[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:
It would not be a bad definition of math to call it the study of terms that have precise meanings.
Sort of. I started writing a this as a reply to that comment, but it grew into a post.
We've all heard of the story of epicycles and how before Copernicus came along the movement of the stars and planets were explained by the idea of them being attached to rotating epicycles, some of which were embedded within other larger, rotating epicycles (I'm simplifying the terminology a little here).
As we now know, the Epicycles theory was completely wrong. The stars and planets were not at the distances from earth posited by the theory, or of the size presumed by it, nor were they moving about on some giant clockwork structure of rings.
In the theory of Epicycles the terms had precise mathematical meanings. The problem was that what the terms were meant to represent in reality were wrong. The theory involved applied mathematical statements, and in any such statements the terms don’t just have their mathematical meaning -- what the equations say about them -- they also have an ‘external’ meaning concerning what they’re supposed to represent in or about reality.
Lets consider these two types of meanings. The mathematical, or ‘internal’, meaning of a statement like ‘1 + 1 = 2’ is very precise. ‘1 + 1’ is defined as ‘2’, so ‘1 + 1 = 2’ is pretty much the pre-eminent fact or truth. This is why mathematical truth is usually given such an exhaulted place. So far so good with saying that mathematics is the study of terms with precise meanings.
But what if ‘1 + 1 = 2’ happens to be used to describe something in reality? Each of the terms will then take on a second meaning -- concerning what they are meant to be representing in reality. This meaning lies outside the mathematical theory, and there is no guarantee that it is accurate.
The problem with saying that mathematics is the study of terms with precise meanings is that it’s all to easy to take this as trivially true, because the terms obviously have a precise mathematical sense. It’s easy to overlook the other type of meaning, to think there is just the meaning of the term, and that there is just the question of the precision of their meanings. This is why we get people saying "numbers don’t lie".
‘Precise’ is a synonym for "accurate" and "exact" and it is characterized by "perfect conformity to fact or truth" (according to WordNet). So when someone says that mathematics is the study of terms with precise meanings, we have a tendancy to take it as meaning it’s the study of things that are accurate and true. The problem with that is, mathematical precision clearly does not guarantee the precision -- the accuracy or truth -- of applied mathematical statements, which need to conform with reality.
There are quite subtle ways of falling into this trap of confusing the two meanings. A believer in epicycles would likely have thought that it must have been correct because it gave mathematically correct answers. And it actually did . Epicycles actually did precisely calculate the positions of the stars and planets (not absolutely perfectly, but in principle the theory could have been adjusted to give perfectly precise results). If the mathematics was right, how could it be wrong?
But what the theory was actually calcualting was not the movement of galactic clockwork machinery and stars and planets embedded within it, but the movement of points of light (corresponding to the real stars and planets) as those points of light moved across the sky. Those positions were right but they had it conceptualised all wrong.
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? 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?
The moral is, when we're looking at mathematical statements, numbers are representations, and representations can lie.
If you're interested in knowing more about epicycles and how that theory was overthrown by the Copernican one, Thomas Kuhn's quite readable The Copernican Revolution is an excellent resource.
"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:
... (read more)"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)
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)
Grahams point s straightforward if expressed as 'pure maths is the study of terms with precise meannigs'.
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.
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.