Please note: I am speaking strictly as a Non-M, which is obviously the reason I ask.
I’ve been attempting to digest the following web page. Needless to say, for me, it’s a rather large pill to swallow (someone with no technical background in the field to speak of).
However, what does interest me are the possible overlaps between the different areas of logic.
More specifically, are people who specialize in mathematical logic familiar with the logical structure of arguments?
For example, are they familiar with (do they encounter - or have any need or use for)
Law of the excluded middle
Modal logic
Techniques and rules listed on the linked article (e.g. affirming the consequent, deduction and induction, modus ponens etc.)
Can mathematical logicians involve themselves in debates discussing valid ontological proofs?
Apologies if this seems a bit naive since I’m just beggining the topic, but I figure the experts on this board will be able to guide me a bit better.
Basically are the two areas of logic completely seperate? If you have grasped the foundations of mathematical logic than does it provide you with enough insight to argue/understand those other aspects of logic (the logic used to produce a valid argument, or an ontological proof)?
It really dpends if they’ve taken a course in symbolic logic, which is ceratinly on offer on some math curiculums. Of course whilst mathematical induction is very useful to a mathaematcian inductive reasoning is of very little use.
Mathematicians are certainly familiar with symbolic logic, which is used to describe the various logical fallacies. We didn’t analyze philosophical arguments in my logic class, but we did analyze proofs and whatnot. In philosophy classes we did much the same thing, though because it was a philosophy class, it would take the professor half an hour to explain it instead of 30 seconds.
People who specialize in mathematics are. Mathematics is really more about the logical structure of proof than it is about an given mathematical object. I avoid numbers whenever possible, for instance.
Not only are mathematicians familiar with this, but I’d hesitate to say anyone else is more familiar. A huge number of mathematical proofs are reductios, which implicitly invoke the excluded middle. Further, there have been full-scale academic wars waged over whether or not this “law” is valid or not, entirely within the mathematical community.
Ontological proofs of what? I wouldn’t take seriously the opinion of a nonmathematician in a discussion about realism-in-ontology concerning mathematical structures, for instance.
I’d like to second Mathochist’s post. I’m a CS person and we use a lot of heavy logic, almost all coming thru the Math world.
With respect to some of the topics mentioned in the OP:
Law of the excluded middle. Yep, both Math and CS people argue about this. (In CS these used to be kinda weird arguments, but with Quantum Computing a hot topic, not so weird anymore.)
Modal logic. We even use this in CS, I actually wrote a paper once… . Anyway, my Math and modal logic story: At one college, one of the Math profs had a “paradox” involving negating a simple quantified formula. Using naive negation rules, it comes out wrong. But when he tried it on me I said “Hey, there’s a modal quantifier in there, it goes like this instead…” Floored him.
Modus ponens. See previous paragraph. Used a lot in that world.
As to von Neumann and theology, I knew he said: “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.” I wasn’t aware he had done more significant research in the area.
I don’t see why you need to go as far as introducing something as subtle as AC to show an intersection between mathematics and philosophy. Frankly, most of the relevant issues about AC are already apparent within the philosophical treatments of the natural numbers.
