Too many responses for me to comment on everything! So, sorry to those I don't address...
Ben, When I claim a mathematical entity exists, I'm saying loosely that meaningful statements can be made using it. So, I think "meaning" is more basic. I mentioned already what my current definition of meaning is: a statement is meaningful if it is associated with a computable rule of deduction that it can use to operate on other (meaningful) statements. This is in contrast to positivist-style definitions of meaning, that would instead require a computable test of truth and/or falsehood. So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. There is of course some basic level of meaningful statements, such as sensory observations, so that this is a working recursive definition of meaning and truth. By this definition of meaning, any statement in the arithmetical hierarchy is meaningful (because each statement can be represented by computable consequences on other statements in the arithmetical hierarchy). I think some hyperarithmetical truths are captured as well. I am more doubtful about it capturing anything beyond the first level of the analytic hierarchy, and general set-theoretic discourse seems far beyond its reach. Regardless, the definition of meaning makes a very large number of uncomputable truths nonetheless meaningful. Russel, I think both Ben and I would approximately agree with everything you said, but that doesn't change our disagreeing with each other :). Mark, Good call... I shouldn't be talking like I think it is terrifically unlikely that some more-intelligent alien species would find humans mathematically crude. What I meant was, it seems like humans are "logically complete" in some sense. In practice we are greatly limited by memory and processing speed and so on; but I *don't* think we're limited by lacking some important logical construct. It would be like us discovering some alien species whose mathematicians were able to understand each individual case of mathematical induction, but were unable to comprehend the argument for accepting it as a general principle, because they lack the abstraction. Something like that is what I find implausible. --Abram ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
