Pi and the collapse of peer review

The 1897 Indiana pi episode

Many of us have heard of the Indiana pi episode, where a bill submitted to the Indiana legislature, written by one Edward J. Goodwin, claimed to have squared the circle, yielding a value of pi = 3.2. Although the bill passed the Indiana House, it narrowly failed in the Senate and never became law, due largely to the intervention of Prof. C.A. Waldo of Purdue University, who happened to be at the Indiana legislature on other business. The story is always good for a laugh to lighten up a dull mathematics lecture.

It is worth pointing out that Goodwin’s erroneous value was ruled out by mathematicians ranging back to Archimedes, who showed that 223/71 < pi < 22/7, and by the third century Chinese mathematician Liu Hui and the fifth century Indian mathematician Aryabhata, both of whom found pi to at least four digit accuracy. In the 1600s, Isaac Newton calculated pi to 15 digits, and since then numerous mathematicians have calculated pi to ever-greater accuracy. The most recent calculation of pi, by Peter Trueb, produced over 22 trillion decimal digits, carefully double-checked by an independent calculation.

The question of whether pi could be written as an algebraic formula or as the root of some algebraic equation with integer coefficients was finally settled by Carl Louis Ferdinand von Lindemann, who in 1882 proved that pi is transcendental. That was 135 years ago, 15 years prior to Goodwin’s claims!

Pi pseudoscience in the 21st century

Aren’t we glad we live in the 21st century, with iPhones, Teslas, CRISPR gene-editing technology, and supercomputers that can analyze the most complex physical, biological and environmental phenomena? and where our extensive international system of peer-reviewed journals produces an ever-growing body of reliable scientific knowledge? Surely incidents such as the Indiana pi episode are well behind us?

Not so fast! Consider the following papers, each of which was published within the past five years in what claim to be reputable, peer-reviewed journals:

Papers asserting that pi = 17 – 8 sqrt(3) = 3.1435935394…:

Papers asserting that pi = (14 – sqrt (2))/4 = 3.1464466094…:

This listing is by no means exhaustive — numerous additional items from peer-reviewed journals could be listed. Some additional variant values of pi (which thankfully have not yet appeared in peer-reviewed venues) include a claim that pi = 4 / sqrt(phi) = 3.1446055110…, where phi is the golden ratio = 1.6180339887…, and a separate claim that pi = 2 * sqrt (2 * (sqrt(5) – 1)) = 3.1446055110…

Along this line, the present author wonders whether the above authors have mobile phones. These phones contain the numerical value of pi (or values computed based on pi), in binary, typically to 7-digit accuracy, as part of their digital signal processing facility, and would certainly would not work properly with a different value of pi. The same can be said about the GPS facility in most mobile phones, which relies critically on equations involving general and special relativity. For that matter, the electronics of mobile phones are engineered based on principles of quantum mechanics, some of which involve pi. If these authors truly believe pi to be in error, they should not use their phones (or any other high-tech device).

Archimedes’ scheme to compute pi

Before continuing, it is worth asking how one might justify the value of pi to a lay reader who is not a mathematician. Arguably the simplest and most direct method is Archimedes’ method, which computes the perimeters of circumscribed and inscribed polygons, beginning with a hexagon and then doubling the number of sides with each iteration. The scheme may be presented in our modern notation as follows: Set a1 = 2 * sqrt(3) and b1 = 3. Then iterate

a2 = 2 * a1 * b1 / (a1 + b1); b2 = sqrt (a2 * b1); a1 = a2; b1 = b2

At the end of each step, a1 is the perimeter of the circumscribed polygon, and b1 is the perimeter of the inscribed polygon, so that a1 > pi > b1. Successive values for 10 iterations are as follows:

0: 3.4641016151 > pi > 3.0000000000

1: 3.2153903091 > pi > 3.1058285412

2: 3.1596599420 > pi > 3.1326286132

3: 3.1460862151 > pi > 3.1393502030

4: 3.1427145996 > pi > 3.1410319508

5: 3.1418730499 > pi > 3.1414524722

6: 3.1416627470 > pi > 3.1415576079

7: 3.1416101766 > pi > 3.1415838921

8: 3.1415970343 > pi > 3.1415904632

9: 3.1415937487 > pi > 3.1415921059

10: 3.1415929273 > pi > 3.1415925166

Note that the two proposed values of pi mentioned in the papers above, namely 3.1464466094 and 3.1435935394, are excluded even by iteration 4. A similar calculation with areas of circumscribed and inscribed polygons, which is an even more direct and compelling demonstration, yields a similar result.

In recent years mathematicians have discovered much more rapidly convergent schemes to compute pi. With the Borwein quartic iteration for pi, for example, each iteration approximately quadruples the number of correct digits. Just three iterations of yields

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193,

which agrees with the classical value of pi to 171 digits (i.e. to the precision shown).

These and numerous other formulas for pi are listed in a collection of pi formulas assembled by the present author.

Pay-to-publish journals and the collapse of peer review

Peer review is the bedrock of modern science. Without rigorous peer review, by well-qualified reviewers, modern mathematics and science could not exist. Reviewers typically rate a submission on criteria such as:

Relevance to the journal or conference’s charter. Clarity of exposition. Objectivity of style. Acknowledgement of prior work. Freedom from plagiarism. Theoretical background. Validity of reasoning. Experimental procedures and data analysis. Statistical methods. Conclusions. Originality and importance.

Needless to say, the papers listed above should never have been approved for publication, since such material immediately violates item 7, not to mention items 3, 4, 6 and others. Keep in mind that no editor or reviewer with even an undergraduate degree in mathematics could possibly fail to notice the claim that the traditional value of pi is incorrect. Indeed, it is hard to imagine a comparable claim in other fields: A claim that Newton’s gravitational constant is incorrect? or that atoms and molecules do not really exist? or that evolution never happened? or that the earth is only a few thousand years old?

At the very least, even to an editor without advanced mathematical training, the assertion that the traditional value of pi is incorrect would certainly have to be considered an “extraordinary claim,” which, as Carl Sagan once reminded us, requires “extraordinary evidence.” And it is quite clear that none of the above papers have offered compelling arguments, presented in highly professional and rigorous mathematical language, to justify such a claim. Thus these manuscripts should have either been rejected outright, or else referred to well-qualified mathematicians for rigorous review.

Also, the fervor with which some of these authors address their work should raise a red flag. There is simply no place in modern mathematics and science for fervor in presenting research work (see item #3 in the list of peer review standards above), since any good scholar should be prepared to discard his or her pet theory, once it has been clearly refuted by more careful reasoning or experimentation. Such problems are part of the explanation for the persistence of young-earth creationism, for instance.

So how could such egregious errors of manuscript review have occurred? The present author is regrettably forced to “follow the money” (as the shadowy informant Deep Throat in the movie All the President’s Men recommended). Indeed, all of the above journals listed above are on Beall’s list of pay-to-publish journals. Many of these journals have acquired a reputation of loose standards of publication, with only a superficial review, in return for charging a fee to authors for having their papers published on the journal’s website.

Conclusions

Obviously the mathematical community, and in fact the entire scientific community, needs to tighten standards for peer review and to oppose any form of “peer-reviewed” publication that involves only a perfunctory review.

Along this line, some say that we should simply ignore papers that claim incorrect values of pi, or even all articles in pay-to-publish journals, in the same way that mathematicians typically ignore email messages from writers who claim to have proven the Riemann hypothesis, or that computer scientists typically ignore writers claiming to have proven that P = NP, or that physicists typically ignore writers claiming to have devised a “theory of everything.” But in that case many legitimate papers would be excluded. Indeed, it is a grave disservice to the quality papers published in these journals for the editors’ loose standards to allow poor quality and clearly erroneous manuscripts to also appear.

In any event, there is a real danger that as a growing number of papers are published with erroneous or questionable results, other papers may cite them, thus starting a food chain of scholarship that is, at its base, mistaken. Such errors may only be rooted out years after legitimate mathematicians and scientists have cited and applied their results, and then labored in vain to understand paradoxical conclusions.

So what will the future bring? Increasing confusion, resulting from growing numbers of questionable and false published results, many in presumably peer-reviewed sources? We all have a stake in this battle.