One of the questions I've been wanting to think about (in part because of my interest in the Quine-Carnap relationship) but haven't really got around to yet is: Is there any important sense in which arithmetic is empirical? I know there is some good literature on the subject, but I've thus far only perused it without really digging into it.
For me, one consideration that makes me think it might not be crazy to think of arithmetic as empirical is what happened with geometry and general relativity. If Einstein can show that the space in which we live is non-Euclidean, isn't it at least imaginable that some future scientist will show us that the 'true' arithmetic of our physical world is non-classical (which I suppose means: it does not obey the Peano axioms). [There could still be a mathematical structure that obeys classical arithmetic, just as Euclidean space is still a mathematical object that obeys all five of Euclid's axioms.]
However, I've always had a hard time imagining what possible observation could cast doubt on classical arithmetic. In last week's Science, there's a report that at least might merit consideration as a candidate. Researchers found that if you add one photon to a light beam and then take one away, you observe a different end-state than if you reverse the order of operations, i.e., first remove one and then add one. In other words, x + 1 - 1 does not equal x - 1 + 1. Even stranger, the authors find that "under certain conditions, the removal of a photon from a light field can lead to an increase in the mean number of photons in that light field," that is, (roughly) that x-1>x. The summary and background for non-specialists is here, and the full technical report is here (both behind subscription walls).
Now, this effect depends on the failure of commutation relations ubiquitous in quantum mechanics, so it is quite possible that this in no sense makes arithmetic look empirical. But I'm not 100% sure about that. Any thoughts?
9 comments:
I think two things should be distinguished:
a)a state of affairs falling under a certain mathematical description, but mathematical truths of that description not holding
b)a state of affairs not falling under a certain mathematical description that we thought it is.
Only if there are cases like a) possible, I think we could talk about the empiricism of math.
But I think all 'problematic' examples are of type b).
Einsteinian Relativity showed us not a), but b). We just applied wrong mathematical description to space. It didn't mean that Euclidean geometry is wrong by itself, but that it is wrong one to apply to space/time. After all Non-Euclidean geometry is not created on base of empirical research, measurement and so on... It is as a priori as Euclidean one.
And if really math was empirical then Euclidean theory would be wrong everywhere. And that is not the case. If you have an Euclidean system (e.g. system with bodies which don't move relatively one to another), Euclidean truths will hold for this system still.
So, what might be empirical is which a priori system we can apply to some system (and that in my humble opinion is a metaphysical issue too, but I see Relativity as very metaphysical theory).
You also say:
"Researchers found that if you add one photon to a light beam and then take one away, you observe a different end-state than if you reverse the order of operations, i.e., first remove one and then add one. In other words, x + 1 - 1 does not equal x - 1 + 1."
I think you are mixing a) and b) again. It doesn't mean that x-1+1 does not equal x-1+1. (as I said, this would mean that it won't be true everywhere, which is obviously not true). It just means that classical arithmetic can't be applied to what happens with photons.
How can this be?
The most obvious answer is because the system is more complex that given mathematical model.
If you want more simpler example, if you put male and female rabbit in a room, and give it some time, you will get not two but multiple rabbits. Does that invalidate 1+1=2. No, just that simple arithmetic isn't sufficient to cover what is happening with the system of two rabbits.
(I've tried to post this a few times, so sorry if it turns out to be a repeat.)h
The problem is that different physical systems will correspond to different types of mathematics (often inconsistent). So for example the intensity of light does not obey the laws of arithmetic (at least on a certain level). But then there are many other physical quantities that obey the laws of arithmetic. A lot depends on which physical terms you are deciding to 'subject' to arithmetic. For example, if you view QM as certain axioms acting on a Hilbert Space with certain properties, blah blah, there is nothing in there that 'contradicts' arithmetic (the commutation relations are relations of functions, not of real numbers). It's only when you choose to interpret certain quantities as numbers that non-commutativity violates classical arithmetic.
I think Tanasije Gjorgoski's distinction (if I understand it correctly) necessarily collapses. The criterion for a state of affairs S' falling under a mathematical description D is that D "holds" in all respects relevant to S. If it doesn't, some other description D* is required. Conversely, to see that D is not the right description for S, we have to see that D in some relevant respect doesn't hold.
Anyway, thought I'd quote Einstein himself on the matter: "To the extent math refers to reality, we are not certain; to the extent we are certain, math does not refer to reality."
Hi Drake,
If you are saying that by thing falling under mathematical description, it is necessary that all truths that go with that description are truths about that system, I agree.
But I think that nothing less is claimed by a claim that "math is empirical". i.e. that a math description can be applied to a system, and yet (somehow magically) it will not obey the rules of the system.
That's why I pointed that what we have is actually (b), and that those cases don't make the math empirical, but just our knowledge of which mathematical description fits the given system.
Ponder --
You're exactly right about the commutation relation failing between operations, not numbers; in the present case, what I was roughly calling '+1' is really represented in QT as the creation operator, and '-1' is the annihilation operator. But I still wonder whether the deeper worry (viz., is math empirical in some important sense) is there, after we start speaking strictly instead of loosely. Is your statement "the intensity of light does not obey the laws of arithmetic" enough to show that arithmetic is in SOME sense empirical?
(This 'in some sense' business is, I think, related to the other issue that Drake and Tanasije are discussing: as I sort of said in the original post, if you believe in a realm of abstract objects (perhaps residing in Plato's intelligible realm), then it seems likely every consistent mathematical system will be true of SOMEthing. That sort of math -- God created the integers, and we're studying them in number theory -- will probably never be reasonably construed as empirical. But if the laws of arithmetic fail of certain physical systems -- or even just can be coherently imagined to fail -- then it seems to me that there is some non-trivial sense in which arithmetic is empirical.
Here's an argument someone might try to use to show that relativity implies that not just Euclidean geometry, but Peano arithmetic, is empirically false. In a Newtonian world, we thought that there were unique parallel lines, and that distances had a Euclidean metric. We also thought that if two things are moving in opposite directions from me with velocities A and B, then they are moving apart from each other with velocity A+B. However, with relativity, we discover not just that the lines and distances work differently, but that these two objects are now moving apart from each other at a rate of (A+B)/sqrt(1-(A+B)^2/c^2), or something like that. So addition doesn't work the way we thought.
Of course, this assumes that addition is defined by an operation on velocities, just as we assumed geometry was defined by relations among things in space.
I suspect that arithmetic is used in so many different ways in different domains, that failure of the usual laws in any one of them would cast no doubt on the notions of arithmetic - arithmetic is inherently abstract, rather than being about any physical thing. And whatever more complicated operation the physical things do obey is often eventually understood in mathematical terms based on the arithmetical operations, so they end up supporting it anyway.
As kenny says, arithmetic is inherently abstract. If some stuff did not obey arithmetic (at any level), it would not be composed of objects. Or, to think of something as a thing is to think of it as one thing, and to make the thought of N similar things possible.
Maybe there are no such things, and maybe arithmetic does not apply empirically; but arithmetic only requires the possibility that there are, that it does.
Still, maybe our concept of arithmetic is not actually coherent (for all that we believe, and must assume, that it is). What if we got excellent empirical evidence that (our conception of) arithmetic was the product of an evil scientist's (or demon's) hypnotic powers? Would that not refute arithmetic empirically? As greg says, we only need that to be remotely possible (imaginable)...
Kenny and Enigman --
What makes arithmetic "inherently abstract" (and geometry perhaps not)?
This is meant as a sincere question, not a thinly veiled challenge. I'm genuinely curious what your answers are, in part because Carnap says a similar thing in 1939's Foundations of Logic and Mathematics, and I've never felt like I understood where he is coming from. (The way he puts the point there is that the intended interpretation of geometry is for physical lines etc., while the intended interpretation of arithmetic is not for any physical thing. And he just sort of leaves it at that, as far as I can tell.)
I wrote a post extending on my comment here
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