Tag Archives: Mark Turner

math, proof, understanding math

Those monks

Leave a comment

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.
Send to Kindle
exposition, language, language of math, math, Mathematica, understanding math

Case Study in Exposition: Secant

3 Comments

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code comes from several Mathematica notebooks lists in the References. The notebooks are available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:


The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

  • The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
  • You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
  • The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

Secant on circle

Another use of the word “secant” is the red line in this picture:


This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

To the ancient Greeks, a (positive) number was the length of a line segment.

The Definition

The secant of an angle $\theta$ is usually defined as $\frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

In many parts of the world, trig students don’t learn the word “secant”. They simply use $\frac{1}{\cos\theta}$.

This illustrates important facts about definitions:

  • Different equivalent definitions all make the same theorems true.
  • Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $\pi/2$, for example).

The definition $\sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $\sec\theta$.  It also reduces the definition of $ \sec\theta$ to a previously known concept.

It used to be common to give only the $ \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.


References

Mathematica notebooks used in this post:

 

Send to Kindle