Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “Aristotle”

Infinitesimal Calculus 1: The Numbers Between Numbers

If I were to ask a person to name a number which comes between 1 and 3, everyone from a three-year-old child to a white-bearded great-grandfather is likely to respond by saying, “2.” If I rephrase the question to ask about a number between 1 and 2, then the young child might be confused, but a fourth-grader might be able to respond with 1\frac{1}{2}. We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, n, there are an infinite number of Rational numbers which are greater than n and yet less than any other Integer which is greater than n.

There are numbers in between the Rational numbers, too. We can define some number, r, which is not equal to any Rational number. There are Rational numbers which are greater than r, and those which are less than r, but somehow our number r squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

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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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On the Kalam Cosmological Argument

One of the most popular arguments for the existence of God is known as the Kalam Cosmological Argument. In general, the “cosmological” family of arguments attempt to show that some initial condition necessarily pre-exists the universe, and declare this initial condition (or its cause) to be God. There have been many different versions of the cosmological argument, but the Kalam is particularly popular because it is composed of a very simple syllogism with premises that many people find self-evident. This simplicity makes the KCA very easy for laymen to remember and explain, while professional philosophers love the hidden nuances and depth which underlie the seemingly simple premises. The KCA was first developed and refined by medieval Muslim thinkers like Al-Kindi, Al-Ghazali, and Averroes in the time when the Arab world stood at the pinnacle of Western philosophy and science. Today, arguably the most avid and scholarly proponent of the KCA is Christian apologist, William Lane Craig (whose work has been a frequent focus of this blog), and it will be Dr. Craig’s particular formulation of the KCA which I will be discussing.

The argument is as follows:

  1. Anything that begins to exist has a cause.
  2. The universe began to exist.
  3. Therefore, the universe has a cause.

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