“You have an irrational belief in rational thought.” ~Dr David Scholes, directed towards me.

“Humankind cannot bear very much reality.” ~T.S. Eliot

I just finished the book Mathematical Cranks by Underwood Dudley, part of a trifecta of skeptical mathematics books.

Doctor Dudley is a professor of mathematics at Depau University and a connoisseur of cranks with a mathematical bent.

What is a mathematical crank?

Mathematics is a peculiar field. Whether or not some aspects of mathematics exist independent of humans is an ongoing debate, but within its axioms and proofs is a consistent body of well defined, internally consistent knowledge.

Within that knowledge, ideas can, under the rules of mathematics and logic, can be proved or disproved, to be absolutely true or false or to be impossible.

No prior plausibility that pester the world of scientific medicine and the evaluation of woo. No borderline p values that hint at effects. No biologic variability. No placebo effect. No investigator or patient bias. No placebo effect. No N rays. No unproven water memory or meridians or subluxations.

Just clean, beautiful, mathematics. True or false. Possible or impossible. I simplify a bit, but mathematics, especially at the lower levels, is an internally consistent field of study. What happens in the math of 11 dimension string theory is beyond my puny intellect.

In mathematics there are things that are impossible. Absolutely impossible. No ifs, ands, or buts. Impossible. Can’t be done no how no way. In the world of mathematics, things are not only impossible, they are proven truly impossible within the boundaries of the mathematical discipline.

An example of mathematical impossibility is the quadrature of the circle, also called squaring the circle.

It is impossible, using only a straight edge ruler and a compass, to construct a square with the same area as a given circle. It was proved to be impossible in 1882 by Lindeman. Not improbable or unlikely or very, very, very difficult. With in mathematical reality, it is impossible.

Just because it is impossible does not prevent people from attempting to square the circle. They send these ‘proofs’ to mathematicians for comment.

“Few people have the imagination for reality.” ~Johann Wolfgang von Goethe

Mathematical Cranks is an examination of these, and other, pseudo-proofs and the accompanying correspondence between the authors and mathematicians. The mathematics is sometimes obtuse, but the interactions between the mathematicians and those with various unique mathematical ‘analysis’ are interesting and resonates with experience in the skeptical and medical fields.

These are the features of mathematical cranks.

1) They are convinced that their opinion is superior to the accumulated opinion of 2000 years of mathematics and mathematicians. That hundreds of mathematicians have worked for hundreds of years on these problems and found no errors in the proof that it is impossible to square a circle is of no consequence. Despite the accumulated mathematical knowledge of uncounted mathematicians, they are convinced that their solution is the right solution. Everyone else for all of history has been wrong. There is a tinge of megalomania in all the correspondence, and some appear to me to be clinically insane.

2) To accommodate their solutions, they are willing to alter reality to fit their proofs. There are solutions to squaring the circle, but they require a value of pi that is different that 3.14159265… Pi, for those that have forgotten, is the ratio of the circumference to the diameter of a circle and is a constant of the universe. For some circle squarers, Pi has a different value and all the mathematics that has confirmed the current value of pi is wrong. Others deny that pi exists or that the definition is meaningless, since they can construct a squared circle with pencil and paper, and send in the (flawed) construction.

3) When errors of math or logic are pointed out, they respond not with understanding, but a redoubling of efforts to prove that their erroneous solution to the problem is actually correct. They are incapable of recognizing flaws in logic, or mathematics, or flaws that are in opposition to mathematical consistency. A crank cannot recognize their error because they cannot recognize that their reality differs from mathematical reality.

4) Cranks are impervious to arguments based on mathematical reality. They do not recognize or understand that their solutions are in error because the solution contradicts known mathematical reality. They do not base their solutions on known mathematics, but on their own flawed understanding of mathematics.

5) Cranks evidently send their ‘solutions’ to multiple mathematical departments and rarely receive a reply. This silence from academia is interpreted not that their solution is worthless, but that there is a conspiracy of Professors of Mathematics to keep their solution secret, to the detriment of human kind. Big Math, out to suppress the truth THEY don’t not want you to know.

“To treat your facts with imagination is one thing, but to imagine your facts is another.” ~John Burroughs

Seem familiar to anyone on this blog?

Parallels are obvious between mathematical cranks and proponents of alternative medicine.

Many alternative proponents value their experience and understanding over the accumulated scientific knowledge of the last 200 years. HIV denial is the most depressing example. There are over 194,000 articles on HIV and AIDS. Even accounting for redundancy, there are over 250,000 people who have published on HIV and the detailed understanding of the disease and the resultant success of HAART had been a triumph of modern medicine.

Yet I read an HIV denier who said that what ever killed all those gay men in the 90’s is gone and now they are suffering from the HAART toxicity for no reason except to make big pharma and doctors rich.

“The formula “two and two make five” is not without its attractions.” ~Fyodor Dostoevsky, Notes from the Underground.

Most alternative proponents either alter or deny one or more components of objective reality as science understands it.

Homeopaths deny chemistry. Acupuncturists and chiropractors deny anatomy. Energy therapies deny our understanding of physics and energy. The all deny the validity of the scientific method.

If the basic precepts of any alternative therapy is correct, then 1000 years of knowledge gained in physics, chemistry, anatomy, physiology, biochemistry, embryology, pathology, biology are fundamentally wrong and the discoverer deserves multiple Nobel prizes. Or they are deluded (in the case of homeopaths, diluted).

I have a two part review of alternative medicine I occasionally give to the residents. Part two is a review of the reasons people are involved in alternative medicine.

“An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it.” ~Mahatma Gandhi

I have attributed the use and belief in alternative medicine to three things.

The first is lack of knowledge. Scientific medicine is complicated and difficult and takes years of work to understand. Most people do not have the time to devote to acquiring the necessary knowledge to become fluent in medicine. Alternative medicine is simple. Each alternative discipline produces an uncomplicated theory of everything based on one or two magical precepts. Easy to learn compared to the 9 plus years after college it takes to learn my field, Infectious Diseases. Once learned, alternative medicine requires no ongoing education. The lure of alternative medicine is understandable since everyone prefers simple solutions.

The mantra of the skeptical world is that knowledge changes behavior. I have always assumed that most people, if given sufficient knowledge and understanding, would abandon alternative therapies. All they need to do is learn to understand objective reality.

“There are lots of people who mistake their imagination for their memory.” ~Josh Billing

The second reason that people are involved with alternative medicine is that they do not understand how memory works and how easy it is to be fooled. I recommend the book The Seven Sins of Memory: How the Mind Forgets and Remembers by Daniel Schacter for a good review of the topic.

The five of the seven sins that apply to alternative medicine, for those who are interested, are

1) Memory Fades. After a month 75% of memory of an event fades. This sets us up for the other bias of memory. Most people think of memory as a videotape that perfectly records the past. Most recollections of past events are reconstructions based on current expectations and knowledge: people remember the past not how it was but how they think it should be.

2) Misattribution. We remember events that never happened or attribute events to people and things that were not there, or recall what happened but it occurs at the wrong time and place. One of the many reasons “anecdotal evidence” of therapeutic efficacy is suspect.

3) Memory is suggestible. More than one-third of subjects recalled being hugged by Bugs Bunny at Disneyland – impossible because Bugs is not a Disney character – after a researcher planted the false memory.” If you don’t accurately remember whether an alternative therapy worked, and you think it should, and someone tells you it worked, you will remember that it worked.

4) Memory is Biased: we remember events as to how we think they should have been.

We met at nine.
I was on time.
No, you were late.
Ah yes! I remember it well.

We dined with friends.
We dined alone.
A tenor sang.
A baritone.
Ah yes! I remember it well.

That dazzling April moon!
There was none that night,
And the month was June.
That’s right! That’s right!
It warms my heart to know that you remember still the way you do.
Ah yes! I remember it well.

How often I’ve thought of that Friday,
Monday night, when we had our last rendezvous.
And somehow I’ve foolishly wondered if you might by some chance be thinking of it too?
That carriage ride.
You walked me home.
You lost a glove.
I lost a comb.
Ah yes! I remember it well.

That brilliant sky.
We had some rain.
Those Russian songs.
From sunny Spain.
Ah yes! I remember it well.

You wore a gown of gold.
I was all in blue.
Am I getting old?
Oh no! Not you!
How strong you were, how young and gay;
A prince of love in every way.
Ah yes! I remember it well.

Hermione Gingold and Maurice Chevalier in Gigi. “These days Thank Heaven for Little Girls” just seems creepy.

5) Memory has persistence when associated with stressors. Intense experiences imprint memory and give them disproportionate importance.

“The difference between false memories and true ones is the same as for jewels: it is always the false ones that look the most real, the most brilliant.” ~Salvador Dali

I also like to refer to the book “Don’t Believe Everything You Think. The 6 basic Mistakes We Make in Thinking” by Thomas Kida, which dovetails nicely with the “Seven Sins of Memory,” and lists are nice for lectures.

1) We prefer stories to statistics.
2) We seek to confirm, not to question, our ideas
3) We rarely appreciate the role of chance and coincidence in shaping events.
4) We tend to oversimplify our thinking.
5) We sometimes misperceive the world around us.
6) We have faulty memories.

“I’m always fascinated by the way memory diffuses fact.” ~Diane Sawyer

Knowing how thinking is flawed combined with knowledge was, I thought, the first step in learning to think clearly.

The third reason I thought people were involved in alternative medicine are the numerous logical fallacies that we all participate in. I have 25 in my lecture, and if you are a splitter, there are many more. Humans do not naturally think clearly and rationally. Most of the time we are not aware we are using poor reasoning. Our every day thinking is flawed.

Combine a lack of knowledge with the vagaries of human experience and the stress of illness, and you have a potent woo flavored kool aid. Faulty thinking is a learned behavior and could be changed with education and awareness.

The examples provided by mathematical cranks gives me pause. There is no experience, no illness, no stress, no vagaries of experience that would lead one to misinterpret the world to make someone think they could square a circle or solve Fermat’s last theorem. Taking echinacea when you think you might have a cold and getting better is an experience that lends to the belief that echinacea prevents colds. What equivalent event leads someone to think they can square the circle?

It makes me wonder. Given the ubiquity of cranks of all kinds, of sins of memory, of logical fallacies, is being a crank in fact the norm? Tote up all the proponents of alternative medicine, astrology, UFO’s, Big foot, Atlantis, and that elephant in the room, and you are left with a paltry number of those who even try to think critically. Are the few who ascribe to a rational assessment of the natural world the deluded, the true cranks?

I am not inclined to consider various manifestation of reality denial as a mental illness, a delusion of some sort. I talked with some psych professionals trying to get a hint as to what, if any, label I could apply to those who think they can square the circle or is convinced that homeopathy makes sense. What do you call someone who consistently denies one or more aspects of objective reality? They, and I, were stymied. The archetype behavior that manifests in mathematical crankiness is too widespread to be abnormal. The closest term we could come up with was human.

It is why we may be doomed.

That’s my story and I’m sticking with it. It is the only reality based thing to do.

“Reality is that which, when you stop believing in it, doesn’t go away.” ~Philip K. Dick



  • Mark Crislip, MD has been a practicing Infectious Disease specialist in Portland, Oregon, from 1990 to 2023. He has been voted a US News and World Report best US doctor, best ID doctor in Portland Magazine multiple times, has multiple teaching awards and, most importantly,  the ‘Attending Most Likely To Tell It Like It Is’ by the medical residents at his hospital. His multi-media empire can be found at

Posted by Mark Crislip

Mark Crislip, MD has been a practicing Infectious Disease specialist in Portland, Oregon, from 1990 to 2023. He has been voted a US News and World Report best US doctor, best ID doctor in Portland Magazine multiple times, has multiple teaching awards and, most importantly,  the ‘Attending Most Likely To Tell It Like It Is’ by the medical residents at his hospital. His multi-media empire can be found at