I guess my original idea (i.e., the idea I had in my very first question in the open thread) was that the physical systems can be phrased in the form of tautologies. Now, I don’t know enough about mathematical logic, but I guess my intuition was/is telling me that if you have a system which is completely described by tautologies, than by (hypothetically) fine-graining these tautologies to cover all options and then breaking the tautologies into alternative theorems, we have an entire “mathematical structure” (i.e., propositions and relations between propositions, based on logic) for the reality. And this structure would be consistent, because we had already shown that the tautologies could be formed consistently using the (hypothetically) available data. Then physics would work by seizing on these structures and attempting to figure out which theorems were true, refining the list of theorems down into results, and so on and so forth.
I’m beginning to worry I might lose the reader do to the impression I am “moving the goalpost” or something of that nature… If this appears to be the case, I apologize and just have to admit my ignorance. I wasn’t entirely sure what I was thinking about to start out with and that was really why I made my post. This is really helping me understand what I was thinking.
Tell me whether the following seems to capture the spirit of your observation:
Let C be the collection of all propositional formulas that are provably true in the propositional calculus whenever you assume that each of their atomic propositions are true. In other words, C contains exactly those formulas that get a “T” in the row of their truth-tables where all atomic propositions get a “T”.
Note that C contains all tautologies, but it also contains the formula A ⇒ B, because A ⇒ B is true when both A and B are true. However, C does not contain A ⇒ ~B, because this formula is false when both A and B are true.
Now consider some physical system S, and let T be the collection of all true assertions about S.
Note that T depends on the physical system that you are considering, but C does not. The elements of C depend only on the rules of the propositional calculus.
Maybe the observation that you are getting at is the following: For any actual physical system S, we have that T is closed under all of the formulas in C. That is, given f in C, and given A, B, . . . in T, we have that the proposition f(A, B, . . .) is also in T. This is remarkable, because T depends on S, while C does not.
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’ve also recently found something interesting where people denote the criterion of mathematical existence as freedom from contradiction. This can be found on pg. 5 of Tegmark here, attributed to Hilbert.
This looks disturbingly similar to my root idea and makes me want to do some reading on this stuff. I have been unknowingly claiming the criterion for physical existence is the same as that for mathematical existence.
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’m inclined to think that it doesn’t really show anything metaphysically significant. When we encode facts about S as propositions, we are conceptually slicing and dicing the-way-S-is into discrete features for our map of S. No matter how we had sliced up the-way-S-is, we would have gotten a collection of features encoded as proposition. Finer or coarser slicings would have given us more or less specific propositions (i.e., propositions that pick out minuter details).
When we put those propositions back together with propositional formulas, we are, in some sense, recombining some of the features to describe a finer or coarser fact about the system. The fact that T is closed under all the formulas in C just says that, when we slice up the-way-S-is, and then recombine some of the slices, what we get is just another slice of the-way-S-is. In other words, my remark about T and C is just part of what it means to pick out particular features of a physical system.
Though the word “tautology” is often used to refer to statements like (A v ~A), in mathematical logic any true statement is a tautology. Are you talking about the distinction between axioms and derived theorems in a formal system?
Little note to self:
I guess my original idea (i.e., the idea I had in my very first question in the open thread) was that the physical systems can be phrased in the form of tautologies. Now, I don’t know enough about mathematical logic, but I guess my intuition was/is telling me that if you have a system which is completely described by tautologies, than by (hypothetically) fine-graining these tautologies to cover all options and then breaking the tautologies into alternative theorems, we have an entire “mathematical structure” (i.e., propositions and relations between propositions, based on logic) for the reality. And this structure would be consistent, because we had already shown that the tautologies could be formed consistently using the (hypothetically) available data. Then physics would work by seizing on these structures and attempting to figure out which theorems were true, refining the list of theorems down into results, and so on and so forth.
I’m beginning to worry I might lose the reader do to the impression I am “moving the goalpost” or something of that nature… If this appears to be the case, I apologize and just have to admit my ignorance. I wasn’t entirely sure what I was thinking about to start out with and that was really why I made my post. This is really helping me understand what I was thinking.
Tell me whether the following seems to capture the spirit of your observation:
Let C be the collection of all propositional formulas that are provably true in the propositional calculus whenever you assume that each of their atomic propositions are true. In other words, C contains exactly those formulas that get a “T” in the row of their truth-tables where all atomic propositions get a “T”.
Note that C contains all tautologies, but it also contains the formula A ⇒ B, because A ⇒ B is true when both A and B are true. However, C does not contain A ⇒ ~B, because this formula is false when both A and B are true.
Now consider some physical system S, and let T be the collection of all true assertions about S.
Note that T depends on the physical system that you are considering, but C does not. The elements of C depend only on the rules of the propositional calculus.
Maybe the observation that you are getting at is the following: For any actual physical system S, we have that T is closed under all of the formulas in C. That is, given f in C, and given A, B, . . . in T, we have that the proposition f(A, B, . . .) is also in T. This is remarkable, because T depends on S, while C does not.
Does that look like what you are trying to say?
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’ve also recently found something interesting where people denote the criterion of mathematical existence as freedom from contradiction. This can be found on pg. 5 of Tegmark here, attributed to Hilbert.
This looks disturbingly similar to my root idea and makes me want to do some reading on this stuff. I have been unknowingly claiming the criterion for physical existence is the same as that for mathematical existence.
I’m inclined to think that it doesn’t really show anything metaphysically significant. When we encode facts about S as propositions, we are conceptually slicing and dicing the-way-S-is into discrete features for our map of S. No matter how we had sliced up the-way-S-is, we would have gotten a collection of features encoded as proposition. Finer or coarser slicings would have given us more or less specific propositions (i.e., propositions that pick out minuter details).
When we put those propositions back together with propositional formulas, we are, in some sense, recombining some of the features to describe a finer or coarser fact about the system. The fact that T is closed under all the formulas in C just says that, when we slice up the-way-S-is, and then recombine some of the slices, what we get is just another slice of the-way-S-is. In other words, my remark about T and C is just part of what it means to pick out particular features of a physical system.
Though the word “tautology” is often used to refer to statements like (A v ~A), in mathematical logic any true statement is a tautology. Are you talking about the distinction between axioms and derived theorems in a formal system?