Caleb Stanford Blog

Note: This post is copied over from an old WordPress blog, with two posts. The posts were moved here on June 23, 2018.

Let \(X\) be a set, and let \(\mathcal{U}\) be a family of subsets of \(X\). Prove that \(\mathcal{U}\) is an ultrafilter on \(X\) if and only if for every partition of \(X\) into three sets \(S_1\), \(S_2\), and \(S_3\), exactly one \(S_i\) lies in \(\mathcal{U}\).

The point, I guess, is that the most fundamental property of ultrafilters is their ability to select one set out of any partition to be “significant” and declare the rest unimportant. This matches the intuition of an ultrafilter being a “perfect locating scheme” (Wikipedia). Except for the very surprising fact that it really only needs to be a perfect scheme; the “locating” part is implied!

Of course, a disadvantage of defining ultrafilters in this way is that I can’t come up with any analogous definition of a filter. Filters, being imperfect locating schemes, need to be explicitly specified to be closed under intersection and so on.

Thanks to Marion Scheepers for the math problem (a variant of what I presented above).

Note: This post is copied over from an old WordPress blog, with two posts. The posts were moved here on June 23, 2018.

I have always wanted to have a blog. At least, since sometime in high school. Several things, however, have stopped me from actually writing one.

The biggest roadblock I always ran into was a tendency to get stuck worrying about the details. What should my blog be called? All of the names I could come up with seemed stupid, pretentious, arbitrary, or all of the above. What can I put in the first post? I had post ideas but they would always seem strange to start with. Basically, I have a tendency to worry too much about whether or not my writing sucks, and that would stop me from actually writing anything.

The other roadblock was less problematic, but it would usually come up a day or two later when I still had not acted at all on my impulse to start blogging. I would think, “hey, is this actually worth my time? Let’s be honest, no one wants to read random shit about my life.” And so I would think, okay, maybe I don’t need to start a blog. And I would forget about it until months or years later when the excitement came up again.

Today, I am happy to announce that I am putting aside (confronting?) both of these roadblocks and starting a blog anyway. Why have I decided to do this?

1. The excited I-want-to-be-a-blogger impulse came again and I don’t think this impulse will ever go away.
2. I am bored and I might as well.
3. Why do we decide to do anything?

Well, okay, what I really wanted to explain was, how am I able to put aside the roadblocks that stopped me so many times before?

With respect to the first roadblock, I realized what should have been obvious from the start: my writing WILL suck, in the way that all impromptu, unrevised writing does. It will probably very often fail to make all the right points, and will feature a lot of embarrassing sentences and stupid ideas. I have to tell myself to explicitly drop all expectations I have for myself, because the only way to write is to just sit down and write. Maybe the actual point of a blog is to have a very low signal-to-noise ratio, and to explore a medium of writing that is more stream-of-consciousness, unplanned, yet more tangible and versatile (and more permanent!) than oral conversation. So: this isn’t meant to be good writing, and I do apologize in advance for everything that I will write that sucks. (Also: using my name as a title for the blog is lame, but again, I’m explicitly dropping the expectation of having a good name for this blog.)

Re: the second roadblock, I actually don’t expect anyone (or at least hardly anyone) to read this. The purpose is more for myself: to have a place to collect thoughts, to organize the ideas that I have had and currently have, and to explore various topics. Presently, I have folders of folders of notes on my computer with various math problems, philosophical ideas, programming projects, and so on–organized so that I can access them, but not available to anyone else. I figure I might as well have a place to talk about these in a way that involves other people (or at least hypothetical other people) and with more versatility in tagging and linking things around the web. And maybe, if I am lucky, some others will read some things I write and there will be some collaboration or discussion. Also, I think this offers more capacity to organize ideas and link back later than, say, Facebook posting (which is currently my main way of sharing ideas publicly).

So, that’s it. I’m actually writing a blog! Hooray.