[edit: sorry, the formatting of links and italics in this is all screwy. I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]
In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham:
It would not be a bad definition of math to call it the study of terms that have precise meanings.
Some senses of "erroneous" that might be involved here include (this list is not necessarily intended to be exhaustive):
Mathematically incorrect -- i.e. the proofs contain actual logical inconsistencies. This was argued by some early skeptics (such as Kronecker) but is basically indefensible ever since the formulation of axiomatic set theory and results such as Gödel's on the consistency of the Axiom of Choice. Such a person would have to actually believe the ZF axioms are inconsistent, and I am aware of no plausible argument for this.
Making claims that are epistemologically indefensible, even if possibly true. E.g., maybe there does exist a well-ordering of the reals, but mere mortals are in no position to assert that such a thing exists. Again, axiomatic formalization should have meant the end of this as a plausible stance.
Irrelevant or uninteresing as an area of research because of a "lack of correspondence" with "reality" or "the physical world". In order to be consistent, a person subscribing to this view would have to repudiate the whole of pure mathematics as an enterprise. If, as is more common, the person is selectively criticizing certain parts of mathematics, then they are almost certainly suffering from map-territory confusion. Mathematics is not physics; the map is not the territory. It is not ordained or programmed into the universe that positive integers must refer specifically to numbers of elementary particles, or some such, any more than the symbolic conventions of your atlas are programmed into the Earth. Hence one cannot make a leap e.g. from the existence of a finite number of elementary particles to the theoretical adequacy of finitely many numbers. To do so would be to prematurely circumscribe the nature of mathematical models of the physical world. Any criticism of a particular area of mathematics as "unconnected to reality" necessarily has to be made from the standpoint of a particular model of reality. But part (perhaps a large part) of the point of doing pure mathematics (besides the fact that it's fun, of course), is to prepare for the necessity, encountered time and time again in the history of our species, of upgrading -- and thus changing --our very model. Not just the model itself but the ways in which mathematical ideas are used in the model. This has often happened in ways that (at least at the time) would have seemed very surprising.
Well, if the AI is doing mathematics, then it needs to reason about the very same entities that human mathematicians reason about.
Maybe that sounds like begging the question, because you could ask why humans themselves need to reason about those entities (which is kind of the whole point here). But in that case I'm not sure what you're getting at by switching from humans to AIs.
Do you perhaps mean to ask something like: "What kind of mathematical entities will be needed in order to formulate the most fundamental physical laws?"
Why do you think that the axiomatic formulation of ZFC "should have meant an end" to the stance that ZFC makes claims that are epistemologically indefensible? Just because I can formalize a statement does not make that statement true, even if it is consistent. Many people (including me and apparently Eliezer, though I would guess that my views are different from his) do not think that the axioms of ZFC are self-evident truths.
In general, I find the argument for Platonism/the validity of ZFC based on common acceptance to be problematic because I j... (read more)