Boxing Pythagoras

Philosophy from the mind of a fighter

03 Dec 2014
5 Comments

Proof that π=2√3

There is an inherent danger attached to blindly accepting the word of someone who sounds like they are presenting a rational, scientific claim. Too many people are willing to accept a proposition solely because they’ve heard it from someone who bears the appearance of intelligence. The line of thought seems to be, “Well, he’s smarter than me, so he must be right!” Unfortunately, this sort of fallacious reasoning goes largely unchecked, and often becomes formative in the common understanding of entire groups of people.

For almost the entirety of your mathematical education, you have been taught that the ratio of a circle’s circumference to its diameter, which we affectionately refer to as π, is something close to 3\frac{1}{7}, or about 3.14; however, today I’m going to show you that your math teachers were wrong. In actuality, the value of π is exactly 2√3, or about 3.46.

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Posted in Epistemology, Mathematics, Science and tagged , , , , , , , ,
20 Nov 2014
2 Comments

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.

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Posted in History, Martial Arts, Mathematics and tagged , , , , , , , , , ,
10 Nov 2014
3 Comments

For the unwise man ’tis best to be mute

For the unwise man ’tis best to be mute
when he come amid the crowd,
for none is aware of his lack of wit
if he wastes not too many words;
for he who lacks wit shall never learn
though his words flow ne’er so fast.

Wise he is deemed who can question well,
and also answer back:
the sons of men can no secret make
of the tidings told in their midst.

–Hávamál 27 & 28

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Posted in Heathenry, Philosophy and tagged , , ,
07 Nov 2014
1 Comment

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

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 c 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.

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Posted in History, Mathematics and tagged , ,
27 Oct 2014
3 Comments

On the Continuum and Indivisibles

Εἰ δ’ ἐστὶ συνεχὲς καὶ ἁπτόμενον καὶ ἐφεξῆς, ὡς διώρισται πρότερον, συνεχῆ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενα δ’ ὧν ἅμα, ἐφεξῆς δ’ ὧν μηδὲν μεταξὺ συγγενές, ἀδύνατον ἐξ ἀδιαιρέτων εἶναί τι συνεχές, οἷον γραμμὴν ἐκ στιγμῶν, εἴπερ ἡ γραμμὴ μὲν συνεχές, ἡ στιγμὴ δὲ ἀδιαίρετον. Οὔτε γὰρ ἓν τὰ ἔσχατα τῶν στιγμῶν (οὐ γάρ ἐστι τὸ μὲν ἔσχατον τὸ δ’ ἄλλο τι μόριον τοῦ ἀδιαιρέτου), οὔθ’ ἅμα τὰ ἔσχατα (οὐ γάρ ἐστιν ἔσχατον τοῦ ἀμεροῦς οὐδέν· ἕτερον γὰρ τὸ ἔσχατον καὶ οὗ ἔσχατον).

–Aristotle, Physics 6.1

There is a concept which is absolutely intrinsic to all of geometry and mathematics. This particular concept is utilized by every single High School student that has ever graphed a line, and yet this concept is so incredibly difficult to understand that most people cannot wrap their heads around it. I’m talking about the concept of the continuum. Basically, the idea is that geometric geometrical objects are composed of a continuous group of indivisibles, objects which literally have no size, but which cannot be considered “nothing.” Despite the fact that these individual objects have no size, they form together into groups which, as a whole, can be measured in length or height or breadth. In mathematics, objects such as lines, planes, volumes, and all other sorts of space are considered to be continua, continuous and contiguous collections of these indivisibles into a unified whole. Because these infinitesimals have no size, themselves, even finite spaces contain an infinite number of these points.

Nearly every mathematician on the planet subscribes to this point of view. However, this was not always the case. Only a little more than 100 years ago, this view was considered extremely controversial and was only held by a fringe minority of scholars. Four centuries before that, this concept was nearly unthinkable. Though it has become, without question, the prevailing view of mathematicians, even today there remain a tiny handful of scholars who object to the use of the infinitesimal, the infinite, the individible, and the continuum in modern math. One such person is Dr. Norman Wildberger, an educator and mathematician for whom I have the utmost respect.

Still, I disagree with Dr. Wildberger’s philosophy on this particular issue.

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Posted in Mathematics, Philosophy and tagged , , , , , , ,
13 Oct 2014
21 Comments

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.

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Posted in History, Philosophy and tagged , , , , , ,
01 Oct 2014
2 Comments

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.

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Posted in History, Mathematics and tagged , , ,
23 Sep 2014
2 Comments

A Finely-Tuned Deception

William Lane Craig’s Reasonable Faith website released a new video, yesterday, highlighting the Fine-Tuning Argument, another extremely popular topic which is quite commonly discussed in modern apologetics circles. If you are unfamiliar with the argument, feel free to watch Craig’s video, below. You can also read the transcript for the video here, if you (like me) would like to digest its claims in a more easily referenced format.

https://www.youtube.com/watch?v=5okFVrLdADk

I’m sure this won’t come as much of a surprise to anyone familiar with this blog, but I find that the video is wholly unconvincing. In fact, the entire Fine-Tuning argument is nothing more than a God-of-the-Gaps which has been camouflaged behind a screen of pseudoscience.

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Posted in Philosophy, Science and tagged , , , , ,
17 Sep 2014
6 Comments

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.

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Posted in History, Philosophy and tagged , , ,
15 Sep 2014
3 Comments

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.

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Posted in History, Mathematics and tagged , , , ,

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