Caleb Stanford Blog

On the practically true but theoretically false

math research

The round table theorem says that for all \(n\), if you can fit \(n\) people around a table, then you can fit \(n+1\) 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 \(n\). 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 \(n\) 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.

Edited on .