Developing questions

271 days ago by calkin

Viewing mathematics as an experimental science.  What is mathematics?

Patterns in numbers, shapes, functions --- patterns in connections between sets.

Devlin:  "the science of patterns".

Does mathematics exist?  Let's assume that *we* exist, and have been doing math.

If math does exist, where?   Does it only exist in our heads?

Assuming we are not all alone in the universe: if we encounter an alien intelligent 

lifeform, will it understand $\pi$?  Not in base 10, necessarily.

Will they understand prime numbers?  Will they understand integers?

Mathematics is also a language for describing science, for understanding the worlds, and to enable us to create, to engineer.

Kronecker: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk".

God made the integers.  All else is the work of humanity.

Experimental mathematics: a voyage of discovery, exploring areas of mathematics *looking* for patterns.

Asking questions.  How do we come up with interesting questions?

Don't ask how to come up with interesting questions.   Ask instead, how to ask questions.

Always ask:  "how many?"  "what happens if?"  "where?"  "why?" "how are these phenomena connected?" "Does there exist?"  "Can we prove it?"

Questions lead to paths of discovery.  Rabbit holes.

We compute to explore. 

A case study.

How many ways to put non-attacking kings on a chess board?

Start with 8x8.  How to do it?  

Start with one dimensional boards instead?   Discover our old friends, the Fibonacci numbers!

Develop techniques to work with $k \times n$ boards.

Can we prove that certain limits exist as $k$ and $n$ go to $\infty$?  Related to questions about entropy of physical systems.

Define $F(n,k)$ to be the number of non-attacking configurations of Kings.  What is 

\[ \lim_{n,k \rightarrow \infty} \frac{1}{nk} \log F(n,k)? \]

What happens in 3 dimensions?  What does a board look like?

What happens in 2 dimensions if we only attack horizontally and vertically?  What happens if we generalize this to 3 dimensions?

What happens if we give each king a weight $q$, and count configurations by total weight = product of the weights of the kings?