Stephen A. Meigs
My main mathematical interest nowadays is logic. I believe ordinary math should be based on a logic that has predicate symbols and function symbols that don't make sense globally, and that specific domains of definition should be associated with such symbols individually. It follows that some terms are undefined. Predicate symbols are silly on undefined things (and things outside their domains of definition). Ordinarily mathematicians have need to assert that statements are silly or true (i.e., that they are not false), which basically means that the subjunctive should be the ordinary mood for math. I believe entailment, giving a preordering, is more fundamental than truth, so axioms of logic should look like rules for lattices (or better yet, once implication is included, Heyting algebras). To talk about silliness, a three-valued logic is useful, which is best dealt with in my opinion by an involutive negation "not" that supplements "false" (the latter defined by implying the false truth value). True should be defined as "false not". Actually, occasionally it is useful for a few silly things to not be assertable, which leads to refinement by distinguishing assertable silly things and non-assertable silly things, which leads to a six-valued logic (and four variants of silliness).
I am not interested much in deep hard to prove logical results per se (and my mind is unsuited to finding such), but rather to understanding very carefully the shallow. E.g., the philosophical implications to what I am studying seem to concern shallow things like laughter and the subjunctive. From my point of view, I am not concerned with alternative three-valued logics, there being only one reasonable choice.
A statement like $1/0 = 2$ should be assertable, though silly; it would be something one could laugh at, only it is so obviously silly there be no communicative value to laughing at it. Such statements are akin to farce that is so farcical (silly) as to not be funny. Math should not be a completely serious subject but rather a very serious subject. Only silly things that are exceedingly silly (too silly to be funny) should be allowed to be asserted in mathematical context.
I think models are overrated and interpretations into complete Henkin theories are underrated. Syntax over semantics, as it were. I don't think subcollections of sets should necessarily be considered sets; in other words, I believe in semisets, which I don't think are presented well, at least not that I have seen. I think if people believed in semisets more, they would believe in models less.
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Last seen Feb 19 at 22:47