Bottom Up, 0: Motivation

What is there? What is reality? How can we be so sure that what we know is true? How do we know if truth is even a valid concept? What does "valid" mean? What is it to "mean" something? "Is"?

I hope I haven't scared you off yet. I don't plan on going into many philosophical debates.

Still, one wonders, how can one know for sure? Is it reasonable to assume anything? Are our current reasoning methods always appropriate? How does one reconcile math and science -- two very different approaches to problem-solving, considered by many to be equally valid. Even within math, there are N ways of solving the same problem. Conflicting axioms, and rules. Different geometries...

Mathematical rigor has proven to be very useful in discovering the innate properties of logic and arithmetic and geometry and so forth, but how can we always be sure that some theorem is correct? How can we know if the proof is correct?

We can generally trust that, after centuries of standing uncorrected, a certain theorem shall remain so. This can be good and bad -- a theorem that is correct should stay correct, but a theorem that has even a minor technical flaw should be debugged. Theorems using outdated terminology or making assumptions that later turn out to be unwarranted ought to be revised. In fact, every now and then this is exactly what happens. Some PhD student finds an error in some ancient (by my standards) proof, and fixes it, and poof, instant degree.

Now, this is all well and good. Progress in math is good for everyone. But what about me? And what about you? How do we get access to all of this information? More importantly, how do we make it efficient and educational? How likely are you to learn a lot from someone else's dissertation? Further, how can you be sure that your foundations, your axioms, are really all that necessary, without tinkering with them yourself?

For this reason, I have decided to commence a little side-side-project, Bottom Up. Through it, I will attempt to define small pieces of math starting from absolutely nothing.

This is not something that's never been done before. In fact, it's been done many, many times, and I certainly won't do a great job at it. Why should I do it, then?

There are two reasons. Firstly, I believe I can learn a great deal from this project. As a student, learning is a big deal for me. I think this project will give me further insight into how or why math has turned out the way it is today, or, failing that, how it could be instead. Coming from a computer science background, I think this will also give me insight into certain aspects of functional programming. (And of course, I'll eventually want to define computation, and that's exciting.)

As I said before, there were two reasons to start this project, and the second reason is, you. By formally declaring this to be a project, I'm giving you the chance to participate, to learn from, to contribute, to discuss or to disparage, and that, in turn, benefits me and everyone else. If you're interested in participating in Bottom Up with some articles of your own, do not hesitate to contact me.

Of course, Blogger is hardly the ideal technology to start such a project. Blogger is great for blogging, not for proving theorems. Therefore, I don't plan on staying on Blogger. Once I feel that Bottom Up is large enough that it warrants a step up to, say, a wiki, I shall migrate.

So, as of today, I declare Bottom Up to have commenced.