Scio me nihil scire, and some thoughts about mathematical paradoxes.

Socrates said (translated into latin) "Scio me nihil scire" ("I know that I know nothing"). And I feel that this is a good place to start. After two and a half years at Uppsala University, and AARMS Summer School in Mathematics I am quite confident that I know nothing about Mathematics (or Physics). People seem suprised when I tell them this, and some people (especially engineers) seem to think that I am saying that they know more than me then. I feel that we should have this humble approach to any subject we study, even the social sciences. If nothing else, it is at least a more gentle way of interacting with others.

To explain the more philosophical nature of this statement, is that if we assume that I have some amount of information in a set B. And the total amount of information, finite or not is denoted by a set A. Then B is a subset of A, and if we know the size (ie. the cardinality) of both set A and set B, we can get an idea of how much of A is covered by set B. If however, we only know the size of set B, and not of A, this is of course impossible. Many people that assume that they know a lot of Mathematics think that their cover of set A is big, mostly because they feel that their covering is big compared to some "average person measure".

A person that knows what an overwhelming amount of knowledge lies within Mathematics, realizes that if the set A is finite at all, their covering of it is very, very small. Therefore, I believe that many great Mathematicians are very humble of their knowledge, and even though they are the masters of their fields, they still believe that they know nothing about it.

Something else that fascinates me about the statement "I know that I know nothing", could provide us with a simple, very intuitive way of characterizing (in a way) what a paradox looks like.

If we assume, according to the statement that the set A of everything I know only contains one element, and that is that I know nothing. It would make that set empty, contradicting our assumption that I know nothing, which then assumes that A is empty. This is really equivalent to Russel's Paradox, and we are tempted to make the following definition.

Definition: If the intersection between a set A and itself is not the full set, then the definition of the sets are paradoxal.

This really assumes that we are given two ways to define the sets. For instance, let us apply this to the famous "Barber paradox". Assume that there is a small town with exactly one (male) barber, and that he shaves all men that don't shave themselves. We must also assume that all men has to shave. The paradox arises when we consider who shaves the barber?

Let us assume that the barber shaves everyone. Let us also regard the necessary condition that he doesn't shave himself, meaning that he is the same set except one. This means that the intersection between all men who shave and the set where the barber doesn't shave himself but everyone else does, is not the same, so we have arrived at a paradox by the definition. By using this definition we can develop it further and in an obvious manner find many other equivalent ways to express it.