What is a dead end? DIY Mathematics, and Graph Theory.

This post is going to be part diary of my experience thinking about cul de sacs, and part meditation on mathematics. Skip to the second part if me recounting doing a poor, rudimentary job of inventing a graph theory doesn't sound interesting.

I was walking down a street one evening, in a place I wasn't too familiar with. I had driven past there several times before, but this was my first time walking there. It was full of row after row of cream-stone terraced houses, the kind that get filled, due to their location and their fanciness, with small offices of lawyers, accountants, and the likes. You don't really see that type of house in this part of England, it's a style more reserved for the south. I was wondering about roads and paths, and how one would define a dead end. I thought, you could trace it down infinitely. Once you got to a dead end, you could turn into a house, and then down a hallway, and then into a room, and then into a box, never meeting a dead end. You could carry on down to the subatomic level, at which the very concept of a dead-end seems absurd. So you need to drag yourself out of the realm of the physical to understand what a dead end is. A dead end can only be properly defined for a fixed set of roads*. So now say we have just a fixed set of roads, how do we define the dead end then? At first, I thought that the dead-end is simply a road that connects to only one other road. However, what about a road that connects to only one other road, but in two places? A road that loops back around to rejoin the road it branched off from. So a dead end is a road that connects to only one other road, in only one place. And at this point I realised that it does not have to be a connection to only one other road. If two roads came together at a crossroads, with a third dead-end branching off, then this would still be incorrect. So a dead-end is a road which connects to another road or roads, in only one place. This is where the nature of the problem reveals itself! It's not a question of roads — it's a question of junctions! A dead end is not a road that connects to only one other road, but a node that connects only to one other node (with only one route). And suddenly, from that, graph theory and why we use it made sense to me!

(the second part i was talking about starts here)

This is really a post about my thoughts on mathematics. I read, in "how to read a book", on the difference between science and philosophy. To compare the two, they describe philosophy as something that you could be thoroughly convinced of, just by sitting in your armchair, and refering to no more than your own memory of normal life experience. To be convinced fully of some scientific literature before you, you would need to repeat the experiment the scientists repeated yourself. So, really, a piece of scientific literature trying to convince one of something is a record of an experiment, and some philosophy on what has occurred. Anyway, that's not too much to do with the difference between philosophy and mathematics. By this definition, all mathematics is philosophy, but not all philosophy mathematics. You can read mathematics, and be fully convinced of it, by drawing on no more than basic experience and other mathematics.

In the same way that if I locked you in a room for the next ten thousand years you could remake all of mathematics (I beleive in you!), you could probably do the same for a lot of philosophy (depending on how much you've experienced in the world (for both actualy! If you had never experienced a road, then you may never come up with graph theory)). And I think there's something to be said for locking yourself in a room and trying to derive some mathematics yourself. It's just so incredibly rewarding, to have created a universal truth from inside your head. Remember mathematics is not something in a vacuum aside from human experience. It is determined by the universe in more ways we would like to admit. The square packing in a square, for example, is musings on squares and packing. Graph theory is musings on roads and routes. Number theory is all from the simplest of counting. Mathematics builds upon itself, until it reaches heights that make us think it's something that exists on its own, but it doesn't. Every piece of mathematics says that "if x is true then that means y", but x always at some level has to relate to something in the human experience. My point is that it is not impossible or even hard to do some mathematical musings. If you're having difficulty, then do some mechanics (not physics). Use your existing understanding of the world (not doing experiments), to come up with something mechanical. Happy mathing!

(bonus - do some amateur philosophy, too! My top tip for this is to not try and tackle the meaning of life, but instead do some philosophy on some sort of system where people interact, perhaps a game or an industry or simmilar. You'll be surprised what you can find sitting down and considering the matter deeply. If this interests you, then I'll soon post a simmilar anecdote to the one above on investigation of known quanitities.)