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But Did Homer Catch the Bus?

Before we get on to the continuum hypothesis, let’s talk briefly about what we did in the last post because it’s a nice introduction to an important idea. We saw that it was possible to add up infintely many things. Let’s discuss this in the context of one of the paradoxes we introduced last time.

Let’s think back to Homer running for the bus (introduced at the start of the previous post). Suppose the distance to the bus is 100 metres. The way the paradox is phrased makes it seem as though, by splitting up 100 into infinitely many bits, it is impossible for Homer to traverse the entire distance because he cannot do infinitely many journeys in a finite amount of time. The mathematical definition we discussed last time (that of a series summing up to a finite quantity – one of the basic ideas of mathematical analysis) ignores this potential problem.

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I thought I’d do a quick note just saying what’s next for Blame It On The Analyst.

Soon I will finish the current series on Infinity and The Continuum Hypothesis and term will begin! For me, this means I’ll be taking  graduate courses, learning as much relevant mathematics as I can, teaching undergraduates and thinking about certain mini-projects. I expect that the content of the blog will shift slightly. It will have to in order to accurately represent what I’ll be spending my time thinking about. Unfortunately, this means that the ‘Mathematics For Everyone’ content is likely to slow down and there’ll be some of something along the lines of  ‘Mathematics for Mathematics Students’.

Imagine an arrow in flight. At any given instant in time, the arrow is not moving, because an instant is a snapshot. If it cannot move in an instant then it cannot move in any instant and motion is therefore impossible.

This is one of Zeno’s Paradoxes!  Here is another:

Suppose that Homer is running for a bus. Before he gets to the bus he must get halfway there. Before he gets halfway there he must get a quarter of the way there and before that an eighth of the way there and so on. Completing the journey requires completion of an infinite number of tasks. Is this possible? Zeno maintained that it was not.

Mathematical thought has something to say about these paradoxes. We will by no means lay to rest every possible concern of the philosopher but in this post and the next we will investigate the mathematical ideas which are brought to light by these things and then move on to discussing the continuum hypothesis.

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We’ll now discuss a classic example of a set which is larger than the set of natural numbers, in the sense that it is infinite but not countable. Such sets are called (sarcastic drum-roll) uncountable. The existence of such sets shows that there really are different sizes of infinity.

Power

The set which we are going to think about is a large collection of sets, so, a set of sets, if you will. Each element of our big collection will be a set of natural numbers. In fact, I want to consider every possible set of natural numbers, and gather all these sets together in one big collection.

Therefore, our big collection includes, for example, the set containing one, two and twenty and nothing else. The (self-explanatory) notation for this set would be {1,2,20}.  Our big collection also contains sets like {1} and {17}, as well as things like {1,2,3,4,5,6,7} and {,2,40,644,9999}, as well as infinite sets like the set of prime numbers, or square numbers, or multiples of 19 and so forth. It is a vast collection. The set I’m describing is the set of all subsets of the natural numbers and is called the power set of the natural numbers.

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In 1877, a mathematician called Georg Cantor put forward a simple hypothesis called the continuum hypothesis. It was a statement about infinity which he believed to be true, but was unable to prove. With hindsight, this was nothing to be ashamed of; it would be 100 years before the mystery surrounding the difficulty of its proof would be understood.

Throughout the 1870’s Cantor had been working on set theory and the theory of the infinite but his work was not received well. People debated about whether or not his ‘theory of the infinite’ was mathematics at all: Did his work really belong to philosophy? Should it be discarded because it challenged the uniqueness of the absolute infinity of God? Infinity had sometimes been seen as a convenient tool to be used when reasoning informally but ultimately when reasoning about the finite world. The ‘actual infinity’ of Cantor’s work was too alien for many mathematicians and philosophers of the time. So, what was his bold hypothesis? And what was so weird and abhorrent about infinity? In this first post on the topic I will introduce infinite sets and we’ll think about how to think about them mathematically.

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Now, we shift our attention to mathematics and logic and in doing so, we get to work on the other end of the big picture; hopefully we’ll be able to link back up with Inception somewhere in the middle.

Fallacy

Many of you will have, at some point, seen a demonstration of a classic mathematical fallacy. For example, a ‘proof’ that one is equal to two.

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It goes without saying that this post contains ‘spoilers’ for Inception.

After I’d first seen Inception, I thought that it was good, but not brilliant. Roughly speaking, I’d taken it at face value. I’d taken it as a sci-fi action movie with a nice idea at its core about ‘dreams within dreams’. There were a few plot points it had glossed over, but then every action movie does that. I remember thinking that I’d actually expected it to be cleverer, given all the hype I’d heard. Sure, the idea of a dream within a dream is quite a cool one, but it isn’t a complicated one and the movie itself wasn’t difficult to follow, as a lot of complex thrillers are.

Slowly, however, the ideas began working on me.

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I have made progress with the rest of my musings about Inception and they are nearly ready to go up! The series has become a three-post series because the second post was becoming enourmous, but I’ll be posting the next two posts in fairly rapid succession. The first of them will be coming up very soon.