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 Theorems using outdated terminology or making assumptions that later turn out to be unwarranted ought to be revised.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.

Short: The Swing

It was always worth a shot.

The playground swing, the one swing meant to entertain sixty children for half an hour. If you stayed on it for too long, the supervisor would drag you off and let someone else take over.

I'd never been on this swing before, you see. I tried every single break, and every single break I got to watch the cool children take turns swinging their legs back and forth on their oscillating body, like a pendulum on a pendulum.

Every time I failed to get on the swing, I got more jealous of the cool children, and more determined to one day be just like them. To be the ruler of all the playground, sitting on his sturdy mobile throne.

We had this rule -- whoever sat on the swing had supreme rule. He (and yes, it was always a he) gets to pick teams, even though he doesn't play. If he asks for your chips, you better give them.

It didn't used to be like this. We used to have no swing in the playground. Granted, there were still the cool kids, the girls, and me, but no one was in charge. The grown-ups had pity on our flat playground, and so they imposed the tyranny of the swing. We were just fine without it here, thank you.

Now, it's become an obsession. How can I have fun in my playground knowing that the swing was never used by me? The only feature of this playground, and I never got to try it.

One day, I felt more determined than ever. I could feel it behind my eyes, a sort of burning desire guiding me to achieve my goals, my dreams. I was visualizing how it would be like to be one of the cool kids, swinging madly back and forth and barking out orders like a frantic principal. This was going to be me. I was going to become one with the swing, with blissful highs and turbulent lows. Just watch me.

I approached the swing. I recognized the kid, he took the school bus home with me sometimes. I think his name was Tim or something. He was a little bit taller than me.

I approached the swing, and Tim, and asked of him very softly, "Can I play?"

He did not reply, and my face burned with shame. I had come all this way only to be ignored by the cool kids. I would have to go back to my little corner and reformulate my plan of attack.

Instead of giving up, however, I pressed on, a little louder, and with a little more confidence, "Can I play on the swing, Tim?"

At the mention of his name, his head snapped toward me. It frightened me for a minute, but then he spoke, "Oh, Matt! Hi! What?"

He knew my name! I wasn't expecting somebody like him to know who I was. And it turns out that he just didn't hear me, so I repeated my request, "Can I play on the swing Tim, just for a little bit?"

He smiled at me and said, "Sure! I'm kinda bored with it anyway." And Tim stopped the swing and got off. He even held the swing in place while I tried to sit down on it.

I was on the swing! I had finally achieved my goal! I was a cool kid now!

Something was not quite right, however. I didn't seem to be having quite as much fun as the cool kids were, and Tim was just staring at me blankly. Oh! I had forgotten to swing! In my excitement to be sitting on the swing, I had forgotten to use it!

Now, I wasn't quite used to being on a swing, but I'd watched the other kids do it all the time, so it couldn't be too hard. You just stretch your legs when going forward, and bend them when going back. I tried doing it, but wasn't really getting anywhere. Tim snapped out of his reverie and started pushing me, and soon, I was swinging.

From that day on, Tim became my best friend. Every day, we took turns between swinging and pushing. Soon, Tim's friends joined in, and eventually they too became my friends. The moment I decided to ask if I could play, I became a cool kid. Let that be a lesson to all would-be swingers -- don't be shy.

Lexicographically Speaking: Recloving.

Words are meant to be discovered.

What does "recloving" mean? I wager its meaning to be the repetition of "cloving," perhaps after a failed attempt. What is cloving? Well, it almost sounds like clothing, so cloving might mean a more infantile version of clothing. I prefer a different definition though: cloving is what clovers do.

What do clovers do, then? According to folk legends, if you find a four-leaf clover, you are extremely lucky. Three-lead clovers, which are ten-thousand times more abundant than four-leaf clovers, can still represent that potential for luck. I suggest then that cloving is to wish someone luck, especially if that person is up against immeasurable odds. For example, "I cloved him to win the lottery."

Recloving, then, must mean either to clove someone a second time (which I find to be a somewhat lame definition), or to clove someone after a round of bad luck (which is a definition I love).

So, if you're feeling down, I reclove you to get back up on your own two feet and take to your new life -- the one starting today -- into the realm of the sublime. May your hapinesses be plenty.

If you aren't feeling down, I clove you to find it in yourself to help others who are less fortunate than you. If you believe a good or doable way to achieve this is by writing for New Happinesses, drop me a comment.

Tick Tock

The watch runs backwards.
The poet talks backwards.
The earth spins clockwise.

What difference is there, then,
between then and now?

We know less every day.
Our lives are shorter every year.
The pendulum swings both ways.

Every day is further away from
the day you took me aside,
and declined.