Caleb Stanford

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

math topology

New posts every Sunday!

Only if I didn’t post earlier in the week.

Even if I have urgent deadlines?!

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.


life-goals sleep this-blog

Life goal: Don’t pull any more all-nighters for work.

Sleep has always been a sort of ongoing personal experiment for me. In high school I wanted to try out a 28-hour sleep schedule (see 1 2 3), but never succeeded. In college, I like to believe I perfected the art of staying up all night. Specifically, I mean staying up all night and then timing my bed-time perfectly to set myself in course for an earlier sleep schedule than I had before — a schedule I desperately wanted, because I was always missing class and other events due to my inability to get up. In semesters with 9am classes and noon classes alike, I always missed a lot of class and sleep was usually the reason. It was so hard for me to get up in the morning that I found it easier — often much easier — to stay up all night (with plenty of food and caffeine, and other tactics) rather than have to go to bed and try to get myself up in the morning. If this doesn’t strike you as absurd, it should! Of course, these all-nighters were often more or less forced by the work I had to do, but they were also usually exciting and energizing. I have always loved the novelty of doing a project or activity late into the night, until the sun rises and longer.

Since then, my sleep habits have slowly gotten less maintainable, and I don’t feel energized to pull all-nighters the way I used to. But at the same time, I still find myself caught in late sleep schedules, unable to get up in the morning, and unable to power through a full all-nighter to reset things. Staying up all night for work has, as a result, been considerably more stressful and miserable. After this semester, I finally decided to see a doctor about my sleep patterns.

It seems that I have Delayed Sleep Phase Disorder. The strategy for correcting (or just investigating) this starts by controlling for all possible negative influences (caffeine, inconsistent sleep, lack of association between bed and sleep, electronic screen time, and so on). In the absence of these factors, you try to obtain a “normal” 24-hour schedule, and you push wake-up time back slowly over a period of weeks. Finally, you start to experiment with which factors have the biggest effect by reintroducing them if necessary. Since classes are over and there are no deadlines, this is the perfect time to be working on sleep, and it’s an exciting project.

Perhaps in a few months I will be back to my usual patterns. That’s OK — it may be how I am happiest. Staying up all night for fun or with the right work drive can be really wonderful. But what I can’t do, and am not going to let myself do, is fail in planning my time so that staying up all night becomes the only possibility, instead of a decision. So, I’ve made that a life goal.

I want to start writing in my blog more. I’m still a bit undecided on whether this is really a better idea than posting to Facebook. Especially since Facebook posts would probably get 10 to 100 times more views. But it’s a different kind of outlet, with different expectations and norms, and I want to try out using it consistently. I think it may have the possibility of being more free-form and more honest. When I post to Facebook I think too much about what my readers will think.

So my goal with the blog is to write down whatever I’ve been thinking about. And I also want to write more. I don’t want it to be polished, I just want to work on my writing and express my thoughts on things. Hopefully, by focusing on quantity instead of quality, I will start writing every week instead of every couple of months, and over time the quality will get better.

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