Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com
We make many decisions during the course of the day. Sometimes these decisions are guided by emotion, sometimes we just rely on a hunch, sometimes we rely on experience, and sometimes we analyze a situation logically and make a decision according to this logical analysis. But very few things in life are easy to analyze in a completely logical way; in most cases, our actual decisions are based on a combination of emotion, experience, and a little bit of logic.
However, when we want a conclusion that isn’t based on any emotion, or hunch, we want a conclusion that is arrived at purely by means of logical argument. This site is devoted to showing how many results that are commonly accepted as being the result of a completely logical argument are in fact flawed because of the failure to acknowledge the significance of the way language is used in the argument. When a statement itself refers to some aspect of language, a seemingly innocuous statement can contain subtle errors which render the statement illogical. Unless every aspect of a statement which refers to some aspect of language itself is very carefully analyzed, an ostensibly logical statement may actually contain subtle errors of logic.
This site explains how such errors may occur; in most cases it is because insufficient attention has been given to the way in which such statements refer to language. If you are visiting this site for the first time, I suggest these pages as suitable starting points:
Most of this site is, naturally enough, based on logical and factual analysis. To provide some contrast, I decided to include some viewpoint based material here - this is where I get an opportunity to voice my opinion on various matters. Feel free to disagree.
Currently we hear a lot about fake news. What we don’t hear much about is fake mathematics. At this point you might be wondering what I mean by fake mathematics.
Fake news might be described as material that is fabricated without any supporting evidence, and which is presented in such a way that naive observers are willing to believe the material without subjecting it to any detailed examination, especially if it concurs with their underlying philosophy.
In a similar vein, fake mathematics might be described as material that is fabricated without any supporting evidence, and which is presented in such a way that naive observers are willing to believe the material without subjecting it to any detailed examination, especially if it concurs with their underlying philosophy.
While we don’t hear much about it, fake mathematics has been prevalent for a great many years. To show that this is the case, we only have to carry out a simple thought experiment. In this thought experiment, we imagine an alternative mathematical world that the one we see today. In our thought experiment, the only proofs accepted by the mathematical community are proofs that have been logically proved, and no proof steps are allowed to be assumed to be correct rather than proven. We now suppose that in this mathematical world (as in our actual world) Gödel submitted his paper on Incompleteness (Footnote: Gödel’s paper was written in German, viewable online Gödel’s original proof in German: here PDF. The English translation of the paper is entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, viewable online Gödel’s Proof - English translation: here.) to various journals. Unfortunately for Gödel, in this mathematical world, all the reviewers rejected his paper because (as in our actual world (Footnote: Peter Smith, although a staunch advocate of Gödel’s proof, acknowledges this in his paper, Expounding the First Incompleteness Theorem (PDF), that, “Gödel only sketches a proof… The crucial step is just asserted.”)) it failed to prove a crucial step in the proof, and Gödel merely assumed that the crucial step (the Proposition V in his paper) was correct. This was completely unacceptable to the reviewers, and Gödel’s paper was never published in this hypothetical mathematical world.
But, as the years rolled on in this mathematical world, large numbers of people still attempted to prove what Gödel tried to prove, but what he never actually did prove. And all these people either tried to rely on an unproven assumption - just like Gödel did - or else they made basic logical errors. (Footnote: See, for example:
The Flaw in Gödel’s Proof of his Incompleteness Theorem
Paper(PDF): The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem
Analysis of Other Incompleteness Proofs
Common Errors in Incompleteness Proofs
Yet another flawed incompleteness proof) In this alternative mathematical world, such people are ridiculed and are called cranks - because what they are doing strikes against the fundamental ethos of this mathematical world, where the establishment of a logical proof of any claim is of paramount importance.
Now, let us look instead at the mathematical world that we actually inhabit. In our actual mathematical world, such people aren’t called cranks. No, often they are professors and have prestigious positions within our mathematical world. Yes, in our current mathematical world, people that should be called cranks and who should be reprimanded for promoting fake mathematics are accepted and even applauded for what they do. In the actual mathematical world that we inhabit, fake mathematics is sitting alongside normal mathematics, instead of being banished forever from it. Surely this is unacceptable in a community in the 21st century that claims to be based on rationality?
In a paper (Psychologism and behaviourism, Ned Block, 1981, Philosophical Review, 90, 5-43, available online Psychologism and behaviourism), Ned Block conceives of a theoretical computer system (now commonly referred to as Blockhead) as part of a thought experiment. Block argues that the internal mechanism of a system is important in determining whether that system is intelligent, and claims that he can show that a non-intelligent system could pass the Turing Test.
Block asks us to imagine a conversation lasting any given amount of time. He argues that, there are only a finite number of syntactically and grammatically correct sentences that can be used to start a conversation. And from this point on there is a limit to how many valid responses can be made to this first sentence, and then again to the second sentence, and so on until the conversation ends.
Block then asks us to imagine a computer which had been programmed with every one of all these possible sentences. Although it has been claimed that the number of sentences required for an hour long conversation would be greater than the number of particles in the universe, Block argues that hypothetically such a machine could exist, so that his argument is still valid as a theoretical argument rather than one which can be applied in practice.
Given this hypothetical machine, Block invites us to agree that such a machine could continue a conversation with a person on any topic, because the computer would be programmed for not only every sentence but for every sequence of sentences that might be inputted to it. On this basis, Block claims that the hypothetical machine would be able to pass the Turing test despite the machine having no attributes that we would assign as indicative of intelligence.
Block claims that his argument shows that the internal composition of the machine must be considered in any assessment of whether that machine can be considered to be intelligent, and that the Turing test on its own cannot suffice. That is to say, his claim is:
Premise: the Blockhead machine does not use any intelligence to produce its response, yet it can pass the Turing test
Conclusion: the Turing test is not a sufficient condition for intelligence.
There are two principal flaws in Block’s argument:
The principal assumption in Block’s argument is that the Blockhead machine, although impossibly large, is not infinitely large, and can contain all of what are considered to be intelligent responses that a human might make. Now, the Turing test is a test of whether a machine can emulate a human. The longer the period of the Turing test, the more difficult it is for a machine to pass the test. Block claims that his argument and a Blockhead machine that he describes, as long as the machine is large enough, suffices for a Turing Test of any duration.
However, Block makes no mention of the time taken for his finite but impossibly huge Blockhead machines to produce a response. He admits that his machine may be larger than the observable universe, but insists that it is a valid theoretical concept since it is nevertheless finite. But if the hypothesis results in an imaginary machine that is larger than the observable universe, it follows that there would be physical limitations with this imaginary machine. For example, the response time to at least some of the questions will be in terms of time greater than 24 hours, so that there will always be a possibility that the response time to a least one question will be greater than the time allocated for the test (It might be mentioned that Block says that there may be new discoveries in sub-atomic physics that would enable his Blockhead machine to be made on a human scale, but this is mere speculation which is completely at odds with scientific opinion at the present time).
As is well known the Turing test is, above all, a behavioural test, a assessment of the interaction of the entity being measured with a specific environment. A human judge will decide that a response time over a certain time interval (which will measured in seconds or minutes, depending on the question, rather than millions of years) is too long for it to be a normal valid human response time, and on that basis can decide that the Blockhead machine is not a human (at least not a normal one), nor a human-like intelligence. After all, we do not find it surprising that IQ tests are time-limited tests; we would class something that takes thirty minutes to solve a puzzle as being less intelligent than one that takes one minute to perform the same task.
In any conversation, a human can introduce new words, and use such new words in subsequent conversation to refer to things that he might otherwise use natural language to refer to. A human could easily invent words of 60 characters or more; using the English alphabet of 26 letters, the number of possible words of up to 60 characters would be greater than the number of atoms estimated to be in the universe. Hence no machine of finite size could cope with every possible new word of up to 60 characters. In other words, such a machine could never exist, even hypothetically, in our universe.
Block’s description of his machine is that it will only deal with “sensible strings”, and that the machine will be programmed using “imagination and judgment about what is to count as a sensible string”. A sensible string would, of course, be any string that a human might use to test the respondent in a Turing test! That is, all strings that would include words of 60 characters or more! The size of the machine is now ridiculously enormous.
But even beyond that, Block’s argument looks even more preposterous when one can envisage that a human can also introduce new symbols and use such new symbols in subsequent conversation. The only limitations on the use of new symbols would be the overall size and that it must be possible to easily distinguish different symbols. There are thousands of Chinese characters, and a human could easily invent new ones. After all, humans must have invented all our language symbols at some point in time, so there is no reason to suppose that a human could not introduce a new symbol in a conversation.
Block’s machine is now becoming more and more ludicrously massive.
Block responds to the type of criticisms above by pleading (his reply to objection 6 in his paper) that:
“My argument requires only that the machine be logically possible, not that it be feasible or even nomologically possible.”
This, of course, is absurdity masquerading as meaningful philosophy. The reality is that humans are physical entities that are subject to the limitations of the physical world. The Turing test is a test envisaged to be applied to physical entities that are also subject to physical limitations.
When Block claims that a hypothetical non-physically realizable machine could pass a test that is designed to be applied in the real physical world to real physical entities, he is simply imagining a magic machine that happens to posses some physical attributes (such as the ability to manipulate symbols) but also possesses magical properties that have no possible physical realization. So the conclusion is?
A magic machine can do magic things that no physically realizable thing can do.
Eh - didn’t we already know that?
I was recently reading a book entitled The Eerie Silence - Are we alone in the Universe? by Paul Davies, when I came to the chapter musing on what humans might include in a message to aliens. In this chapter Davies says:
“The deepest products of the human mind are arguably the mathematical theorems that have been constructed by some of the world’s most brilliant thinkers. Gödel’s incompleteness theorem, for example, is so profound that it is possible that no theorem in the universe can trump it.”
No. Rather than being profound, Gödel’s incompleteness proof is a demonstration of how to wrap up trite banality in so much superficial glitter that it appears profound to the simple-minded, whereas an intelligent analysis reveals it to be erroneous, as is demonstrated in various pages on this website. Similar methods are used by creationists and intelligent design fanatics to impress those who are have limited intellectual capacity or who are simply too intellectually lazy to evaluate the claims.
I wonder would an advanced civilization be more amused or appalled by the claim that the incompleteness ‘theorem’ might be the most important mathematical theorem that there can ever be. Would an advanced civilization even want to communicate with primitive beings that are so steeped in the promotion of such illogical statements? Even if they did commence a communication, it would seem quite possible that human mathematicians, logicians and philosophers would actually try to argue their case against the superior intelligence, being so entrenched in their herd mentality in the correctness of incompleteness proofs.
Apart from anything else, there is no real world application of an incompleteness proof, so why should such a proof be adjudged as superior to all the mathematical advances that have led to the development of modern technology? It is very difficult to fathom why, even given that a person has a belief that the incompleteness proof is correct, that a person should find it profound in any way. After all, all it supposedly proves is that if a formal system could refer to its own statements, then it would not be able to prove certain self-referential statements. But it says absolutely nothing about whether the formal system could prove or not prove other statements that are not self-referential. After all, if a mathematical system could logically state the equivalent of “This statement is not provable”, what would be so profound about not being able to prove it?
It is noteworthy that the notion of incompleteness has been used to promote all sorts of wacky claims, such as that it demonstrates the existence of god, see Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century, and Gödel’s Incompleteness Theorem. It has also been claimed that it proves the non-existence of god, see Incompleteness and God, and that it proves the existence of spirituality, see Either God exists or one must accept mystery. Of course, the original ‘proof’ of incompleteness was written by Kurt Gödel, who, according to his wife “was religious and read the Bible in bed every Sunday morning”, and who attempted an ontological proof of god’s existence. He also thought that his incompleteness ‘proof’ helped to prove Platonism, that is, that mathematical concepts are actually real but non-physical things. Davies cites Gödel as one of the “world’s most brilliant thinkers”, but Gödel was actually a believer in intelligent design, and said:
“I don’t think the brain came in the Darwinian manner. In fact, it is disprovable. Simple mechanism can’t yield the brain. I think the basic elements of the universe are simple. Life force is a primitive element of the universe and it obeys certain laws of action. These laws are not simple, and they are not mechanical.”
“I believe that mechanism in biology is a prejudice of our time which will be disproved. In this case, one disproof, in my opinion, will consist in a mathematical theorem to the effect that the formation within geological times of a human body by the laws of physics (or any other laws of a similar nature), starting from a random distribution of the elementary particles and the field, is as unlikely as the separation by chance of the atmosphere into its components.”
(Sections 6.2.11 and 6.2.12 in the book A Logical Journey by Hao Wang MIT Press, 1996.)
Here Gödel is just one of the first in a long line of creationists and intelligent design fanatics who think that it is possible to produce a meaningful probability value that suggests a designer, despite our having no means of calculating such a probability (for example, if there are many universes, then we have no way of knowing how many universes are like ours).
People say that incompleteness proofs prove that there are limitations to what formal systems can do. But that isn’t profound, since we knew that anyway. We know that there are physical limitations to any system of expressing mathematical notions; it is not possible to prove all possible mathematical proofs by less than a given number of symbols. That means that there is a practical limit to the length of mathematical proofs - so that regardless of any incompleteness proof, there will be mathematical statements that we cannot prove to be correct or incorrect. But that is hardly a profound realization.
Mark Chu-Carroll is a computer scientist and software engineer. He writes a blog Good Math, Bad Math which has the headline:
“Good Math, Bad Math: Finding the fun in good math. Squashing bad math and the fools who promote it.”
I have read a few of Chu-Carroll’s blogs, and generally I found them interesting; he writes well and writes sensible articles about people who make erroneous claims regarding mathematics. Several of his articles are about people who have published mathematical claims that are clearly erroneous, and in general he doesn’t dismiss anyone as a crank unless he can provide a reasoned explanation as to why they are wrong. But, like anyone else, he is not infallible. On his blog, he devotes five pages to an ‘explanation’ of Gödel’s proof of incompleteness. He illustrates various functions and relations in Gödel’s proof by reference to computer programs, following an English translation of Gödel’s proof by Martin Hirzel, available online Gödel’s Proof - Hirzel’s translation: here PDF. It is in his final section, Gödel Part 4, that Chu-Carroll makes serious errors which mean that his purported explanation of Gödel’s proof is not an explanation at all. As Chu-Carroll remarks, the objective of the proof is to obtain a self-referential statement. But if the explanation is wrong at the crucial steps where the proof claims self-reference, then it’s not explaining the proof at all.
I e-mailed Chu-Carroll and pointed out the errors in his article. After waiting a considerable time and obtaining no response, I posted a comment on his article pointing out the principal error in his article (the text of that comment is given below). That was several months ago, but Chu-Carroll has not corrected the error.
Since Chu-Carroll’s blog is supposedly promoting “good math” and “squashing bad math”, I had expected a positive response from Chu-Carroll to the demonstration of an error in one of his articles.
But no, it seems that he is quite content to proudly proclaim that his site is promoting good math, and squashing bad math - at the same time as continuing to publish content which, mathematically speaking, is absolute drivel.
It doesn’t seem to be simply a case of Chu-Carroll over-simplifying the argument, since at other points where he does so, he says quite clearly that he is doing so. It’s not for me to say if Chu-Carroll fully understands Gödel’s proof or not, but I can say that he has made a very poor attempt at explaining it. Surprising, because Chu-Carroll says in The Danger When You Don’t Know What You Don’t Know:
“As any long-time reader of this blog knows, I’m absolutely fascinated by Kurt Gödel, and his incompleteness theorem. Incompleteness is, without a doubt, one of the most important, most profound, most surprising, and most world-changing discoveries in the history of mathematics. It’s also one of the most misunderstood.”
so one would expect that Chu-Carroll would be very careful when he attempts to assist people in understanding it. He also says, in Gödel (Reposts):
“Incompleteness is one of the most beautiful and profound proofs that I’ve ever seen. If you’re at all interested in mathematics, it’s something that’s worth taking the effort to understand.”
So if it’s worth taking the effort to understand, then surely it’s worth it for Chu-Carroll to make his explanation of the Gödel’s proof of incompleteness error-free and clear and understandable? Moreover, besides the errors, Chu-Carroll’s article is very difficult to read even for anyone familiar with Gödel’s proof, because, among other things, he fails to clearly differentiate between the names for symbol strings of the formal system and the corresponding numbers that are given by the application of Gödel numbering to those symbol strings - Chu-Carroll often refers to formulas of the formal system as numbers; this is completely wrong and very confusing.
The result is that anyone who is not already familiar with Gödel’s proof will find it hard going trying to follow Chu-Carroll’s ‘explanation’. It is not surprising that a comment posted says (posted to an earlier edition of the last part, Gödel Part 4 but the later version is almost identical): “I read the words in this post backwards and it made more sense to me.”
Chu-Carroll’s complete article is given by the following links:
This is the text of the comment that I posted on Chu-Carroll’s site:
There are several errors in this part 4 (The Payoff) of your explanation of Gödel’s proof which render your account almost impenetrable as well as being completely wrong. The principal error is described below.
When you state r = subst(q, 19, p) and i = subst(p, 19, r), such use of the subst function does not correspond to a valid substitution of a variable of the formal system. A careful examination of the definition of the function shows that the subst function corresponds (by Gödel numbering) to the concept in the meta-language of the substitution of a free variable in a formula of the formal system by any string of symbols of the formal system. Since p corresponds (by Gödel numbering) to a string of symbols that is a formula, not a natural number, your use of the subst function corresponds to the substitution of a variable of a formula of the formal system by a formula of the formal system, whereas it should correspond to the substitution of a variable of a formula of the formal system by a natural number. Perhaps you actually intended:
r = subst(q, 19, GN(p))
where GN is a function that gives the Gödel number of the formula p, and
i = subst(p, 19, GN(r)) which gives i = subst(p, 19, GN(GN(r)))
But if that was your intention, then your account gives no explanation as to how the Gödel numbering function GN might be included within the formal system itself. Since the Gödel numbering function requires a variable whose domain is all symbol strings of the formal system, and no variable of the formal system has such a domain, this is no trivial matter.
As such, when you say “This is what we’ve been trying to get at all this time: self-reference!”, surely if the point of the exercise is to show how self-reference is generated, then one shouldn’t fudge the crucial step in the process? When you fail to address the crucial point of how the formal system supposedly includes the Gödel numbering function within itself, your account is not an explanation at all. There seems to be little point in going into fine detail in the previous sections if you are simply going to fudge a crucial detail later on.
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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain, I developed a different system for footnotes, and 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.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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