Pseudomathematics

Gabrielle Birchak/ January 11, 2022/ Ancient History, Modern History

Pod­cast transcript

Three is a mag­ic num­ber. And I’m not just say­ing that because it’s a fan­tas­tic School House Rock song. The rule of three is often used in screen­writ­ing with the three-act struc­ture and three char­ac­ters. A table with four legs is not near­ly as sta­ble as a table with three legs, which is, as we all know, a tri­pod. Three con­nect­ed orthog­o­nal rec­tan­gles indi­cate the ver­tices of a reg­u­lar icosahedron.

The num­ber three is an approx­i­ma­tion of Pi. It is also the first Fer­mat Prime, which is 2k+1=3 (where k>0).

In pho­tog­ra­phy, the rule of thirds is an effec­tive tech­nique that places the sub­ject of the object in the left or right third of the image. Most songs use the rule of three. This struc­ture includes the verse, the cho­rus, and the bridge. Also, you can make a major chord using three notes that are each three keys apart. C, E, G.

So, obvi­ous­ly, the num­ber three is an effec­tive num­ber. Numerol­o­gists believe that three rep­re­sents cre­ativ­i­ty, ener­gy, and pos­si­bil­i­ty, as ref­er­enced by the past, the present, and the future. But the truth is, numerol­o­gy is a pseu­do­science, and three is not mag­i­cal. It’s just a num­ber. And its mag­ic can­not be proved. And even though many believe that pseu­do­science is based on the sci­en­tif­ic method, it is not. Sci­en­tif­ic mir­a­cles do not occur. 

The same goes for math. There are math­e­mat­i­cal equa­tions that are com­plete­ly unsolv­able and can­not be proved, despite inex­pe­ri­enced math­e­mati­cians who mirac­u­lous­ly solve them. Some of these unsolv­able math­e­mat­i­cal prob­lems have been around for thou­sands of years. How­ev­er, they have not been solved because they are quite lit­er­al­ly math­e­mat­i­cal­ly impos­si­ble to solve with­in the con­structs that they are pre­sent­ed. This activ­i­ty of solv­ing unsolv­able prob­lems using basic, non-rig­or­ous, non-peer-reviewed meth­ods is known as pseudo­math­e­mat­ics or math­e­mat­i­cal crankery.

There have been pseu­do-math­e­mat­i­cal appli­ca­tions to the Mil­len­ni­um prob­lems, Fermat’s Last The­o­rem, the Col­latz Con­jec­ture, the P vs. NP prob­lem, and the three pop­u­lar prob­lems that include the Squar­ing of the Cir­cle, the Dou­bling of the Cube, and tri­sect­ing the angle. The list is long. Those who have claimed to solve these impos­si­ble math prob­lems and can­not show the rig­or of the math­e­mat­i­cal frame­work behind solv­ing these prob­lems are known as pseudo­math­e­mati­cians or math­e­mat­i­cal cranks. This is not to be con­fused with a cranky math­e­mati­cian. We’ve all met those from time to time. 

The Quad­ra­ture of the Circle

In 1851 a gen­tle­man by the name of Mr. John R Park­er pub­lished a book called The Quad­ra­ture of the Cir­cle, and in this book, he claimed to have solved the squar­ing of the cir­cle. The thing is, the squar­ing of the cir­cle is impos­si­ble to solve. The chal­lenge behind this prob­lem is to con­struct a square that has the same area as a cir­cle using a com­pass and a straight­edge and solve it in only a few steps. And using these para­me­ters, it is impos­si­ble to solve the squar­ing of the cir­cle. The rea­son is because π is an irra­tional num­ber. Also, pi is a tran­scen­den­tal num­ber. As a result, it is not an alge­bra­ic num­ber, which means that it can­not be con­struct­ed. It’s not a con­structible num­ber. But I don’t want to digress. The point of me telling you this is that squar­ing the cir­cle using a com­pass and a straight­edge is impos­si­ble. How­ev­er, John Park­er claimed that it was pos­si­ble and held stead­fast to his belief even when oth­er math­e­mati­cians showed how his work was flawed.

Anoth­er impos­si­ble prob­lem is the Dou­bling of the Cube. I pre­vi­ous­ly men­tioned this prob­lem in Math! Sci­ence! His­to­ry! Pod­cast num­ber Nine called Nigel Tufnel and Dou­bling the Cube. I post­ed it on my blog at mathsciencehistory.com along with the video of Nigel Tufnel singing Stone­henge because noth­ing says, “this math is impor­tant,” like watch­ing an 18-inch Stone­henge fig­ure drop­ping from the ceil­ing dur­ing a rock con­cert. You can find that at MathScienceHistory.com.

If you are inter­est­ed in see­ing the math behind the Dou­bling of the Cube and a more exten­sive expla­na­tion, please come vis­it me at https://Patreon.com/MathScienceHistory and sign up for a tier. When you become a patron, you get ear­ly access to my pod­casts, videos, as well as more math and science!

The dou­bling of the cube has been pro­posed since 400 BCE. Pos­si­bly even longer. So, this prob­lem is over 2000 years old. And by the 17th and 18th cen­turies, uni­ver­si­ties were flood­ed with cube dou­blers and cir­cle squar­ers so much so that in the 18th cen­tu­ry, the French Acad­e­my announced that they would no longer exam­ine any solu­tions that “solved” the dou­bling of the cube, the squar­ing of the cir­cle, and the tri­sec­tion of the angle. With­out exact­ly using the term pseudo­math­e­mat­ics, this was one of the first indi­ca­tions by acad­e­mia and aca­d­e­m­ic math­e­mati­cians that pseudo­math­e­mat­ics is a preva­lent activ­i­ty among non-mathematicians.

Anoth­er fun chal­lenge is the Col­latz Con­jec­ture, which starts out by pick­ing a num­ber. It can be any num­ber. I’m going to go with the num­ber 42, which is an excel­lent num­ber, by the way. It’s the answer to the ulti­mate ques­tion. In this chal­lenge, if the num­ber is odd, you mul­ti­ply it by three and add 1. If the num­ber is even, you divide it by two.

So, the con­jec­ture is as follows:

Math­e­mati­cians 

I want to clar­i­fy that this does­n’t mean that any­one who attempts to solve an impos­si­ble prob­lem is a pseudo­math­e­mati­cian. On Sep­tem­ber 8, 2019, the bril­liant math­e­mati­cian Dr. Ter­rence Tao pre­sent­ed a par­tial proof on the Col­latz Con­jec­ture stat­ing that the con­jec­ture is “almost” true for “almost” all num­bers. How­ev­er, he did not solve it. And even though he came close, the solu­tion nev­er pre­sent­ed itself. Thus, even he knew that this con­jec­ture could not be solved.

So, let’s talk about the his­to­ry of pseudomathematics.

The first ref­er­ence to the word pseu­do­science began in the ear­ly 17th cen­tu­ry when sci­en­tists dis­cussed the rela­tion­ship between reli­gion and impe­ri­al­ism. Then, this word pseu­do­science popped up in 1796 when the his­to­ri­an James Pet­tit Andrew wrote that alche­my was a “fan­tas­ti­cal pseu­do­science,” which it is because it opened the door to chem­istry, which is a ver­i­fi­able sci­ence.[1]

And this pow­er­ful word “pseu­do” was used to define many things. But it was­n’t until 1915 that the word pseudo­math­e­mat­ics was first used by the great math­e­mati­cian Augus­tus de Mor­gan. In his book, A Bud­get of Para­dox­es, de Mor­gan defined a pseu­do math­e­mati­cian as “a per­son who han­dles math­e­mat­ics as the mon­key han­dles the razor.” Where the mon­key observes his own­er using the razor to shave, attempts to do the same thing but does­n’t know how to hold the razor. In the process ends up cut­ting his own throat. De Mor­gan then went on to call out the pseu­do math­e­mati­cian James Smith who claimed that Pi is exact­ly 3 and 1/8.[2] Thus, and as a result, in math cir­cles, pseudo­math­e­mati­cian became the ulti­mate burn.

So how do we define a true pseu­do math­e­mati­cian? After all, what is wrong with some­one being enthu­si­as­tic about math­e­mat­ics? Well, let’s talk about peer review. Peer review is the process of hav­ing your work reviewed by a peer. And for those who have solved some of these poten­tial­ly unsolv­able prob­lems, their work has been reviewed, eval­u­at­ed, and dupli­cat­ed by oth­er bril­liant math­e­mati­cians to ver­i­fy that the math­e­mat­ics was rig­or­ous and found­ed in log­ic. What is won­der­ful about the process of peer review is that it not only allows the math­e­mati­cian, the sci­en­tist, or the his­to­ri­an, to see the flaws in their work so that they can cor­rect it, peer review also estab­lish­es a foun­da­tion for oth­er math­e­mati­cians, sci­en­tists, or his­to­ri­ans to step in and build upon those the­o­ries. Thus, if the work is pre­sent­ed as truth with­out the process of peer review and adjust­ment, it is just sim­ply math­e­mat­i­cal crankery.

So back to Mr. Park­er and his book, The Quad­ra­ture of the Cir­cle. Accord­ing to Petr Beck­mann in his book The His­to­ry of Pi (which is a phe­nom­e­nal book), Mr. Park­er was not hap­py with the rejec­tion of his the­o­ries by oth­er math­e­mati­cians. He was very defen­sive and resort­ed to name call­ing and accus­ing those who had eval­u­at­ed his work as being “among the least com­pe­tent to decide on any new­ly dis­cov­ered prin­ci­ple.” Beck­mann also ref­er­ences Carl Theodore Heisel, who claimed to have squared the cir­cle. Heisel addi­tion­al­ly reject­ed dec­i­mal frac­tions as inex­act and “demon­strat­ed” how he “utter­ly” dis­proved the absolute truth of the infa­mous Pythagore­an Theorem.

But don’t get me wrong, there is noth­ing wrong with try­ing to solve the unsolv­able. The most beau­ti­ful thing about try­ing to solve the unsolv­able math­e­mat­i­cal equa­tion is that it opens your mind to a world of math­e­mat­ics that can alter your view­point and the per­spec­tive of others.

These are puz­zles. And the beau­ti­ful thing about puz­zles is that they can engage a com­mu­ni­ty of oth­er math­e­mati­cians and sci­en­tists. These puz­zles cre­ate dis­course and a lev­el of com­mu­ni­ca­tion that can be absolute­ly excit­ing. And you don’t need an advanced degree to sit down and give it a go. Who knows, maybe by try­ing to solve the unsolv­able, you may find your­self so enam­ored with the math­e­mat­ics that you might just find your­self in col­lege pur­su­ing a degree in math, veer­ing away to study physics or chem­istry, or land­ing in a his­to­ry course, real­iz­ing that there is a mas­sive world of stuff to learn! That’s the pow­er of math, sci­ence, and his­to­ry! Until next time, carpe diem!

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[1] “Sci­ence and Pseu­do-Sci­ence,” Stan­ford Ency­clo­pe­dia of Phi­los­o­phy, accessed Jan­u­ary 3, 2022, https://plato.stanford.edu/entries/pseudo-science/.

[2] Petr Beck­mann, A His­to­ry of Pi (Lon­don: Macmil­lan, 1971), 178.