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math, proof, understanding math

Those monks

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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.
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category theory, exposition, math, understanding math

More about defining “category”

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In a recent post, I wrote about defining “category” in a way that (I hope) makes it accessible to undergraduate math majors at an early stage.  I have several more things to say about this.

Early intro to categories

The idea is to define a category as a directed graph equipped with an additional structure of composition of paths subject to some axioms.  By giving several small finite examples of categories drawn in that way that gives you an understanding of “category” that has several desirable properties:

  • You get the idea of what a category is in one lecture.
  • With the right choice of examples you get several fine points cleared up:
    • The composition is added structure.
    • A loop doesn’t have to be an identity.
    • Associativity is a genuine requirement —  it is not automatic.
  • You get immediate access to what is by far the most common notation used to work with a category — objects (nodes) and arrows.
  • You don’t have to cope with the difficult chunking required when the first examples given are sets-with-structure and structure-preserving functions.  It’s quite hard to focus on a couple of dots on the paper each representing a group or a topological space and arrows each representing a whole function (not the value of the function!).

Introduce more examples

Then the teacher can go on with the examples that motivated categories in the first place: the big deal categories such as sets, groups and topological spaces.   But they can be introduced using special cases so they don’t require much background.

  • Draw some finite sets and functions between them.  (As an exercise, get the students to find some finite sets and functions that make the picture a category with $f=kh$ as the composite and $f\neq g$.)
  • If the students have had calculus,  introduce them to the category whose objects are real finite nonempty intervals with continuous or differentiable mappings between them.  (Later you can prove that this category is a groupoid!)
  • Find all the groups on a two element set and figure out which maps preserve group multiplication.  (You don’t have to use the word “group” — you can simply show both of them and work out which maps preserve multiplication — and discover isomorphism!.)  This introduces the idea of the arrows being structure-preserving mape. You can get more complicated and use semigroups as well.  If the students know Mathematica you could even do magmas.  Well, maybe not.

All this sounds like a project you could do with high school students.

Large and small

If all this were just a high school (or intro-to-math-for-math-majors) project you wouldn’t have to talk about large vs. small.  However, I have some ideas about approaching this topic.

In the first place, you can define category, or any other mathematical object that might involve a proper class, using the syntactic approach I described in Just-in-time foundations.  You don’t say “A category consists of a set of objects and a set of arrows such that …”.  Instead you say something like “A category $\mathcal{C}$ has objects $A,\,B,\,C\ldots$ such that…”.

This can be understood as meaning “For any $A$, the statement $A$ is an object of  $\mathcal{C}$ is either true or false”, and so on.

This approach is used in the Wikibook on category theory.  (Note: this is a permanent link to the November 28 version of the section defining categories, which is mostly my work.  As always with Wikimedia things it may be entirely different when you read this.)

If I were dictator of the math world (not the same thing as dictator of MathWorld) I would want definitions written in this syntactic style.  The trouble is that mathematicians are now so used to mathematical objects having to be sets-with-structure that wording the definition as I did above may leave them feeling unmoored.  Yet the technique avoids having to mention large vs. small until a problem comes up. (In category theory it sometimes comes up when you want to quantify over all objects.)

The ideas outlined in this subsection could be a project for math majors.  You would have to introduce Russell’s Paradox.  But for an early-on intro to categories you could just use the syntactic wording and avoid large vs. small altogether.

 

http://en.wikibooks.org/w/index.php?title=Category_Theory/Categories&stableid=2221684

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