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The following is the response sent to me by Dr. Aloisius Louie,
regarding the questions raised earlier on the mathematical basis of some of my
father's statements in his published work:
----- Original Message -----
From: Dr. Aloisius H.
Louie
To: Judith Rosen
Sent: Wednesday, August 10, 2005 4:34 PM
Subject: Re: Fw: [ROSEN] Impredicativity in Rosennean
parlance 1. Let me first get those initialisms out of the way. PA = Peano axioms (or Peano arithmetic) ZF = Zermelo-Franco set theory ZFC = ZF + Axiom of Choice 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. We only need to know they are examples of purely syntactic formal systems. 2. Of course a lot of mathematics are captures by PA and ZFC. But that's not Rosen's point. The issue at hand here is really what Rosen meant by those quantifiers "an infinitesimal part", "very little", "almost every", etc. The concept is "genericity". Mathematically, to say a property is "generic" means it is the dominant characteristic. For example, in the set of real numbers, irrational numbers are generic. That means if an arbitrary real number is chosen, the overwhelming odd (in a precise mathematical sense) is that it is irrational. That is NOT to say, however, that a generic object is most commonly encountered in everyday occurrence. Everyday experience is a completely different animal, as it were. Although irrationals are generic, what we almost always use are rationals. This is because we simply cannot practically handle non-terminating decimals. Neither can computers, for that matter. In the classic Star Trek episode "Wolf in the Fold", for example, Spock forced an evil entity (composed of pure energy and which fed on fear) out of the Starship Enterprise's computer by commanding the computer to "compute to the last digit the value of pi", thus sending the computer into an infinite loop. When we compute with pi (which is an irrational), we are only using its "simulation" (in the Rosen sense of LI Ch.7), which is a truncated (hence rational) approximate value, e.g. 3.1416. Indeed, almost all the numbers we use in everyday life, in measurements or whatever, are rationals (read terminating decimals). But the rationals form a set of measure zero, hence an infinitesimal part, in the real numbers. So mathematically they are very rare, but mundanely they are very common. What Rosen meant by "infinitesimal part" was that syntactic rules only capture a "nongeneric part" of real mathematics. The fact that a great deal of useful mathematics has been done syntactically is precisely the point discussed in the Praeludium of LI. Similarly, mechanisms form a nongeneric part of natural systems, but all of physics (so far) is embedded in it (the point of LI Ch.9). Rosen was not saying that those "nongeneric parts" are not useful. He simply said that we need the "generic parts" to do biology (indeed real science). 3. Anybody who claims that Rosen has not presented his arguments clearly, precisely, and formally must not have studied the Rosen Oeuvres too carefully. -- Dr. Aloisius H. Louie Mathematical Biologist (613)749-8238 *** |