Friday, November 28, 2014

The History of Pi by Petr Beckmann

Source of Book: Borrowed from the library

At the outset, let me note that this is not the same book as The Life of Pi, a work of fiction. This is a non-fiction book about the real life history of π, the mathematical constant.

I picked this book up on an utter whim, with a camping trip coming up, and a potential shortage of non-fiction reading material suitable for reading in awkward spurts. Also, I really loved The Nothing That Is, so why not? While this book didn’t rise to that lofty level, it was interesting, and a bit provocative in its gratuitous musings on the relationships between science, politics, and religion over the course of recorded history.

So first, a bit about the actual topic of the book. Man’s fascination with the mathematics of the circle date back thousands of years. Relatively early on, it became clear to ancient engineers that the ratio of the diameter of a circle to its circumference was a constant, regardless of the size of the circle, and that this knowledge was highly useful. Picture, for example, the question of how much material would be needed for the walls of a circular building. Initially, the value was calculated to a single digit: 3:1, or 3.00. This was good enough for really rough construction, but not for more intricate work.

While the process initially involved simple measurement, the ancient Greeks (and later other civilizations independently of the Greeks) devised means of calculating a more accurate figure.

Much of this part of the book contains math. Lots of math. To a degree, I find this fun. I pursued the law and related topics after high school, so my math knowledge stops at trigonometry and coordinate geometry. For practical purposes 20 years later, I would say my functional ability is limited to algebra and Euclidian geometry. I never learned calculus, although, in a perfect world, I would have the time to learn it now. So, this book got out of my mathematical depth pretty darn quickly. I hung in there pretty well through the proofs of Euclid and Archimedes, and kind of understood some of the alternate geometric approaches. However, once Newton came along, I was done. No chance of understanding the calculations. Ah well.

Because of the nature of the math, the author couldn’t keep the later chapters quite as non-technical as in The Nothing That Is. Still, the book remained interesting even as I had to increasingly gloss over the math problems.

So much for the story itself, which is pretty cool, if technical. It is the other stuff in the book that got interesting really fast.

First, it was interesting that Pi doesn’t have to be accurate to very many decimal places in order to be useful. It is easy to forget, but each additional decimal place - by definition - make the number 10 times as accurate as the last. Thus, very quickly one finds that the degree of error is insignificant for calculating human-scale distances and areas. Beyond 5 or 6 decimal places, one is talking about millimeters of error in solar-system sized circles. For the average earth-bound engineer, this is more than sufficient. Now, of course, the calculation has so far exceeded the needs of practicality as to be an esoteric exercise in pure mathematics, rather than a practical necessity. The book itself (published in 1971) prints the first 10,000 digits of Pi just for chuckles.

Now, about the philosophy. The author, born in Czechoslovakia, makes clear in his introduction that, being neither a mathematician (he is a professor of engineering) nor an historian, he felt no need to attempt a neutral and unbiased approach to the story of the development of Pi. And he definitely has a few axes to grind.

Let’s see, he is particularly negative toward (in rough historical order), Aristotle, the Roman Empire, Aquinas, Medieval Christianity, and religious interference with science in general.

One may or may not agree with the overall idea, but it is pretty difficult to argue with the specifics of historical fact that he discusses.

At the outset, the author makes a great point that originally, in the very early history of mankind, religion and science were intertwined. Those who were the priests were also the first mathematicians and engineers, being those with the leisure to pursue intellectual and academic enterprises. And also, with the incentive to have impressive practical knowledge that would awe the masses. Thus, the priests once held the knowledge of the movement of the stars and planets, could keep the calendar, and could eventually predict eclipses and other events with great accuracy. (Hence, the indisputable fusion of astrology and astronomy at one time.) The two separated later, and religion became a field completely separate from - and all too often opposed to - science.

The author’s bold assertion as to the cause is provocative, but it unfortunately fits all too well with human nature. In the author’s view, eventually, the priests realized that “mumbo-jumbo” sold better than genuine science - and cost far less to produce. As I said, agree or not. I find it less compelling in the sense that it doesn’t explain why science ever got connected with religion, because it seems that mumbo-jumbo has always sold better than the real thing. But, I certainly can see that selling formulas and talismans and other trappings of unprovable and all-too-convenient “mumbo-jumbo” does tend to be a faster route to riches for the practitioner than genuine, hard won discovery. (Witness, among many other things, the multi-millionaire televangelists…)

Eventually, this split led to inevitable conflict. They were, if you will, competitors in the same marketplace: the explanation of “life, the universe, and everything.” And, sadly, from the beginning, each side encroached on the other’s turf. Religion has a regrettable history of trying to horn in on the realm of science, while science has tended to promise to explain more than it can. A great example of the latter would be the quest to reduce ethics and morality to a purely mathematical calculation a la Jeremy Bentham, to pick an easy example. Each over-promises and under-delivers outside of its domain.

Lest one think that the author’s ire is completely for religion, let me assure you that it actually extends far more to the realms of philosophy and politics.

Aristotle does not appear to best advantage in this book. While he may have contributed much to the study of ethics and to philosophy in general, his influence on science and how we think of it caused damaging reverberations for literally millennia - and probably continues today. In Aristotle’s view, empirical research, scientific observation, and really the entire scientific method, was useless. More than that, it was the domain of the common laborer. The real progress needed to be made through pure thought, not experimentation. Thus, one could arrive at the truths of the universe by thinking logically, rather than by observation and testing of the real world. Unfortunately, his philosophy became the dominant one in Greco-Roman society, and indeed in much of the Western World until modern times. Aristotle and Galen so dominated thought that when later physicians would dissect corpses, they would find that the corpses didn’t match Galen, and conclude - not that Galen was wrong - but that something was wrong with the corpse. Likewise, Aquinas (who had some admirable contributions to theology) adopted the Aristotelian opposition to science as a less pure form of truth.

I wish I could say that this is a vestige of the Roman Empire or even the Dark Ages. But it persists today. I wrote a bit more about this in my post on Young Earth Creationism. Many still believe that one can determine the details of the history of the world through pure intellectual effort applied to an ancient text that was likely never intended to be a scientific description. Likewise, many today and in the recent past continue to pretend that one can arrive at an ideal system of morality, government, and societal structure through pure intellectual theology and the interpretation of an ancient text without any reference to anything outside that system. (As the most obvious example, slavery wasn’t ended because the parties concluded that the Bible forbade it. Indeed, the most literalist of the interpreters found that it justified their ownership of fellow humans. Rather, it was through the exercise of military force that made the change. And this is just one egregious example of the failure of pure theology or pure intellectualism to come to a just result.) Aristotle continues to influence this discussion. So many still believe that theology is all we need to know the truths of the universe, and that anything outside the preferred theological system must - by definition - be false, despite the evidence from the real physical world.

The author certainly had no love for Rome either. As he notes, the best known early clash between the “thinkers” and the “thugs” happened in the wars between Athens and Sparta. As he puts it, “the thugs always win, but the thinkers last longer.” And Rome was in many ways worse than Sparta. Starting with the senseless slaughter of Archimedes, who wanted to finish his math problem before going with the Roman soldier, Rome had a history of violence. As the author notes, nobody idolizes the huns or the nazis, but the “pax Romana” remains somehow admirable, even though it was held by brutalism. The author muses that if the same standards were applied, the “pax sovietica” would have at least as many admirers. And, believe me, the author has no love for the Soviets.

The worst result, though, came when the forces of religion and politics united. It was the politicians and the priests in their joint quest for power that irrevocably damaged the scientific quest for knowledge. I already mentioned in my review of On The Map, the amazing library at Alexandria was destroyed at the order of Caliph Omar, who studied under Mohammed himself.

“If the contents of the books are in accordance with the book of Allah, we may do without them, for in that case, the book of Allah more than suffices. If, on the other hand, they contain matter not in accordance with the book of Allah, there can be no need to preserve them. Proceed then, and destroy them.”

This is the most striking instance, but the author goes on to list the extensive book burnings at the insistence of politically empowered religious leaders, from the days of the Roman Empire under Constantine through the crusades and even up to the Spanish Inquisition. I would love to quote entire passages of the book, but it would take too much space and time. However, I did look up the sources he cited, and he is correct in the details.

To give a modern example, during the late middle ages, ships would employ a Jewish navigator, because the mathematics necessary to navigate by the stars were considered to be occult, and thus sinful for a Christian.

Whatever your view of the theology, the author makes a compelling argument that the Middle Ages were truly a dark age when it came to science and mathematics, wherein most of what had been discovered prior to the age of Rome was forgotten, and not really rediscovered until the generation of Newton.

Again, don’t think that the author is specifically anti-Christian. He lists his big trifecta of “religions” that had the will and the power to convert by force, and these are Medieval Christianity, Medieval (through modern) Islam, and modern Communism. I believe he is correct in this assessment, particularly in the sense of the Stalinist purges, the Inquisition, and the ongoing attempts of radical Islam to convert or slaughter the “infidels.” (One of the best books on the essentially religious nature of Communism is The Opium of the Intellectuals by Raymond Aron. )

I’ve gotten a bit off the main trail here, but I do want to point out these facets of the book, as I believe they will be of interest to many who do not care about the mathematics of Pi. The divergence of religion and science is nothing new, and religion has a lot of explaining to do as to why it allied with politics to exterminate the scientific threat to its power. It is that particular unholy alliance that has caused much grief and has led to the jihads of the past and the present.

Note on Archimedes:

He really was one of the most fascinating early scientists, and much more brilliant than most realize. His death really illustrates that the whole “defer and obey authority” thing is more about protecting power structures than about anything wholesome or healthy. His death was senseless, useless, and tragic. But it fits all too well with the “show deference to the cops” thing that seems to be popular these days in certain circles. Sure, he disobeyed the guy with the sword. Never mind that he was one of the most brilliant men of all time. He should have kissed the soldier’s sandal better.

One final bit: the story about how he discovered the laws of displacement gave rise to the California State Motto. Allegedly, he ran naked through the streets yelling “Eureka!” “I have found it!”

Note on an Old Testament “error”:

The author notes early on that there is an approximation of Pi in the Old Testament. It is found in I Kings 7:23 and II Chronicles 4:2. Here is the Kings version (NASB):

23 Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference.

This is a clear mistake, if you will, according to modern knowledge. The value of Pi is clearly not 3, and that fact was known by the time that the Talmud was written in 500 C.E. It wasn’t until later that a Jewish commentator made a creative attempt to reconcile this wrong value to reality. (It really was a great attempt, using the inner and outer diameters of the giant bowl.)

However, all this misses the point. At the time the original event took place, the ancient Hebrews were using the old version of Pi, and were not aware of the developments in Egyptian mathematics making Pi equivalent to 3 1/7. The original story was written using the science of the time and place, and did not anticipate later discoveries.

Now, to be fair, most of us modern Christians just gloss over this, and realize that the mathematical value isn’t exact. It was for its time, and isn’t ultimately a big issue. Now, we have computers, and can calculate Pi to millions of digits. We don’t expect that a 4000 year old text will get modern science correct. We don’t expect the story to be a science
text. And yet, we do for some reason expect that in other areas, this text got science, medicine, and engineering correct in other areas. Or, to be more controversial, we are certain that it got psychology, sociology, and politics absolutely more correct than all the discoveries since that time. I don’t think this is really an issue of “inerrancy” so much as it is a tendency to interpret the text as completely self-sufficient, and impervious to considerations outside the text and theological structures. In other words, Aristotle continues to dominate the terms of the discussion.

One bit of weird history:

There have been several “modern” attempts to prove that Archimedes was wrong - more than one of which ended up as legislation, believe it or not. The book mentions the bill in Indiana put forward by a certain Edward J. Goodwin, which only went down to defeat because of the timely intervention of a professor who just happened to be present when the bill came up for a vote. You can read about the whole thing here. 

The whole thing would have been laughable, except that the bill would have required the teaching of the erroneous value of Pi in schools throughout the state.

I am reminded of this every time I see a political move to dictate the teaching of a dubious “fact.”

Even more interesting, perhaps, though, was the quote from John A. Parker, one of the other supposed “refuters of Archimedes,” who laid out a remarkably familiar conspiracy theory regarding established science. Any of us familiar with both Young Earth Creationism and the modern “Alternative Medicine” snake oil industry will recognize his arguments about why the entirety of the scientific and mathematical establishment wasn’t open to his ideas. No, it wasn’t that he might be freaking wrong, provably wrong, or anything like that. No, it was that they would only believe what they had been taught and what was acceptable within the establishment of science and academia.

He wrote this in 1897. The is nothing freaking new under the sun, is there?

A video: 

Come on, you knew this was coming... 



  1. If the contents of the books are in accordance with the book of Allah, we may do without them, for in that case, the book of Allah more than suffices. If, on the other hand, they contain matter not in accordance with the book of Allah, there can be no need to preserve them. Proceed then, and destroy them.

    Change this to "God" and "the Bible" and it's way too easy to imagine a nouthetic counselor saying it.

    1. Absolutely. In fact, in Jonathan Lindvall's home church (which my wife's family attended when she was a teen), this philosophy was explicitly held. Many of the families were opposed to fiction (because it was false - a lie) and let their children only read missionary biographies. (A few more allowed Elsie Dinsmore - a different topic altogether.)

      I would also add that Bill Gothard's homeschool curriculum was based on this idea as well - that everything needed for all knowledge of everything could be found between the pages of the Bible.