Because I’m not trying to make any assertion about the necessity or contingency of the truth of math.
Then I do not follow your original point.
[Math and physics] are both different from the statement that the sky is blue.
Presumably because there needs to be an observer to see the blue, and different people may draw different delineations between what is blue vs. purple vs. teal? If you describe it in wavelengths, is it the same as physics? If not, what is it that makes it not?
Personally, I think math (but definitely not physics) is objective, because there are important similarities between the statement of modus ponens and the statement that 2 + 2 = 4.
This seems to just be a fact about the territory. Presumably, there are similar similarities between a perfectly accurate and completely specified quantum theory and chemical reactions, they are just harder to work out.
[Math and physics] are both different from the statement that the sky is blue.
That is not a correct edit of what I said. At the level of generality I’m talking, “The sky is blue” is a physics statement. A better summary would be “[Math and logic] are both different from the statement that the sky is blue.
Personally, I think math (but definitely not physics) is objective, because there are important similarities between the statement of modus ponens and the statement that 2 + 2 = 4.
This seems to just be a fact about the territory. Presumably, there are similar similarities between a perfectly accurate and completely specified quantum theory and chemical reactions, they are just harder to work out.
Fair enough. But I assert that the particular similarity we are discussing does not exist between math and physics, no matter how hard we look.
Fair enough. But I assert that the particular similarity we are discussing does not exist between math and physics, no matter how hard we look.
Can you clarify the grounds for that assertion? It seems to me that in both cases the territory contains facts and rules, and combinations of these lead to other consequences. The big difference seems, to me, to be a historical question of where we started our mapping; with physics, we started looking at consequences very far removed from the facts and rules that are rigidly true, and so our map was by necessity fuzzier. This is a fact about the map, not the territory, though.
I have heard a Strong Reductionist theory, which goes as follows:
History is Psychology. Psychology is Biology. Biology is Chemistry. Chemistry is Physics. Physics is Math.
In other words, you can (in principle) derive the higher order theory from the more fundamental theory.
If for no other reason, I reject that theory because Physics is not Math. Physics is inherently about making statements that have empirical content. Math is inherently about making statements that lack empirical content.
I have heard a Strong Reductionist theory, which goes as follows: History is Psychology. Psychology is Biology. Biology is Chemistry. Chemistry is Physics. Physics is Math. In other words, you can (in principle) derive the higher order theory from the more fundamental theory.
For any deterministic interpretation of quantum mechanics, it seems that this would necessarily have to hold (though for MWI, it would not tell you which universe you happen to be in).
As a completely irrelevant aside, the first link is tenuous—it’s awfully hard to explain Pompeii in terms of just psychology—but it would still be true that you could derive all of history from sufficient knowledge of starting conditions, and could possibly work backwards from sufficient knowledge of the present (which, for MWI, would include sufficient knowledge of the state of all worlds).
If for no other reason, I reject that theory because Physics is not Math. Physics is inherently about making statements that have empirical content. Math is inherently about making statements that lack empirical content.
Would you say that you are expressing a difference in the territories covered by physics and math, in our existing maps of physics and math, in any potential map of physics and math, or in our methods of constructing the maps?
Would you say that you are expressing a difference in the territories covered by physics and math, in our existing maps of physics and math, in any potential map of physics and math, or in our methods of constructing the maps?
I think the answer to your question is “method of construction.”
In principle, a Cartesian skeptic should be able generate the same map of “Math” that we use. In contrast, there is no reason that a Cartesian skeptic’s map of Physics would have any resemblance to the territory at all. (I accept that it is hard to see what could motivate a Cartesian skeptic to generate Math).
I’m almost tempted to say that for a priori truths, the map is the territory. I hesitate because I’m not confident I am using the metaphor correctly.
In principle, a Cartesian skeptic should be able generate the same map of “Math” that we use. In contrast, there is no reason that a Cartesian skeptic’s map of Physics would have any resemblance to the territory at all. (I accept that it is hard to see what could motivate a Cartesian skeptic to generate Math).
I am not convinced this is the case. How do we pick which axioms to use, except by comparison to reality? Certainly, this is how it historically happened.
After describing all possible physics models, the skeptic still has no idea which is right.
In contrast, all the math articulated is right. (There are probably some caveats I should make, like no inconsistent axioms, and some reference to Godel, but I suspect I don’t know enough actual math to make all the necessary caveats).
I’m almost tempted to say that for a priori truths, the map is the territory. I hesitate because I’m not confident I am using the metaphor correctly.
This definitely seems to be a slip up in application of the metaphor. The map is my beliefs, the territory is truth. I know from experience that I can believe something to be a mathematical truth, only to thereafter find a mistake in my proof—I don’t expect “the statement was true for as long as your map represented it” is actually your position.
That is not a correct edit of what I said. At the level of generality I’m talking, “The sky is blue” is a physics statement. A better summary would be “[Math and logic] are both different from the statement that the sky is blue.
Ah, “they” was “modus ponens and 2 + 2 = 4″; completely missed that interpretation, sorry.
Then I do not follow your original point.
Presumably because there needs to be an observer to see the blue, and different people may draw different delineations between what is blue vs. purple vs. teal? If you describe it in wavelengths, is it the same as physics? If not, what is it that makes it not?
This seems to just be a fact about the territory. Presumably, there are similar similarities between a perfectly accurate and completely specified quantum theory and chemical reactions, they are just harder to work out.
That is not a correct edit of what I said. At the level of generality I’m talking, “The sky is blue” is a physics statement. A better summary would be “[Math and logic] are both different from the statement that the sky is blue.
Fair enough. But I assert that the particular similarity we are discussing does not exist between math and physics, no matter how hard we look.
Can you clarify the grounds for that assertion? It seems to me that in both cases the territory contains facts and rules, and combinations of these lead to other consequences. The big difference seems, to me, to be a historical question of where we started our mapping; with physics, we started looking at consequences very far removed from the facts and rules that are rigidly true, and so our map was by necessity fuzzier. This is a fact about the map, not the territory, though.
I have heard a Strong Reductionist theory, which goes as follows: History is Psychology. Psychology is Biology. Biology is Chemistry. Chemistry is Physics. Physics is Math. In other words, you can (in principle) derive the higher order theory from the more fundamental theory.
If for no other reason, I reject that theory because Physics is not Math. Physics is inherently about making statements that have empirical content. Math is inherently about making statements that lack empirical content.
For any deterministic interpretation of quantum mechanics, it seems that this would necessarily have to hold (though for MWI, it would not tell you which universe you happen to be in).
As a completely irrelevant aside, the first link is tenuous—it’s awfully hard to explain Pompeii in terms of just psychology—but it would still be true that you could derive all of history from sufficient knowledge of starting conditions, and could possibly work backwards from sufficient knowledge of the present (which, for MWI, would include sufficient knowledge of the state of all worlds).
Would you say that you are expressing a difference in the territories covered by physics and math, in our existing maps of physics and math, in any potential map of physics and math, or in our methods of constructing the maps?
I think the answer to your question is “method of construction.”
In principle, a Cartesian skeptic should be able generate the same map of “Math” that we use. In contrast, there is no reason that a Cartesian skeptic’s map of Physics would have any resemblance to the territory at all. (I accept that it is hard to see what could motivate a Cartesian skeptic to generate Math).
I’m almost tempted to say that for a priori truths, the map is the territory. I hesitate because I’m not confident I am using the metaphor correctly.
I am not convinced this is the case. How do we pick which axioms to use, except by comparison to reality? Certainly, this is how it historically happened.
What can’t the skeptic say “If you accept the Axiom of Choice, here’s what follows. If you reject the Axiom of Choice, this follows instead.”
And you are right that I misused the map/territory metaphor.
Why can’t the skeptic similarly say, “If you accept that there is a particle at..., here is what follows; otherwise, this follows instead”?
After describing all possible physics models, the skeptic still has no idea which is right.
In contrast, all the math articulated is right. (There are probably some caveats I should make, like no inconsistent axioms, and some reference to Godel, but I suspect I don’t know enough actual math to make all the necessary caveats).
I am not certain that the same objection cannot be made to math, but I at least follow what your objection is.
This definitely seems to be a slip up in application of the metaphor. The map is my beliefs, the territory is truth. I know from experience that I can believe something to be a mathematical truth, only to thereafter find a mistake in my proof—I don’t expect “the statement was true for as long as your map represented it” is actually your position.
Ah, “they” was “modus ponens and 2 + 2 = 4″; completely missed that interpretation, sorry.