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Curiosity

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Science Daily recently reported on a new study [1] that shows that intellectual curiosity is a good predictor of academic performance.  A few days ago I published the post Liberal-Artsy people.  Now I know that what I was talking about are people with intellectual curiosity!  In the earlier post, I contrasted them with what I called “B.Sc.” types, who are narrowly focused and are not interested in asides in math class about the connections with some concept and other concepts, stories about the discoverer of the concept, the meaning of the name of the concept, and so on.

So better names would be “IC people” instead of Liberal Artsy people and “NF people” (Narrow Focus people) instead of B.Sc.  This is better terminology because it isn’t the type of undergraduate degree they have that matters but their attitude toward knowledge of the world.

There are things to say about these concepts with respect to research mathematicians.  I have known a good many over the years.  (My advice to young people who want to do math research is: Hang around people who know more than you do.)  My impression is that most of the very best mathematicians are IC people who are interested in all sorts of things, not just their branch of math.

Even so, some of the best mathematicians are narrowly focused.  This has always been the case.  Isaac Newton was evidently IC but Kurt Gödel was apparently NF.  (He had no interest in things outside math.  On the other hand, he did find a new model of general relativity, so he was willing to look at others parts of math besides logic.)

I have known some NF mathematicians.  When I wanted to tell them about something they might say, “I have enough trouble keeping up with my field”.  The ones that I knew were mediocre and rarely published much beyond writing up their dissertation.  I suspect that the famous NF mathematicians were simply brilliant enough to get away with being NF.

Perhaps the sort of NF student whose eyes glaze over when

  • you mention Evariste Galois’s tough and short life, or
  • talk about how group theory can be used to classify crystals, or
  • mention that “tangent” comes from the Latin word for “touching”

are doomed to the same mediocrity.  But undoubtedly some of those NF students will turn out to do great things, and some of the IC students will wind up dilettanting through life and never coming close to achieving their potential.

Don’t prejudge students.

[1] S. von Stumm, B. Hell, T. Chamorro-Premuzic. The Hungry Mind: Intellectual Curiosity Is the Third Pillar of Academic Performance. Perspectives on Psychological Science, 2011; 6 (6): 574 DOI: 10.1177/1745691611421204

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Thinking about abstract math

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The abstraction cliff

In universities in the USA, a math major typically starts with calculus, followed by courses such as linear algebra, discrete math, or a special intro course for math majors (which may be taken simultaneously with calculus), then go on to abstract algebra, analysis, and other courses involving abstraction and proofs.

At this point, too many of them hit a wall; their grades drop and they change majors.  They had been getting good grades in high school and in calculus because they were strong in algebra and geometry, but the sudden increase in abstraction in the newer courses completely baffles them. I believe that one big difficulty is that they can't grasp how to think about abstract mathematical objects.  (See Reference [9] and note [a].)   They have fallen off the abstraction cliff.  We lose too many math majors this way. (Abstractmath.org is my major effort to address the problems math majors have during or after calculus.)

This post is a summary of the way I see how mathematicians and students think about math.  I will use it as a reference in later posts where I will write about how we can communicate these ways of thinking.

Concept Image

In 1981, Tall and Vinner  [5] introduced the notion of the concept image that a person has about a mathematical concept or object.   Their paper's abstract says

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.

The concept image you may have of an abstract object generally contains many kinds of constituents:

  • visual images of the object
  • metaphors connecting the object to other concepts
  • descriptions of the object in mathematical English
  • descriptions and symbols of the object in the symbolic language of math
  • kinetic feelings concerning certain aspects of the object
  • how you calculate parameters of the object
  • how you prove particular statements about the object

This list is incomplete and the items overlap.  I will write in detail about these ideas later.

The name "concept image" is misleading [b]), so when I have written about them, I have called them metaphors or mental representations as well as concept images, for example in [3] and [4].

Abstract mathematical concepts

This is my take on the notion of concept image, which may be different from that of most researchers in math ed. It owes a lot to the ideas of Reuben Hersh [7], [8].

  • An abstract mathematical concept is represented physically in your brain by what I have called "modules" [1] (physical constituents or activities of the brain [c]).
  • The representation generally consists of many modules.  They correspond to the list of constituents of a concept image given above.  There is no assumption that all the modules are "correct".
  • This representation exists in a semi-public network of mathematicians' and students' brains. This network exercises (incomplete) control over your personal representation of the abstract structure by means of conversation with other mathematicians and reading books and papers.  In this sense, an abstract concept is a social object.  (This is the only point of view in the philosophy of math that I know of that contains any scientific content.)

Notes

[a]  Before you object that abstraction isn't the only thing they have trouble with, note that a proof is an abstract mathematical object. The written proof is a representation of the abstract structure of the proof.  Of course, proofs are a special kind of abstract structure that causes special problems for students.

[b] Cognitive science people use "image" to include nonvisual representations, but not everyone does.  Indeed, cognitive scientists use "metaphor" as well with a broader meaning than your high school English teacher.  A metaphor involves the cognitive merging of parts of two concepts (specifically with other parts not merged). See [6].

[c] Note that I am carefully not saying what the modules actually are — neurons, networks of neurons, events in the brain, etc.   From the point of view of teaching and understanding math, it doesn't matter what they are, only that they exist and live in a society where they get modified by memes  (ideas, attitudes, styles physically transmitted from brain to brain by speech, writing, nonverbal communication, appearance, and in other ways).

References

  1. Math and modules of the mind (previous post)
  2. Mathematical Concepts (previous post)
  3. Mental, physical and mathematical representations (previous post)
  4. Images and Metaphors (abstractmath.org)
  5. David Tall and Schlomo Vinner, Concept Image and Concept Definition in Mathematics with particular reference to limits and continuity, Journal Educational Studies in Mathematics, 12 (May, 1981), no. 2, 151–169.
  6. Conceptual metaphor (Wikipedia article).
  7. What is mathematics, really? by Reuben Hersh, Oxford University Press, 1999.  Read online at Questia.
  8. 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005.
  9. Mathematical objects (abstractmath.org).

 

 

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Liberal-artsy people

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I graduated from Oberlin College with a B.A. as a math major and minors in philosophy and English literature, with only three semesters of science courses.  I was and am "liberal-artsy".   As professor of math at Case Western Reserve University,  I had lots of colleagues in both pure and applied math who started out with B.Sc. degrees. We did not always understand each other very well!

Caveat: "Liberal-artsy" and "Narrowly Focused B.Sc. type" (I need a better name) are characteristics that people may have in varying amounts, and many professors in science and math have both characteristics.   I do, myself, although I am more L.A. that B.Sc.  Furthermore, I know nothing about any sociological or cognitive-science research on these characteristics.  I am making it all up as I write.  (This is a blog post, not a tome.)

I recently posted on secants and  tangents.  These articles were deliberately aimed to tickle the interests of L.A.  students.

Liberal-artsy types want to know about connections between concepts.  In each post, I wrote on both common meanings of the words (secant line and function, tangent line and function) and the close connections between them.  Some trig teachers / trig texts tell students about these connections but too many don't.   On the other hand, many B.Sc. types are left cold by such discussions.  B.Sc. types are goal-oriented and want to know a) how do I use it? b) how do I calculate it?  They get impatient when you talk about anything else.  I say point out these connections anyway.

L.A. types want to know about the reason for the name of a concept.  The post on secants refers to the metaphor that "secant" means "cutting". This is based on the etymology of "secant", which is hidden to many students  because it is based on Latin.  The post makes the connection that the "original" definition of "secant" was the length of a certain line segment generated by an angle in the unit circle. The post on tangents makes an analogous connection, and also points out that most tangent lines that students see touch the curve at only a single point, which is not a connotation of the English word "touch".

Many people think they have learned something when they know the etymology of a word.  In fact, the etymology of a word may have little or nothing to do with its current meaning, which may have developed over many centuries of metaphors that become dead, generate new metaphors that become dead, umpteen times, so that the original meaning is lost.  (The word "testimony" cam from a Latin phrase meaning hold your testicles, which is really not related to its meaning in present-day English.)

So I am not convinced that etymologies of names can help much in most cases.  In particular, different mathematical definitions of the same concept can be practically disjoint in terms of the data they use, and there is no one "correct" definition, although there may be only one that motivates the name.  (There often isn't a definition that motivates the name.  Think "group".)  But I do know that when I mention the history of a name of a concept in class, some students are fascinated and ask me questions about it.

L.A. types are often fascinated by ETBell-like stories about the mathematician who came up with a concept, and sometimes the stories illuminate the mathematical idea.  But L. A. types often are interested anyway.  It's funny when you talk about such a thing in class, because some students visibly tune out while others noticeably perk up and start paying attention.

So who should you cater to?  Answer:  Both kinds of students.  (Tell interesting stories, but quickly and in an offhand way.)

The posts on secants and tangents also experimented with using manipulable diagrams to illustrate the ideas.  I expect to write about that more in another post.

For more about the role of definitions, check out the abmath article and also Timothy Gowers' post on definitions (one of a series of excellent posts on working with abstract math).


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Tangents

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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 Tangent Line.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.

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

  1. The line is a straight line through the point.
  2. The curve goes through that point.
  3. The curve is differentiable in a neighborhood of the point.
  4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $ y=x^3-x$ and the tangent is shown in red. You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”. (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else). But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$. With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $ \theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:


This defines the tangent function for $ -\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by \tan\theta=\frac{\sin\theta}{\cos\theta}, which is easily seen by similar triangles to be the same on -\frac{\pi}{2} < x < \frac{\pi}{2}.

We can now see the relationship between the geometric and the $ \frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:


The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $ \frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $ \frac{\sin\theta}{\cos\theta}$ definition.

References

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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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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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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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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, 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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