|
Here are the latest clarifications from Dr.
Aloisius Louie:
Torkel Franzen wrote: > It would be a great deal more illuminating to have a couple of > examples of why questions that can be answered from within the system, > and why questions that cannot be answered from within the system. Isn't this what Rosen did in LI Ch.10? For a mechanism, ultimately some efficient cause has to be entailed from the outside; but for an organism, all the efficient causes are entailed within. For example, in a mechanism a "repair the repair" iteration will ultimately lead to the environment, whereas in an organism, the "repair the repair" iterative loop is closed. >> ZF = Zermelo-Franco set theory Calvin Ostrum wrote: > Zermelo-Fraenkel set theory, actually AL: This was an inadvertent typo. But it in fact
illustrates
the point of impredicativity. Googling "Zermelo-Franco" only yielded one hit (somebody made the same quick-typing-by-phonetics typo). The inference "by Zermelo-Franco I mean Zermelo-Fraenkel" cannot be generated within an algorithmic machine. Google would prompt me for "Did you mean: "robert rosen"" when I searched "robert rosn". But this is only because there is some built-in algorithm to recognize "common typos". Any kind of "smart checking" has to be externally programmed somehow. It can only nitpick on syntax, but cannot verify semantics. Similarly: >> This is because we simply cannot practically handle non-terminating >> decimals. Calvin Ostrum wrote: > Sure we can. It is easy to deal with fractions, and we also perform > lots of exact computation with all sorts of irrational numbers like > pi and e all the time. AL: I mean "cannot practically handle numerically" of course,
as would be clear from the context. So the point is that semantics cannot be deduced from syntax. "Intuition", "heuristics", etc. are impredicative. Syntax can only recognize "what I say"; semantics is necessary to get "what I mean". This is why the 2001: A Space Odyssey computer HAL (Heuristic-ALgorithmic) remains science fiction. Heuristics is noncomputable. A simulation of intelligence is not a model of intelligence. Read Rosen's discussions on the Turing test. >> PA is a formal basis of number theory, and ZFC is a formal >> basis of (naive) set theory. There is no need to go further >> than this. >Calvin Ostrum wrote: What is "naive" about the set theory formalized by ZFC? > What is an example of a set theory that is not naive in > comparison to ZFC, and why is it not naive? This is "naive set theory" in the sense of Paul Halmos. It is the standard mathematical usage. "Naive" here most certainly does not mean "innocent and inexperienced". It means formal set theory without worrying about foundation issues such as Russell's paradox. i.e. we don't talk about "set of all sets with a certain property", but a lot of good mathematics can still be done. >> The fact that a great deal of useful mathematics has been done >> syntactically is precisely the point discussed in the Praeludium >> of LI. > Calvin Ostrum wrote: What useful mathematics cannot be done "purely syntactically". AL: We cannot syntactically come up with intuition. In
other
words, "how to come up with the Eureka! to prove a theorem" is impredicative. Once the idea is formed, the proof itself is of course done syntactically and formally. The proof of Fermat's Last Theorem was completely rigorous; but how Andrew Wiles came up with the way he used to do it is semantics, not syntax. >> are not useful. He simply said that we need the "generic parts" >> to do biology (indeed real science). > Calvin Ostrum wrote: Why? For any "real science"? What about modern physics, > chemistry, and social and behavioral sciences? They seem to > get on quite well with mathematics that can be "captured" in > ZFC. That was slightly flippant on my part. Of course a lot of good science came out of pure syntax. But Rosen's point was that so much more could be done if we go beyond the machine metaphor. "Molecular biology" was his favorite oxymoron. Once we go "molecular" it is no longer "biology" -- "molecular biophysics" or "molecular biochemistry" perhaps, and the subject yields useful data, but it is not biology. "It's dead, Jim." Aloisius |