Why combinatorics is neat

Because it allows you to reason outside of regular mathematical domain.
By extracting the "meaning" of different concepts and combining them, you can derive theorems of arithmetic. I suppose this is not too different from the concepts where all mathematical reasoning came, but here we have a parallel system of reasoning consistent with the existing system.

I suppose it's analogous to reasoning with graphs, diagrams or other visual devices.


CONCEPT : ARITHMETIC EXPRESSION

# of binary strings of length n with 0 "1"s: C(0,n)
+
# of binary strings of length n with 1 "1"s: C(1,n)
+
# of binary strings of length n with 2 "1"s: C(2,n)
+
# of binary strings of length n with 3 "1"s: C(3,n)
+
.
.
.
+
# of binary strings of length n with n-1 "1"s: C(n-1,n)
+
# of binary strings of length n with n "1"s: C(n,n)

=

# of binary strings of length n: 2^n


So we have SUM_i(C(i,n)) = 2^n
because we have the concept that the sets with constant numbers of "1"s are a partition of the large set (of size 2^n)

The equation of CONCEPTS entails an equation of ARITHMETIC EXPRESSIONS.

This theorem can be proven using the axioms of arithmetic, or it can be proven using the reasoning I just described.
The point is that there exist "common sense" tools for mathematical reasoning which are not formally axiomatized. Is this related to diagrammatic reasoning?

There are a bunch of examples like this in combinatorics.

For similar examples of combinatorial identities, look at the many properties of Pascal's triangle.

Exceedingly Silly Personal Statement

(I also gave them an abridged version, in case they didn't accept this because it exceeds 100 words by a lot)

Personal Statement - FULL VERSION

I wish to be a student at the ILLC because it is the best place I know to explore my interdisciplinary interests.

If I had to summarize my strongest interests in a few words, I would say "Cybernetics and Philosophy of Science".

I love to study the logical structure of systems such as natural languages, music and even soccer. I have been fascinated by theories of information, and have read some philosophy from J. Pierce's "An Introduction to Information Theory" and G. Chaitin's "The Limits of Mathematics".

I have a strong interest in Bayesian probability theory, and the philosophy of what constitutes a reasonable prior. I am intrigued by philosophical questions like "Why are some distributions more 'natural' than others?", and "How can one infer causality from non-experimental data?". In this area, I have read from E.T. Jaynes's "Probability: The Logic of Science" and J. Pearl's "Causality".

In my quest for correct reasoning, and for the elimination of meaningless arguments, I decided to investigate the formalization of knowledge. I have read and thought about verification / proof systems such as NuPRL. I dream of a project for formalizing (thereby justifying) all scientific knowledge. I believe that the computer can be a great theoretical aid to scientists, if only we can bridge the gap between formal and informal theories.

I am also very interested in Cognitive Science and cognitive modeling.

After spending a year at an uninteresting job as a programmer, I decided to take a break. So I went to ESSLLI 2002, and I discovered epistemic logic. It was very interesting by itself, and it also provided fuel for my thoughts about rationality and game theory. Since then, I have bought two books on the subject, and have tried to study it in more depth.

Unfortunately, despite spending so much time reading and thinking about these interesting questions, I have made little concrete progress. Since all this was done by me alone, I lacked foundational knowledge, guidance from experts and the discipline to investigate the problems rigorously. In fact, there are severe gaps in my knowledge. I need to fill these gaps so that I can produce good research.

While the root of my interests is mostly philosophical, I feel an urge to tackle them at a technical level. Besides that, most "mono-disciplinary" departments tend to be restricted in their focus and approach. Thus I believe that no Philosophy, Economics, Cognitive Science or Computer Science department can satisfy my interests better than the ILLC, which is interested in questions across all of these fields. Not only that, but it would allow me to study logic in depth, and apply it to my problems of interest. I am excited about such a prospect, and this is why I believe that the ILLC is the best choice for me.