In my long post, Proofs without dry bones, I discussed the Monk Theorem (my name) in the context of my ideas about rigorous proof. Here, I want to amplify some of my remarks in the post.

This post was stimulated by Mark Turner’s new book on conceptual blending. That book has many examples of conceptual blending, including the monk theorem, that go into deep detail about how they work. I highly recommend reading his analysis of the monk theorem. Note: I haven’t finished reading the book.

The Monk Theorem

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the original monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

The proof

One of the points in Proofs without dry bones was that the proof above is a genuine mathematical proof, in spite of the fact that it uses no recognizable math theorems or math objects. It does contain unspoken assumptions, but so does any math proof. Some of the assumptions:

• A path has the property that if two people, one at each end, start walking to the opposite end, they will meet each other at a certain time.
• A day is a period of time which contains a time “dawn” and a later time “dusk”.

From a mathematician’s point of view, the words “people”, “walking”, “meet”, “path”, “day”, “dawn” and “dusk” could be arbitrary names having the properties stipulated by the assumptions. This is typical mathematical behavior. “Time” is assumed to behave as we commonly perceive it.

If you think closely about the proof, you will probably come up with some refinements that are necessary to reveal other hidden assumptions (particularly about time). That is also typical mathematical behavior. (Remember Hilbert refining Euclid’s postulates about geometry after thousands of year of people not noticing the enthymemes in the postulates.)

This proof does not require that walking on the path be modeled by a function \[t\mapsto (x,y,z):\mathbb{R}\to\mathbb{R}\times\mathbb{R}\times\mathbb{R}\] followed by an appeal to the intermediate value theorem, which I mentioned in “Proofs without dry bones”.

You could simply proceed to make your assumptions about “path”, “meet”, “time”, and so on more explicit until you (or the mathematician you are arguing with) is satisfied. It is in that sense that I claim the proof given above is a genuine mathematical proof.

References

• The Origin of Ideas: Blending, Creativity, and the Human Spark, by Mark Turner. Oxford University Press, 2014.
• The Way We Think: Conceptual Blending And The Mind’s Hidden Complexities, by Giles Fauconnier and Mark Turner. Basic Books, 2003.
• Proofs without dry bones. Blog post.
• The rigorous view: inertness. Article on abstractmath.org.
• Conceptual blending in Wikipedia.