I would describe first-order logic as “a formal encapsulation of humanity’s most fundamental notions of how the world works”. If it were shown to be inconsistent, then I could still fall back to something like intuitionistic logic, but from that point on I’d be pretty skeptical about how much I could really know about the world, beyond that which is completely obvious (gravity, etc.).
What did I say that implied that I “think that there is no notion of logic beyond the syntactic”? I think of “logic” and “proof” as completely syntactic processes, but the premises and conclusions of a proof have to have semantic meaning; otherwise, why would we care so much about proving anything? I may have implied something that I didn’t believe, or I may have inconsistent beliefs regarding math and logic, so I’d actually appreciate it if you pointed out where I contradicted what I just said in this comment (if I did).
Looking back, it’s hard to say what gave me that impression. I think I was mostly just confused as to why you were spending quite so much time going over the syntax stuff ;) And
First of all, the dichotomy between “logic” and “mathematics” can be dissolved by referring to “formal systems” instead.
made me think that you though that all logical/mathematical talk was just talk of formal systems. That can’t be true if you’ve got some semantic story going on: then the syntax is important, but mainly as a way to reach semantic truths. And the semantics don’t have to mention formal systems at all. If you think that the semantics of logic/mathematics is really about syntax, then that’s what I’d think of as a “formalist” position.
Oh, I think I may understand your confusion, now. I don’t think of mathematics and logic as equals! I am more confident in first-order logic than I am in, say, ZFC set theory (though I am extremely confident in both). However, formal system-space is much larger than the few formal systems we use today; I wanted to emphasize that. Logic and set theory were selected for because they were useful, not because they are the only possible formal ways of thinking out there. In other words, I was trying to right the wrong question, why do mathematics and logic transcend the rest of reality?
Definitely position two.
I would describe first-order logic as “a formal encapsulation of humanity’s most fundamental notions of how the world works”. If it were shown to be inconsistent, then I could still fall back to something like intuitionistic logic, but from that point on I’d be pretty skeptical about how much I could really know about the world, beyond that which is completely obvious (gravity, etc.).
What did I say that implied that I “think that there is no notion of logic beyond the syntactic”? I think of “logic” and “proof” as completely syntactic processes, but the premises and conclusions of a proof have to have semantic meaning; otherwise, why would we care so much about proving anything? I may have implied something that I didn’t believe, or I may have inconsistent beliefs regarding math and logic, so I’d actually appreciate it if you pointed out where I contradicted what I just said in this comment (if I did).
Looking back, it’s hard to say what gave me that impression. I think I was mostly just confused as to why you were spending quite so much time going over the syntax stuff ;) And
made me think that you though that all logical/mathematical talk was just talk of formal systems. That can’t be true if you’ve got some semantic story going on: then the syntax is important, but mainly as a way to reach semantic truths. And the semantics don’t have to mention formal systems at all. If you think that the semantics of logic/mathematics is really about syntax, then that’s what I’d think of as a “formalist” position.
Oh, I think I may understand your confusion, now. I don’t think of mathematics and logic as equals! I am more confident in first-order logic than I am in, say, ZFC set theory (though I am extremely confident in both). However, formal system-space is much larger than the few formal systems we use today; I wanted to emphasize that. Logic and set theory were selected for because they were useful, not because they are the only possible formal ways of thinking out there. In other words, I was trying to right the wrong question, why do mathematics and logic transcend the rest of reality?