a list compiled by Alex Kasman (College of Charleston)
|This novel intersperses a fictionalized account of the life of Kurt Gödel with the modern tale of an engineer who realizes (and eventually convinces the world) that Gödel's proof was flawed and that his Incompleteness theorems are false.
Though I appreciate his attempt to present Gödel as being more human than the standard portrayals -- which show him as being simultaneously unbelievably brilliant and pathetic -- the main goal of this book appears to be sharing with the world what the author himself believes to be a flaw in Gödel's work, and so it is more an attempt at popular non-fiction than fiction. There is no problem with that in general. Quite a few works on this site are didactic and merely use fiction as a way to better convey a mathematical idea. The problem is, I believe that Meyer is wrong.
The author of this self-published novel is certainly not shackled by modesty. In the preface, he he offers this explanation for why he wrote the book:
It is his contention that all of those who came before him were taken in by their intuition and a desire to believe in Gödel's work and missed what he has now discovered to be its fatal flaw. In fact, I believe that Gödel's theorem is one of the most carefully scrutinized mathematical results precisely because it is not intuitive and because many people objected to it. (It is also probably the mathematical result most frequently misquoted and misused by non-mathematicians...but in general I have found that mathematicians do seem to understand it well.)
A very brief description of the result is this: Given any axiomatic system which includes arithmetic (technically, any decidable axiomatic system which includes not necessarily all of but at least a large enough part of arithmetic to regulate addition and multiplication, but I'm trying to get the main idea here), there is a meaningful mathematical statement which is false if and only if it can be proved to be true from the axioms. Consequently, if we believe mathematics itself to be such an axiomatic system, we can conclude that there is either a meaningful statement which can neither be proved nor disproved or that it is inconsistent. (Moreover, an additional result shows that we cannot ever prove that it is the former and not the latter.)
It seems to me that one piece of Meyer's misunderstanding is his belief that the mathematical statement is assumed to be true. In fact, most people assume only that the statement must be either true or false (a seemingly reasonable assumption) and the theorem retains its power. (One could "solve" the problem by claiming that any statement which cannot be resolved using the axioms is neither true nor false, but that is only a semantic solution. We generally assume that a statement like 80532=64810809 or "every even number greater than two is a sum of two primes" is either true or false, regardless of whether we are actually able to determine it.) Additionally, he focuses a great deal on the question of whether statements in the meta-mathematical language can have mathematical meaning. However, Gödel set up his correspondence explicitly so that this would be the case, and I do not see any contradictions or gaps in his having done so. Moreover, although he seems quite focused on showing that the original proof published by Gödel is not valid, he seems to completely ignore the fact that there are now many different proofs of the same result, all reaching the same conclusions.
I do not want to be seen as rejecting Meyer's conclusions out of hand. Certainly, there is always the possibility that some famous mathematical result is invalid and that many great mathematicians have been misled. (In fact, according to Gödel there is the possibility that all of math is wrong!) However, there is a burden on the part of the person making such a claim to point out this error in a convincing and irrefutable way. I am sure Meyer believes he has done so in the form of this novel. However, I remain unconvinced. Having read his argument, I still believe that it is Meyer, and not the rest of the mathematical community, who is mistaken. He has not even convinced me that he has found a flaw in the original proof, but even if he has the other proofs of the same result would still stand and so such a flaw would not be as spectacularly exciting as he suggests when he describes this novel as "a book that is set to become one of the most talked about books of the year."
You may also wish to visit Meyer's Website which includes a brief biography of the author and a paper that presents the same ideas as the book in a non-fictional setting. However, as a contrast to the opinion presented by Meyer, please consider also looking at this response from a logic student who explains both the problems with Meyer's claims and the story of his attempts to discuss the topic with him.
BTW: If you are interested in this work of fiction, you may also be interested in Cantor's War, a short story by an author who seems to argue that there was a flaw in Georg Cantor's comparison of the "sizes" of infinite sets.
Additional information added on April 6 2009: I have exchanged some e-mails with Meyer. The literary aspects of the book were not really a topic of conversation. Rather, the discussion was about the mathematics behind it. The e-mails were polite and full of serious discussion, but in my opinion not very productive. Here are some comments from the author that I agreed to post:
Meyer is correct in saying that I was not able to point out any error in his book, but that is because I do not see the argument in it. Certainly, Gödel made use of the meta-mathematical technique by which mathematical expressions were used to say things about mathematics. Meyer sees a logical flaw in that procedure, and I cannot point out where Meyer is mistaken because I do not even see what flaw he thinks he sees. Rather, most of our discussion was focused on what he says was his first clue that Gödel must have been wrong. There, at least, I believe I see what he is trying to say. However, I still disagree with his conclusions there. In particular, it is quite significant that the statement that forms the basis of Gödel's theorem, the one which is equivalent to its own unprovability, is not proved to be true. In fact, a key part of the proof is that he proves the statement can neither be proved to be true or false. (In particular, it is true unless the system is inconsistent. So, if one could write a proof showing that the system is consistent then it would prove that the statement is true...but Gödel also famously proved that you cannot prove consistency from within the system.) Many mathematicians (and perhaps Douglas Hofstadter, who is not a mathematician) do assume that statement is true in at least some reasonable axiomatization of arithmetic, because if it is not then mathematics is inconsistent which would be difficult to believe. However, even they would admit that though they believe it to be true, they do not have a mathematical proof that it is. Meyer is correct that if they believed they had a mathematical proof that the statement was true but unprovable, that would be an obvious contradiction. (Too obvious to have gone unnoticed by so many smart people for so many years.) But, whether others believe that it is true (as Meyer claims) or that it is either true or false (as I maintain), the fact that we all agree it has not been proved saves it from being as obviously wrong as Meyer suggests.
As to whether there is some deeper problem that only Meyer sees, I suppose you will have to determine that for yourselves. Meyer claims that the mathematicians he has tried to talk to are not interested in rational argument. They (and I) would counter that it is Meyer whose arguments are irrational. That there are more of us than him should not be seen as an important factor in deciding who is right here. Truth is not a democracy in which the majority rules. It could be that he sees something that the rest of us have been missing for over half a century. It could be, but I cannot simply take his word for it. In my discussions with him I have been unable to see any point behind his discussion of the role of the interpretations of expressions in the mathematical and meta-mathematical systems, and believe I can see the errors in his "initial clues" that there was an error in this famous proof.
Lest we get into any conspiracy theories, let me point out that I have no particular attachment to Gödel's theorem. None of my published research would be at risk if it turned out to be wrong. In fact, if I thought Meyer had a point, I would be happy to spread the word and would try to make my own contributions. In fact, I think many researchers would be happy to get in on the "ground floor" of the new foundations of mathematics that would grow out of Meyer's research if it were correct. That I am not is simply an indication of the fact that I have not found his argument at all convincing.
If you are interested, please do read his arguments for yourself (either in the form of a novel or an article).
|More information about this work can be found at .|
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(Maintained by Alex Kasman, College of Charleston)