# Cranks and Crackpots

Page last updated 19 Dec 2023

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/

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 (How I found glaring errors in Einstein’s calculations) by Pascal Boyer

Cranks at Wikipedia

The Prime Numbers’ Crackpot Index by Chris K. Caldwell

And an old site which now seems to be defunct, and in any case, when it is operational, most of the links are broken, `http://www.crank.net`

by Erik Max Francis.

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. Calling someone a crank without finding any error in their claims is rather like a jury finding someone guilty because they have shifty looking eyes and ignoring all the hard evidence.

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.

As long ago as 1973, Amos Tversky & Daniel Kahneman wrote a paper PDF Availability: A Heuristic for Judging Frequency and Probability, (Footnote:
Tversky, Amos, and Daniel Kahneman. "Availability: A heuristic for judging frequency and probability." Cognitive psychology 5.2 (1973): 207-232.)
where the gist of the paper is that people are actually very poor at estimating probabilities, because they don’t actually calculate probabilities - what they actually do, when trying to assign a probability to a scenario, is that they try to recall a similar scenario that they have encountered in the past - and the more available it is to them, they conclude that the scenario that are examining is more probable simply on that account. There is no logical evaluation. Tversky & Kahneman call this the “availability heuristic”, and remark that, “… the use of the availability heuristic leads to systematic biases.” The upshot is that, if you don’t examine a text in detail, then not only do you not know some of the detailed information, you don’t know what you don’t know, which means that you have no way of judging whether you have sufficient knowledge of the text. And Michael Lewis remarks: “What people remember about the past is likely to warp their judgment of the future. We often decide that an outcome is extremely unlikely or impossible, because we are unable to imagine any chain of events that could cause it to occur. The defect, often, is in our imagination.” (Footnote:
Michael Lewis, *The undoing project: A friendship that changed the world*. Penguin UK, 2016)
It is on such grounds that people decide to dismiss an article as the fabrication by a crank, rather than taking the time to actually logically analyze what the article says. For them it is not sufficient to say that simply that they have not examined the article in depth - they feel that they have, nevertheless, to pass judgment on it.

Here is a summary of some cognitive biases:

*Confirmation bias:*The human mind is bad at seeing things it does not expect to see, and rather too eager to see what it expects to see. It is insidious because you don’t even realize it is happening, and it can skew our ability to see the world objectively.*Belief perseverance:*emotional investment in an argument or a belief is powerful and human minds are not predisposed to abandoning that commitment - this applies even more so when the belief is challenged, see opinion polarization.*Opinion polarization:*when presented with evidence that counters an ingrained belief, humans are inclined to double down and believe the opinion with even more zeal than before.*Theory attachment:*when people become overly attached to a theory, they fit the evidence to the theory rather than devising a theory that best fits the evidence.*The Law of Conservation of Ignorance:*A false conclusion once arrived at and widely accepted is not easily dislodged, and the less it is understood, the more tenaciously it is held. As Charles Darwin said: “Ignorance more frequently begets confidence than does knowledge.” (Footnote: In the introduction of*The descent of man, and selection in relation to sex*. Vol. 1. pub John Murray, 1871.)

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, PDF Unskilled and Unaware of it … Inflated Self-Assessments.

For 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.

(Update: at How to tell if someone is a crackpot and Sennimalai Kalimuthu - Crackpot or Joker? I have an analysis of two crackpot notions.)

#### Misrepresentation

One thing that is really irksome is when people misrepresent others on Internet posts. I discovered one such post recently at Reddit: badmathematics …. For a time this Reddit section was closed and off limits, because *“This community has become something of a shitshow”* - not really surprising, when it is moderated by moderators who have limited abilities in what they are moderating.

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 claimed that his 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.)

That contributor goes on to say that he has read my PDF 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.

The contributor says of Gödel’s Proposition V that the fact that it is only a sketch of how one might proceed with a proof:

*“doesn’t seem to be a flaw in the proof, merely a bit in the exposition that isn’t entirely transparent”*

thereby totally ignoring the fact that this part of Gödel’s proof is not actually proved, but simply assumed to be correct. Apparently this contributor thinks it is okay to assume that a part of a proof is correct even though this makes a mockery of the principles of mathematical proof.

Furthermore, the contributor then admits that he hasn’t actually read Gödel’s paper (this admission was later deleted). He then has the audacity to suggest that my paper is difficult to follow. He doesn’t appear to consider that if he has difficulty in following a paper that analyzes Gödel’s paper, then perhaps that might be because he would also have difficulty in following the original paper. Many people consider Gödel’s paper to be difficult to follow, which is why I created A step-by-step guide to Gödel’s incompleteness proof to help understand it.

It’s very easy to criticize someone else’s analysis of Gödel’s paper at the same time as not having any real knowledge of that paper - such criticism is utterly worthless.

The contributor goes on to that while he agrees with my statement:

That means that the language PV of Gödel’s Proposition V is a meta-language to both the formal language and to ‘number-theoretic relations’.

he says that he doesn’t see *“any evidence that Gödel failed to distinguish”* the meta-language and the sub-languages

Well if he didn’t see any evidence, that must be because he completely ignored the many instances where this was demonstrated in my paper, including the very next section after that statement.

And this same contributor who clearly likes to jump to conclusions also wrote that my PDF 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?

Tellingly, he finishes up by saying:

*“I feel like if this guy learned about Skolem’s paradox it would blow his mind.”*

thus making the completely erroneous assumption that I am not aware of Skolem’s contradiction - otherwise known by the euphemism “Skolem’s paradox”. I am well aware of it, and the reason it does not *“blow my mind”* is simply because it is obviously a contradiction that is directly attributable to the fact that there has never been any proof of the existence of transfinite numbers, see Proof of more Real numbers than Natural numbers?

A 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.

You can also see that the same commenter also simply assumes that most of my criticisms are regarding `explanatory papers", whereas the truth is that nearly all my material on incompleteness is with respect to papers published in mathematical journals, a fact that the commenter could easily have discovered by examining my site.

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.

*Other Posts*

Footnotes:

Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.Site MissionPlease see the menu for numerous articles of interest. Please leave a comment or send an email if you are interested in the material on this site.

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