Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “mathematics”

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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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 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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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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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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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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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

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The Elements of Geometry

Some time ago, I wrote about Alexandria, the most important city in history, briefly discussing the lives of just 17 of the men and women that made it so. Prime to that list, both in sequence and in importance, was Euclid of Alexandria, a personal hero of mine who I consider to be one of the most inspirational and influential people in all of human history. We know next to nothing about Euclid’s life– we do not know where or when he was born, where or when he died, and extremely little about the time between those events. We know that he lived in Alexandria at roughly the same time as Ptolemy I, circa 300 BCE, and we know that he wrote prolifically about mathematics. Yet, even with so very little information as this, I would strongly argue that Euclid contributed far more to the world than did much more well-known figures like the great historian, Herodotus; or the conquering emperor, Julius Caesar; or even the revolutionary preacher, Jesus of Nazareth. What could Euclid have possibly done that outshines these other, great men? Euclid of Alexandria wrote the Elements.

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Be smart. Use tau.

For anyone who didn’t know, this past Saturday was Tau Day, a celebration of the proper circle constant!

A couple weeks ago, I told all of you about how π is stupid, and urged everyone to be smart and use τ, instead. However, you might be surprised to learn that this is not the end of the debate, when it comes to angles. While I argue that people should measure angles in terms of τ, many traditionalists argue that they should be measured in terms of π, our grammar schools are still intent on teaching the incredibly archaic degrees of arc, and if you’ve ever fiddled with a scientific calculator, you might have learned that some backwards people prefer gradians. But that’s still not the end of the debate. According to a video by Dr. David Butler of the University of Adelaide, “π may be wrong, but so is τ!”

I’m going to celebrate Tau Day, belatedly, by rebutting Dr. Butler’s presentation. I’m going to show that degrees, gradians, η, and π are all stupid, and that the only smart choice in this debate is τ.

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WLC on Time, Part 5: More Mathematical Misconceptions

After my last installment of this series, I had thought that I would be done critiquing Dr. William Lane Craig’s misunderstandings of the science and mathematics regarding time. After all, I’ve already shown that his arguments in support of the archaic Tensed Theory of Time are unfalsifiable, fallacious, ill-conceived, and self-contradictory. What more could there be for me to say? Well, in this week’s Reasonable Faith Podcast, Dr. Craig gifts me with more of his misconceptions about time. Starting at the 13:15 mark and lasting through the rest of the podcast, Dr. Craig addresses a question posed to him about the implications of the Tenseless Theory of Time on the theory of Evolution by Natural Selection, which the questioner refers to as “the holy grail of atheism.” I’ll note that this questioner doesn’t seem to realize that even a great many devout Christians completely accept the veracity of Evolution by Natural Selection, and that it is no more an “atheist” theory than is the Pythagorean Theorem. However, the particular implications on evolutionary biology will take a back seat, today, to the more general implications which Dr. Craig claims are made by the Tenseless Theory of Time. Specifically, Dr. Craig asserts that nothing actually changes over time, on the Tenseless Theory. Read more…

Pi is Stupid

I teach Brazilian Jiu-Jitsu to people of all ages, from preschoolers to middle-aged parents. While BJJ, in itself, is not necessarily the most academic of pursuits, I also happen to be a huge nerd. So while teaching some of my 8 to 13 year-old students, it sometimes happens that I overhear them talking about their math classes, often to complain about ideas that they’re struggling to grasp. Being a huge nerd, and also a delighted teacher, I do my best to help them through these issues. If I can teach a kid how to find the length of the hypotenuse of a right triangle at the same time as teaching her how to finish a Triangle Choke, I become pretty much the proudest martial arts instructor you could hope to meet.

One of the things that my kids often use in their math classes, but almost never really understand, is the constant π (pi). They are taught π in class to help learn things like how to calculate the area of a circle, but they usually don’t really know what π actually is. They just think of it as some number that they have to memorize, never thinking about where the number comes from, or why it is what it is. Sometimes, I’ll tell the kids that they can earn their way out of doing push-ups if anyone can tell me what π is. Most often– after the jokes about desserts are made– I’ll hear someone say, “Coach, π is three-point-one-four!” Every now and again, one of the kids is clever enough to say, “Coach, π is three-point-one-four-on-into-infinity!” They get confused when I tell them that’s the value of π, but that is not what π actually is. It’s not their fault that they get confused by this; they were usually taught about π all wrong. I don’t even blame their math teachers, because most of the time, those math teachers were also taught about π in the wrong way. For a very long time, math classes have been teaching that π is a number, instead of teaching that π is the relationship between a circle’s circumference and its diameter. There is a reason it has been taught this way.

Ladies and gentlemen, π is just plain stupid.

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