# Boxing Pythagoras

## On Aquinas’ Five Ways

In his seminal work, Summa Theologica, the celebrated Christian philosopher, Thomas Aquinas, engages with the question of the existence of God. He notes that there are certainly objections to the claim that there exists such a divinity, but Aquinas believes that these objections can be overcome and that this existence can be shown to be well-founded. The eminent philosopher then lays out a list of arguments which he supposes to make this case. These arguments have come to be known as Aquinas’ Five Ways, and they have been so influential in philosophy that many theologians and apologists still cite them as if they are authoritative logical proofs, more than 700 years after the Italian priest set them to page. On the contrary, however, it seems that there are a number of issues which prevent Aquinas’ Five Ways from being quite so powerful, now, as they may have been in his own day.

Translating ancient documents into modern languages carries with it more difficulties than most people realize. Pretty much anyone who has ever taken a foreign language class in high school understands that it can often be quite hard to find a word which corresponds exactly between two tongues. Those who have studied outside of the modern Romance languages– classes like Arabic, Chinese, or Japanese– often realize that there are subtleties in grammatical constructions which can convey a great deal more than can be expressed in English. It is a very frequent occurrence that a phrase from one language cannot be rendered with 100% accuracy in another language. In English, this has led to the popular idiom that “something has been lost in translation.”

Ancient languages maintain these problems, but add an entirely new layer of obfuscation which is not found even in most culturally distinct modern languages. Over the past few thousands of years, human understanding of the world around us has changed quite significantly. Just one hundred years ago, no one had ever viewed the ground from five miles up in the air. Two hundred years ago, we had no idea that microscopic organisms cause disease. Three hundred years ago, humanity had no idea that oxygen exists. Four hundred years ago, the world was shocked to learn the the planet Jupiter has moons. The manner in which religion, philosophy, and science have discussed a myriad of things about reality has changed so greatly in recent millennia that very often even one word in a single language can mean something exceedingly different to people living in different periods of time.

The documents which comprise the New Testament of the Christian Bible were written 2000 years ago. In those ensuing twenty centuries, many of the words used by the original authors and many of the concepts which they espoused have engendered incredible amounts of revision, alteration, and nuance by subsequent philosophers and theologians which would have been wholly alien to those initial ancient writers. The vast majority of modern readers– including an embarassingly large number of modern scholars of the text– seem wholly ignorant of this fact when they read a passage from their Bibles.

## A Nativity Story

The man stood, dumbfounded, attempting to comprehend the news he had just been given. The Messenger of God had just informed him that his wife– his Virgin– was now with child, and would present him with a son who would surpass all who had ever lived in beauty and wisdom. The boy to come would be a divine gift, the greatest beneficence that the human race would ever know. Though coming into this world by birth through woman, the infant boy had in fact pre-existed his human form, and was wholly divine in nature. The man rushed home to his wife to find that the Messenger of God had spoken true. She was, indeed, with child.

Thus, Pythagoras was born into this world.

What? You thought I was talking about somebody else? This account is from the Life of Pythagoras, written by the great Neoplatonist philosopher Iamblichus in the 3rd Century, CE. Now, I will completely admit that I paraphrased the story in order to obfuscate it, a bit. The “Messenger of God” that spoke to Mnesarchus (Pythagoras’ father) was the Oracle at Delphi. And the reason that I capitalized “Virgin” in my paraphrase was because that was the name of Mnesarchus’ wife, before he received this news from the Oracle– Parthenis, in Greek. As soon as he received the announcement from the Oracle, Mnesarchus changed his wife’s name to Pythais, in honor of Pythian Apollo (which, Iamblichus tells us, was also the source for Pythagoras’ name). Still, the paraphrase stands: a Messenger of God informed a man that his wife’s womb had been filled by deity, and that the child would be divine, the greatest gift the world could hope to receive.

## Euclid and the Sword

I have written, often, about one of my personal heroes from history, Euclid of Alexandria, who wrote a textbook called Elements which would serve as the foundation for all Western mathematics for 2000 years. You may recall that, outside of his name and a list of his writings, we know almost nothing about Euclid. We know nothing of his birth, or his schooling, or his politics. We don’t know if he traveled extensively or if he was relatively sedentary. We don’t know if he was tall, short, fat, skinny, handsome, or ugly. However, one thing we do know is that Euclid’s work, though purely mathematical, bore a tremendous influence on a wide variety of fields of knowledge.

Euclid’s Elements set out to prove the whole of mathematics deductively from very simple definitions, axioms, and postulates. Deductive logic provided a sound and absolute basis by which mathematics operated for every man, whether rich or poor, high-born or peasant, male or female, famous or obscure. During the 16th and 17th Centuries, this strong foundation became lauded and sought after by philosophers, who began attempting to provide all philosophy with the rigor one found in the Elements. The appeal was obvious: if one could deductively prove his philosophical system, in the manner that Euclid had proved his geometry, then one would be left with incontrovertible conclusions to questions which had previously been highly disputed. Such extremely notable philosophers as Thomas Hobbes and Baruch Spinoza, amongst countless others, attempted to replicate the Definitions-Postulates-Proofs format Euclid had employed in order to settle questions of morality and ethics and governance.

Martial philosophy was no less affected, in that period. The sword and swordplay, especially, underwent a dramatic evolution during that same time. Just as Hobbes and Spinoza attempted to replicate Euclid for ethics, fencing masters similarly moved toward a more rigorous and geometric approach towards understanding combat. And, in my opinion, they were far more  successful in that endeavor than the philosophers had been.

## On the Pythagorean Theorem

In right-angled triangles, the square on the side subtending the right-angle is equal to the (sum of the) squares on the sides containing the right-angle.

Euclid’s Elements, Book 1, Proposition 47 (R. Fitzpatrick, trans.)

Figure 1: A right triangle with squares on its sides

The Pythagorean Theorem is my favorite math problem of all time. I feel so strongly about this particular bit of geometry that I have the theorem tattooed on my chest. Over my heart. In the original Greek. Yeah, I’m that kind of nerd. Most people have some vague recollection from their high school math classes that the Pythagorean Theorem is $a^2+b^2=c^2$; and a few even remember that the in that equation refers to the hypotenuse of a right triangle, while the a and b refer to the other two legs. However, most of the time, people were just taught to memorize this theorem– they weren’t taught how to prove that it was actually true. Now, the Internet is full of all kinds of really clever visual proofs involving rearranging copies of the triangle in order to form the different squares, but I’m not really a huge fan of these. They make it very easy to see that the Pythagorean Theorem is true, but they don’t really make it easy to see why the Pythagorean Theorem is true. So, today, I wanted to discuss my favorite proof for the Pythagorean Theorem, which comes to us by way of Euclid’s Elements, which was the standard textbook for math in the West for around 2000 years.

## On Carrier’s pre-Christian Jesus Myth

Richard Carrier is a freelance historian with a PhD in Ancient History from Columbia University. He is arguably the most prominent proponent of the Christ Myth hypothesis, today, and one of the few historical scholars with actual qualifications in history that holds to such a position. If you are unaware, the Christ Myth hypothesis argues that there never existed an actual, historical Jesus of Nazareth upon whom the Christian faith eventually became focused. Instead, the Jesus of Nazareth presented in the gospels is a deliberate attempt to tie myths about a celestial being into history. This view is generally dismissed, panned, and ignored by the vast majority of mainstream scholarship, and one could quite rightly describe Richard Carrier as a fringe scholar. However, the simple fact that Carrier is a fringe scholar is not a very good reason for dismissing his work, out of hand. The man is actually a qualified historian, with a PhD from a respected university, who has had articles published in respected academic journals. The fact that his hypothesis goes against mainstream scholarship does not invalidate the rest of his qualifications.

Carrier recently published a book entitled On the Historicity of Jesus: Why We Might Have Reason to Doubt which lays out his views and arguments. I have been meaning to purchase, read, and review that book for this site since it was released, but I refuse to pay \$85 for the hardcover or \$35 for the paperback version– I find such prices to be wholly excessive. Unfortunately, the book has not yet received an eBook release, which I might be more inclined to purchase (though not if the price is similarly high). Still, Carrier has engaged in a number of debates and public presentations, and it is easy to find at least an overview of his position. For example, he recently gave a talk at Zeteticon which outlines his view.

One of the major points that Carrier alleges, in his presentation, is that we have evidence that there was a pre-Christian, Jewish belief in a celestial being which was actually named Jesus, and was the firstborn son of God, in the celestial image of God, who acted as God’s agent of creation, and was God’s celestial high priest. I have seen Carrier present this information numerous times, in different talks, including the one which I linked above, and he always presents it without actually quoting from the sources which he cites. Now, as I’ve said, I haven’t yet read On the Historicity of Jesus, and it is fully possible that Carrier addresses some of my contentions there, but I find his entire claim that there was a pre-Christian, Jewish belief in a celestial Jesus to be almost entirely unsupportable.

## Egyptian Math: Multiplication and Division

A little while back, I wrote up a little post on basic Egyptian mathematics, concentrating on how using Egyptian numerals in addition and subtraction can aid students in understanding our base-10 number system. I wanted to continue that discussion, today, by looking at how the ancient Egyptians performed Multiplication and Division. Unlike my discussions of addition and subtraction, I am not advocating Egyptian multiplication and division as a means for teaching the Common Core standards.

To be honest, I just find the system to be really interesting.

## You must stand clear, Mr. Holmes, or be trodden underfoot.

“That is not danger,” said he. “It is inevitable destruction.  You stand in the way not merely of an individual, but of a mighty organisation, the full extent of which you, with all your cleverness, have been unable to realise.  You must stand clear, Mr. Holmes, or be trodden underfoot.”

The Final Problem, by Sir Arthur Conan Doyle

A few days ago, I was reading a post from fellow blogger, Andrew Crigler, who writes Entertaining Christianity. He had written a fun little post, jovially comparing blind-faith beliefs to clothing for puppies, which I enjoyed and with which, for the most part, I agreed. However, at the end of the article, Andrew recommended his readers to J. Warner Wallace’s book Cold Case Christianity. If you have been reading my blog for a while, you might remember that I am no fan of J. Warner Wallace and, in fact, I think he is more akin to a crooked cop than an honest detective. I commented on Andrew’s post to convey this, and that began a nice back-and-forth conversation between us regarding Wallace and his claims. At one point, Andrew suggested that Wallace had written other articles which were more convincing, and formed on better logic, than the ones which I had critiqued. I asked him to suggest one, for me, so that I could read and review it here. Andrew provided me with a link to one of Wallace’s posts entitled, “The Case for the Eyewitness Status of the Gospel Authors.”

Unfortunately, I find this article to be just as poor as Wallace’s others.

## Egyptian Math for the Common Core

A short while back, one of my friends posted a series of videos on Facebook complaining about the Common Core standards which are being rolled out in the United States. Unsurprisingly, not a single one of the videos actually addresses the standards laid out by the Common Core– despite their being freely available on the Internet— and instead the videos display knee-jerk reactions to specific teaching methodologies which are not understood by the complainants. Generally, these sorts of arguments against the Common Core focus on the methods of early, basic arithmetic taught in the 3rd and 4th grades. At this stage, the Core requires that students become familiar with the nature of a base-10 counting system, such as the one we utilize. The Indian-Arabic number system which we have adopted for mathematics has the benefit of simplifying these base-10 properties, but unfortunately that comes at the cost of obfuscation.

Teaching the base-10 system as it ought to be initially taught– without the shortcuts inherent in Indian-Arabic numerals– is a very alien procedure to most people. Because it is new and strange and takes more steps to accomplish than the familiar method of arithmetic, parents are frightened and confused; and when parents are frightened and confused, they tend to lash out rather than taking the time to actually learn the purpose and reasoning behind the methodology.

It occurs to me that a possible solution might be found in Egyptian arithmetic.

## The Death of Dignity and Virtue

There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner which she had acquired in consequence of the cultivation of her mind, she not infrequently appeared in public in the presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.

–Socrates Scholasticus, Ecclesiastical History