Aside from my finding Goedel's theorem false, I see this book as dismally failing in its purport to be written for a general audience, also contended in the two blurbs on the back cover, stating that the book "explain[s] clearly and thoroughly just what the theorems really say" and "With exceptional clarity...gives careful, non-technical explanations..."
The book instead indulges in such a profusion of technical language that it appears only suitable for discussions in specialized journals, and indeed there seems to be a polemic going on in it with many fellow-professionals, including well known scientists like Hawking, Dyson or Penrose. In the process the author doesn't as much as give a clear form of the basic "Goedel sentence", around which the theorem revolves, although he refers to it numerous times, notably on page 55. Likewise he doesn't make clear how "Goedel numbering" is performed, claiming it a "technicality that will be avoided" (p.35), despite its frequent and simple explanation by others, and to which he attaches great importance for its "arithmetization of syntax" (same page). Therein lies the objectionable connection of the theorem with mathematics.
"Arithmetization of syntax" discloses that "Goedel numbering" is performed on linguistic components of the arguments. That is to say, these components, from single letters to series of sentences are each assigned numbers, in impressively intricate arrangements, and then it is said the contents of those arguments somehow apply to mathematics. It is not recognized that the numbering merely concerns the language in which the arguments are couched, not the contents of that language.
To support their reasoning, the proponents offer all kinds of analogies. In this book the author uses (p.36) the comparison of binary data, the mathematical collection of bits 0 and 1, as representing sounds and pictures in computer games. Here, however, the subject is physics, to which mathematics applies, even if not observable in the result. Similarly there are other examples in which the subject matter has some connection with mathematics. But the subject matter in Goedel's arguments, the content of the "Goedel sentence" and of the logic applied, does, again, not concern mathematics. Only the language, by being designated with numbers, does.
It should be appropriate here to go back to that Goedel sentence and the associated logic. I discussed these in other reviews, and I might now first provide a simple form of that sentence again, looked for in vain in the book reviewed. It is:
THIS STATEMENT IS UNPROVABLE (IN THE SYSTEM).
As noted previously, Goedel's alleged proof of this statement is said derived from outside the system and accordingly not to be contradicting the statement. But I pointed out that the rules of logic used can be incorporated into any system, a procedure that should be allowed in order to find what is or isn't logically possible, and therefore the proof would indeed be a contradiction. I noted in fact another contradiction resulting via simple logic: THE STATEMENT CAN BE PROVED UNPROVABLE, since if provable it would be contradicted; AND THE STATEMENT CANNOT BE PROVED UNPROVABLE, since if proved it would again be contradicted.
What is significant is that the statement, thus harboring contradictions, cannot be added to the axioms of the system as suggested by the discussants, because that would make the system inconsistent, with consistency vehemently, and justly, insisted on by all authors. The discussants believe that the statement is legitimate, because it is a "well formed formula", i.e. it abides by grammatical rules. But contradictions are possible within the best of grammar. The problem is better attributed to positing "formal systems", ones without meaning, since in those cases one has no content to fall back on for the search of hidden contradictions.
Hidden contradictions are the province of paradoxes, to which the Goedel sentence can be relegated, and which I also consider, but elsewhere: On Proof for Existence of God, and Other Reflective Inquiries.
|
home
