In mathematical systems, axioms and proven implications of axioms (theorems) are tautological. On the other hand, unproven mathematical hypotheses may or may not have a known (or knowable) truth value, though they may have meaning. For example, the Goldbach Conjecture ("every even number greater than 2 can be written as the sum of two primes") is meaningful, though it may be unprovable. It is meaningful because we might find some even number that would violate the proposition.

Alone, empirical facts form a system of axioms and tautologies. If the mass, m, of a cylinder remains the same over some time period, we can restate these facts by saying that between time T1 and time T2 M(t) = m. This isn't a theory, just a tautological restatement of our observations.

A physical theory combines empirical facts (empirical axioms) with some set of mathematical axioms by creating a correspondence between the mathematics and the empirical axioms. Physical theories generate scientific hypotheses about which empirical facts can be added to the system with consistency. Experiments that confirm the theory are able to add the new empirical facts to the system as axioms of that system without breaking the consistency of the theory. Experiments that falsify the theory show that there are absolutely true empirical axioms that are inconsistent with the theory.

But where lies metaphysics? I see only two options:

1) Metaphysics might be purely mathematical, there being no special axioms of metaphysics except, perhaps, those axioms that we require for reasoning about the consistency of systems.

2) Metaphysics might somehow be grafted onto physical theories. If we add mathematics to a physical theory such that the mathematics has no implications for experiment, what have we done?

If case 1, then metaphysics would not be about the world, only a statement about possible mathematically consistent systems.

In case 2, we have more questions to ask.

Let's assume that the same metaphysical system is claimed to be true independent of the mathematical theory of the day. Yet, the mathematical theory is a function of empirical fact. As new empirical facts are discovered, the mathematical theory has to be modified. Empirical facts tell use what (infinite) set of possible mathematical theories are consistent with those facts ({E} constrains {T}).

We know that a metaphysical system does not imply anything about empirical facts ({M} does not constrain {E}). If it did, it would be empirically testable.

Therefore, to be about the world, metaphysical claims must have something to say about the allowed subset of physical theories we can use to explain empirical facts ({M} constrains {T}). The principle of causality might be such a metaphysical claim. Without causality, a mathematical theory might predict two outcomes for the same experiment. Causality limits the possible mathematical theories we can cook up.

However, if a metaphysical system does not restrict our physical theories, then that metaphysical system would be wholly detachable from physical theories, and we revert to case 1.

Propositions about God (e.g., 'God exists' and 'God is good') do not constrain empirical facts we might observe, i.e., they are not scientific. No surprise there. However, they also fail to constrain scientific theory-building. Therefore, it is totally separable from physical theories, so it must be pure mathematics, and not about the world.

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## 7 comments:

Hi Doc, you know what I like best about your blog? You always make me think! And that is something more Americans need to do--use their brain and think.

And the second thing I like about your blog? It doesn't matter that I disagree with you or not, it is that you have always treated me with respect in your comments to me even though I know deep down you probably would like to hit me over the head with your physics books!! ;-0}

Been great debating these days with you! Hope for more!

Propositions about God (e.g., 'God exists' and 'God is good') do not constrain empirical facts we might observe, i.e., they are not scientific. No surprise there. However, they also fail to constrain scientific theory-building. Therefore, it is totally separable from physical theories, so it must be pure mathematics, and not about the world.As I pointed out before on

Tu Quoque, you're setting up a false dilemma. You're saying: either metaphysics is bound up with (not totally separable from) physical theories, or it must be pure mathematics and thus not about the world. But as far as I can tell, you haven't given a reason for thinking that these are the only two possibilities. Why can't there be a third category, which has in common with mathematics that it is entirely separable from physical theories, and which has in common with physical theories that it is about the world (construed broadly as more than just the physical world)? If there's more to the world than just what is physical, then why can't we articulate theories of non-physical aspects of the world which would be about these aspects and yet would not constrain or be constrained by physical reality?Hi Peg,

Thank you for your kind words.

I'm very pleased to have you as a regular, here!

P.S. I'll get back to your FDR comment later today. :)

leibniz,

Why can't there be a third category, which has in common with mathematics that it is entirely separable from physical theories, and which has in common with physical theories that it is about the world (construed broadly as more than just the physical world)?Well, I think the burden of proof is on the metaphysician to show that metaphysical statements are about the world. If there is no process, mental or physical, for distinguishing metaphysics from mathematics, then I can't see how we can be justified in claiming they are different in anything but pretense.

The metaphysician must decide on some criteria for deciding whether a system is about the world or not in the absence of any experiential test.

You might claim that criteria is that there be certain axioms of the metaphysical system that are absolutely true across all similar systems. However, I see no mechanism for identifying such axioms (as opposed to scientific propositions which can be tested by any of us in the laboratory). If there is no way to distinguish metaphysical axioms as being about the world, then we are not justified in drawing such a distinction.

My personal theory is that mathematical statements are mistaken for propositions about the world (i.e., metaphysical) when they borrow symbols from physical theories.

P.S. I didn't see a new post at Tu Quoque, but I'm happy to continue here.

I think the burden of proof is on the metaphysician to show that metaphysical statements are about the world.The argument of your original post takes the form of a dilemma: either metaphysics is purely mathematical or it can be "grafted onto physical theories" (whatever exactly that means). But, you go on to argue, the second option isn't plausible for claims such as 'God exists' or 'God is good', so these claims must belong to pure mathematics, in which case they are not statements about the world but only statements about "possible mathematically consistent systems."

In response to this, I pointed out that you have overlooked a third possibility, namely, that metaphysics belongs to a kind of hybrid category. This is at least a prima facie possibility. After all, the vast majority of us believe that when we make claims about God (that he exists, or doesn't exist, or is good, etc.), we're making substantive claims about reality, i.e., the world broadly construed, and not just claims of pure mathematics. Since you are claiming that there are only two possibilities, but it is generally accepted that there is this third possibility, the burden of proof is actually on you to show that this third option isn't really available.

But does it not seem as though your claim rests on our emotional response to propositions? We

feelthat 'God is immortal' is about the world, so it must be.The problem with this thinking is that reason is no longer required, either to give meaning to propositions or to assert that they are absolutely true.

We can only rationally claim this third category if we have some formal mechanism for deciding whether a proposition is about the world or not.

I'll give you two more slightly-different phrasings of my argument.

1. A physical theory is a mathematical model of reality. It consists of mathematical relations and a map from the mathematics to empirical facts. For example, according to Hooke's Law, the force due to the compression of a spring is F = kx, where k is a constant and x is the displacement of the spring. There are an infinite number of mathematical systems that might include such a relation, but they are not about the world until the have a correspondence map to the world. Metaphysical theories are just like physical theories, but they lack a correspondence map. That makes them indistinguishable from mathematics.

2. Suppose I take a physical theory like Hooke's Law, and add some additional propositions that are both beyond measurement and unnecessary for prediction. For example, I state that there is some new quantity known as God which is some function of time and space G(x, t). However, I am not given any recipe for measuring G. Not only is it impossible to measure the value of G, or form any theories about it, but the G function has no effect on any other measurements of the system. Because it is asserted that G is not empirically measurable, it has to drop out of any formula for a physical quantity. It would be like stating that the force due to compression of a spring were

F = kx + G(x, t) - G(x, t)

= kx

This means that G is totally superfluous to the theory. It is no more about the world than any other mathematical formula that we might add to the theory.

Hello again, Dr. Logic.

But does it not seem as though your claim rests on our emotional response to propositions? We feel that 'God is immortal' is about the world, so it must be.No, it doesn't seem this way to me at all. You made an argument that relied on a certain dilemma. In my response, I pointed out that the dilemma overlooks a third option, and that since this

tertium quidis generally accepted as a possibility, the burden of proof is on you, the one setting up the dilemma, to rule it out. In no way does this reply rest on any emotional response. Nor does it involve the suggestion that it would be acceptable to infer that God is immortal from the fact that we feel that he is immortal. The point, again, is that since this third option is generally thought to be a possibility (by people who have thought about this issue), the burden is on the person setting up the dilemma to rule it out. I am therefore not recommending the inferencePeople generally feel that God is immortal. Therefore God is immortal.but rather:

People generally believe that metaphysics can be about the world, even though it cannot be "somehow grafted onto physical theories." Therefore, in setting up your dilemma, you cannot simply assume that this isn't a possibility; you must provide an argument showing it to be impossible.As to your most recent (1) and (2), you characterize them as restatements of your argument. But in truth they are only restatements, and elucidations, of your conclusion, without any argument for that conclusion being provided. In (1), for example, "Metaphysical theories are just like physical theories, but they lack a correspondence map" is a restatement of the conclusion of your original argument. But you have provided no reason in (1) (or (2), for that matter) for accepting this conclusion. So in all this we have been given no good reason for accepting your conclusion about metaphysics.

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