Can't square a circle

Before reading this post, it might help going through the article on irrational numbers, if the reader is not familiar with them already. Irrational numbers are important to understand when talking about circles. This is because of the irrational number pi (3.1415....). Pi defines the relationship between a circle's radius and it's circumference and area. A circle has a circumference of length pi x diameter, and an area of pi x (radius squared), so all circles have a circumference and an area which is irrational.
Mathematicians and philosophers since ancient Greek times have tried to square a circle. Squaring a circle simply means constructing a square with the same area of any circle. Now here's the weird part. Because the circle has these irrational properties, you can't construct a square out of any circle.

Consider the circle with radius of length 1 unit. The circle has an area of pi (pi x radius squared) and so the square that you would construct out of this circle would have a side length of the square root of pi. Now, since pi is irrational, the square root of pi is irrational. But the length of a square is measurable - rational. Lindemann proved the impossibility of squaring a circle in 1882, putting to rest the famous classical problem.

Food for thought:

A right-angled triangle with two side lengths of 1 unit has a hypotenuse of length square root 2. This is an irrational, straight length.

Transcendental Numbers

Maths is a philosophical concern for us, because it tells us of the many mysteries in the universe. One of these mysteries is transcendental numbers. They're the ones that come up on the calculator and you can't write down because they go on forever. They're also known as irrational numbers, so called because they seem to defy reason.

The discovery of irrational numbers has a curious story about to it. Apparently, it was discovered by someone who belonged to the school of Pythagoras (the well known ancient Greek mathematician). When the person who discovered it told the other Pythagoreans, they did not want anyone else to know about it, so they drowned him out at sea. I have heard many different reasons as to why they might have done this, but I think that it might have been because it went against everything they believed in. The Pythagoreans studied and worshipped Mathematics like a religion. To them, (and to the few who appreciate it nowadays), it was perfect and beautiful and complete. The idea that there were numbers that you couldn't express as fractions would have horrified them.

Here is the proof of the irrationality of the sqrt(square root) of 2. It is a reductio ad absurdum (latin for "reduction to absurdity") , which is an argument that shows that something cannot be true because it leads to a contradiction. In this case, we show that the root of 2 cannot be expressed as a fraction, by first assuming that it can:

Assume that the sqrt2 can be expressed as a fraction, say a/b (where a, b are whole numbers)
Let a/b be irreducible so that a and be have no common divisors (like 1/3 but not 2/6)

Now, we calculate:

sqrt2 = a/b

2 = (a squared) / (b squared) ________ (squaring both sides)

2 x (b squared) = (a squared) _________ (multiplying both sides by "b squared")

Since the left side is divisible by 2, the right side must be divisible by 2. The square root of a positive number is also a positive number, so we can now express "a" as "2c" for some integer c:

2 x (b squared) = (2c) squared ________ (by substitution)

2 x (b squared) = 4 x (c squared) __________ (expanding RHS)

(b squared) = 2 x (c squared) ___________ (dividing throughout by 2)

Since the right side is divisible by 2, the left side must be divisible by 2. The square root of a positive number is also a positive number, so b is divisible by 2.

But this contradicts the fact that a and b have no common divisors! It means that sqrt2 cannot be expressed as a fraction, which in turn means that it has an infinite number of decimal places.
The beauty of the classical proof is that we can know that such numbers go on forever, without actually checking that they do (which is impossible anyway).

For your interest - sqrt2 looks like this:

sqrt2 = 1.4142135623730950488......

On a line that is two units long, sqrt 2 is impossible to find. You can find an estimate, but you can't pin-point it because it exists way too deeply within the line. Now you can see why they're called transcendental numbers: they seem to transcend the number line. These sorts of numbers exist between every mentionable number. Other well known irrationals include pi, the exponential e, and other square roots like sqrt 3.

Questions to ponder

1. Why am I me, rather than someone else? (this question does not stop at the writer- it is for everyone to ponder)

This question has been bugging me lately and I find it very intriguing. What is consciousness? The more you think about it, the more you realise that this is not a question of science, but a question of philosophy. Anyone like to have a go?

2. Is the glass half full or half empty? Or are these descriptions an example of the way language cannot capture truth in its objective form?

3. Is there really such a thing as good and bad or are these just meanings that we assign to things in order to understand them better?

Gettier clocks

The part of philosophy called Epistemology is concerned with knowledge. What is knowledge? When can we say that we know something? Philosophers since ancient times have been pondering these sorts of questions.

In the post on Descartes, we saw that the only thing that we could be sure of knowing was that we had a mind. Descartes discovered this by deductive reasoning. Deducting reasoning, when correct, is absolute. The following argument is an example of deductive reasoning:
(1) Bachellors are single, middle-aged men.
(2) Bob is a single, middle-aged man.
(3) Therefore, Bob is a bachellor.
We deduced from the fact that Bob was a single, middle-aged man, that he must be a bachellor. Bishop George Berkely(1685 - 1753) used similar reasoning to show that the world didn't exist:
(1) We percieve ordinary objects (mountains, houses, etc.)
(2) We percieve only ideas
(3) Therefore, ordinary objects are ideas
That is, what we see of this world is only in our minds - it isn't really there. Descartes also showed that we couldn't be positive about physical reality.

The problem is, when we want to show that what we see is real, we have to use abductive arguments. Abductive arguments are arguments to the best explanation:
(1) Things fall downwards
(2) The world looks flat
(3) Therefore, the world is flat
As you can see, abductive arguments can sometimes be wrong. We don't have the certainty that comes with deductive reasoning. But, nevertheless, there are good abductive arguments:
(1) Both you and I percieve a red chair
(2) Therefore, there is a red chair before us, causing us to perceive it (rather than it existing only in our minds)

So now the challenge is to come up with a theory which accurately describes how we can know that there is a red chair before us. We come up with conditions that we need to satisfy. These were the conditions that were stated by the ancients:
If p is any statement (eg. there is a red chair before us), then we know that p only when:
1. p is true
2. p is justified
3. we believe that p

These three conditions were widely accepted to be sufficient for having knowledge. However, it was only forty years ago that Edmund Gettier found a serious problem with it. He asks us to imagine that a person, let's call him Bob, who wants to know the time. It is 10:00. Bob looks at the clock and it says that it is 10:00. Naturally, he believes that it is 10:00. So Bob has a true, justified belief; according to the above theory, he has knowledge.

But let us also imagine that the clock had stopped, and it was only by sheer chance that it had shown the right time. Does Bob know that it is 10:00? Intuitively, it would seem that he did not: if the clock had shown that it was 10:30, then he would have thought that it was 10:30. So the classical theory is not an accurate theory of knowledge. For decades now, epistemologists have been trying to successfully overcome the problem of Gettier clocks.


Further Reading:

• Edmund, L. G., 1963, "Is Justified True Belief Knowledge?", Analysis Vol. 23, pp. 121 - 23

Are we really that selfish?

There is a well known theory called Psychological Egoism. Most people don't know that it by this name, but it has been discussed many times before, even in the TV show, 'Friends'. It's the episode where Joey challenges Phoebe by saying that there is no such thing as a selfless deed. In other words, every deed is selfish. Phoebe tries to prove him wrong by getting stung by a bee and saying that she did it so that the bee's friends would be impressed. Joey points out that the bee would have died and that would have done him little good. But he also fails to point out that even this deed is selfish: the only reason Phoebe let herself get stung was to prove Joey wrong. So ta daa! Not a self-less deed at all.

You can think of any charitable action as a selfish deed. Look at Mother Teresa. According to the psychological egoist, the only reason that she helped all those people was to make herself feel better. But is this an accurate picture? Many people have sacrificed their lives for their country, or to save a loved one. But does that mean that they did it for self-glory?

As of yet, there hasn't been one counter-example found to this theory. But that doesn't necessarily make it correct. In this case, it makes it suspicious. It seems like every situation has some selfshiness in it, and the Psychological Egoist says 'aha!' - this must have been the prime motive of the action. Not only is this pessimistic, it's unjustified. If we only cared about our self-gratification, then the following situation would be true: Bob is only interested in feeling happy. He doesn't care about art, music, other people, and anything that you can imagine being worthy of value. He only cares about his own happiness. So selfish Bob does whatever he can to make himself happy. But he doesn't care about anything else, so making someone laugh wouldn't make him happy. Saving the lives of a hundred people wouldn't make him feel happy. In short, caring only about happiness is the fastest way of losing it.

So the idea that Mother Teresa cared only about herself when she helped others makes no sense. She must have cared about the people in order to have felt any sense of satisfaction.

Feeling good about helping others makes it clear that we can't be that selfish.


Further Reading:

• Feinberg, J., 1995, "Psychological Egoism", Ethical Theory, Belmont, Wadsworth, pp. 62 - 72