ScottMessick

200

Suggestions for Slytherin: Sun Tzu's *Art of War* and some Nietzsche, maybe *The Will to Power*?

Suggestion for Ravenclaw: *An Enquiry Concerning Human Understanding*, David Hume.

30

The post seems to confuse the law of non-contradiction with the principle of explosion. To understand this point, it helps to know about minimal logic which is like intuitionistic logic but even weaker, as it treats ((false)) the same way as any other primitive predicate. Minimal logic rejects the principle of explosion as well as the law of the excluded middle (LEM, which the main post called TND).

The law of non-contradiction (LNC) is just ). (In the main post this is called ECQ, which I believe is erroneous; ECQ should refer to the principle of explosion (especially the second form).) The principle of explosion is either %20\to%20Q) or . These two forms are equivalent in minimal logic (due to the law of non-contradiction). As mentioned above, minimal logic has the law of non-contradiction, but not the principle of explosion, so this shows that they're not equivalent in every circumstance. Rejecting the principle of explosion (especially the second form) is the defining feature of paraconsistent logics (a class into which many logics fall). Some of these still have the validity of the law of non-contradiction. Anti-intuitionistic logic does not, because LNC is dual to LEM, which is invalid intuitionistically.

Ok, so I ended up taking a lot of time researching that nitpick so I could say it correctly. Anyway, I'm curious to see where this is going.

10

Super-upvoted.

60

I'm not going to say they haven't been *exposed* to it, but I think quite few mathematicians have ever developed a basic appreciation and working understanding of the distinction between syntactic and semantic proofs.

Model theory is, very rarely, successfully applied to solve a well-known problem outside logic, but you would have to sample many random mathematicians before you could find one that could tell you exactly how, even if you restricted to only asking mathematical logicians.

I'd like to add that in the overwhelming majority of academic research in mathematical logic, the syntax-semantics distinction is not at all important, and syntax is suppressed as much as possible as an inconvenient thing to deal with. This is true even in model theory. Now, it is often needed to discuss formulas and theories, but a syntactical proof need not ever be considered. First-order logic is dominant, and the completeness theorem (together with soundness) shows that syntactic implication is equivalent to semantic implication.

If I had to summarize what modern research in mathematical logic is like, I'd say that it's about increasingly elaborate notions of complexity (of problems or theorems or something else), and proving that certain things have certain degrees of complexity, or that the degrees of complexity themselves are laid out in a certain way.

There are however a healthy number of logicians in computer science academia who care a lot more about syntax, including proofs. These could be called mathematical logicians, but the two cultures are quite different.

(I am a math PhD student specializing in logic.)

30

The explanation "number of partners" question is problematic right now. It reads "0 for single, 1 for monogamous relationship, >1 for polyamorous relationship" which makes it sound like you must be monogamous if you happen to have 1 partner. I am polyamorous, have one partner and am looking for more.

In fact, I started wondering if it really meant "ideal number of partners", in which case I'd be tempted to put the name of a large cardinal.

40

I continue to be surprised (I believe I commented on this last year) that under "Academic fields" pure mathematics is not listed on its own; it is also not clear to me that pure mathematics is a hard science; relatedly, are non-computer science engineering folk expected to write in answers?

I second this: please include pure mathematics. I imagine there are a fair few of us, and there's no agreed upon way to categorize it. I remember being annoyed about this last year. (I'm pretty sure I marked "hard sciences".)

130

I wonder how it would be if you asked instead "When should we say a statement is true?" instead of "What is truth?" and whether your classmates would think them the same (or at least closely related) questions.

I was disappointed to see my new favorite "pure" game Arimaa missing from Bostrom's list. Arimaa was designed to be intuitive for humans but difficult for computers, making it a good test case. Indeed, I find it to be very fun, and computers do not seem to be able to play it very well. In particular, computers are nowhere close to beating top humans despite the fact that there has arguably been even more effort to make good computer players than good human players.

Arimaa's branching factor dwarfs that of Go (which in turn beats every other commonly known example). Since a super-high branching factor is also a characteristic feature of general AI test problems, I think it remains plausible that simple, precisely defined games like Arimaa are good test cases for AI, as long as the branching factor keeps the game out of reach of brute force search.