Mathematics and Pseudoscience

First, please don't panic! This article is about pseudoscience and those who practice it, and does not require knowledge of mathematics.

There are two types of pseudoscience. The first type is a deliberate marketing ploy: using the language of science to add credibility to products. As skeptics we are very aware of how common this is. This cynical ‘blinding with science' not only means that consumers are exploited, but it potentially weakens the public's attitude to science.

However, this article is not about that form of pseudoscience. It is about unintentional pseudoscience. Here the practitioner believes that (s)he is doing science and, furthermore, that (s)he has made a significant contribution to the field. If the methodology is sound and the finding is true, then of course the practitioner has been doing valid science. There are examples of original scientific work being done by ‘amateurs', and again I will say little about this topic.

My main interest is when the methodology and/or findings are wrong, but the practitioner is unable to recognise this. This is the territory of ‘cranks.' Of course, this situation can happen with professionals (one is reminded of Linus Pauling and his promotion of vitamin C), but is more commonly associated with amateurs, by which I mean people who are not employed in a university or research laboratory.

I remember receiving my first letter from a mathematical crank. I had only finished my PhD about one year previously and I was employed as a post-doctoral researcher in a mathematics department in the UK. I received a hand-addressed envelope containing some pages purporting to prove Fermat's Last Theorem. My first thought was that I must have really ‘made it' as a mathematician if my name was becoming known to such people. Now I realise that, since their letters are ignored by the overwhelming majority of mathematicians, such cranks are so desperate to have an audience that they send their papers to everyone they can think could read it. So the threshold to receive such letters is not high.

Since that time I have received around a dozen letters or emails from amateur mathematicians. Unlike the majority of mathematicians, I usually do not delete these immediately. Instead, I am inclined to engage with the senders. This is, in part, due to my interest in skepticism. But before we get into that, first I want to make some comments about mathematics so that we can distinguish mathematics from ‘pseudomathematics'.

What is mathematics?

Mathematics is quite hard to define. As a subject it bears little resemblance to the school subject with the same name. According to Wikipedia “Mathematics is the abstract study of quantity, structure, space, change, and many other topics”. I prefer the following definition, due to the famous geometer Bill Thurston: mathematics is the smallest subject satisfying the following:

• Mathematics includes the natural numbers and plane and solid geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human understanding of mathematics.

It is more useful to consider the characteristic features of mathematics (as compared with the physical sciences). Mathematics is based on rigid logical rules and a notion of ‘proof'. Mathematical theorems are ‘true', and remain true forever (unlike theories in the physical sciences, which are always models or approximations to reality that can be improved in future). Interestingly, mathematics contains the tools to analyse itself. For example, it is possible to prove that certain problems are impossible to solve within a given mathematical system.

People often wonder (mistaking ‘mathematics' with the subject studied at school) why it is necessary or possible to do research in mathematics. Surely, they think, we know how to add and multiply and so there is nothing to be done?

But I hope most readers will know that mathematics, like science, is perpetually generating new questions for itself, as well as finding new applications.

My PhD was in a branch of mathematics called Number Theory, which is one of the most ancient parts of mathematics. This is a subject that is mainly about problems regarding the integers (rather than the real numbers). For example, prime numbers are part of Number Theory.

Number Theory was considered to be an entirely ‘pure' (meaning, studied for its own beauty, rather than because it has applications) branch of mathematics for over 2000 years, but nowadays is recognised as fundamental to cryptography and communications (which are my own main research areas).

There are many famous problems in number theory. For example, Fermat's last theorem; Riemann hypothesis (million dollar prize for solving it!); Goldbach conjecture (every even integer is sum of two primes); and Twin primes (there are infinitely many integers p, p+2 that are both prime).

The first of these is now proved, but the other three are still unsolved.

Number Theory is very attractive to amateurs because the problems can be very easily stated (for example, the Golsbach and Twin primes problems skeptics.nz | 9 “ can be understood by 10 year olds). Hence it is known that number theorists (like me) are more likely to receive letters from cranks than researchers in other branches of mathematics.

Paradoxers and cranks

Having said some words about the nature of mathematics, I can say what I mean by ‘pseudomathematics'. It is any purported solution to a mathematical problem that does not meet the accepted standards of logic and rigour. Sometimes it is the mathematics that contains a mistake, but more often it is the writing that has a superficial resemblance to genuine mathematical writing, but that does not follow the usual laws of mathematical logic.

You might be surprised to learn that this is not a recent phenomenon. The distinguished mathematician Augustus De Morgan (1806- 1871), first president of the London Mathematical Society, collected a large body of works by cranks and wrote a number of short articles about the phenomena. These were collected posthumously under the title A Budget of Paradoxes (1872).

De Morgan used the word paradoxer to mean a freethinker who challenges orthodoxy (what we would nowadays call a maverick): “a paradox is something which is apart from general opinion, either in subject-matter, method, or conclusion”. He notes that the progress of science requires paradoxers, but also that many who believe they are paradoxers are actually wrong. The word crank means a paradoxer who is wrong and is unable to see that they are wrong. So there is a subtle dividing-line between being a maverick and a crank, and the difference may only become clear with hindsight.

De Morgan writes “The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it.” In other words, the mark of a crank is to not understand the subject fully, to not use the language correctly, and to not follow the accepted methodology of the subject. A consequence is that most writings by cranks are incomprehensible and would take much more time to evaluate than work by professionals. US Judge Richard Posner writes “To call a person a crank is to say that, because of some quirk of temperament, he is wasting his time pursuing a line of thought that is plainly without merit.”

Another who has written extensively about cranks is Underwood Dudley, and I have been influenced by his book Mathematical Cranks. He describes mathematical cranks as follows: “A lot of them are amateurs...who like to work on mathematical problems. [They] aren't nuts, they're just people who have a blind spot in one direction.” One particular subclass of mathematical crank are those who try to solve problems that have already been proven to be impossible to solve (such as trisecting the angle using the rules of Euclidean geometry). These are the mathematical equivalents of people who try to build perpetual motion machines.

What is a crank thinking when they contact a mathematician? They have a conviction that they are correct, and a desire to be recognised for their contribution (sometimes this recognition requires prizes and accolades).

A warning!!!!

The hallmarks of cranks in mathematics and science are: enthusiastic amateurs who are well-read about science; often people with some formal academic training (e.g. a bachelors degree in mathematics, engineering, physics); imaginative and intelligent free-thinkers; people with spare time in their later lives.

So you, dear readers of this journal, are an ‘at risk' category. The rest of the article is to give you some friendly advice about how to protect yourself from crankdom.

Steps to avoid crankdom

So, imagine you are a free-thinker with an interest in a problem in mathematics or science. And suppose you have some ideas that you believe to be original and correct. What should you do?

You should first accept the possibility that you might be mistaken. Your thoughts may not be original, or they may not be correct. This is what I am always thinking whenever I think I may have made a breakthrough in research, and it is how all scientifically-minded people should be.

How can one be certain that one is not a crank? Well, you could talk to people in the pub and try to convince them that you have amazing new ideas. Or you could write letters to newspapers or journals. And this may make you feel good. But these avenues are not robust tests of your ideas.

Instead, you should seek professional help! Which means, you should try to talk to professionals in the subject area of your work and ask them to critique your ideas.

Most young mathematicians are advised to ignore emails from cranks. For example, this is the advice given in Underwood Dudley's book:

• “It is almost always a mistake to correspond with trisectors, because it is virtually impossible to convince them that they have made an error.”
• “Some of us are so filled with the urge to educate that we try to reason with the trisector. This is almost always futile.”
• “To the first letter reply politely ... If this technique does not suffice then be brutal. Write a letter that is harsh, scathing ...”

However, I have chosen to ignore this advice. Partly, I think due to my interest in skepticism.

So here are some reasons why I interact with mathematical paradoxers:

• I like to be nice to people who like mathematics. In my opinion it is important that society has a high regard for science and mathematics and so people who are interested in these subjects should be encouraged.
• Who else should do it?
• It is part of my social duty as a professional mathematician and public servant.
• I might learn something. Usually not about mathematics (though occasionally I do), but mainly about human psychology.

Step 1: How should a paradoxer approach a professional?

Keep in mind that almost all mathematicians will delete your email or throw away your letter immediately, and young mathematicians are discouraged from “wasting their time” talking to amateurs. This is not because they are evil, but because they are busy. A very distinguished mathematician, who is an editor of a good journal, told me he immediately rejects any crank paper submitted because “either it is wrong or it is right, in which case it deserves to be published in a much better journal.”

Case study 1: “I have an inquiry about whether you would be willing to review a paper on a number theoretic topic. I am an amateur mathematician with no contacts in academia and no history of published papers. To make matters worse, the problem to which I am proposing a solution is notorious for attracting failed attempts. Consequently my chances of getting my paper reviewed by any journal without supporting expert opinion could not be less favorable.”

This is a good approach. It shows an awareness of reality and suggests that the author is not actually a crank.

Dos and Don'ts:

• Be polite.
• Do not be arrogant.
• Especially, do not compare yourself to Einstein or Galileo or Ramanujan.

Step 2: Try to learn the language

Science and mathematics have their own language. It is not easy to get into. But if your proof will result in a one million dollar prize, is it such a bad hourly rate to learn something about the language and standards of the subject? Here's one email I got from the same person as I quoted above:

“Apologies my paper is painful to read. I do not find it easy myself and I wrote it.”

Step 3: Taking criticism

You asked for an expert opinion, now listen to what they say. In particular, try to understand their argument before responding.

Case study 2: “I'm just a dumb Kiwi physicist, but I have been thinking about factorisation using classical algorithms. The short attached paper discusses an algorithm that is probably well known, but I cannot easily find it in the literature. Could you please take a quick look and give me some literature hints? The physics is very interesting. Thanks.”

Once I replied, pointing out some connections between their work and previous (rather naive) algorithms, I received this: “Thank you for your helpful email. Clearly I do not have any appreciation for the complexity of linear algorithms ... Actually, I was not expecting it to be fast, just different in some way ... because I am more interested in the physics.”

So this is a good outcome for all, and the sender shows themselves to be not a crank.

Step 4: Try to control the anger

You will become impatient and frustrated by the reviewer's insistence on details. You will be disappointed in how many mistakes there are in your work. Please try not to send angry emails telling the reviewer to “destroy all copies of my papers that you have in your possesion” or “I have decided I do not want to continue with the review. Thank you for your feedback. It has been much appreciated.” Because sooner or later you will likely want to re-contact that one person who listened to you.

Step 5: Be gracious in defeat

Inevitably (though it doesn't feel like it at the beginning) your idea will turn out to be either not original or not correct. You will be disappointed. But please be gracious about it.

“Thanks for explaining...I understand your argument now. I am afraid my wish for... undermined my ability to think clearly about it. It makes a proof of...uncertain, to say the least.”

Can cranks be cured?

To earn the name “crank” one needs to be more-or-less impervious to criticism, so in that sense it is a terminal condition. However, the people I have been quoting in this article have had their minds changed by our communications, and so are not fully-fledged cranks. Some of them I have never heard from again, so I presume they are cured.

For at least one them, treatment is ongoing. My feeling is that a full ‘cure' is rare, but that early intervention may have some preventative effect on the worst symptoms (e.g. persecution complex, recourse to conspiracy theories).

How likely is it that an amateur is correct? The case of Kurt Heegner

Heegner was a German high school teacher. He announced a proof of the “class number 1” problem in 1952, but it was not recognised as a correct solution.

Stark and Baker independently solved the problem (and were awarded major prizes) in the 1960s. Birch writes: “Heegner's paper was written in an amateurish and rather mystical style, so perhaps it was not surprising that at the time no-one tried very hard to understand it. It was thought that his solution of the class number problem contained a gap.” In 1967 Birch deduced that Heegner's proof was correct. Stark later recognised that Heegner's approach was essentially equivalent to his own. Heegner died in 1965, before his work was vindicated.

Alf Van der Poorten writes “Fortunately this story is not well known, otherwise it would feed the persecution complexes of amateurs. Is this a disgraceful scandal? I think not. An amateur better have clear arguments to get a proper hearing. That's not unfair; it's our playing the odds. If in consequence great contributions are neglected, that is a misfortune, not a scandal.”

I agree with Alf here. The fact that now and then an amateur turns out to be right, does not imply that the thousands of others deserve careful scrutiny by experts.

There are a lot of paradoxers around (I have been contacted by four or five such people here in little old NZ). Paradoxers should contact professionals for a critique of their ideas, rather than complaining to their mates in the pub.

Most professionals will not respond to requests. But I encourage experts to communicate with paradoxers ... up to a point. The mark of a crank is how they deal with critical comments. Besides, if we can't cure cranks in mathematics, what hope can there be for the rest of (pseudo-)science?