Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com
Dr Kasman of the College of Charleston has posted a review of the book “The Shackles of Conviction” on his web-site for mathematical fiction. While he ostensibly tries to be fair, (Footnote: It might be noted that he begins his review with a prominent ad hominem attack, see the section And a bit more on understanding.) it is clear that he is not a disinterested observer, and he has applied his dogmatic convictions to his review of the book. The review seems be principally an exposition of his deeply held conviction that Gödel’s proof cannot ever be interpreted as giving an expression that can be interpreted to be at the same time ‘true’ and also unprovable in the formal system (also see True but unprovable).
In his review he declares that I do not understand Gödel’s incompleteness proof.
You might expect that he might back up that claim by providing evidence that I have failed to follow Gödel’s argument.
But no, he makes that claim on the basis that his interpretation of Gödel’s result, based on underlying assumptions that he considers reasonable, is the only valid interpretation of Gödel’s result. But having a different interpretation of a result of an argument can never indicate that one does not understand the argument itself. It is quite irrational to assert that someone else does not understand something simply because they have a different interpretation or because they do not share your basic assumptions.
I have created a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside Gödel’s proof. If you are wondering whether there is any substance to Dr. Kasman’s claim that I do not understand Gödel’s proof, I suggest that you take a look at it; it may help you to decide whether it is reasonable to suggest that I do not understand the proof. When I decided to make that guide, I did an intensive search to see if there was anything similar already published anywhere, but I found no detailed guide at all - so I constructed my guide from scratch - which I would claim is hardly indicative of someone who does not understand Gödel’s proof.
And as is shown in True but unprovable, Dr Kasman’s conviction has no rational basis.
Dr Kasman even claims to know that: “In fact, most people assume only that the statement must be either true or false”.
Quite why Dr Kasman believes he is omniscient on the viewpoint of the majority of people on this matter is a mystery to me. The facts are that:
“From the remark that [the unprovable statement] asserts its own unprovability, it follows at once that [the unprovable statement] is correct, since [the unprovable statement] is certainly unprovable (because undecidable). So the proposition which is undecidable in the [formal system] yet turns out to be decided by meta-mathematical considerations.”
But, despite these facts, having decided that his personal conviction is in line with the majority of people, Dr Kasman concludes that because he thinks he does understand Gödel’s proof, and because I do not share his conviction, then it follows that I do not understand Gödel’s proof. This is so even though by his own admission, he is not an expert on Gödel’s proof, and is not even familiar with Gödel’s actual proof. Whereas I am so familiar with the details of Gödel’s proof that I have been able to create a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside the proof.
And in his review, Dr Kasman goes on to use the same old argument (Common Objection 1) that “there are now many different proofs of the same result, all reaching the same conclusions…” And he also says that even if I have found a flaw in the original proof, the other proofs of the same result would still stand. I have examined several of these other proofs and found fundamental flaws in each of them - the details are given at Analysis of Incompleteness Proofs. See also Errors in Incompleteness Proofs.
But, even if we ignore these other proofs, Kasman’s review misses the underlying crucial point. If Gödel’s proof is wrong and if mathematicians have scrutinized Gödel’s proof for eighty-five years and have been quite unable to see the fundamental flaw in it, then how can anyone be confident that all these other proofs (which have been subjected to much less scrutiny) are correct?
As noted above, several of these other incompleteness proofs have been examined it has been found that they are also irredeemably flawed. Kasman simply perpetuates the old myth that there actually is a rigorous proof of Gödel’s incompleteness theorem; even though he admits he has not done any in-depth study of these incompleteness proofs, he assumes that someone else of sufficient competence must have thoroughly checked them.
Dr Kasman also rehashes a familiar old chestnut – the claim that many people objected to Gödel’s proof at the time of its publication.
But when Gödel’s proof appeared, it was not a revolutionary overthrowing of the commonly accepted philosophy that existed at the time of its publication – it was accepted by the vast majority of philosophers, mathematicians and logicians precisely because it appeared to say what they wanted to hear – in effect, that man is capable of a higher level of reasoning than a formal system, that there are realities that cannot be captured by a formal system, and that human intuition is superior to formal reasoning. The fact remains that Gödel’s paper failed to provide any detailed proof of the crucial part of his proof (his Proposition V) but despite that, Gödel’s result was easily accepted. It is difficult to see how that would have been the case if the majority of observers could not accept Gödel’s result as fitting in with their principal philosophical beliefs.
Later in the review, Dr Kasman then tries to be even-handed, stating that he does not want to be seen as rejecting my results out of hand. He then goes on to state that he is not convinced by my argument, but he gives no logical reason as to why my argument might be wrong. The result is that by the end of the review, we have only learned that:
And, given Kasman’s sketchy knowledge of Gödel, it is quite likely that he is unaware that in a different paper, Gödel made a completely unproven assertion which almost everyone assumed to be correct because of Gödel’s esteemed reputation - but which was later found to be wrong. Gödel’s intuitive assertion in that paper was proved wrong some 50 years later - Gödel’s intuitive assumption stated precisely the opposite of what was actually the case. For details of this see Gödel’s Intuitive Error (2). Not many mathematicians are aware of this - Gödel’s reputation is still so honored today that most discussions of Gödel’s work skip over this inconvenient detail.
And we still have no idea why, if Dr Kasman really thinks the demonstration of a flaw in Gödel’s proof is wrong, why he doesn’t simply give a logical explanation of where it is wrong. As shown in Flawed Disproofs, if you really understand Gödel’s proof, it’s not actually very difficult to find an error in a disproof of it - if there actually is an error in the disproof.
But ultimately, what is so disappointing about Dr Kasman’s review is that it tells us more about Dr Kasman’s dogmatic convictions than it tells us about the book. It tells the reader of the review next to nothing about the book, and gives the reader no insight as to whether he should read the book, whether he might enjoy reading it, or whether he would be stimulated by it.
By the way, Dr Kasman mentioned in his review that we had some communication, which he declined to continue. I found it a rather frustrating communication. While I was careful to answer very specifically every point that he would make, he would only ever give a general reply. That meant that he avoided answering many of the specific points and questions that I had raised.
There are some quite remarkable utterances in Dr Kasman’s review that I think are worth singling out for particular attention. Among them are:
“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.”
What a wonderful way of avoiding having to point out an error in someone else’s argument. Dr Kasman’s method of countering an argument is superbly crafted - since he is someone who is capable of great perspicacity, then if he claims that he doesn’t see an argument, then there isn’t one there - and because of his great wisdom, we are all bound to accept his pronouncements without question.
“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.”
Presumably Dr Kasman’s intention here is that if he says that he is ”unable to see any point behind” an argument, then there can’t be any.
Then there is the claim that my arguments are irrational:
“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.”
Here Dr Kasman appears to confirm the very point I am making; instead of providing a logical argument against what I claim, Dr Kasman makes it quite clear that he is not interested in doing so. He attempts to deflect attention away from this fact by simply stating, without providing any evidence whatsoever, that my arguments are irrational.
Finally, he says:
“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.”
No, I’m not expecting anyone to take my word for it. I don’t even want anyone to take my word for it. That’s why I have provided a logically reasoned argument, and provided a simplified version so that it as accessible to as many people as possible.
You only have to look at the facts; it is not I, but it is Dr Kasman who expects people to simply take his word for it — he wants people to accept his assertion that he is correct without providing any rational argument to support what he says.
In common with many others, Dr Kasman in his review draws particular attention to my claim in the preface of my book that I do fully understand Gödel’s proof of his incompleteness theorem, whereas previously it has not been understood. He does so by highlighting it in a box all on its own, and states that I am certainly not lacking in modesty.
In common with much of Dr Kasman’s ‘review’, it gives the reader no information as to whether he should read the book; it simply serves to tell us about Dr Kasman’s opinion that I cannot possibly be correct – an opinion based not on any intensive study of Gödel’s proof on his part, but simply because it is beyond the bounds of his imagination that he could be wrong.
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There is now a new page on a contradiction in Lebesgue measure theory.
There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
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