Good Math, Bad Math?

Page last updated 21 Dec 2023

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. (Update: Since then I discovered that he does not in fact play fair when he is having difficulty defending his position, see the page The duplicity of Mark Chu-Carroll).

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 at PDF 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.

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:

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