All posts by SixWingedSeraph

Charles Wells. Professor Emeritus at Case Western Reserve University. Now living in Minneapolis, Minnesota, USA. CWRU provides support for this blog with software and library privileges.
exposition, language, language of math, math, Mathematica, understanding math

Case Study in Exposition: Secant

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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:

 

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exposition, Handbook, language, language of math, Uncategorized, understanding math

Etymology

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Retire

I was recently asked about the etymology of the English word “retire”(in connection with quitting work).  It comes from Old French “retirer”, compounded from “re” (meaning “back”, a prefix used in Latin) and the Old French verb “tirer” meaning something like “pull” (which comes from a Germanic language, not Latin, and is related to “tier”, but not apparently to “tire”).

Its earliest citations in the Oxford English Dictionary show meanings such as

  • Pull back or retreat from the enemy.
  • To move back for safety or storage (“they retired to their houses”).
  • Leave office or work permanently.

All these meanings appear in print in the 16th century.

What good does it do to know this?  Not much.  You can’t explain the modern meaning of a word knowing the meaning of its ancient roots.

In the case of “retire”, I can make up a story of meanings changing using a chain of metaphors.

  1. “Retirer” in French meant literally “pull back” in the physical sense, for example pulling on a dog’s leash to drag it back so it won’t get into a fight with another dog. This literal meaning has not survived in the English word “retire” (nor, I think, in the French word “retirer”).
  2. In the 12th century (sez the OED without citation) the French word was used to refer to an army pulling back from a battle.  This is clearly a metaphor based on the literal meaning.  In a phrase such as “The Army retired from battle” it has become intransitive, but perhaps people once said things like “The General retired the Army from battle”.  Note that in modern English we could use the exact same metaphor with “pull back”: “The General pulled the Army back from battle”, although “withdrew” would be more common.
  3. Now someone comes along and uses the metaphor “going to work is like being in a battle”, and says things like “He retired from his job”.   This happened in English before 1533 and the usage has survived to this day.  It is probably the commonest meaning of the word “retire” now.

Now all that is a story I made up.  It is plausible, but it might have happened in a different way.  It is not at all likely we will discover the workings of metaphors in the minds of people who lived 600 years ago.  (Conceivably someone could have written down their thoughts about the word “retire” and it will be discovered in an odd subcrypt of Durham Cathedral and some linguist would get very excited, but I could win the lottery, too).

That’s why knowing the original literal meaning of the roots of a modern English word really means nothing about the modern meaning.  There could have been many steps along the way where a metaphorical usage became the standard meaning, then someone took the standard meaning and used it in another metaphor, maybe many times.  And metaphors aren’t the only method.  Words can change meaning because of misunderstanding, specialization, generalization, use in secret languages that become public, and so on.

I didn’t include etymology in the Handbook, mainly for this reason.  But there are certain mathematical words where knowing the metaphor or even the literal meaning can be of help.  I’ll write about that in a separate article.

 

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exposition, math, Mathematica, Uncategorized, understanding math

Some demos of families of functions

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I have posted on abstractmath.org a CDF file of families of functions whose parameters you can control interactively. It is fascinating to play with them and see phenomena you (or at least I) did not anticipate.  Some of them have questions of the sorts you might ask students to discuss or work out.  Working out explanations for many of the phenomena demand some algebra skills, and sometimes more than that.

The Mathematica command that sets up one of the families looks like this:

Manipulate[
Plot[{Sin[a x], a Cos[a x]}, {x, -2 Pi, 2 Pi},
PlotRange -> {{-4, 4}, {-4, 4}}, PlotStyle -> {Blue, Red},
AspectRatio -> 1], {{a, 1}, -4, 4, Appearance -> “Labeled”}]

It would be straightforward to make a command something like

PlotFamily[functionlist, domain, plotrange]

with various options for colors, aspect ratio and so on that would do these graphs.  But I found it much to easy to simply cut and paste and put in the new inputs and parameters as needed.

This sort of Mathematica programming is not hard if you have an example to copy, but you do need to get over the initial hump of learning the basic syntax.   I know of no other language where it would be as easy as the example above to produce an interactive plot of a family of functions.

But many people simply hate to learn a new language.  If this sort of interactive example turns out to be worthwhile, someone could design an interface that would allow you to fill in the blanks and have the command constructed for you.  (I could say the same about some of other cdf files I have posted on this blog recently.) But that someone won’t be me.  I have too much fun coming up with new ideas for math  exposition to have to spend time working out all the details.  And all my little experiments are available to use under the Creative Commons License.

I would appreciate comments and suggestions.

 

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math, understanding math

Prechunking

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The emerging theory of how the brain works gives us a new language to us for discussing how we teach, learn and communicate math.

Modules

Our minds have many functionalities.  They are implemented by what I called modules in Math and modules of the mind because I don’t understand very much about what cognitive scientists have learned about how these functionalities are carried out.  They talk about a particular neuron, a collection of neurons, electrical charges flowing back and forth, and so on, and it appears there is no complete agreement about these ideas.

The functions the modules implement are physical structures or activities in the brain.  At a certain level of abstraction we can ignore the mechanism.

Most modules carry out functionalities that are hidden from our consciousness.

  • When we walk, the walking is carried out by a module that operates without our paying (much) attention to it.
  • When we recognize someone, the identity of the person pops into our consciousness without us knowing how it got there.  Indeed, we cannot introspect to see how the process was carried out; it is completely hidden.

Reasoning, for example if you add 56 and 49 in your head, has part of the process visible to your introspection, but not all of it.  It uses modules such as the sum of 9 and 6 which feel like random access memory.  When you carry the addition out, you (or at least I) are conscious of the carry: you are aware of it and aware of adding it to 9 to get 10.

Good places to find detailed discussion of this hiddenness are references [2] and [4] below.

Chunking

Math ed people have talked for years about the technique of chunking in doing math.

  • You see an algebraic expression, you worry about how it might be undefined, you gray out all of it except the denominator and inspect that, and so on.  (This should be the subject of a Mathematica demo.)
  • You look at a diagram in the category of topological spaces.  Each object in the diagram stands for a whole, even uncountably infinite, space with lots of open and closed subsets and so on, but you think of it just as a little pinpoint in the diagram to discover facts about its relationship with other spaces.  You don’t look inside the space unless you have to to verify something.

Students have a hard time doing that.  When an experienced mathematician does this, they are very likely to chunk subconsciously; they don’t think, “Now I am chunking”.  Nevertheless, you can call it to their attention and they will be aware of the process.

There are modules that perform chunking whose operation you cannot be aware of even if you think about it.  Here are two examples.

Example 1. Consider these two sentences from [2], p. 137:

  • “I splashed next to the bank.”
  • “There was a run on the bank.”

When you read the first one you visualize a river bank.  When you read the second one you visualize a bank as an institution that handles money.  If these two sentences were separated by a couple of paragraphs, or even a few words, in a text you are likely not to notice that you have processed the same word in two different ways.  (When they are together as above it is kind of blatant.)

The point is the when you read each sentence your brain directly presents you with the proper image in each case (different ones as appropriate).  You cannot recover the process that did that (by introspection, anyway).

Example 2. I discussed the sentence below in the Handbook.  The sentence appears in references [3].

…Richard Darst and Gerald Taylor investigated the
differentiability of functions f^p (which for our
purposes we will restrict to (0,1)) defined for
each p\geq1 by

In this sentence, the identical syntax (a,b) appears twice; the first occurrence refers to the open interval from 0 to 1 and the second refers to the GCD of integers m and n.  When I first inserted it into the Handbook’s citation list, I did not notice that (I was using it for another phenomenon, although now I have forgotten what it was).  Later I noticed it.  My mind preprocessed the two occurrences of the syntax and threw up two different meanings without my noticing it.

Of course, “restricting to (0, 1)” doesn’t make sense if (0, 1) means the GCD of 0 and 1, and saying “(m, n) = 1” doesn’t make sense if (m, n) is an interval.  This preprocessing no doubted came to its two different conclusions based on such clues, but I claim that this preprocessing operated at a much deeper level of the brain than the preprocessing that results in your thinking (for example) of a topological space as a single unstructured object in a category.

This phenomenon could be called prechunking.  It is clearly a different phenomenon that zooming in on a denominator and then zooming out on the whole expression as I described in [1].

This century’s metaphor

In the nineteenth century we came up with a machine metaphor for how we think.  In the twentieth century the big metaphor was our brain is a computer.  This century’s metaphor is that of a bunch a processes in our brain and in our body all working simultaneously, mostly out of our awareness, to enable us to live our life, learn things, and just as important (as Davidson [4] points out) to unlearn things.  But don’t think we have Finally Discovered The Last Metaphor.

References

  1. Zooming and chunking in abstractmath.org.
  2. Mark Changizi, The vision revolution.  Benbella Books, 2009.
  3. Mark Frantz, “Two functions whose powers make fractals”.  American Mathematical Monthly, v 105, pp 609–617 (1998).
  4. Cathy N. Davidson, Now you see it.  Viking Penguin, 2011.  Chapters 1 and 2.
  5. Math and modules of the mind (previous post).
  6. Cognitive science in Wikipedia.
  7. Charles Wells, The handbook of mathematical discourse, Infinity Publishing Company, 2003.
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exposition, language of math, math, understanding math

Unless

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Mark Meckes recently wrote (private communication):

I’m teaching a fairly new transition course at Case this term, which involves explicitly teaching students the basics of mathematical English along with the obvious things like logic and proof techniques.  I had a student recently ask about how to interpret “A unless B”.  After a fairly lively discussion in class today, we couldn’t agree on the truth table for this statement, and concluded in the end that “unless” is best avoided in mathematical writing.  I checked the Handbook of Mathematical Discourse to see if you had anything to say about it there, but there isn’t an entry for it.  So, are you aware of a standard interpretation of “unless” in mathematical English?

I did not consider  “unless” while writing HMD.   What should be done to approach a subject like this is to

  • think up examples  (preferably in a bull session with other mathematicians) and try to understand what they mean logically, then
  • do an extensive research of the mathematical literature to see if you can find examples that do and do not correspond  with your tentative understanding.  (Usually you find other uses besides the one you thought of, and sometimes you will discover that what you came up with is completely wrong.)  

What follows is an example of this process.

I can think of three possible meanings for “P unless Q”:

1.  “P if and only if not Q”,
2.  “not Q implies P”
3.  “not P implies Q”.

An example that satisfies (1) is “x^2-x is positive unless 0 \leq x \leq 1“.  I have said that specific thing to my classes — calculus students tend not to remember that the parabola is below the line y=x on that interval. (And that’s the way you should show them — draw a picture, don’t merely lecture.  Indeed, make them draw a picture.)

An example of (2) that is not an example of (1) is “x^2-x is positive unless x = 1/2“.  I don’t think anyone would say that, but they might say “x^2-x is positive unless, for example, x = 1/2“.  I would say that is a correct statement in mathematical English.  I guess the phrase “for example” translates into telling you that this is a statement of form “Q implies not P”, where Q is now “x = 1/2”.   Using the contrapositive, that is equivalent to “P implies not Q”, but that is neither (2) nor (3).

An example of (3) that is not an example of (1) is “x^2-x is positive unless -1 < x < 1“.  I think that any who said that (among math people) would be told that they are wrong, because for example (\frac{-1}{2})^2-\frac{-1}{2} = \frac{3}{4}.  That reaction amounts to saying that (3) is not a correct interpretation of “P unless Q”.

Because of examples like these, my conjecture is that “P unless Q” means “P if and only if not Q”.  But to settle this point requires searching for “unless” in the math literature and seeing if you can find instances where “P unless Q” is not equivalent to “P if and only if not Q”.  (You could also see what happens with searching for “unless” and “example” close together.)

Having a discussion such as the above where you think up examples can give you a clue, but you really need to search the literature.  What I did with the Handbook is to search JStor, available online at Case.  I have to say I had definite opinions about several usages that were overturned during the literature search. (What “brackets” means is an example.)

My proxy server at Case isn’t working right now but when I get it repaired I will look into this question.

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category theory, representations, Sketches and forms

Turning definitions into mathematical objects

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When G&G was moved to this current location, most of the links were trashed, so I have been repairing them a bit at a time. There are still some broken links from 2009 and before but I am working on them.  Honest.

G&G contained a series of posts about turning definitions into mathematical objects, mostly written in 2009. Not only were their links broken (and they used many links to each other), but two of the articles were trashed.  I have now removed them from this website. They are all still at the old website: http://sixwingedseraph.wordpress.com/ and as far as I know all the links to each other work.

When I have time I will combine them into one long article.  Until then, the old website will remain.

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category theory, math, Mathematica, Sketches and forms, understanding math

Showing categorical diagrams in 3D

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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 is the Mathematica Notebook algebra1.nb, which is 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.

In Graph-Based Logic and Sketches, Atish Bagchi and I needed to construct a lot of cones based on fairly complicated diagrams. We generally show the base diagram and left the reader to imagine the cone. This post is an experiment in presenting such a diagram in 3D, with its cone and other constructions based on it.

To understand this post, you need a basic understanding of categories, functors and limit cones (see References).

The notebook and CDF files that generate this display may be downloaded from here:

These files may be used and modified as you wish according to the Creative Commons rule listed under “Permissions” (at the top of the window).

The sketch for categories: composition

A finite-limit sketch (FL sketch) is a category with finite limits given by specifying certain  nodes and arrows, commutative diagrams using these nodes and arrows, and limit cones based on diagrams using the given nodes and arrows.  A model of an FL sketch is a finite-limit-preserving functor from the FL sketch into some category \mathcal{C}.  Detailed descriptions of FL-sketches are  in References [1], [2] and [3] (below).

Categories themselves may be sketched by FL-sketches. Here I will present the part of the sketch that constructs (in a model) the object of composites of two arrows.  This is the specification for composite:

  1. The composite of two arrows f:A\to B and g:B'\to C is defined if and only if B=B'.
  2. The composite is denoted by gf.
  3. The domain of gf is A and the codomain is C.

We start with a diagram in the FL sketch for categories that gives the data corresponding to two arrows that may be composed.  This diagram involves nodes ob and ar, which in a model become the object of objects and the object of arrows of the category object in \mathcal{C}.  (Suppose \mathcal{C} is the category of sets; then the model is simply a small category.  The node ob goes to the set of objects of the small category and ar goes to the set of arrows.)  The arrows labeled dom and cod take (in a model) an arrow to its domain and codomain respectively. Here is the diagram:

You can move the diagram around in three dimensions to see it from different perspectives. (Of course it isn’t really in three dimensions. Your eyes-to-brain module reconstructs the illusion of three dimensions when you twirl the diagram around.)

Note that this is a diagram, not a directed graph (digraph). (In the paper, Atish and I, like most category theorists, say “graph” instead of “digraph”.) It has an underlying digraph (see Chapter 2 of Graph-Based Logic and Sketches), but the labeling of several different nodes of the underlying digraph by the name of the same node of the sketch is meaningful. 

Here, the key fact is that in the diagram there are two arrows, one labeled dom and the other cod, to the same node labeled ob, and two other arrows to two different nodes labeled ob. 

Now click c1.

This shows a cone over the diagram.  One of the nodes in the sketch must be cp (in other words given beforehand; that is, we are specifying not only that the blue stuff is a limit cone but that the limit is the node cp.)   In a model, this cone must become a limit cone.  It follows from the properties of limits that the elements of cp in the model in Sets are pairs of arrows with the property that one has a codomain that is the same as the domain of the other.  The label “cp” stands for “compatible pairs”.

Now click c2.

The green stuff is a diagram showing two arrows from the node labeled ar to the left and right nodes labeled ob in the original black diagram.  This is not a cone; it is just a diagram.  In a model, any arrow in the vertex must have domain the same as the domain of one of the arrows in the compatible pair, and codomain the same as the codomain of the other arrow of the pair.  Thus in the model, an arrow living in the set labeled with “ar” in green must satisfy requirement 3 in the specification for composition given above.

Note that the requirement that the green diagram be commutative in a model is vacuous, so it doesn’t matter whether we specify it specifically as a diagram in the sketch or not.

Now click c3.

The arrow labeled comp must be specified as an arrow in the sketch.  We want its value to be the composite of an element of cp in a model, in other words a compatible pair of arrows.  At this point that will not necessarily be true.  But all can be saved:

Now click c4.

We must specify that the diagram given by the thick arrows must be a diagram of the sketch.  The fact that it must become commutative in a model means exactly that the red arrow comp from cp to ar takes a compatible pair to an arrow that satisfies requirements 1–3 of the specification of composite shown above.

References

  1. Peter T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Volume 2 (Oxford Logic Guides 44), by Oxford University Press, ISBN 978-0198524960.
  2. Michael Barr and Charles Wells, Category theory for computing science (1999).    (This is the easiest to start with but it doesn’t get very far.)
  3. Michael Barr and Charles Wells, Toposes, Triples and Theories (2005).  Reprints in Theory and Applications of Categories 1.

 
\mathcal{C}

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Mathematica

Modification of “Picturing derivatives”

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There have been problems with viewing the CDF file in my post “Picturing derivatives” (see the post “Problems with picturing derivatives” for details.) I have replaced the embedding code


WolframCDF source="http://www.abstractmath.org/Mathematica/New5thDegreePolynomial.cdf" CDFwidth="531" CDFheight="620" altimage="http://www.abstractmath.org/Mathematica/New5thDegreePolynomial.pdf"

with the Javascript code
var cdf = new cdfplugin();
cdf.embed('http://www.abstractmath.org/Mathematica/New5thDegreePolynomial.cdf', 531, 690);
cdf.setDefaultContent('http://www.abstractmath.org/Mathematica/New5thDegreePolynomial.pdf');

The first version uses the CDF Plugin for WordPress provided by Wolfram. The second is a javascript-enabled code suggested by Wolfram’s document on how to embed CDF files. About this code they said

For greater flexibility we recommend using the Wolfram CDF Embed Script, which is a free open source JavaScript library. It requires no other libraries and guarantees cross-browser compatibility, provides a way to check that the CDF plugin is installed, displays a CDF Player logo and link when the plugin is missing, and offers a means to display static images in place of the interactive content.

Well it certainly doesn’t guarantee browser compatibility.

The new version works on my copy of IE8 running on an XP computer and on Mozilla Firefox. The bottom is cut off on Chrome. (This behavior was reported for Firefox as well by someone else running IE8 on an XP computer.) This is exactly the behavior I had with the previous code.

I would like to hear from readers about problems with the new version. I will experiment with changing the CDF file.

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Uncategorized

Welcome to the new site

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Welcome to the new site for Gyre&Gimble.  It now resides on my own website, abstractmath.org.  This will enable me to post interactive documents using Mathematica CDF files, and gives me more control in other ways as well.

Warning:  The links to other G&G posts in pre-2010 posts are wrong.  They still take you to the old site.  The old site will remain available until I can repair this problem.

If you are using an RSS feed, you need to change it (see  META to the right).  Keep in touch!

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