Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the month “October, 2014”

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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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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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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