A Holiday Thought on Actuality

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Today is Pi Day, because 3/14 is the first 3 digits of the illustrious ratio. Next year it will be the most ultimate Pi day because 3/14/15. I had a brief conversation with my roommate about Pi, and how, while it is a potential infinite, there is no such thing as an actual infinite. He did point out to me that it is a ratio between two points, and that there are plenty of instances where in 10-digit math, that the exact number is infinite. A good example of this is 2 divided by 3 which one will get .666666666 (ad infinitum). Now in actuality, there is a limit between two things. This is a problem that goes back to ancient Greece, known as Zeno’s paradox.

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Another example of this paradox is if I drew a line, with two distinctive points, and had you go half the distance of that line.

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Then I tell you to go half of half your distance. Then if we keep doing this, as the story goes, we will never run out of spaces. Now, I don’t like to be a pragmatist, but when we look at the line that I drew above, we KNOW that there is a limit, the limit is somewhere between point A and point B.

Another fun problem is that of the Infinite Hotel:

Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests

It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

While this appears possible, it is unfathomable. It’s absurd. Why? Because a hotel is actually, and when we envision a hotel, we can’t fathom one which can accommodate infinity. Just like Pi, the ratio between the circumference and diameter in any given circle, when we see a circle, we KNOW there is only so much matter within that circle, but we can imagine an abstract idea like Pi and call ourselves smart so that we have a quirky socialite moment.

The whole number idea, as the idea of infinity, is abstract and adjustable to each culture, and each individual. Hope this gets your brain working for a nice weekend. Here is the first 100,000 digits of Pi.

*P.S. My roommate is cleaver and shared this joke that he had heard with me after this post:

A mathematician and an engineer agreed to take part in an experiment. They were both placed in a room and at the other end was a beautiful naked woman on a bed. The experimenter said every 30 seconds they would be allowed to travel half the distance between themselves and the woman. The mathematician said “this is pointless” and stormed off. The engineer agreed to go ahead with the experiment anyway. The mathematician exclaimed on his way out “don’t you see, you’ll never actually reach her?” To which the engineer replied, “so what? Pretty soon I’ll be close enough for all practical purposes!”

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