Mathematics (cont.)

The Problem of Consistency; Beauty and Elegance; Mathematics in the Universe

Although Math seems like the most provable, definite area of knowledge, it has its flaws and paradoxes. 

One example given in the lecture was Gottlob Frege's work, which involved trying to reduce all theories into simple arithmetic. Lord Russell challenged Frege's work at a fundamental level, concerning the 'set paradox'. This paradox was about a set containing all sets that are not members of themselves.

What happens with the set of the "sets that do not contain themselves as elements"? Now it seems that there are knot theories (diagram below) which can get around the set paradox.

With the problem showed in the lecture of whether N=0.999... is equal to 1, the 'paradox' created here depends upon our definition of infinity (quoted from my mom :) ). We can prove N to be 1 with simple algebra:

N = 0.99999

10N = 9.9999

Therefore 

10N-N = 9.9999 - 0.9999

9N = 9

N = 1

The simplicity in this logic is satisfying. However if we were to really look into whether 0.999 does equal 1, we would look at if we define 0.9999 approaching 1 to be 1, thus how we define the infinite number of nines. When talking about beauty and elegance in math, I feel this type of discussion is what attracts people to pure maths. 

Maths does have flaws, but mathematicians work on its flaws to correct them and explain how things work. In fact without its paradoxes and flaws, I'm not sure if there would be that many mathematicians in the world, because everything would work in perfect simple logic. 

Then there is the question, is maths just a coincidence that math is so prevalent in reality? Was maths invented by people or was it discovered? Maths is a pure science, that is systematic and logical, as well as an art to create fundamental logic. It definitely to some point has been defined by people, as we can see in ancient civilizations where maths that began in different areas of the world came about differently. 

I think it makes sense that maths works, because the universe was not created by pure randomness. When things happen there is normally a cause and effect, and if the universe was created by a chain of cause and effects, then it seems reasonable that things fell into place in a somewhat structured way. The coincidence of the Golden Ratio may not be explainable by human minds (at least currently). But the brain is a biological object which has cells, structure, science to it, and it seems logical that we react to certain ratios naturally.