Those monks

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.

Math majors attacked by cognitive dissonance

In some situations you may have conflicting information from different sources about a subject.   The resulting confusion in your thinking is called cognitive dissonance.

It may happen that a person suffering cognitive dissonance suppresses one of the ways of understanding in order to resolve the conflict.  For example, at a certain stage in learning English, you (small child or non-native-English speaker) may learn a rule that the past tense is made from the present form by adding “-ed”. So you say “bringed” instead of “brought” even though you may have heard people use “brought” many times.  You have suppressed the evidence in favor of the rule.

Some of the ways cognitive dissonance can affect learning math are discussed here

Metaphorical contamination

We think about math objects using metaphors, as we do with most concepts that are not totally concrete.  The metaphors are imperfect, suggesting facts about the objects that may not follow from the definition. This is discussed at length in the section on images and metaphors here.

The real line

Mathematicians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. Between any two points there are uncountably many other points. See density of the reals.

Infinite math objects

One of the most intransigent examples of metaphorical contamination occurs when students think about countably infinite sets. Their metaphor is that a sequence such as the set of natural numbers $\{0,1,2,3,4,\ldots\}$ “goes on forever but never ends”. The metaphor mathematicians have in mind is quite different: The natural numbers constitute the set that contains every natural number right now.

Example

An excruciating example of this is the true statement
$.999\ldots=1.0$.” The notion that it can’t be true comes from thinking of “$0.999\ldots$” as consisting of the list of numbers $0.9,0.99,0.999,0.9999,0.99999,\ldots$ which the student may say “gets closer and closer to $1.0$ but never gets there”.

Now consider the way a mathematician thinks: The numbers are all already there, and they make a set.

The proof that $.999\ldots=1.0$ has several steps. In the list below, I have inserted some remarks in red that indicate areas of abstract math that beginning students have trouble with.

1. The elements of an infinite set are all in it at once. This is the way mathematicians think about infinite sets.
2. By definition, an infinite decimal expansion represents the unique real number that is a limit point of its set of truncations.
3. The problem that occurs with the word “definition” in this case is that a definition appears to be a dictatorial act. The student needs to know why you made this definition. This is not a stupid request. The act can be justified by the way the definition gets along with the algebraic and topological characteristic of the real numbers.

4. It follows from $\epsilon-\delta$ machinations that the limit of the sequence $0.9,0.99,0.999,0.9999,0.99999,\ldots$ is $1.0$
5. That means “$0.999\ldots$” represents $1.0$. (Enclosing a mathematical expression in quotes turns it into a string of characters.)
6. The statement “$A$” represents $B$ is equivalent to the statement $A=B$. (Have you ever heard a teacher point this out?)
7. It follows that that $0.999\ldots=1.0$.

Each one of these steps should be made explicit. Even the Wikipedia article, which is regarded as a well written document, doesn’t make all of the points explicit.

Semantic contamination

Many math objects have names that are ordinary English words.
(See names.) So the person learning about them is faced with two inputs:

• The definition of the word as a math object.
• The meaning and connotations of the word in English.

It is easy and natural to suppress the information given by the definition (or part of it) and rely only on the English meaning. But math does not work that way:

If another source of understanding contradicts the definition
THE DEFINITION WINS.

“Cardinality”

The connotations of a name may fit the concept in some ways and not others. Infinite cardinal numbers are a notorious example of this: there are some ways in which they are like numbers and other in which they are not.

For a finite set, the cardinality of the set is the number of elements in the set. Long ago, mathematicians started talking about the cardinality of an infinite set. They worked out a lot of facts about that, for example:

• The cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers.
• The cardinality of the number of points on the real line is the same as the cardinality of points in the real plane.

The teacher may even say that there are just as many points on the real line as in the real points. And know-it-all math majors will say that to their friends.

Many students will find that totally bizarre. Essentially, what has happened is that the math dictators have taken the phrase “cardinality” to mean what it usually means for finite sets and extend it to infinite sets by using a perfectly consistent (and useful) definition of “cardinality” which has very different properties from the finite case.

That causes a perfect storm of cognitive dissonance.

Math majors must learn to get used to situations like this; they occur in all branches of math. But it is bad behavior to use the phrase “the same number of elements” to non-mathematicians. Indeed, I don’t think you should use the word cardinality in that setting either: you should refer to a “one-to-one correspondence” instead and admit up front that the existence of such a correspondence is quite amazing.

“Series”

Let’s look at the word “series”in more detail. In ordinary English, a series is a bunch of things, one after the other.

• The World Series is a series of up to seven games, coming one after another in time.
• A series of books is not just a bunch of books, but a bunch of books in order.
• In the case of the Harry Potter series the books are meant to be read in order.
• A publisher might publish a series of books on science, named Physics, Chemistry,
Astronomy, Biology,
and so on, that are not meant to be read in order, but the publisher will still list them in order.(What else could they do? See Representing and thinking about sets.)

Infinite series in math

In mathematics an infinite series is an object expressed like this:

$\sum\limits_{k=1}^{\infty }{{{a}_{k}}}$

where the ${{a}_{k}}$ are numbers. It has partial sums

$\sum\limits_{k=1}^{n}{{{a}_{k}}}$

For example, if ${{a}_{k}}$ is defined to be $1/{{k}^{2}}$ for positive integers $k$, then

$\sum\limits_{k=1}^{6}{{{a}_{k}}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}=\frac{\text{5369}}{\text{3600}}=\text{ about }1.49$

This infinite series converges to $\zeta (2)$, which is about $1.65$. (This is not obvious. See the Zeta-function article in Wikipedia.) So this “infinite series” is really an infinite sum. It does not fit the image given by the English word “series”. The English meaning contaminates the mathematical meaning. But the definition wins.

The mathematical word that corresponds to the usual meaning of “series” is “sequence”. For example, $a_k:=1/{{k}^{2}}$ is the infinite sequence $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}\ldots$ It is not an infinite series.

“Only if”

“Only if” is also discussed from a more technical point of view in the article on conditional assertions.

In math English, sentences of the form $P$ only if $Q$” mean exactly the same thing as “If $P$ then $Q$”. The phrase “only if” is rarely used this way in ordinary English discourse.

Sentences of the form “$P$ only if $Q$” about ordinary everyday things generally do not mean the same thing as “If $P$ then $Q$”. That is because in such situations there are considerations of time and causation that do not come up with mathematical objects. Consider “If it is raining, I will carry an umbrella” (seeing the rain will cause me to carry the umbrella) and “It is raining only if I carry an umbrella” (which sounds like my carrying an umbrella will cause it to rain).   When “$P$ only if $Q$” is about math objects,
there is no question of time and causation because math objects are inert and unchanging.

Students sometimes flatly refuse to believe me when I tell them about the mathematical meaning of “only if”.  This is a classic example of semantic contamination.  Two sources of information appear to contradict each other, in this case (1) the professor and (2) a lifetime of intimate experience with the English language.  The information from one of these sources must be rejected or suppressed. It is hardly surprising that many students prefer to suppress the professor’s apparently unnatural and usually unmotivated claims.

These words also cause severe cognitive dissonance

• “If” causes notorious difficulties for beginners and even later. They are discussed in abmath here and here.
• A, an
and the implicitly signal the universal quantifier in certain math usages. They cause a good bit of trouble in the early days of some students.

The following cause more minor cognitive dissonance.

References for semantic contamination

Besides the examples given above, you can find many others in these two works:

• Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms.  Routledge & Kegan Paul.
• Hersh, R. (1997),”Math lingo vs. plain English: Double entendre”. American Mathematical Monthly, vol 104,pages 48-51.

A very early satori that occurs with beginning abstract math students

In the previous post Pattern recognition and me, I wrote about how much I enjoyed sudden flashes of understanding that were caused by my recognizing a pattern (or learning about a pattern). I have had several such, shall we say, Thrills in learning about math and doing research in math. This post is about a very early thrill I had when I first started studying abstract algebra. As is my wont, I will make various pronouncements about what these mean for teaching and understanding math.

Cosets

Early in any undergraduate course involving group theory, you learn about cosets.

1. Every subgroup of a group generates a set of left cosets and a set of right cosets.
2. If $H$ is a subgroup of $G$ and $a$ and $b$ are elements of $G$, then $a$ and $b$ are in the same left coset of $H$ if and only if $a^{-1}b\in H$. They are in the same right coset of $H$ if and only if $ab^{-1}\in H$.
3. Alternative definition: $a$ and $b$ are in the same left coset of $H$ if $a=bh$ for some $h\in H$ and are in the same right coset of $H$ if $a=hb$ for some $h\in H$
4. One of the (left or right) cosets of $H$ is $H$ itself.
5. The relations
$a\underset{L}\sim b$ if and only if $a^{-1}b\in H$

and

$a\underset{R}\sim b$ if and only if $ab^{-1}\in H$

are equivalence relations.

6. It follows from (5) that each of the set of left cosets of $H$ and the set of right cosets of $H$ is a partition of $G$.
7. By definition, $H$ is a normal subgroup of $G$ if the two sets of cosets coincide.
8. The index of a subgroup in a group is the cardinal number of (left or right) cosets the subgroup has.

Elementary proofs in group theory

In the course, you will be asked to prove some of the interrelationships between (2) through (5) using just the definitions of group and subgroup. The teacher assigns these exercises to train the students in the elementary algebra of elements of groups.

Examples:

1. If $a=bh$ for some $h\in H$, then $b=ah’$ for some $h’\in H$. Proof: If $a=bh$, then $ah^{-1}=(bh)h^{-1}=b(hh^{-1})=b$.
2. If $a^{-1}b\in H$, then $b=ah$ for some $h\in H$. Proof: $b=a(a^{-1}b)$.
3. The relation “$\underset{L}\sim$” is transitive. Proof: Let $a^{-1}b\in H$ and $b^{-1}c\in H$. Then $a^{-1}c=a^{-1}bb^{-1}c$ is the product of two elements of $H$ and so is in $H$.
• Which exercises are used depends on what is taken as definition of coset.
• In proving Exercise 2 at the board, the instructor might write “Proof: $b=a(a^{-1}b)$” on the board and the point to the expression “$a^{-1}b$” and say, “$a^{-1}b$ is in $H$!”
• I wrote “$a^{-1}c=a^{-1}bb^{-1}c$” in Exercise 3. That will result in some brave student asking, “How on earth did you think of inserting $bb^{-1}$ like that?” The only reasonable answer is: “This is a trick that often helps in dealing with group elements, so keep it in mind.” See Rabbits.
• That expression “$a^{-1}c=a^{-1}bb^{-1}c$” doesn’t explicitly mention that it uses associativity. That, too, might cause pointing at the board.
• Pointing at the board is one thing you can do in a video presentation that you can’t do in a text. But in watching a video, it is harder to flip back to look at something done earlier. Flipping is easier to do if the video is short.
• The first sentence of the proof of Exercise 3 is, “Let $a^{-1}b\in H$ and $b^{-1}c\in H$.” This uses rewrite according to the definition. One hopes that beginning group theory students already know about rewrite according to the definition. But my experience is that there will be some who don’t automatically do it.
• in beginning abstract math courses, very few teachers
tell students about rewrite according to the definition. Why not?

• An excellent exercise for the students that would require more than short algebraic calculations would be:
• Discuss which of the two definitions of left coset embedded in (2), (3), (5) and (6) is preferable.
• Show in detail how it is equivalent to the other definition.

A theorem

In the undergraduate course, you will almost certainly be asked to prove this theorem:

A subgroup $H$ of index $2$ of a group $G$ is normal in $G$.

Proving the theorem

In trying to prove this, a student may fiddle around with the definition of left and right coset for awhile using elementary manipulations of group elements as illustrated above. Then a lightbulb appears:

In the 1980’s or earlier a well known computer scientist wrote to me that something I had written gave him a satori. I was flattered, but I had to look up “satori”.

If the subgroup has index $2$ then there are two left cosets and two right cosets. One of the left cosets and one of the right cosets must be $H$ itself. In that case the left coset must be the complement of $H$ and so must the right coset. So those two cosets must be the same set! So the $H$ is normal in $G$.

This is one of the earlier cases of sudden pattern recognition that occurs among students of abstract math. Its main attraction for me is that suddenly after a bunch of algebraic calculations (enough to determine that the cosets form a partition) you get the fact that the left cosets are the same as the right cosets by a purely conceptual observation with no computation at all.

This proof raises a question:

Why isn’t this point immediately obvious to students?

I have to admit that it was not immediately obvious to me. However, before I thought about it much someone told me how to do it. So I was denied the Thrill of figuring this out myself. Nevertheless I thought the solution was, shall we say, cute, and so had a little thrill.

A story about how the light bulb appears

In doing exercises like those above, the student has become accustomed to using algebraic manipulation to prove things about groups. They naturally start doing such calculations to prove this theorem. They presevere for awhile…

Scenario I

Some students may be in the habit of abandoning their calculations, getting up to walk around, and trying to find other points of view.

1. They think: What else do I know besides the definitions of cosets?
2. Well, the cosets form a partition of the group.
3. So they draw a picture of two boxes for the left cosets and two boxes for the right cosets, marking one box in each as being the subgroup $H$.
4. If they have a sufficiently clear picture in their head of how a partition behaves, it dawns on them that the other two boxes have to be the same.
• Not many students at the earliest level of abstract math ever take a break and walk around with the intent of having another approach come to mind. Those who do Will Go Far. Teachers should encourage this practice. I need to push this in abstractmath.org.
• In good weather, David Hilbert would stand outside at a shelf doing math or writing it up. Every once in awhile he would stop for awhile and work in his garden. The breaks no doubt helped. So did standing up, I bet. (I don’t remember where I read this.)
• This scenario would take place only if the students have a clear understanding of what a partition is. I suspect that often the first place they see the connection between equivalence relations and partitions is in a hasty introduction at the beginning of a group theory or abstract algebra course, so the understanding has not had long to sink in.

Scenario II

Some students continue to calculate…

1. They might say, suppose $a$ is not in $H$. Then it is in the other left coset, namely $aH$.
2. Now suppose $a$ is not in the “other” right coset, the one that is not $H$. But there are only two right cosets, so $a$ must be in $H$.
3. But that contradicts the first calculation I made, so the only possibility left is that $a$ is in the right coset $Ha$. So $aH\subseteq Ha$.
4. Aha! But then I can use the same argument the other way around, getting $Ha\subseteq aH$.
5. So it must be that $aH=Ha$. Aha! …indeed.
• In step (2), the student is starting a proof by contradiction. Many beginning abstract math students are not savvy enough to do this.
• Step (4) involves recognizing that an argument has a dual. Abstractmath.org does not mention dual arguments and I can’t remember emphasizing the idea to my classes. Tsk.
• Scenario 2 involves the student continuing algebraic calculations till the lightbulb strikes. The lightbulb could also occur in other places in the calculation.

Pattern recognition and me

Recently, I revised the abstractmath.org article on pattern recognition. Doing that that prompted me to write about my own experiences with patterns. Recognizing patterns is something that has always delighted me: it is more of a big deal for me than it does for many other people. That, I believe, is what led me into doing research in math.

I have had several experiences with déjà vu, which is the result of pattern recognition with one pattern hidden. That will be a separate post. I expect to post about my experiences in recognizing patterns in math as well.

Patterns in language

As a teenager I was a page in the Savannah Public Library. There I discovered grammars for many languages. The grammars of other languages are astonishingly different from each other and are full of obscurities that I love to detect. Until I went to college, I was the only person I knew who read grammars for fun.

I am using the word “grammar” in the sense that linguists use it: patterns in our speech and writing, mostly unnoticed, that help express what we want to say)

The word “grammar” is also used to mean rules laid down by the ruling classes about phrases like “between you and I” and the uses of “whom”. Such rules primarily divide the underprivileged from the privileged, and many will disappear when the older members of the privileged class die (but they will think of new ones).

Grammar-induced glee

Russian

I got pretty good at reading and speaking Russian when I was a student (1959-62), but most of it has disappeared. In 1990, we hosted a Russian cello student with the Soviet-American Youth Orchestra for a couple of days. I could hardly say anything to him. One time he noticed one of our cats and said “кошка”, to which I replied “два кошки” (“two cats”). He responded by correcting me: “две кошки”. Then I remembered that the word for “two” in Russian is the only word in the language that distinguishes gender in the plural. I excitedly went around telling people about this until I realized that no one cared.

Spanish

Recently I visited a display about the Maya at the Minnesota Science Museum that had all its posters in English and Spanish. I discovered a past subjunctive in one of the Spanish texts. That was exciting, but I had no one to be excited with.

The preceding paragraph is an example of a Pity Play.

Just the other day our choir learned a piece for Christmas with Spanish words. It had three lines in a row ending in a past subjunctive. (It is in rhyming triples and if you use all first conjugation verbs they rhyme.) Such excitement.

Turkish

During the Cold War, I spent 18 months at İncirlik Air Base in Turkey. Turkish is a wonderful language for us geeks, very complicated yet most everything is regular. Like a computer language.

I didn’t know about computer languages during the Cold War, although they were just beginning to be used. I did work on a “computer” that you programmed by plugging cables into holes in various ways.

In Turkish, to modify a noun by a noun, you add an ending to the second noun. “İş Bankası” (no dot over the i) means “business bank”. (We would say “commercial bank”.) “İş” means “business” and “bank” by itself is “banka”. Do you think this is a lovably odd pattern? Well I do. But that’s the way I am.

A spate of spit

We live a couple blocks from Minnehaha Falls in Minneapolis. Last June the river flooded quite furiously and I went down to photograph it. I thought to my self, the river is in full spate. I wondered if the word “spate” came from the same IE root as the word “spit”. I got all excited and went home and looked it up. (No conclusion –it looks like it might be but there is no citation that proves it). Do you know anyone who gets excited about etymology?

Secret patterns in nature

All around us there are natural patterns that people don’t know about.

Cedars in Kentucky

For many years, we occasionally drove back and forth between Cleveland (where we lived) and Atlanta (where I had many relatives). We often stopped in Kentucky, where Jane grew up. It delighted me to drive by abandoned fields in Kentucky where cedars were colonizing. (They are “red cedars,” which are really junipers, but the name “cedar” is universal in the American midwest.)

What delighted me was that I knew a secret pattern: The presence of cedars means that the soil is over limestone. There is a large region including much of Kentucky and southern Indiana that lies over limestone underneath.

That gives me another secret: When you look closely at limestone blocks in a building in Bloomington, Indiana, you can see fossils. (It is better if the block is not polished, which unfortunately the University of Indiana buildings mostly are.) Not many people care about things like this.

The bump on Georgia

The first piece of pattern recognition that I remember was noticing that some states had “bumps”. This resulted in a confusing conversation with my mother. See Why Georgia has a bump.

Maybe soon I will write about why some states have panhandles, including the New England state that has a tiny panhandle that almost no one knows about.

Minnesota river

We live in Minneapolis now and occasionally drive over the Mendota Bridge, which crosses the Minnesota River. That river is medium sized, although it is a river, unlike Minnehaha Creek. But the Minnesota River Valley is a huge wide valley completely out of proportion with its river. This peculiarity hides a Secret Story that even many Minnesotans don’t know about.

The Minnesota River starts in western Minnesota and flows south and east until it runs into the Mississippi River. The source of the Red River is a few miles north of the source of the Minnesota. It flows north, becoming the boundary with North Dakota and going by Fargo and through Winnipeg and then flows into Lake Winnipeg. Thousands of years ago, all of the Red River was part of the Minnesota River and flowed south, bringing huge amounts of meltwater from the glaciers. That is what made the big valley. Eventually the glaciers receded far enough that the northern part of the river changed direction and started flowing north, leaving the Minnesota River a respectable medium sized river in a giant valley.

The Mendota Bridge is also one of the few places in the area where you can see the skyscrapers of Minneapolis and of St Paul simultaneously.

Music

Baroque music

I love baroque music because of patterns such as fugues, which I understood, and the harmony it uses, which I still don’t understand. When I was 10 years old I had already detected its different harmony and asked my music teacher about it. She waved her hands and declaimed, “I don’t understand Bach.” (She was given to proclamations. Once she said, “I am never going out of the State of Georgia again because in Virginia they put mayonnaise on their hamburgers!”)

Some baroque music uses a ground bass, which floored me when I first heard it. I went on a rampage looking for records of chaconnes and passacaglias. Then I discovered early rock music (Beatles, Doors) and figured out that they sometimes used a ground bass too. That is one of the major attractions of rock music for me, along with its patterns of harmony.

Shape note music

Some shape note tunes (for example, Villulia), as well as some early rock music, has a funny hollow sound that sounds Asian to me. I delight in secretly knowing why: They use parallel fifths.

The Beatles have one song (I have forgotten which) that had a tune which in one place had three or four beats in a row that were sung on the same pitch — except once, when the (third I think) beat was raised a fourth. I fell in love with that and excitedly pointed it out to people. They looked at me funny. Later on, I found several shape note tunes that have that same pattern.

Sets

I have been working my way through abstractmath.org, revising the articles and turning them into pure HTML so they will be easier to update. In some cases I am making substantial revisions. In particular, many of the articles need a more modern point of view.

The math community’s understanding of sets and structures has changed because of category theory and will change
because of homotopy type theory.

This post considers some issues and possibilities concerning the chapter on sets.

The references listed at the end of the article include several about homotopy type theory. They provide different viewpoints and require different levels of sophistication.

A specification of the concept of set

The abmath article Specification of sets specifies what a set is in this way:

A set is a single math object distinct from but completely determined by what its elements are.

I have used this specification for sets since the eighties, first in my Discrete Math lecture notes and then in abstractmath.org. It has proved useful because it is quite simple and the statement implies lots of immediate consequences. Each of the first four consequences in this list below exposes a confusion that some students have.

Consequences of the specification

1. A set is a math object. It has the same status as the number “$143$” and the sine function and the real line: they are all objects of math. A set is not merely a typographically convenient way to define a certain collection of things.
2. A set is a single object. Many beginners seem to have in their head that the set $\{3,4\}$ is two things.
3. A set is distinct from its elements. The set $\{3,4\}$ is not $3$, it is not $4$, it is not a number at all.
4. The spec implies that $\{3,4\}$ is the same set as $\{4,3\}$. Some students think they understand this but some of their mistakes show that they don’t really understand it.
5. On the other hand, $\{3,5\}$ is a different set from $\{3,4\}$. I haven’t noticed this bothering students but it bothers me. See the discussion on ursets below.

Those consequences make the spec a useful teaching tool. But if a beginning abstract math student gets very far in their studies, some complications come up.

Defining “set”

In the late nineteenth century, math people started formally defining particular math structures such as groups and various
kinds of spaces. This was normally done by starting with a set and adding structure.

You may think that “starting with a set and adding structure” brushes a lot of complications under the rug. Well, don’t look under the rug, at least not right now.

The way they thought about sets was a informal version of what is now called naive set theory. In particular, they freely defined particular sets using what is essentially setbuilder notation, producing sets in a way which (I claim) satisfies my specification.

Bertrand Russell wakes everyone up

Then along came Russell’s paradox. In the context of this discussion, the paradox implied that the spec for sets is not a definition.The spec provides a set of necessary conditions for being a set. But it is not sufficient. You can say “Let $S$ be the set of all sets that…[satisfy some condition]” until you are blue in the face, but there are conditions (including the empty condition) that don’t define a set.

The Zermelo-Fraenkel axioms

The Zermelo-Fraenkel axioms were designed to provide a definition that didn’t create contradictions. The axioms accomplish this by creating a sort of hierarchy that requires that each set must be defined in terms of sets defined previously. They provide a good way (but not the only one) of providing a way of legitimizing our use of sets in math.

Observe that the “set of all sets” is certainly not “defined” in terms of previously defined sets!

Sets as a foundation

During those days there was a movement to provide a solid foundation for mathematics. After Zermelo-Fraenkel came along, the progress of thinking seemed to be:

1. Sets are in trouble.
2. Zermelo-Fraenkel solves our set difficulties.
3. So let’s require that every math object be a set.

That list is oversimplified. In particular, the development of predicate logic was essential to this approach, but I can’t write about everything at once.

This leads to monsters such as the notorious definition of ordered pair:

The ordered pair $(a,b)$ is the set $\{a,\{b\}\}$.

This leads to the ludicrous statement that $a$ is an element of $(a,b)$ but that $b$ is not.

By saying every math object may be modeled as a set with structure, ZF set theory becomes a model of all of math. This approach gives a useful proof that all of math is as consistent as ZF set theory is.

But many mathematicians jumped to the conclusion that every math object must be a set with structure. This approach does not match the way mathematicians think about math objects. In particular, it makes computerized proof assistance hard to use because you have to translate your thinking into sets and first order logic.

Sets by category theory

“A mathematical object is determined by the role it plays in a category.” — A. Grothendieck

In category theory, you define math structures in terms of how they relate to other math structures. This shifts the emphasis from

What is it?

to

What are its properties?

For example, an ordered pair is a mathematical object $p$ determined by these properties:

• It determines mathematical objects $p_1$ and $p_2$.
• $p$ is completely determined by what $p_1$ is and what $p_2$ is.
• If $p$ and $q$ are ordered pairs and $p_1=q_1$ and $p_2=q_2$ then $p=q$.

Categorical definition of set

“Categorical” here means “as understood in category theory”. It unfortunately has a very different meaning in model theory (set of axioms with only one model up to isomorphism) and in general usage, as in “My answer is categorically NO” said by someone who is red in the face. The word “categorial” has an entirely different meaning in linguistics. *Sigh*.

William Lawvere has produced an axiomatization of the category of sets.
The most accessible introduction to it that I know of is the article Rethinking set theory, by Tom Leinster. This axiomatization defines sets by their relationship with each other and other math objects in much the same way as the categorical definition of (for example) groups gives a definition of groups that works in any category.

“Set” means two different things

The word set as used informally has two different meanings.

• According to my specification of sets, $\{3,4\}$ is a set and so is $\{3,5\}$.
• $\{3,4\}$ and $\{3,5\}$ are not the same set because they don’t have the same elements.
• But in the category of sets, any two $2$-element sets are isomorphic. (So are any two seven element sets.)
• From a categorical point of view, two isomorphic objects in a category can be be thought of as the same object, with a caveat that you have better make it clear which isomorphism you are thinking of.

One of the great improvements in mathematics that homotopy type theory supplies is a systematic way of keeping track of the isomorphisms, the isomorphisms between the isomorphisms, and so on ad infinitum (literally). But note: I am just beginning to understand htt, so regard this remark as something to be suspicious of.

• But $\{3,4\}$ and $\{3,5\}$ may not be thought of as the same object according to the spec I gave, because they don’t have the same elements.
• This means that the traditional idea of set is not the same as the strict categorical idea of set.

I suggest that we keep the word “set” for the traditional concept and call the strict categorical concept an urset.

A traditional set is a structure on an urset

The traditional set $\{3,5\}$ consists of the unique two-element urset coindexed on the integers.

A (ur)set $S$ coindexed by a math structure $A$ is a monic map from $S$ to the underlying set of $A$. In this example, the map has codomain the integers and takes one element of the two-element urset to $3$ and the other to $5$.

Note added 2014-10-05 in response to Toby Bartels’ comment: I am inclined to use the names “abstract set” for “urset” and “concrete set” for coindexed sets when I revise the articles on sets. But most of the time we can get away with just “set”.

There is clearly no isomorphism of coindexed sets from $\{3,4\}$ to $\{3,5\}$, so those two traditional sets are not equal in the category of coindexed sets.

I made up the phrase “coindexed set” to use in this sense, since it is a kind of opposite of indexed set. If terminology for this already exists, lemme know. Linguists will tell you they use the word “coindexed” in a different sense.

Elements

The concept of “element” in categorical thinking is very different from the traditional idea, where an element of a set can be any mathematical object. In categorical thinking, an element of an object $A$ of a category $\mathbf{C}$ is an arrow $1\to A$ where $1$ is the terminal object. Thus $4$ as an integer is the arrow $1\to \mathbb{Z}$ whose unique value is the number $4$.

An object is an element of only one set

In the usage of category theory, the arrow $1\to\mathbb{R}$ whose value is the real number $4$ is a different math object from the arrow $1\to\mathbb{Z}$ whose value is the integer $4$.

A category theorist will probably agree that we can identify the integer $4$ with the real number $4$ via the well known canonical embedding of the ring of integers into the field of real numbers. But in categorical thinking you have to keep all such embeddings in mind; you don’t say the integer $4$ is the same thing as the real number $4$. (Most computer languages keep them distinct, too.)

This difference is actually not hard to get used to and is in fact an improvement over traditional set theory. When you do category theory you use lots of commutative diagrams. The embeddings show up as monic arrows and are essential in keeping the different objects ($\mathbb{Z}$ and $\mathbb{R}$ in the example) separate.

The paper Relating first-order set theory and elementary toposes, by Awodey, Butz, Simpson and Streicher, introduces a concept of “structural system of inclusions” that appears to me to restore the idea of object being an element of more than one set for many purposes.

Homotopy type theory allows an object to have only one type, with much the same effect as in the categorical approach.

Variable elements

The arrow $1\to \mathbb{Z}$ that picks out the integer $4$ is a constant function. It is useful to think of any arrow $A\to B$ of any category as a variable element (or generalized element) of the object $B$. For example, the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^2$ allows you to
think of $x^2$ as a variable number with real parameter. This is another way of thinking about the “$y$” in the equation $y=x^2$, which is commonly called a dependent variable.

One way to think about $y$ is that some statements about it are true, some are false, and many statements are neither true nor false.

• $y\geq 0$ is true.
• $y\lt0$ is false.
• $y\leq1$ is neither true nor false.

This way of thinking about variable objects clears up a lot of confusion about variables and deserves to be more widely used in teaching.

The book Category theory for computing science provides some examples of the use of variable elements as a way of thinking about categorical ideas.

References

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Script and calligraphic styles in math writing

This is a draft of an addition to the entry Alphabets in abstractmath.org.

Mathematicians use the word script to refer to two rather different styles. Both of them apply only to uppercase letters.

Script

 $A$: $\scr{A}$ $H$: $\scr{H}$ $O$: $\scr{O}$ $V$: $\scr{V}$ $B$: $\scr{B}$ $I$: $\scr{I}$ $P$: $\scr{P}$ $W$: $\scr{W}$ $C$: $\scr{C}$ $J$: $\scr{J}$ $Q$: $\scr{Q}$ $X$: $\scr{X}$ $D$: $\scr{D}$ $K$: $\scr{K}$ $R$: $\scr{R}$ $Y$: $\scr{Y}$ $E$: $\scr{E}$ $L$: $\scr{L}$ $S$: $\scr{S}$ $Z$: $\scr{Z}$ $F$: $\scr{F}$ $M$: $\scr{M}$ $T$: $\scr{T}$ $G$: $\scr{G}$ $N$: $\scr{N}$ $U$: $\scr{U}$

Calligraphic

 $A$: $\cal{A}$ $H$: $\cal{H}$ $O$: $\cal{O}$ $V$: $\cal{V}$ $B$: $\cal{B}$ $I$: $\cal{I}$ $P$: $\cal{P}$ $W$: $\cal{W}$ $C$: $\cal{C}$ $J$: $\cal{J}$ $Q$: $\cal{Q}$ $X$: $\cal{X}$ $D$: $\cal{D}$ $K$: $\cal{K}$ $R$: $\cal{R}$ $Y$: $\cal{Y}$ $E$: $\cal{E}$ $L$: $\cal{L}$ $S$: $\cal{S}$ $Z$: $\cal{Z}$ $F$: $\cal{F}$ $M$: $\cal{M}$ $T$: $\cal{T}$ $G$: $\cal{G}$ $N$: $\cal{N}$ $U$: $\cal{U}$

Using script

• In LaTeX, script letters are obtained using “\scr” and calligraphic using “\cal”. For example, “{\scr P}” gives ${\scr P}$. The file Script fonts for LaTeX shows how to get variations other than the ones shown above.
• Both script and calligraphic are used to provide yet another type style for naming mathematical objects.
• One of the most common uses is to refer to the powerset of a set $S$: ${\scr P}(S)$, ${\scr P}S$, ${\cal P}(S)$, ${\cal P}S$.
• There may be some tendency to use script or cal to name objects that are in some way high in the hierarchy of objects or else a space that contains a lot of the stuff you are talking about. In most of the paper I found in a cursory exam of Jstor shows only a couple of exceptions (in Lie algebra). This is one of many places in abmath where I throw out examples of usages in math that I have found but have not verified through serious lexicographical research.
• The names of categories are commonly denoted by script or calligraphic. Some authors have trouble because they want to put names of categories such as “Set” and “Grp” in cal or scr but don’t have lower case letters in those styles. In Toposes, Triples and Theories the online version went through several changes over the years. Category Theory for Computing Science uses bold for category names.
• I have never run across a paper that used both script and calligraphic to mean two different things.

Acknowledgments

Thanks to JM Wilson for suggesting this topic and to the various people on Math Stack Exchange and Math Educators Stack Exchange who discussed script and cal.

Improving abstractmath.org

This post discusses some ideas I have for improving abstractmath.org.

Handbook of mathematical discourse

The Handbook was kind of a false start on abmath, and is the source of much of the material in abmath. It still contains some material not in abmath, parti­cularly the citations.

By citations I mean lexicographical citations: examples of the usage from texts and scholarly articles.

I published the Handbook of mathe­ma­tical discourse in 2003. The first link below takes you to an article that describes what the Handbook does in some detail. Briefly, the Handbook surveys the use of language in math (and some other things) with an emphasis on the problems it causes students. Its collection of citations of usage could some day could be the start of an academic survey of mathematical language. But don’t expect me to do it.

The Handbook exists as a book and as two different web versions. I lost the TeX source of the Handbook a few years after I published the book, so none of the different web versions are perfect. Version 2 below is probably the most useful.

1. Handbook of mathe­ma­tical discourse. Description.
2. Handbook of mathe­ma­tical discourse. Hypertext version without pictures but with active internal links. Some links don’t work, but they won’t be repaired because I have lost the TeX input files.
3. Handbook of mathe­ma­tical discourse. Paperback.
4. Handbook of mathematical discourse. PDF version of the printed book, including illustrations and citations but without hyperlinks.
5. Citations for the paperback version of the Handbook. (The hypertext version and the PDF version include the citations.)

Abmath

Soon after the Handbook was published, I started work on abstractmath.org, which I abbreviate as abmath. It is intended specifically for people beginning to study abstract math, which means roughly post-calculus. I hope their teachers will read it, too. I had noticed when I was teaching that many students hit a big bump when faced with abstraction, and many of them never recovered. They would typically move into another field, often far away from STEM stuff.

These abmath articles give more detail about the purpose of this website and the thinking behind the way it is presented:

Presentation of abmath

Informal

Abmath is written for students of abstract math and other beginners to tell them about the obstacles they may meet up with in learning abstract math. It is not a scholarly work and is not written in the style of a scholarly work. There is more detail about its style in my rant in Attitude.

Scholarly works should not be written in the style of a scholarly work, either.

To do:

Every time I revise an article I find myself rewriting overly formal parts. Fifty years of writing research papers has taken its toll. I must say that I am not giving this informalization stuff very high priority, but I will continue doing it.

No citations

One major difference concerns the citations in the Handbook. I collected these in the late nineties by spending many hours at Jstor and looking through physical books. When I started abmath I decided that the website would be informal and aimed at students, and would contain few or no citations, simply because of the time it took to find them.

Boxouts and small screens

The Handbook had both sidebars on every page of the paper version containing a reference index to words on that page, and also on many pages boxouts with comments. It was written in TeX. I had great difficulty using TeX to control the placement of both the sidebars and especially the boxouts. Also, you couldn’t use TeX to let the text expand or contract as needed by the width of the user’s screen.

Abmath uses boxouts but not sidebars. I wrote Abmath using HTML, which allows it to be presented on large or small screens and to have extensive hyperlinks.
HTML also makes boxouts easy.

The arrival of tablets and i-pods has made it desirable to allow an abmath page to be made quite narrow while still readable. This makes boxouts hard to deal with. Also I have gotten into the habit of posting revisions to articles on Gyre&Gimble, whose editor converts boxouts into inline boxes. That can probably be avoided.

To do:

I have to decide whether to turn all boxouts into inline small-print paragraphs the was you see them in this article. That would make the situation easier for people reading small screens. But in-line small-print paragraphs are harder to associate to the location you want them to refer, in contrast to boxouts.

Abmath 2.0

For the first few years, I used Microsoft Word with MathType, but was plagued with problems described in the link below. Then I switched to writing directly in HTML. The articles of abmath labeled “abstractmath.org 2.0″ are written in this new way. This makes the articles much, much easier to update. Unfortunately, Word produces HTML that is extraordinarily complicated, so transforming them into abmath 2.0 form takes a lot of effort.

Illustrations

Abmath does not have enough illustrations and diagrams. Gyre&Gimble has many posts with static illustrations, some of them innovative. It also has some posts with interactive demos created with Mathematica. These demos require the reader to download the CDF Player, which is free. Unfortunately, it is available only for Windows, Mac and Linux, which precludes using them on many small devices.

To do:

• Create new illustrations where they might be useful, and mine Gyre&Gimble and other sources.
• There are many animated GIFs out there in the math cloud. I expect many of them are licensed under Creative Commons so that I can use them.
• I expect to experiment with converting some of the interactive CFD diagrams that are in Gyre&Gimble into animated GIFs or AVIs, which as far as I know will run on most machines. This will be a considerable improvement over static diagrams, but it is not as good as interactive diagrams, where you can have several sliders controlling different variables, move them back and forth, and so on. Look at Inverse image revisited. and “quintic with three parameters” in Demos for graph and cograph of calculus functions.

Abmath content

Language

Abmath includes most of the ideas about language in the Handbook (rewritten and expanded) and adds a lot of new material.

1. The languages of math. Article in abmath. Has links to the other articles about language.
2. Syntactic and semantic thinkers. Gyre&Gimble post.
3. Syntax trees in mathematicians’ brains. Gyre&Gimble post.
4. A visualization of a computation in tree form.Gyre&Gimble post.
5. Visible algebra I. Gyre&Gimble post.
6. Algebra is a difficult foreign language. Gyre&Gimble post.
7. Presenting binops as trees. Gyre&Gimble post.
8. Moths to the flame of meaning. How linguistics students also have trouble with syntax.
9. Varieties of mathematical prose, by Atish Bagchi and Charles Wells.

To do:

The language articles would greatly benefit from more illustrations. In parti­cular:

• G&G contains several articles about using syntax trees (items 3, 4, 5 and 7 above) to understand algebraic expressions. A syntax tree makes the meaning of an algebraic expression much more transparent than the usual one-dimensional way of writing it.
• Several items in the abmath article More about the language of math, for example the entries on parenthetic assertions and postconditions could benefit from a diagrammatic representation of the relation between phrases in a sentence and semantics (or how the phrases are spoken).
• The articles on Names and Alphabets could benefit from providing spoken pronunciations of many words. But what am I going to do about my southern accent?
• The boxed example of change in context as you read a proof in More about the language of math could be animated as you click through the proof. *Sigh* The prospect of animating that example makes me tired just thinking about it. That is not how grasshoppers read proofs anyway.

Understanding and doing math

Abmath discusses how we understand math and strategies for doing math in some detail. This part is based on my own observations during 35 years of teaching, as well as extensive reading of the math ed literature. The math ed literature is usually credited in footnotes.

Math objects and math structures

Understanding how to think about mathematical objects is, I believe, one of the most difficult hurdles newbies have to overcome in learning abstract math. This is one area that the math ed community has focused on in depth.

The first two links below are take you to the two places in abmath that discuss this problem. The first article has links to some of the math ed literature.

To do: Everything is a math object

An important point about math objects that needs to be brought out more is that everything in math is a math object. Obviously math structures are math objects. But the symbol “$\leq$” in the statement “$35\leq45$” denotes a math object, too. And a proof is a math object: A proof written on a blackboard during a lecture does not look like it is an instance of a rigorously defined math object, but most mathe­maticians, including me, believe that in principle such proofs can be transformed into a proof in a formal logical system. Formal logics, such as first order logic, are certainly math objects with precise mathematical definitions. Definitions, math expressions and theorems are math objects, too. This will be spelled out in a later post.

To do: Bring in modern ideas about math structure

Classically, math structures have been presented as sets with structure, with the structure being described in terms of subsets and functions. My chapter on math structures only waves a hand at this. This is a decidedly out-of-date way of doing it, now that we have category theory and type theory. I expect to post about this in order to clarify my thinking about how to introduce categorical and type-theoretical ideas without writing a whole book about it.

Particular math structures

Abmath includes discussions
of the problems students have with certain parti­cular types of structures. These sections talk mostly about how to think about these structure and some parti­cular misunder­standings students have at the most basic levels.

These articles are certainly not proper intro­ductions to the structures. Abmath in general is like that: It tells students about some aspects of math that are known to cause them trouble when they begin studying abstract math. And that is all it does.

To do:

• I expect to write similar articles about groups, spaces and categories.
• The idea about groups is to mention a few things that cause trouble at the very beginning, such as cosets, quotients and homomorphisms (which are all obviously related to each other), and perhaps other stumbling blocks.
• With categories the idea is to stomp on misconceptions such as that the arrows have to be functions and to emphasize the role of categories in allowing us to define math structures in terms of their relations with other objects instead of in terms with sets.
• I am going to have more trouble with spaces. Perhaps I will show how you can look at the $\epsilon$-$\delta$ definition of continuous functions on the reals and “discover” that they imply that inverse images of open sets are open, thus paving the way for the family-of-subsets definition of a topoogy.
• I am not ruling out other particular structures.

Proofs

This chapter covers several aspects of proofs that cause trouble for students, the logical aspects and also the way proofs are written.

It specifically does not make use of any particular symbolic language for logic and proofs. Some math students are not good at learning languages, and I didn’t see any point in introducing a specific language just to do rudimentary discussions about proofs and logic. The second link below discusses linguistic ability in connection with algebra.

I taught logic notation as part of various courses to computer engineering students and was surprised to discover how difficult some students found using (for example) $p+q$ in one course and $p\vee q$ in another. Other students breezed through different notations with total insouciance.

To do:

Much of the chapter on proofs is badly written. When I get around to upgrading it to abmath 2.0 I intend to do a thorough rewrite, which I hope will inspire ideas about how to conceptually improve it.

More alphabets

This post is the third and last in a series of posts containing revisions of the abstractmath.org article Alphabets. The first two were:

Addition to the listings for the Greek alphabet

Sigma: $\Sigma,\,\sigma$ or ς: sĭg'mɘ. The upper case $\Sigma$ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function. The ς form for the lower case has not as far as I know been used in math writing, but I understood that someone is writing a paper that will use it.

Hebrew alphabet

Aleph, א is the only Hebrew letter that is widely used in math. It is the cardinality of the set of integers. A set with cardinality א is countably infinite. More generally, א is the first of the aleph numbers $א_1$, $א_2$, $א_3$, and so on.

Cardinality theorists also write about the beth (ב) numbers, and the gimel (ג) function. I am not aware of other uses of the Hebrew alphabet.

If you are thinking of using other Hebrew letters, watch out: If you type two Hebrew letters in a row in HTML they show up on the screen in reverse order. (I didn't know HTML was so clever.)

Cyrillic alphabet

The Cyrillic alphabet is used to write Russian and many other languages in that area of the world. Wikipedia says that the letter Ш, pronounced "sha", is the only Cyrillic letter used in math. I have not investigated further.

The letter is used in several different fields, to denote the Tate-Shafarevich group, the Dirac comb and the shuffle product.

It seems to me that there are a whole world of possibillities for brash young mathematicians to name mathematical objects with other Cyrillic letters. Examples:

• Ж. Use it for a ornate construction, like the Hopf fibration or a wreath product.
• Щ. This would be mean because it is hard to pronounce.
• Ъ. Guaranteed to drive people crazy, since it is silent. (It does have a name, though: "Yehr".)
• Э. Its pronunciation indicates you are unimpressed (think Fonz).
• ю. Pronounced "you". "ю may provide a counterexample". "I do?"

Type styles

Boldface and italics

A typeface is a particular design of letters.  The typeface you are reading is Arial.  This is Times New Roman. This is Goudy. (Goudy may not render correctly on your screen if you don't have it installed.)

Typefaces typically come in several styles, such as bold (or boldface) and italic.

Examples

 Arial Normal Arial italic Arial bold Times Normal Times italic Times bold Goudy Normal Goudy italic Goudy bold

Boldface and italics are used with special meanings (conventions) in mathematics. Not every author follows these conventions.

Styles (bold, italic, etc.) of a particular typeface are supposedly called fonts.  In fact, these days “font” almost always means the same thing as “typeface”, so I  use “style” instead of “font”.

Vectors

A letter denoting a vector is put in boldface by many authors.

Examples
• “Suppose $\mathbf{v}$ be an vector in 3-space.”  Its coordinates typically would be denoted by $v_1$, $v_2$ and $v_3$.
• You could also define it this way:  “Let $\mathbf{v}=({{v}_{1}},{{v}_{2}},{{v}_{3}})$ be a vector in 3-space.”  (See parenthetic assertion.)

It is hard to do boldface on a chalkboard, so lecturers may use $\vec{v}$ instead of $\mathbf{v}$. This is also seen in print.

Definitions

The definiendum (word or phrase being defined) may be put in boldface or italics. Sometimes the boldface or italics is the only clue you have that the term is being defined. See Definitions.

Example

“A group is Abelian if its multiplication is commutative,” or  “A group is Abelian if its multiplication is commutative.”

Emphasis

Italics are used for emphasis, just as in general English prose. Rarely (in my experience) boldface may be used for emphasis.

In the symbolic language

It is standard practice in printed math to put single-letter variables in italics.   Multiletter identifiers are usually upright.

Example

Example: "$f(x)=a{{x}^{2}}+\sin x$".  Note that mathematicians would typically refer to $a$ as a “constant” or “parameter”, but in the sense we use the word “variable” here, it is a variable, and so is $f$.

Example

On the other hand, “e” is the proper name of a specific number, and so is “i”. Neither is a variable. Nevertheless in print they are usually given in italics, as in ${{e}^{ix}}=\cos x+i\sin x$.  Some authors would write this as ${{\text{e}}^{\text{i}x}}=\cos x+\text{i}\,\sin x$.  This practice is recommended by some stylebooks for scientific writing, but I don't think it is very common in math.

Blackboard bold

Blackboard bold letters are capital Roman letters written with double vertical strokes.   They look like this:

$\mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F}\,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L}\,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R}\,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X}\,\mathbb{Y}\,\mathbb{Z}$

In lectures using chalkboards, they are used to imitate boldface.

In print, the most common uses is to represent certain sets of numbers:

Remarks

• Mathe­ma­tica uses some lower case blackboard bold letters.
• Many mathe­ma­tical writers disapprove of using blackboard bold in print.  I say the more different letter shapes that are available the better.  Also a letter in blackboard bold is easier to distinguish from ordinary upright letters than a letter in boldface is, particularly on computer screens.

The Greek alphabet in math

This is a revision of the portion of the article Alphabets in abstractmath.org that describes the use of the Greek alphabet by mathematicians.

Every letter of the Greek alphabet except omicron is used in math. All the other lowercase forms and all those uppercase forms that are not identical with the Latin alphabet are used.

• Many Greek letters are used as proper names of mathe­ma­tical objects, for example $\pi$. Here, I provide some usages that might be known to undergraduate math majors.  Many other usages are given in MathWorld and in Wikipedia. In both those sources, each letter has an individual entry.
• But any mathematician will feel free to use any Greek letter with a meaning different from common usage. This includes $\pi$, which for example is often used to denote a projection.
• Greek letters are widely used in other sciences, but I have not attempted to cover those uses here.

The letters

• English-speaking mathematicians pronounce these letters in various ways.  There is a substantial difference between the way American mathe­maticians pronounce them and the way they are pronounced by English-speaking mathe­maticians whose background is from British Commonwealth countries. (This is indicated below by (Br).)
• Mathematicians speaking languages other than English may pronounce these letters differently. In particular, in modern Greek, most Greek letters are pro­nounced differ­ently from the way we pronounce them; β for example is pro­nounced vēta (last vowel as in "father").
• Newcomers to abstract math often don’t know the names of some of the letters, or mispronounce them if they do.  I have heard young mathe­maticians pronounce $\phi$ and $\psi$ in exactly the same way, and since they were writing it on the board I doubt that anyone except language geeks like me noticed that they were doing it.  Another one pronounced $\phi$ as  “fee” and $\psi$ as “fie”.

Pronunciation key

• ăt, āte, ɘgo (ago), bĕt, ēve, pĭt, rīde, cŏt, gō, ŭp, mūte.
• Stress is indicated by an apostrophe after the stressed syllable, for example ū'nit, ɘgō'.
• The pronunciations given below are what mathematicians usually use. In some cases this includes pronunciations not found in dictionaries.

Alpha: $\text{A},\, \alpha$: ă'lfɘ. Used occasionally as a variable, for example for angles or ordinals. Should be kept distinct from the proportionality sign "∝".

Beta: $\text{B},\, \beta$: bā'tɘ or (Br) bē'tɘ. The Euler Beta function is a function of two variables denoted by $B$. (The capital beta looks just like a "B" but they call it “beta” anyway.)  The Dirichlet beta function is a function of one variable denoted by $\beta$.

Gamma: $\Gamma, \,\gamma$: gă'mɘ. Used for the names of variables and functions. One familiar one is the $\Gamma$ function. Don’t refer to lower case "$\gamma$" as “r”, or snooty cognoscenti may ridicule you.

Delta: $\Delta \text{,}\,\,\delta$: dĕltɘ. The Dirac delta function and the Kronecker delta are denoted by $\delta$.  $\Delta x$ denotes the change or increment in x and $\Delta f$ denotes the Laplacian of a multivariable function. Lowercase $\delta$, along with $\epsilon$, is used as standard notation in the $\epsilon\text{-}\delta$ definition of limit.

Epsilon: $\text{E},\,\epsilon$ or $\varepsilon$: ĕp'sĭlɘn, ĕp'sĭlŏn, sometimes ĕpsī'lɘn. I am not aware of anyone using both lowercase forms $\epsilon$ and $\varepsilon$ to mean different things. The letter $\epsilon$ is frequently used informally to denoted a positive real number that is thought of as being small. The symbol ∈ for elementhood is not an epsilon, but many mathematicians use an epsilon for it anyway.

Zeta: $\text{Z},\zeta$: zā'tɘ or (Br) zē'tɘ. There are many functions called “zeta functions” and they are mostly related to each other. The Riemann hypothesis concerns the Riemann $\zeta$-function.

Eta: $\text{H},\,\eta$: ā'tɘ or (Br) ē'tɘ. Don't pronounce $\eta$ as "N" or you will reveal your newbieness.

Theta: $\Theta ,\,\theta$ or $\vartheta$: thā'tɘ or (Br) thē'tɘ.  The letter $\theta$ is commonly used to denote an angle. There is also a Jacobi $\theta$-function related to the Riemann $\zeta$-function. See also Wikipedia.

Iota: $\text{I},\,\iota$: īō'tɘ. Occurs occasionally in math and in some computer languages, but it is not common.

Kappa: $\text{K},\, \kappa$: kă'pɘ. Commonly used for curvature.

Lambda: $\Lambda,\,\lambda$: lăm'dɘ. An eigenvalue of a matrix is typically denoted $\lambda$.  The $\lambda$-calculus is a language for expressing abstract programs, and that has stimulated the use of $\lambda$ to define anonymous functions. (But mathematicians usually use barred arrow notation for anonymous functions.)

Mu: $\text{M},\,\mu$: mū.  Common uses: to denote the mean of a distribution or a set of numbers, a measure, and the Möbius function. Don’t call it “u”.

Nu: $\text{N},\,\nu$: nū.    Used occasionally in pure math,more commonly in physics (frequency or a type of neutrino).   The lowercase $\nu$ looks confusingly like the lowercase upsilon, $\upsilon$. Don't call it "v".

Xi: $\Xi,\,\xi$: zī, sī or ksē. Both the upper and the lower case are used occasionally in mathe­matics. I recommend the ksee pronunciation since it is unambiguous.

Omicron: $\text{O, o}$: ŏ'mĭcrŏn.  Not used since it looks just like the Roman letter.

Pi: $\Pi \text{,}\,\pi$: pī.  The upper case $\Pi$ is used for an indexed product.  The lower case $\pi$ is used for the ratio of the circumference of a circle to its diameter, and also commonly to denote a projection function or the function that counts primes.  See default.

Rho: $\text{P},\,\rho$: rō. The lower case $\rho$ is used in spherical coordinate systems.  Do not call it pee.

Sigma: $\Sigma,\,\sigma$: sĭg'mɘ. The upper case $\Sigma$ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function.

Tau: $\text{T},\,\tau$ or τ: tăoo (rhymes with "cow"). The lowercase $\tau$ is used to indicate torsion, although the torsion tensor seems usually to be denoted by $T$. There are several other functions named $\tau$ as well.

Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.

Phi: $\Phi ,\,\phi$ or $\varphi$: fē or fī. Used for the totient function, for the “golden ratio” $\frac{1+\sqrt{5}}{2}$ (see default) and also commonly used to denote an angle.  Historically, $\phi$ is not the same as the notation $\varnothing$ for the empty set, but many mathematicians use it that way anyway, sometimes even calling the empty set “fee” or “fie”.

Chi: $\text{X},\,\chi$: kī.  (Note that capital chi looks like $\text{X}$ and capital xi looks like $\Xi$.) Used for the ${{\chi }^{2}}$distribution in statistics, and for various math objects whose name start with “ch” (the usual transliteration of $\chi$) such as “characteristic” and “chromatic”.

Psi: $\Psi, \,\psi$; sē or sī. A few of us pronounce it as psē or psī to distinguish it from $\xi$.  $\psi$, like $\phi$, is often used to denote an angle.

Omega: $\Omega ,\,\omega$: ōmā'gɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.

Demos for graph and cograph of calculus functions

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers but not necessarily on many smaller devices.

This post provides interactive examples of the endograph and cograph of real functions. Those two concepts were defined and discussed in the previous post Endograph and cograph of real functions.

Such representations of functions, put side by side with the conventional graph, may help students understand how to interpret the usual graph representation. For example: What does it mean when the arrows slant to the left? spread apart? squeeze together? flip over? Going back and forth between the conventional graph and the cograph or engraph for a particular function should make you much more in tune to the possibilities when you see only the conventional graph of another function.

This is not a major advance for calculus teachers, but it may be a useful tool. The source code is the Mathematica Notebook GraphCograph.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License.

Line segment

$y=a x+b$

$y=ax^2+b$

$y=a x^2$ (blue) and $y=2 a x$ (red)

Cubic

$y=a x^3-b x$

Sine

$y=a \sin b x$

Sine and its derivative

$y=\sin a x$ (blue) and $y=a\cos x$ (red)

Quintic with three parameters

$y=a x^5-b x^4-0.21 x^3+0.2 x^2+0.5 x-c$