Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “Set Theory”

What do we mean by Numbers? A simple introduction to Set Theory

The vast majority of people never think about what they mean when they use numbers to describe things. The concept is so ingrained into our early development that we simply take for granted the fact that people understand us when we apply numbers to different things. For most people, numbers are simply numbers, and questions about the meaning of those numbers are confusing and seemingly nonsensical.

Mathematicians are not most people.

For quite a long time, now, mathematicians have recognized that there are at least two very distinct ways in which we use numbers to describe things. Being the scholarly, academic types that they are, mathematicians have assigned names to these two different types of numbers which sound heady and difficult to the average person: ordinal numbers and cardinal numbers. Indeed, even mathematics students sometimes need quite a bit of work and explanation in order to really grasp the difference between these two types of number; but I’m going to do my best to explain these things in a very simple way for a casual audience.

Ordinal and cardinal numbers roughly correspond to the ideas of value and size, respectively.

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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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