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Language, Logic and Understanding

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:



Gödel’s Incompleteness Theorem


Formal Language





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.


8 Apr 2016    Are we alone in the Universe?

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.


13 May 2015     Good Math, Bad Math?

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, smiley facesad face 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:

Gödel (Reposts)

Gödel Numbering

Gödel part 2: Arithmetic and Logic

Gödel part 3: Meta-Logic with Arithmetic

Gödel Part 4: The Payoff


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.


30 Apr 2015    The Chinese Room

In 1980, the philosopher John Searle published a paper that claimed to show that artificial intelligence machines could never have ‘understanding’, regardless of their reasoning abilities. The fundamental idea of John Searle’s argument (commonly called the Chinese Room argument), taken down to its bare bones, is as follows:

Consider a thing with the following properties:Picture: chinese art

  1. The thing can take, as an input, sentences of a given language.
  2. The thing includes the ability to process the input and generate output sentences of that same language according to a set of instructions.
  3. The thing includes as an integral part of itself this set of instructions. The instructions determine what the output will be for any valid input. The instructions are such that, for valid inputs, outputs are generated that would be an appropriate response for a human fluent in that language.
  4. The thing always follows the instructions.

From this Searle claims:

Premise: the processing part of the thing does not require any understanding of the language in order to process the input and generate an output according to the instructions,


Conclusion: the thing does not understand the language.


The absurdity of the argument is obvious. There is no logical basis, given the premise, for inferring the conclusion. There is no reasoned argument at all.


It is quite astonishing that there has been ongoing controversy over Searle’s Chinese room argument for so long when a logical analysis of the argument readily demonstrates the absurdity of the argument.


Perhaps the reason is that the argument which Searle presents dresses up the above fundamental argument by adding in quite extraneous details which do not affect the basic premises. In Searle’s account, the thing is a closed room Picture: Chinese symbols with no windows, the inputs and outputs are slits in the wall through which paper can be pushed, the language is Chinese, Picture: Chinese symbolsthe set of instructions are written on paper, and the processing part of the thing is a human who does not understand Chinese. If the instructions are good enough, the responses of the room will be indistinguishable from the responses of a fluent Chinese human.


Searle concludes that, although the responses are indistinguishable from the responses of a fluent Chinese human, since the human in the room does not understand Chinese, then the entire room does not understand Chinese.


But when the extra baggage is removed from the argument, we see that Searle’s conclusion is simply the conclusion that he wants, reached by an appeal to intuition. Searle’s Chinese Room does not provide any logical basis for his conclusion, but serves as a smoke screen that has obscured the fact that Searle has no logical basis whatsoever for his conclusion.


Searle admits that there might be objections to his arguments on various grounds, and gives counter-arguments to those objections. None of the counter-arguments that he refers to address the issue of the lack of any logical inference of Searle’s conclusion from his premise.


It is unfortunate that responses to Searle seem to have concentrated on every other aspect of his argument other than his failure to provide a logical reason for drawing his conclusion from the premises given. Such responses have unfortunately created the impression that there might be some substance in his argument, whereas a logical analysis shows his argument to be hopelessly subjective and without any logical foundation.


The failure of Searle and others to perceive that the argument fails to logically derive its conclusion from the premise appears to result from the lack of definition of ‘understanding a language’ or any measurement of such. An intuitive notion of ‘understanding’ is applied instead. Nowhere does Searle make any attempt to move towards an objective definition of what he intends ‘to understand a language’ to mean, nor does he give any consideration to the question of formulating an objective method of measurement of ‘understanding’ of a language. Instead Searle boldly asserts that he knows what ‘to understand a language’ means, but refuses to define it, with the result no-one can be sure what he is actually talking about. Searle asserts that his intuition and subjective judgment are to be the criteria by which it is to be known that no machine could ever be made that could understand a language, where ‘to understand a language’ means only what Searle wants it to mean in any given context, and is not be subjected to any objective measurement.



While Searle’s Chinese Room scenario cannot be used to prove that no machine can ever ‘understand’ a human language, it nevertheless raises interesting issues regarding what we mean by ‘understanding’. Searle gives no objective measure of understanding of Chinese that we can apply to the processor and the entire thing. Searle blithely states that the entire thing understands nothing of Chinese, but does not consider how the entire thing might respond to an objective test of understanding of Chinese.


What Searle calls the ‘other minds reply’ is perhaps the closest he comes to considering the definition of understanding. Searle states that an objection might be:

“Searle has not given any information as to how one might determine whether another mind understands Chinese, that one might expect that such determination would have to be by the behaviour of that mind, that Searle has not demonstrated that a machine can never pass such behavioural tests as well as a human.”

Searle dismisses this in a few sentences, without addressing the issue of measurement of understanding, and confuses the issue by referring to ‘cognitive states’ rather than ‘understanding’ (continually changing the terms of reference is a common method of deflecting an inconvenient objection to an argument).


Searle states that (here I have reworded his arguments since his original arguments are so abstruse and poorly worded)

  1. One needs to consider not how I know that people/machines have understanding, but rather what it is that I am attributing to them when I attribute understanding to them.

Here Searle deflects the question of an objective measurement of understanding, and dismisses it as unimportant. He considers his subjective judgment to be sufficient for determining what understanding is.

  1. When one states that a human has ‘understanding’, the attributes that underpin ‘understanding’ cannot be merely computational processes and their output, because the computational processes and their output can exist without understanding.

Here Searle argues that since computational processes and their output can exist without understanding, that no combination of computational processes and their outputs cannot ever be considered to have ‘understanding’. This argument is absurd. One might as well say that since parts of the brain can exist without having understanding, then the brain as a complete entity cannot have understanding. Or that since muscles cannot walk, then legs which use muscles cannot walk.

  1. In ‘understanding’ one must assume that there is a reality and knowability of the mental state in the same way that in physical sciences one has to assume the reality and knowability of physical objects.

Here Searle simply argues that one must assume that there is a ‘mental state’ that is real. But that ‘mental state’ is not itself a physical object, rather it is the property of a physical object, so the ‘mental state’ is dependent for its existence on the physical attributes of the physical object. Searle does not explain how this might refute the objection, unless he is assuming that no machine, regardless of its physical attributes, can have a ‘mental state’. But that is simply introducing another ill-defined term, ‘mental state’, which adds nothing to Searle’s argument and serves only to confuse. Other attempts by Searle to deal with objections to his argument show similar flawed logic and similar appeals to intuition.


Another interesting viewpoint on the Chinese Room argument can also be seen at Conscious Entities - Against the Chinese Room.



31 Mar 2015    Cranks and Crackpots

You will find numerous web-pages and books that refer to cranks and crackpots. Most of them describe a set of characteristics that they consider to be common to these cranks/crackpots. The notion seems to be that if you think that you can ascribe enough of these characteristics to a person, Picture: Cracked Potthen it is almost certain that the person is a crank/crackpot. A selection of web-pages that give lists of characteristics of cranks/crackpots is given below:


The Crackpot Index by John Baez


Ten Signs a Claimed Mathematical Breakthrough is Wrong by Scott Aaronson


The Alternative-Science Respectability Checklist at Discover Magazine


Features of crackpot science by Pascal Boyer


Crank Dot Net by Erik Max Francis


Cranks at Wikipedia


The Prime Numbers’ Crackpot Index by Chris K. Caldwell


Some of these articles are primarily aimed at science cranks; however I’m not going to discuss science cranks here, just mathematical cranks. In the lists that give the characteristics of a mathematical crank, the strange thing is that they almost invariably omit the one factor that is the one salient and essential characteristic of a mathematical crank - that they are wrong and can always be shown to be wrong because there is a logical flaw in their reasoning, or because they have made an untenable assumption, or both.


Picture: Crank

Some of these lists are accompanied by a concession that some mathematicians who are known not be cranks nevertheless exhibit several of the characteristics on such lists. So it should be obvious to anyone that such lists are not conclusive, and that the only conclusive evidence that someone is a mathematical crank is given by the demonstration of errors in their arguments.


But it seems that many people aren’t interested in the possibility that they might be wrong in dismissing someone as a crank. They prefer to believe that commonly accepted mainstream mathematical results are always absolutely correct, and therefore, anyone who questions their correctness must be wrong, and therefore must be a crank. But mathematics is not as precisely certain as many people would like to believe, see for example the page Mathematical Proof.


The Internet seems to be awash with people who think that they are experts in wide range of subjects, and on that basis pass judgment on other people on various web forums, without providing any reasoned argument justifying their conclusions. Studies have been made which show that people who are not competent in a certain field often overestimate their ability in that field, and are overly prone to reaching erroneous conclusions. See, for example, the paper by Kruger and Denning, Unskilled and Unaware of it … Inflated Self-Assessments.


Picture: Book: Mathematical CranksFor a book on the subject, you could read Underwood Dudley’s book Mathematical Cranks. I haven’t read it all, but it seems that Dudley is fair in his appraisals. He isn’t content to simply apply the name crank to the various people mentioned in his book; he examines what they have to say and points out the obvious errors in their pronouncements.



One thing that really irks me is when people misrepresent me on Internet posts. I discovered one such post recently at http://www.reddit … flaw …. (the post is now closed for comments).


The principal contributor to the post, going by the username of completely-ineffable is a Reddit website moderator, apparently moderating on ‘bad philosophy’, ‘bad social science’, ‘bad mathematics’ and others, among them being ‘bad logic’, so one might expect that he is scrupulously logical in his posts.


However, he doesn’t demonstrate that. On the above post he stated that, “According to [James R Meyer], RCF and true arithmetic are incomplete” (Note: RCF is a Real Closed Field and it refers to a field that has certain properties of the field of real numbers).


The facts are that I make no mention of a Real Closed Field or true arithmetic either by name or description anywhere on my site - apart from right here. The contributor bases his claim on a note that I made in an introductory page on the incompleteness theorem regarding the conditions where Gödel’s incompleteness theorem is applicable to a formal system. The note was: “The conditions are basically ….” By the inclusion of the word ‘basically’, the note is clearly not intended to be comprehensive. To treat it as though it was intended to be a comprehensive description of precisely which formal systems Gödel’s incompleteness proof applies to is a gross misrepresentation. (I have now added a link in the note linking to the conditions given in Gödel’s paper.)


The contributor goes on to say that he has read my paper on Gödel’s proof but he can’t find a clear statement and explanation of the supposed error in Gödel’s proof”. And neither could he provide any clear statement and explanation of any error in my analysis. Instead he chose to ignore the detailed analysis in the section of the paper that leads to the “logical absurdity where a proposition is asserted to imply a specific value”, and sidesteps the fact that he has failed to find any error in the analysis that leads to that conclusion.


And this same contributor who clearly likes to jump to conclusions also wrote that my paper on Gödel’s proof “appears to have been created from an MS Word document”. In fact, the PDF was produced from a Latex file (if you don’t believe me, I will email a copy of the Latex file upon request). I wonder how many points he awarded to his crankometer index for the mere suspicion that a document had been created in MS Word?


Finally, the contributor also misrepresented me by saying, ‘he talks about how proofs of the Incompleteness Theorem “based on the halting problem” are worthless because they’re also all “overly vague”.’ whereas what I actually said was, “there are many claims of incompleteness proofs based on the Halting problem and similar notions, but you will find that many details are glossed over, and there are many implicit assumptions”. Besides the obvious indefensible faux pas of attributing a quotation to me that I did not make (“overly vague”), to say that I am claiming that such proofs are vague is untrue, because in general they aren’t vague and it’s usually quite clear what the essence of the argument amounts to. The meaning of what I said is clearly that the arguments skip over important details, or that they involve assumptions that are not explicitly stated.


When one comes across such misrepresentation, false quotations and jumps to erroneous conclusions, one is reminded of the paper by Kruger and Denning, Unskilled and Unaware of it … Inflated Self-Assessments, which describes how some people can be so unaware of their own incompetence that they rate their ability as far higher than it actually is.


Appeals to Authority

I came across an interesting snippet in the book, ‘An Appetite for Wonder’ by Richard Dawkins, where he says:


“Galileo was showing a learned man an astronomical phenomenon through his telescope. This gentleman said, approximately: ‘Sir, your demonstration with your telescope is so convincing that, were it not that Aristotle positively states the contrary, I would believe you.’ Today it amazes us - or ought to - that anybody could possibly reject real observational or experimental evidence in favour of what some supposed authority had simply asserted.”


Similarly, it should amaze us that anyone today rejects reasoned logical argument in favour of what various supposed authorities have asserted over the years.



Previous Posts




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The Lighter Side



Easy Footnotes

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).



O’Connor’s “computer checked” proof

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.



New page on Chaitin’s Constant

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.



Updated: Diagonal Lemma

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma webpage.



New page on Quine’s paradox

There is now a new page discussing Quine’s paradox and similar paradoxes.


Previous Blog Posts  


16th Mar 2015 Bishops Dancing with Pixies?


23rd Feb 2015 Artificial Intelligence




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:

Gödel Links


– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines


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