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Formal first order logic is the foundation of ZFC set theory. This gives me the impression that a theoretical system has to be based on a formal logical system, with its own axioms and deduction rules. However, as I read books on formal logic, I found that many conclusions of the formal system itself are drawn through proof at the level of meta-language, e.g., proof for soundness and completeness of the system itself. This is natural, as a proof system cannot be used to prove conclusions of itself. Since meta-language is not formalized, I am considering, which one is more fundamental, intuitive thinking or formal thinking. Can we develop a formal system without resorting to intuitive thinking, or is it that we have to rely on intuition at the very root of our thinking?

Ziqi Fan
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  • The second one. – Git Gud Oct 04 '21 at 15:13
  • @GitGud I don't think everyone would agree with you on that... At some level, we have to be guided by intuition to decide where to begin formalism. "Ex nihilo nihil fit" – Rushabh Mehta Oct 04 '21 at 15:15
  • @DonThousand This really isn't up for debate, unless we understand the question differently. We always need the metalanguage which is informal. This what I understand here as "intuitive". – Git Gud Oct 04 '21 at 15:17
  • @DonThousand The edit to your comment confused me. I said "the second one", meaning "we have to rely on intuition at the very root of our thinking", but then your edit leaves one thinking that we agree. – Git Gud Oct 04 '21 at 15:18
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    Oh, by second one you mean intuition? Ah, I thought the answer was to the titular question. – Rushabh Mehta Oct 04 '21 at 15:18
  • @DonThousand Of course you wouldn't know this, but I'm huge supporter of "the title isn't the question" :) – Git Gud Oct 04 '21 at 15:19
  • Could someone explain where intuitive language comes from? I have always held the thought that how to generate the so-called intuition is the root problem of AI. If there is no explanation for intuition, then AI can never match human wisdom, no matter how much computational power we have. – Ziqi Fan Oct 04 '21 at 15:21
  • @ZiqiFan That's a question for the cognitive scientists. I'm almost positive there used to be a Cognitive Science Stack Exchange, but now I just get this, which seems appropriate. – Git Gud Oct 04 '21 at 15:24
  • @ZiqiFan It's a bit of a chicken and egg problem. We'd like to avoid intuition in the foundations of mathematics as far as possible, since ill-founded intuition can lead us astray (see the response to Cantor, for example). However, formalism can't begin from nothing. It has to start in an environment we create via our intuition as to what produces an effective formal structure. So how would AI replicate that? Well, our intuition (at least I think) is simply a combination of our experience, our intelligence (i.e., genetics), and our evolutionary instinct. How do you teach that? – Rushabh Mehta Oct 04 '21 at 15:24
  • Maybe useful – Mauro ALLEGRANZA Oct 04 '21 at 15:25
  • You will do far better to ask questions like this on philosophy stack exchange. Note that we can work in a formalised metalanguage but just moves the problem to concerns about that formalisation. – Rob Arthan Oct 04 '21 at 19:48
  • We have to resort in logic and mathematical thinking... When you start studying algebra in middle school you do not use axioms and predicate logic. – Mauro ALLEGRANZA Oct 05 '21 at 06:09

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