planet is constant (i.e. the planet would appear to be revolving at a uniform rate if observed from this third point).

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 classical logic"

• "I am an infinite set atheist - have you ever actually seen an infinite set?"

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 hand, having the ability to articulate that concept in words.

It is more difficult to say what mathematical statements mean.

Often, a mathematical statement will just be about the mathematical concepts it mentions. A statement of geometry, for example, is often just about the concepts of triangle, circle, or whatever geometrical concepts it mentions.

However, in a mathematician's working practice, these statements often stop being about the original mathematical concepts. Rather, they might become about isomorphism classes of concepts (which are themselves concepts), where "isomorphism" is taken in the sense of some mathematical theory. Or the statements might become mathematical concepts in their own right. When the mathematician thinks of them this way, they no longer refer to anything outside of themselves. They are themselves the precise concepts being studied mathematically.

Thus, different utterances of a mathematical statement can mean different things. That is, the things that it refers to, and what it asserts about those things, can change with different utterances (while never being about actual physical things, which is the domain of physics). Usually, mathematicians try to arrange it so that a given statement has (provably) the same truth value in all cases, but not always.

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.

Formatting point: please use the "summary break" button when you have a long post.

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 them that way when talking technically about the meanings of words. They are very useful jargon.

Similarly, please never use 'begs the question' or any form of it when not referring to the logical fallacy.