Several thousand years ago the Greeks made a mathematical discovery that rocked the intellectual world at the time. They believed firmly in logic, that any statement logically derived from a true statement must also be true. To doubt this would be to doubt their entire intellectual community, and in some places their entire culture. Pythagoras was a Greek mathematician, as well as a cultist. His religion was one of logic and reason. He and his followers developed a cult of numbers. They worshipped mathematical beauty, and strongly believed the universe was comprised completely of integer ratios. The patterns of the natural numbers defined the Universe and the world inherited it’s natural order from the natural order inherent to ratios. So it was all the more earth shattering when it was proven that the square root of two is not an integer ratio. Creating in mathematics a classification of numbers know as the ‘irrational numbers’. They were aptly named as they defied the logic of the time, and in some ways they still make little sense now.
Imagine a perfect right angle triangle. Also imagine that the artist of this triangle was absolutely exact when measuring it out. The two sides adjacent to the right angle in this triangle are exactly 1 unit long. The remaining side, the hypotenuse, it’s length can be determined by using the famous Pythagorean theorem which was known at the time.
\(x^2 = 1^2 + 1^2 = 2 \Leftrightarrow x = \sqrt{2}\)
The hypotenuse is exactly \(\sqrt{2}\) units long. So far so good.
Now Pythagoras didn’t know anything about \(\sqrt{2}\) or the irrational numbers, so he did the only thing his mathematical mind could do at the time; he tried to figure out what ratio \(\sqrt{2}\) was. Every ratio is in the form \(\frac{a}{b}\) so equating that to \(\sqrt{2}\) we get the formula
\(\frac{a}{b} = \sqrt{2} \Leftrightarrow \frac{a^2}{b^2} = 2 \Leftrightarrow a^2 =2b^2\)
(note: the symbol \(\Leftrightarrow\) means the two statements on either side are equivalent. In this case basic algebra can take you from one to the other.)
Now before we can properly analyze this equation we need to know something about measurement.
Measurement is a universal system for comparing objects. In this case we will be dealing entirely with length. In our society the meter is used as the primary unit of length. If someone told you they had a stick that was 2 meters long we can visualize that distance by joining two meter sticks end to end. Now if we wanted a distance of 1.5 meters we would then have to cut a meter stick in half and join it end to end with a full meter stick. Alternately, we can define a new unit the ‘half meter stick’ and join three of them together. The half meter stick is then \(\frac{1}{2}\) meters long. Notice the ratio. In fact we can define any unit this way. The centimeter is actually \(\frac{1}{100}\) meters and the millimeter is \(\frac{1}{1000}\) meters. If we joined 60 millimeters together we would get \(\frac{60}{1000}\) meters. Then given any ratio \(\frac{a}{b}\) the number b defines the unit we are using, and the number a defines how many of those units we have. Now \(\frac{1}{2}\) is bigger then \(\frac{1}{4}\), so a bigger b actually defines a smaller unit. If we cut b in half, our unit would end up twice as big as the original, and if we multiply b by two the resulting unit would be half the size. Perfect measurement in our system assumes that given any two sticks of arbitrary length, there exists some unit that can measure both sticks. That means that an integer number of those unit sticks can be joined end to end to form a stick exactly the same length as both the sticks being measured.
Now returning to our equation
\(a^2=2b^2\).
Since \(2b^2\) is an even number \(a^2\) must also be an even number. Now it was known at the time that a square number like \(a^2\) can only be even if a is even. I’m not going to prove it, but feel free to try it. Grab a calculator and try it out. If you square an even number the resulting number is always even, and if you square an odd number the resulting number is always odd. Since a is even, we can factor out the 2 resulting in \(a = 2c\). Our equation then becomes,
\((2c)^2 = 2 b^2 \Leftrightarrow 4c^2 = 2b^2\).
We can divide both sides by two to get
\(2c^2 = b^2\).
Now it is easy to see that b is also even using the exact same argument that we used to determine that a is even. So
\(b=2d \Leftrightarrow 2c^2= (2d)^2 \Leftrightarrow 2c^2=4d^2 \Leftrightarrow c2=2d^2\).
Which is the exact same formula we started with. Solving for \(\sqrt{2}\) gives us
\(\sqrt{2} = \frac{a}{b} = \frac{c}{d}\).
But remember \(b=2d\) so the unit d is twice as big as the unit b. Which means if there exists a unit b that can measure the hypotenuse of our original triangle then there also exists a unit d that is twice as big that can do the same job. Now by the same logic if there is a unit d that can measure the hypotenuse of our triangle then there is another unit twice as big as that, and then another one twice as big as that. Eventually we come up to the completely ludicrous conclusion that I can use a meter stick \(2^{99999}\) times or more larger then our original unit. A number that is clearly larger then the length we are measuring.
People were killed trying to let this secret out, that’s how dangerous this knowledge was at the time. One thing the Greeks worshiped was logic, and here we have a completely logical basis followed by completely logic reasoning, resulting in a totally absurd conclusion. Today we know that \(\sqrt{2}\) is not an integer ratio. \(\sqrt{2} \neq \frac{a}{b}\) for any integer a or b. In mathematics we just accept it as fact and move on. However, for early empiricists this is a catastrophe. Imagine that when this triangle is drawn there is exactly 1 billion evenly spaced molecules inside the unit edge. How many molecules are in the hypotenuse? The answer is approximately 1.41 billion molecules, but the decimal trail never ends. Eventually one of the molecules will have to be split into parts, but we can’t measure how to split it. Even if we displaced the molecules so that they weren’t evenly spaces, we wouldn’t be able to measure their displacement. So we are forced to arrive at the unpleasant conclusion that one of two things cannot be possible. Either it is impossible to measure all lines in a right angle triangle, or the triangle was not really a right angle triangle to begin with. In both cases the conclusion is the same, either it’s impossible to measure some lines, or to draw some angles. In both cases our system of measurement is logically flawed.
Today quantum mechanics has given as a way around this problem, but it is still characteristic of a common problem that crops up again and again in every field I have studied. Before this proof came across, it would have been impossible to convince Pythagoras that portions of the universe were fundamentally unmeasurable.To him the universe had to be measurable, the world just wouldn’t make any sense otherwise. This was a statement that coheres so well with the remainder of his world belief system that it was not only true, but it’s truth was self evident and required no further explanation. However, his beliefs aside, the statement was still wrong.
The remainder of the history of mathematics is nothing more then a long series of extremely intelligent people creating hard and fast rules that were simply ignored by the next generation. The greatest leaps in the field come when old unquestionable systems of truths are questioned and then thrown out. This is most certainly not limited to mathematics. I can’t begin to describe how many conversations I’ve been in where someone has tried to convince me of an utterly wrong statement. Like Pythagoras they do not yet have the knowledge necessary to understand why they are wrong, and like Pythagoras whatever it is that they believed coheres so fundamentally to their worldview that absolutely nothing I say can ever change them of their mind. Likewise, I am aware of the same things withing myself. To then arrive at anything that can even remotely resemble the truth of the universe, it becomes necessary to individually pick out every single assumption, challenge it, then keep it if and only if it survives. Even then it must be acknowledged that it is still possible that the evidence of it’s falsehood simply does not yet exist. To do this, I willingly admit, that I may not have the knowledge necessary to challenge those things which I accept as truth. However, here is where the main difference between mathematics and philosophy aids me. Philosophy tries to understand the universe, without defining anything for fear that those definitions may be wrong. In math, we just define away, and deal with the consequences later. Does that make it true of the universe — absolutely not. However, it makes it true of the ideal universe defined by the assumptions I was forced to make. The question concerning which assumptions correspond to the actual universe we live in, is still an open question.
