Caleb Stanford

Under construction


Presently this blog and site (that is, are “under construction”. What I mean by that is that I’m thinking about the following questions:

  • What is the best way to organize personal thoughts, content, and projects online?

    Blogs are limited in scope because they are sorted chronologically. Most pages reflect thoughts at a particular time and may not stay relevant. Pages are not sorted into a natural hierarchy (by topic?), which limits their utility, and tags are only a limited way to get around this.

  • How can I effectively separate my UPenn page from this one, without overlap?

    I had intended keep my UPenn page minimal. Now, I think it should contain more. For now, I am copying some pages from here to there, but they are still present over here…

  • What do I want out of my personal website?

    I’ve used this as a place to publish and organize my thoughts. But I also like getting comments on my thoughts, and starting online discussions. That’s never going to happen here. I think personal websites may be better suited for sharing projects, downloadable materials, and notes — not for open dialog.

On the practically true but theoretically false

math philosophy

The round table theorem says that for all , if you can fit people around a table, then you can fit people around that table. I learned it at Mathcamp a long time ago. As far as I can tell, it applies to all tables and not just round ones.

Let me ignore the ambiguity in the definition of “fit” and suppose that there is some upper bound on how many people can actually fit around a table comfortably. (Otherwise, it’s really an instance of the Sorites paradox.) In this case, the theorem is certainly false for some . Yet, whenever the situation arises that one more person wants to fit into a table, everyone can always make room (I have yet to see a counterexample). So the for which it is false does not arise in practice. :)

Therefore, this is an example of a statement that is practically true (i.e. true in the situations which actually arise and fit the hypothesis of the theorem), but logically false.

Similar statements occur sometimes in computer science, where a practical success is not technically true theoretically (e.g., reasonable running time on practical inputs despite a theoretical worst-case exponential running time). I am suspicious of such statements. They should be restated in such a way that they are true not just practically, but also theoretically. This way, we have a more accurate mathematical model, which validates the practical result rather than being disjoint from it. And then we know why, instead of just having practical success for an unknown reason.

3 facts about the 4-color theorem


Here are some facts about the four color theorem.

1. There is always a way to color the outer region as well.

Take a look at this 4-coloring of the US map, from the Wikipedia page:

4-colored US map

It’s technically suboptimal because white is a color, too, and they used white for the outside! For some reason, this bothers me. It’s in fact always possible to color the entire map, including the outer region, with 4 colors. Here, I did it in Pinta image editor to show you:

4-colored US map including outer region

(Note that Utah / New Mexico and Arizona / Colorado must be considered non-adjacent for the 4-color theorem to apply.)

2. The regions can have holes!

The 4-color theorem is valid even if not all of the regions are topologically disks; they can have holes. In fact, point #1 was already an example of this, since the outer region has one big hole. But it’s true more generally. However, the regions certainly can’t be “adjacent” to themselves, and also, they must be contiguous regions. Which brings us to:

3. The theorem doesn’t technically apply to real maps, including the US map.

Why not? Because real maps have countries and states that are split into multiple non-contiguous parts. In the US, that’s just the state of Michigan, but there are probably much worse examples world-wide. This blog post discusses it in detail and concludes that, as of July 2016, the globe still happens to be 4-colorable. (I guess that means 5-colorable if you include the color of oceans / water.)

Anyway I would def buy a globe that was 5-colored where blue is reserved for all bodies of water and countries which have multiple pieces are colored the same. That would be a cool thing to have.

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