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Vladimir_Nesov comments on Supernatural Math - Less Wrong
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Comments (56)
Wrong, these are different objects, (universe1 :: 2+2) and (universe2 :: 2+2).
But even saying that implies that there's some meta framework from which we can consider both. From inside our reality, there's no reason to presume that we can know what physical laws, logic, math, or anything holds in another reality (or even that such a thing as an "other reality" exists). I switched to using "reality", by the way, to make it clear that I'm not using this in the way that scientists sometimes talk about "other universes", which, if we could know anything about, would necessarily be part of our reality.
Do you hold that there is a possible world in which 2+2 does not = 4? Just trying to translate your position to my vocabulary.
No, I'm not holding that there actually is such a world, only that there would be no reason to apply our reality's rules to such a world. My real point is that the logical follows in actual historical fact from the physical, rather than being some sort of special knowledge that can be deduced without reference to anything physical.
Really, our beliefs about the logical follow from the physical. Don't confuse the map with the territory.
Heh. This is precisely the question, isn't it? Are logic and mathematics in the territory somewhere or are they the language of the map?
The way I see it is that logic is a territory, my beliefs about logic form a corresponding map, and that map is useful for constructing maps of other territories (and the accuracy of those maps is evidence of the accuracy of the logic map).
If nothing else this is a really well phrased statement of position. Maybe I'm just committing the philosopher's fallacy (deciding things are necessary because I'm not creative enough to think up alternatives) but I really just can't see what it would mean for there to be a world in which A didn't = A, in which the middle wasn't excluded, in which triangles are round, etc. What criteria are you using to decide on one view over the other?
Until the time of writing this comment, I did believe that there would be a meta-logic that governs the math or logic that "can exist". However, I concede that such a statement requires proof and I do not have enough (read: any) background in logic to know if such a proof is forthcoming.
This is the question: if there were two physical universes realizing two distinct sets of mathematical ideas such that at least one mathematical idea is realized in one set and not the other, then would mathematics still provide a meta-framework for both sets of mathematical ideas?
More succinctly: Given any two models, can you always find a meta-model that includes them both? Or maybe it can be proven that this is unprovable within a given model?
A logician's input on this would be helpful.
I think this is a confused question. The answer is yes, but when I tell you what the meta-model is, I don't think your underlying curiosity will be satisfied.
Given 2 models A and B, we can construct a model C. The set of objects in C will be the union of the set of elements of the form (A, a) for all objects a in the model A, and the set of elements of the form (B, b) for all objects b in the model B. Then every proposition about A can can correspond to a proposition about C, particularly about its objects of the form (A, a), by simple substitution of (A, a) for a; and similarly propositions about B correspond to propositions about the objects of the form (B, b). This is, of course, a trivial combination in which the parts that correspond to the original models do not interact with each other at all, hence my belief that you will not be satisfied.
Perhaps a better question to ask would be: is there a meta framework that distinguishes some mathematical systems as "good" and others as "bad"? I think the answer to this question is the criteria that a mathematical system be self-consistent, that is, it does not produce contradictions. Of course, this criteria is not always possible to verify.