= 27d567bc2d48d9f40e9ccc20ea370b15)
chaosmosis comments on Logical Pinpointing - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (338)
Your idea of pinning down the natural numbers using second order logic is interesting, but I don't think that it really solves the problem. In particular, it shouldn't be enough to convince a formalist that the two of you are talking about the same natural numbers.
Even in second order PA, there will still be statements that are independent of the axioms, like "there doesn't exist a number corresponding to a Godel encoding of a proof that 0=S0 under the axioms of second order PA". Thus unless you are assuming full semantics (i.e. that for any collection of numbers there is a corresponding property), there should be distinct models of second order PA for which the veracity of the above statement differs.
Thus it seems to me that all you have done with your appeal to second order logic is to change my questions about "what is a number?" into questions about "what is a property?" In any case, I'm still not totally convinced that it is possible to pin down The Natural Numbers exactly.
I think this is his way of connecting numbers to the previous posts. If "a property" is defined as a causal relation, which all properties are, then I think this makes sense. It doesn't provide some sort of ultimate metaphysical justification for numbers or properties or anything, but it clarifies connections between the two and such a justification isn't really possible anyways.
I don't think that I understand what you mean here.
How can these properties represent causal relations? They are things that are satisfied by some numbers and not by others. Since numbers are aphysical, how do we relate this to causal relations.
On the other hand, even with a satisfactory answer to the above question, how do we know that "being in the first chain" is actually a property, since otherwise we still can't show that there is only one chain.
You just begged the question. Eliezer answered you in the OP:
I can't think of an example, but I'm thinking that if a property existed then it would be a causal relation. A property wouldn't represent a causal relation, it would be one. I wasn't thinking mathematically but instead in terms of a more commonplace understanding of properties as things like red and yellow and blue.
The argument made by the simple idea of truth might be a way to get us from physical states (which are causal relations) to numbers. If you believe that counting sheep is a valid operation, then quantifying color also seems fine. The reason I spoke in terms of causal relations is because I believe understanding qualities as causal relations between things allows us to deduce properties about things through a combination of Salmonoff Induction and the method described in this post.
Are you questioning the idea that numbers or properties are a quality about objects? If so, what are they?
I'm feeling confused though. If the definition of property used here doesn't connect to or means something completely different than facts about objects, then I'm way off base. I might also be off base for other reasons. Not sure.
I am questioning the idea that numbers (at least the things that this post refers to as numbers) are a quality about objects. Numbers, as they are described here, are an abstract logical construction.