Best known for his Incompleteness Theorem, Kurt Gödel (1906-1978) is considered one of the most important mathematicians and logicians of the 20th century. By showing that the establishment of a set of axioms encompassing all of mathematics would never succeed, he revolutionized the world of mathematics, logic, and philosophy.
An Irishman, James R. Meyer, worked first as a veterinarian and, later, as an engineer. He became interested in what he sees as discrepancies in Gödel’s Incompleteness Theorem and set out to investigate the flaws. His novel, The Shackles of Conviction, questions Gödel’s widely accepted theorem.
Simply Charly: What prompted you to write a novel on Kurt Gödel and his Incompleteness Theorem?
James R. Meyer: The idea of a novel began when I was working on Gödel’s theorem when I was reading a lot, not just about the theorem, but articles about Gödel’s life as well. I started to become intrigued when these articles portrayed Gödel as a stereotypical cold and impassive logician—because while he was working towards his famous theorem, he also managed to have two affairs with married women, one of whom he eventually married. So it seemed to me that there was more to Gödel than the conventional “cold fish” image. And the more I read, the more the facts about Gödel’s life just didn’t add up.
One day the thought came to me—what if Gödel had suspected that there was something wrong with his proof of his famous theorem? And suddenly, everything clicked into place. Instead of a cardboard cutout character, I now saw Gödel as a genuinely human character, with human failings, human worries, and subjected both to the triumphs and disappointments of life. I saw a man trapped in a situation from which he could envisage no escape, a situation that made him ever more eccentric and which would eventually drive him to the brink of insanity.
From that moment on, I knew I had to put this idea into words. And when I had discovered the real truth about Gödel’s theorem, I couldn’t resist the opportunity of putting my ideas about Kurt Gödel’s life and his theorem together in a novel.
SC: In your novel, The Shackles of Conviction, you make a compelling claim that Gödel’s theorem is wrong despite its widespread acceptance by the mathematical and philosophical community. How did you arrive at this conclusion?
JM: I first came across Gödel’s proof of his theorem about twenty years ago when I was reading a popular book on curious mathematical ideas. From the description of the theorem as it was given in the book, I felt that there had to be something wrong—either with Gödel’s theorem or the book’s description of it. I started to study Gödel’s actual proof, but I had to put it aside because it was taking up too much of my time. Then, about three years ago, quite by chance, I came across another book about Gödel’s theorem, and I was hooked again, and this time I started to study it in depth.
It didn’t take me long to discover that Gödel’s proof threw up logical contradictions. But it took me a lot longer to work out completely what was going on and to understand fully how these contradictions were coming from a fundamental flaw in Gödel’s proof. Knowing what I know now when I look back, I find it hard to believe that it took me so long. But then I am consoled by the fact that if hundreds of logicians and mathematicians weren’t able to see the flaw for seventy-five years, then it was always going to be difficult.
SC: Your novel seems to be partly autobiographical as it mimics what you have demonstrated in real life, which is, that there is a fundamental flaw in Gödel’s theorem. Would you say this is true?
JM: Oh no, the novel isn’t in any way autobiographical, although I’m sure that I’ve instilled something of myself into the main character of the book—or perhaps that should be how I would like people to imagine what sort of character I am! And, of course, the main character in the book is spurred on by the promise of an enormous sum of money as a reward—unfortunately, this wasn’t the case for me!
SC: Why did you choose the vehicle of fiction to tell this story than as a work of non-fiction?
JM: There were several reasons. The main reason was that once I had the ideas about Kurt Gödel’s life that would show him as a real human character, I knew I wanted to write a novel about it. That fitted in well with my discoveries about Gödel’s theorem. There were other reasons, of course. I hoped that I could reach a wider audience in this way that I could ever hope to do with a non-fictional account. And it was always going to be a battle trying to get any paper published when it flies in the face of conventional mathematical wisdom. Before I had worked out exactly where the flaw was in Gödel’s proof, I had already submitted a paper to several journals where I showed how Gödel’s proof led to logical contradictions. These were all rejected on the basis that “lots of people have looked at Gödel’s proof and found nothing wrong with it—therefore there is nothing wrong with it.” So I knew at that stage that trying to get a paper on the flaw published in a journal could take years, maybe decades, maybe never. I hope that the book shows that its creation was prompted not by a fleeting whimsy, but that this book has something important to say. I hope it can show people how to understand really how Gödel’s proof works (or doesn’t), without the need for any complicated mathematics. And maybe for some of those people, they will have the thrill of experiencing that special moment where you get a sudden flash of insight, and you say, “Oh my God, I see it now!” If I succeed in doing that, then the effort involved in writing the book will have been worthwhile.
SC: On your website, you’ve published a paper that purports to show a comprehensive demonstration of the flaw in Gödel’s proof of his Incompleteness Theorem. Has anyone challenged your claim?
JM: Oh yes, lots of people have challenged it. But no one has come up with any logical reason for why what I have to say about Gödel’s proof is wrong.
I don’t blame anyone for the knee-jerk reaction when they first hear of my paper and my book. There are a lot of cranks and crackpots out there who regularly claim that they have found an error in some area of mathematics, so it’s not surprising that the first reaction is one of skepticism when Gödel’s proof has been accepted as correct for seventy-five years.
What I do get annoyed about is when people, logicians, and mathematicians included, are completely illogical in their responses to my discovery of a flaw in Gödel’s proof. Some people take it upon themselves to show that my demonstration of a flaw in Gödel’s proof is wrong, believing that it will be easy to find where I have “gone wrong.” And then, when they find that there is nothing wrong with it, they resort to some other tactic.
For example, they may say that they know of another incompleteness proof, and say that they’ll accept that Gödel’s proof is wrong if I can find a flaw in that one—which is absurd, since if they won’t accept my discovery of a flaw in Gödel’s proof, they aren’t likely to accept my demonstration of a flaw in any other proof.
Or they say that my argument must be wrong because Gödel’s proof is obviously correct—since if no one has found a flaw in it in seventy-five years, then there can’t be a flaw in it!
None of these responses show any appreciation that the only sure way to refute a logical argument is to show an error in it. Instead of pointing out an error in my argument, they simply resort to arguments that sidestep the real issue.
One professor who is highly respected in the areas of logic, the foundations of mathematics, and computer science wrote:
“I did download your paper and have to say that you describe your arguments clearly, so that for those (essentially all logicians) who hold the validity of Gödel’s argument, one could relatively easily point at a misunderstanding in your writing (as I believe there must be).”
He didn’t point out any error in my paper, but he passed on my paper to a colleague who also could not find any error in my paper and who eventually conceded that: “Perhaps Gödel made a mistake in his proof. I don’t know. I have not read his original (translated) proof in careful detail. I’m not here to defend Gödel’s original proof … Even if Gödel’s proof has an error in it, it would only be of historical interest.”
That’s just one example (by the way, the flaw in Gödel’s proof isn’t some sort of minor problem that can be overcome by rewriting a few lines of the proof. It’s a fundamental flaw that can’t be ignored and from which we can learn so much).
So while one logician says it must be relatively easy to point at the alleged “misunderstanding” in my paper, no one has actually been able to do so. If I’m getting those sort to responses from people who are purportedly experts knowledgeable about Gödel’s theorem, it’s not surprising that it will take a while to overcome the resistance to the idea that there can be any problem with Gödel’s proof.
SC: If what you say is true of Gödel’s proof, then why isn’t it more widely known? After all, your paper would seem to pull the rug from under what has been regarded as the predominant pillar of thinking for the past 75 years. It would be catastrophic, don’t you think?
JM: I think you have to remember that the circle of mathematicians and logicians that are professionally employed and who work in detail on these areas is extremely small, and as such, it is a community in which a wrong move can spell disaster for a future career. And since the ones at the top have got where they are by adhering to the common core of acceptance, anyone who tries to go against the current will have a very hard time doing so. This isn’t anything new. Battles to be recognized as the bearer of the most logical argument in areas of mathematics and logic have been going on like this for well over two hundred years.
SC: Are any renegade mathematicians/logicians disputing the theorem of incompleteness, and if so, on what grounds?
JM: The exact opposite is the case. No one disputes Gödel’s theorem, but people insist that it has no philosophical impact, no bearing on how math should actually be done, and does not change the traditional formalist mathematical stance believing in static black-and-white absolute truth attained through logic and the axiomatic method. So now, in a reversal of fortune, it requires daring to claim that incompleteness is significant. (I myself am such a renegade.)
So although there are many logicians of repute who are well aware of my demonstration of the flaw in Gödel’s theorem, no one wants to be the first to admit it publicly. And although they privately believe that there is something wrong with Gödel’s proof, they are too scared to damage their reputations by stepping out of line. They are scared that perhaps they have missed something and that maybe I am wrong, although they cannot see how that might be the case.
It’s a big problem. I know will take time. The problem is that, because Gödel’s proof has been accepted as completely correct, proofs that are different but somewhat similar have been subject to little scrutiny and criticism. They have been accepted because they appear to fall in line with conventional thinking. And that has reinforced the viewpoint that Gödel’s proof must be correct. I just wonder how long logicians will continue to insist that it must be easy to refute my argument at the same time as not doing so. Isn’t it going to become more and more embarrassing as time passes?
Eventually, people are going to have to accept what logic dictates, and that Gödel’s proof is just the same as all other proofs. And that means that it gives a result that depends completely on all the assumptions and rules that are used to generate it. And if some of those assumptions and rules are not logically acceptable, then you cannot accept the result of the proof. Gödel’s proof includes steps that have been accepted because they appear to be intuitively correct, but those intuitive assumptions turn out to be incorrect. And in a way that is very ironic, because so many philosophers have claimed that Gödel’s result shows that intuition is in some way superior to formal reasoning—but Gödel’s result is actually a result of faulty intuition.
And will it be catastrophic? No, of course not. How could it ever be a catastrophe to discover something that pushes our understanding forward? That can’t be a catastrophe, it is a learning experience. And in the case of Gödel’s theorem, it shows that there is a fundamental error in the way that we think about certain things, and learning that can only be beneficial—it is an opportunity to learn how we can prevent similar errors in reasoning in the future. So rather than being catastrophic, I think the discovery will be of great benefit. People will be forced to rethink many deeply held convictions, and in many cases, these convictions will be found wanting. That can only be a good thing. (I myself am such a renegade.)
When I started to work on Gödel’s proof, I suspected that if I found the flaw in the proof, I would have learned something very important. And that has indeed been the case. The flaw in Gödel’s proof teaches us fundamental things about logic and language that will be of great benefit to logic and mathematics. So it’s not the fact that Gödel’s proof has a flaw in it that is so important, but knowing how that flaw operates because that shows us how we can avoid similar problems in the future.
SC: What exactly is Gödel’s Incompleteness Theorem?
JM: Before you can understand what Gödel’s Incompleteness Theorem is, you have to have some idea of what a formal mathematical system is. And basically, a formal mathematical system consists of three things. First, there is a fully defined language, so that the alphabet of the language is fully determined, and there are definite rules that decide what are valid statements in that alphabet and what aren’t. Secondly, there is a collection of fundamental statements in the language that aren’t proven, but they are taken to be the fundamentally “true” statements of the language. These are called the axioms of the system. And thirdly, there are proof rules—rules that decide how a statement can be proven from one or more other statements in the language.
So, once you have such a system, you have the basis for making a complete proof in that language. And what Gödel’s Incompleteness Theorem says is that it doesn’t matter which formal mathematical language system you use, you can always come up with an actual statement about numbers in that formal language which cannot be proven to be true by that formal system, and nor can it be proven to be false (hence the name “incomplete”).
That in itself may not sound particularly significant, nor very interesting. But it is not the claim of incompleteness in itself that makes Gödel’s proof so interesting. The really intriguing thing about Gödel’s proof is that it also purportedly “proves” that this actual statement about numbers is actually true.
And it is this that has fascinated so many people, including myself—because Gödel’s proof is itself expressed in a language, and it relies on certain assumptions. And if that language and those assumptions can prove this actual statement about numbers to be actually true, I have to ask, what is it about that language system that makes it capable of such things, when no fully defined mathematical language could do so?
And when you come down to it, this is what demonstrates the paradoxical nature of Gödel’s result—that for any formal system there is a formula which is unproven by the formal system but is provable by the language of Gödel’s proof. And if Gödel’s proof was actually correct, wouldn’t that indicate that Gödel’s proof could never be stated in a formal language? But on the other hand, if Gödel’s proof is a logically coherent argument from given first principles, then, ultimately, should it not be possible to translate this logical argument into a precisely defined formal language?
What I find so amazing is that logicians who have studied Gödel’s proof seem to be content to set this question aside. They are content to sit back and simply say that Gödel’s proof proves that formal languages can never prove all “true statements.” But when they do that they are saying in effect that ordinary language must be somehow superior to any formal language, since it can prove a statement of the formal system that the formal system cannot. Surely, I ask myself, any logician would want to understand why this is the case.
SC: Can you summarize the essence of your argument that Gödel’s proof contains a flaw?
JM: Well, this follows on from the last question. The flaw arises from the very fact that Gödel’s proof isn’t expressed with the same strict precision of a fully defined mathematical language. The astonishing thing is that at the crucial point in Gödel’s proof, the point where the flaw occurs, Gödel simply doesn’t bother to give a fully detailed proof. All he does is suggest a rough outline of how you might do a detailed proof. Gödel justifies this by saying that this part of the proof doesn’t need a detailed proof, that it’s all intuitively obvious.
But it’s a fundamental principle in mathematics and logic that you can’t replace a logical argument by intuition. Otherwise, there would be no need for any mathematical proof at all. Intuition is fine as the basis for the idea for a proof, but that intuition has to be backed up by a logical argument. And sometimes intuition is wrong—there have been several other cases in mathematics where this has been the case.
And that is why the crucial part of Gödel’s proof where he simply relies on intuition has to be looked at very carefully. If you go through Gödel’s outline for a proof, you can indeed build a detailed proof that superficially appears to be completely logical, and which gives the same result as Gödel’s intuitive outline. But if you examine it carefully, you see that it involves a confusion of language. Gödel’s proof at the crucial point actually involves not two, but three separate languages. One is the language of the formal system, the second is another language that involves number relationships, and the third language is the language that talks about these two other languages.
The error that Gödel makes is that he confuses the second language and the language that talks about that language. His intuitive outline refers to a statement that is actually a statement in that second mathematical number language, but Gödel makes the error of assuming that is a statement in the language that talks about that second language.
And that’s what is intrinsically wrong with Gödel’s proof. It’s not something that arises from the way that I have filled in the details of the proof. It means that the result that Gödel got hasn’t arisen from any aspect of the formal system. It comes from the ambiguities inherent to the language that Gödel uses for his proof.
By the way, I now have on my website a simplified explanation of Gödel’s proof and the flaw in it that I have written to be easy to follow, but which includes the essentials of the proof.
SC: What are the implications of your discovery? How do you think it will affect mathematics going forward?
JM: I think it will eventually be accepted, more so perhaps by the incoming generation of mathematicians and logicians, rather than those who are inflexible in their fixed beliefs. In my mind, I see two possible scenarios, and I don’t know which will prevail in the short term.
On the one hand, I see the possibility of an exciting new future for mathematics, one that is firmly based on reason and logic and in which many beliefs of the past—those that have no basis in reason—are finally shrugged off as we move into a mathematical land free of such myths. You know, when I look at the whole field of scientific knowledge, I find it somewhat ironic that the area of mathematics and logic should be among the last of the scientific subjects to finally free themselves of myths, even though that area should be the most rigorous of all scientific subjects.
The failure of logicians to see that Gödel’s proof had to be wrong might be seen as an embarrassing failure. How could such a thing happen? Isn’t it worrying that the crowd has followed blindly the decree that Gödel’s proof must be correct? That must stimulate self-examination within the cozy world of mathematics and logic, one which may be uncomfortable for those who feel quite settled in that world. But it will have to come, and the result will be a better environment for new students to work in, a new environment in which everything must be subjected to rigorous logic, where nothing is taken for granted. And that is how it should be.
The other possibility I see is that mathematicians will bury their heads in the sand and ignore any possibilities that their deeply cherished beliefs might be wrong. Currently, I see a problem because new ideas that do not conform to the accepted norm are rejected. There are so many submissions to journals that those who try to review such papers cannot possibly check every such submission in depth. The result is that papers that conform to the accepted norm are published, while those that do not are rejected—not because the reviewer can find anything wrong with the logical argument of the submission, but on the basis that “Your paper cannot be correct because it contradicts [ here you fill in some commonly accepted result].”
The problem is that once a paper is published in a journal, that effectively makes it “correct.” It is rare for a published paper to be later found to be wrong. And that means that the reviewers who look at papers and who have to judge them are under great pressure not to make a mistake. They don’t dare to let through a paper that will later be found to be wrong. And the big problem is that this has become so ingrained that even short, simple submissions that can be easily examined are rejected on that basis. The end effect is that the journals will not publish anything that contradicts the status quo. And this may continue to be the case for some time. But I think eventually the cracks will become so big that they can no longer be ignored, even by those who push their heads in the sand as deeply as possible.
SC: Mathematician Gregory Chaitin has said that “Gödel’s incompleteness theorem is a reductio ad absurdum (reduction to absurdity) of David Hilbert’s traditional formalist view that math is based on logical reasoning and the axiomatic method.” Do you think this still holds true after your discovery?
JM: It was always the case, and it still is, that mathematics is considered to be something that is based on logical reasoning. I think my discovery will finally show the absurdity of the approach of Chaitin and similar people when they say that you can use informal, intuitive language that isn’t fully defined to “prove” that a fully defined formal language is somehow “not as good” as this informal, intuitive language. Chaitin and others like him revel in producing paradoxes, and insist that such things are a fundamental part of all mathematics. They’re not. They are indications that there is something wrong with the system of mathematics you are using. It does not mean that every mathematical system is in some way inherently paradoxical.
SC: Some mathematicians feel that Gödel’s theorem hasn’t really affected how math should actually be done, and does not change the traditional formalist mathematical stance believing in static black-and-white absolute truth attained through logic and the axiomatic method. Do you think this is true?
JM: That’s pretty much the way I think of it—except that I think if the mathematical system that you use is to be able to say anything about the real world, then the axioms of that system have to be seen as actually applying to the real world (the axioms being the statements of the system that are the fundamental “true” statements of the system).
There is a viewpoint is that modern mathematics is divided into two quite distinct sides, where one side is devoted to practical mathematics, such as mathematics for engineering, computing, physics, and so on, and another side which is not interested in whether their mathematics is based on reality, and which creates proofs that say nothing about the real world we live in. The problem is that in practice, there is no clear distinction between the two sides, so that we now have professors in computer science who, contrary to what you might expect, are deeply immersed in working with mathematical systems whose fundamental axioms are not based on any experience of the real world. The problem is that they don’t make their students aware of this distinction.
SC: What are you currently working on?
JM: It’s not easy finding the time to do all that I would like to do. I have to carry on with a life that has some semblance of normality. At the moment, when I do get the time, I am working a number of areas in the foundations of logic and mathematics. Some of them follow directly from my findings of how the flaw in Gödel’s proof operates. I’m trying to bring them all together, and hopefully, I will eventually manage to get it all organized into a book on faulty reasoning in areas of logic and mathematics, which I think will be quite a revelation.
I wish I could devote more time to the foundations of mathematics and logic. But unless some far-sighted benefactor is willing to sponsor me, what I can do is limited. But I will keep at it and get there in the end.
Kurt Gödel – On Formally Undecidable Propositions of Principia Mathematica and Related Systems (English translation by Martin Hirzel)
James R. Meyer – The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem