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Copyright   James R Meyer    2012 - 2020 https://www.jamesrmeyer.com

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Good Math, Bad Math?

13 May 2015

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 at Hirzel’s translation of Gödel’s Incompleteness paper. 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.

 

Note added 21 Feb 2018, Chu-Carroll has written elsewhere on his site (http://www.goodmath.org/blog/2010/02/04/a-crank-among-cranks-debating-john-gabriel/):

“One thing that I’m proud of on this blog is that I’ve got a history of admitting my errors. Just go back and look at the history of the blog. I’ve made my share of mistakes. And I’ve always done my best to admit them, and correct them. And I’ve done it without trying to hide it: I’ve always made the correction, and inserted extra text to explain that the original post contained an error.”

But he still hasn’t addressed the error that I brought to his attention.

 

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.

 

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