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SilasBarta comments on The role of mathematical truths - Less Wrong
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Thanks, that does get to the heart of the matter. To borrow from one of the linked articles, I imagine someone in the role of Eliezer Yudkowsky here, being challenged by "the one" (bold added):
That is, a rationalist could avoid making obvious or large errors, but still believe "2+3=5", above and beyond any physically-verifiable claim between two people, and above and beyond any specific model (map) of reality, physically instantiated in agents. My article says to that rationalist, no, you needn't believe in this platonic "2+3=5" apart from its implication in a commonly used model, and you can still elegantly and consistently handle all of the dilemmas associated with having to classify such abstract statements. In fact, you needn't make a statement about anything non-physical.
Do you believe I've done so, and said something relevant to rationalists?
Which implication is still a fact which seems to be non-physical, seems to have been true before there were any humans to do logic, etc. You've eliminated Platonic numerical entities and metaphysically privileged formal systems - which do seem to be improvements - but not non-physical a priori truths.
It is a counterfactual claim about something physical. You can represent it in a causal diagram with only physical referents.
The causal diagram in your OP contains a node labeled "Integer math implies 2+2 = 4?"
What is the physical referent for "Integer math"?
Any physical system that, as best you can infer, behaves with a known isomorphism to integer math.
**ETA: Oops, see zero_call's correction; following the article, integer math actually corresponds to some widely-held conception -- within human brains -- of how numbers work. Since Tyrrell_McAllister's point was that I was slipping in non-physicality, the rest of the exchange is still relevant, though.
So, to make the pure physicality of all referents clear, should we label that node:
where S is the name of a specific concrete physical system such that the string '2+2=' physically makes S output '4' in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4?
Yes, basically. I mean, I'd tweak it to read something more like
but I don't think that impacts whatever point you were trying to make.
I think that your tweak makes an important difference. And, if I may be so bold, I think that you want something closer to what I wrote :).
I'm trying to make good your claim that the causal diagram refers only to physical things. But your label refers to M, which is an isomorphism. What is the concrete physical referent of "M"?
Why? The constraint that a system output a "string" is too strict; it suffices that they output something interpretable as a string.
An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.
I agree now that "string" is too strict. I should have said "symbol", where a symbol is anything with physical tokens. My proposed label is now
where
"S" is the name of a specific concrete physical system, and
"A" and "B" are the names of specific physically-manifested symbols,
such that a token of the symbol A physically makes S output a token of the symbol B in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4.
I think that the work that you want to do by adding the word "interpretable" to the label is done by my conditions on what S, A, and B are.
Then you should be able to make the label refer directly to that physical encoding of M. That is, instead of mentioning the isomorphism M, you ought to be able to refer just to some specific physical system T that "encodes" M in the same way that my physical system S above encodes the operation of adding 2 to 2.
However, if you're still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.
(A strict platonist would say that even my label refers to nonphysical things, because it refers to symbols, only the tokens of which are physical. I'm happy to ignore this.)
No, I thought the physical referent for the integer math was something like "Human mental instantiation of an idea that is reasonably agreed upon." I believe you are referring to the physical referent of the preimage of the isomorphism (i.e., the physical system itself. A somewhat redundant thing to call a referent, since it is actually the explicit meaning of the statement.)
You're right, I agree. I was being inconsistent with my article there.