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.

References

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.

Links

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.

Links

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.

Links

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.

Link

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.

Links

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.

Links

  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.

Links

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.

Links

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.

Links

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.

Links

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.

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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 use of fraktur in math

This post is a revision of the part of the abmath article on alphabets concerning the fraktur typeface, followed by some corrections and remarks. A revision of the section on the Greek alphabet was posted previously.

Fraktur

In some math subjects, a font tamily (typeface) called fraktur, formerly used for writing German, Norwegian, and some other languages, is used to name math objects.  The table below shows the upper and lower case fraktur letters. 








$A,a$: $\mathfrak{A},\mathfrak{a}$ $H,h$: $\mathfrak{H},\mathfrak{h}$ $O,o$: $\mathfrak{O},\mathfrak{o}$ $V,v$: $\mathfrak{V},\mathfrak{v}$
$B,b$: $\mathfrak{B},\mathfrak{b}$ $I,i$: $\mathfrak{I},\mathfrak{i}$ $P,p$: $\mathfrak{P},\mathfrak{p}$ $W,w$: $\mathfrak{W},\mathfrak{w}$
$C,c$: $\mathfrak{C},\mathfrak{c}$ $J,j$: $\mathfrak{J},\mathfrak{j}$ $Q,q$: $\mathfrak{Q},\mathfrak{q}$ $X,x$: $\mathfrak{X},\mathfrak{x}$
$D,d$: $\mathfrak{D},\mathfrak{d}$ $K,k$: $\mathfrak{K},\mathfrak{k}$ $R,r$: $\mathfrak{R},\mathfrak{r}$ $Y,y$: $\mathfrak{Y},\mathfrak{y}$
$E,e$: $\mathfrak{E},\mathfrak{e}$ $L,l$: $\mathfrak{L},\mathfrak{l}$ $S,s$: $\mathfrak{S},\mathfrak{s}$ $Z,z$: $\mathfrak{Z},\mathfrak{z}$
$F,f$: $\mathfrak{F},\mathfrak{f}$ $M,m$: $\mathfrak{M},\mathfrak{m}$ $T,t$: $\mathfrak{T},\mathfrak{t}$  
$G,g$: $\mathfrak{G},\mathfrak{g}$ $N,n$: $\mathfrak{N},\mathfrak{n}$ $U,u$: $\mathfrak{U},\mathfrak{u}$  
  • Many of the forms are confusing and are commonly mispronounced by younger mathematicians.  (Ancient mathematicians like me have taken German classes in college that required learning fraktur.)  In particular the uppercase $\mathfrak{A}$ looks like $U$ but in fact is an $A$, and the uppercase $\mathfrak{I}$ looks like $T$ but is actually $I$.  
  • When writing on the board, some mathematicians use a cursive form when writing objects with names that are printed in fraktur.
  • Unicode regards fraktur as a typeface (font family) rather than as a different alphabet. However, unicode does provide codes for the fraktur letters that are used in math (no umlauted letters or ß). In TeX you type "\mathfrak{a}" to get $\mathfrak{a}$.
  • In my (limited) experience, native German speakers usually call this alphabet “Altschrift” instead of “Fraktur”.  It has also been called “Gothic”, but that word is also used to mean several other quite different typefaces (black­letter, sans serif and (gasp) the alphabet actually used by the Goths.
  • I have been doing mathematical research for around fifty years. It is clear to me that mathematicians' use of and familiarity with fraktur has declined a lot during that time. But it is not extinct. I have made a hasty and limited search of Jstor and found recent websites and research articles that use it in a variety of fields. There are also a few citations in the Handbook (search for "fraktur").

    • It is used in ring theory and algebraic number theory, in particular to denote ideals.
    • It is use in Lie algebra. In particular, the Lie algebra of a Lie group $G$ is commonly denoted by $\mathfrak{g}$.
    • The cardinality of the continuum is often denoted by $\mathfrak{c}$.
    • It is used occasionally in logic to denote models and other objects.
    • I remember that in the sixties and seventies fraktur was used in algebraic geometry, but I haven't found it in recent papers.

Acknowledgements

Thanks to Fernando Gouvêa for suggestions.

Remarks about usage in abstractmath.org

The Handbook has 428 citation for usages in the mathematical research literature. After finishing that book, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect and very often, perhaps even usually, incomplete.

Corrections to the post The Greek alphabet in math

Willie Wong suggested some additional pronunciations for upsilon and omega:


Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn; (Br) ĭp'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.


Omega: $\Omega ,\,\omega$: ōmā'gɘ, ō'māgɘ; (Br) ōmē'gɘ, ō'mēgɘ. $\Omega$ 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.  

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

$y=a x+b$


Quadratic

$y=ax^2+b$


Quadratic and its derivative

$y=a x^2$ (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

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

Images and metaphors in math

About this post

This post is the new revision of the chapter on Images and Metaphors in abstractmath.org.

Images and metaphors in math

In this chapter, I say something about mental represen­tations (metaphors and images) in general, and provide examples of how metaphors and images help us understand math – and how they can confuse us.

Pay special attention to the section called two levels!  The distinction made there is vital but is often not made explicit.

Besides mental represen­tations, there are other kinds of represen­tations used in math, discussed in the chapter on represen­tations and models.

Mathe­matics is the tinkertoy of metaphor. –Ellis D. Cooper

Images and metaphors in general

We think and talk about our experiences of the world in terms of images and metaphors that are ultimately derived from immediate physical experience.  They are mental represen­tations of our experiences.

See Thinking about thought.

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Examples

Images

We know what a pyramid looks like.  But when we refer to the government’s food pyramid we are not talking about actual food piled up to make a pyramid.  We are talking about a visual image of the pyramid.    

Metaphors

We know by direct physical experience what it means to be warm or cold.  We use these words as metaphors
in many ways: 

  • We refer to a person as having a warm or cold personality.  This has nothing to do with their body temperature.
  • When someone is on a treasure hunt we may tell them they are “getting warm”, even if they are hunting outside in the snow.

Children don’t always sort meta­phors out correctly. Father: “We are all going to fly to Saint Paul to see your cousin Petunia.” Child: “But Dad, I don’t know how to fly!”

Other terminology

  • My use of the word “image” means mental image. In the study of literature, the word “image” is used in a more general way, to refer to an expression that evokes a mental image..
  • I use “metaphor” in the sense of conceptual metaphor. The word metaphor in literary studies is related to my use but is defined in terms of how it is expressed.
  • The metaphors mentioned above involving “warm” and “cold”
    evoke a sensory experience, and so could be called an image as well. 
  • In math education, the phrase concept image means the mental structure associated with a concept, so there may be no direct connection with sensory experience.  
  • In abstractmath.org, I use the phrase metaphors and images to talk about all our mental represen­tations, without trying for fine distinctions.

Mental represen­tations are imperfect

One basic fact about metaphors and images is that they apply only to certain aspects of the situation.

  • When someone is getting physically warm we would expect them to start sweating.
  • But if they are getting warm in a treasure hunt we don’t expect them to start sweating. 
  • We don’t expect the food pyramid to have a pharaoh buried underneath it, either.

Our brains handle these aspects of mental represen­tations easily and usually without our being conscious of them.  They are one of the primary ways we understand the world.

Images and metaphors in math

Half this game is 90% mental. –Yogi Berra

Types of represen­tations

Mathe­maticians who work with a particular kind of mathe­matical object
have mental represen­tations of that type of object that help them
understand it.  These mental represen­tations come in many forms.  Most of them fit into one of the types below, but the list shouldn’t be taken too seriously: Some represen­tations fit more that of these types, and some may not fit into any of them except awkwardly.

  • Visual
  • Notation
  • Kinetic
  • Process
  • Object

All mental represen­tations are conceptual metaphors. Metaphors are treated in detail in this chapter and in the chapter on images and metaphors for functions.  See also literalism and Proofs without dry bones on Gyre&Gimble.

Below I list some examples. Many of them refer to the arch function, the function defined by $h(t)=25-{{(t-5)}^{2}}$.

Visual image

Geometric figures



The arch function

  • You can picture the arch function in terms of its graph, which is a parabola.     This visualization suggests that the function has a single maximum point that appears to occur at $t=5$. That is an example of how metaphors can suggest (but not prove) theorems.
  • You can think of the arch function
    more physically, as like the Gateway Arch. This metaphor is suggested by the graph.

Interior of a shape

  • The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house.
  • Sometimes, the interior can be described using analytic geometry. For example, the interior of the circle $x^2+y^2=1$ is the set of points \[\{(x,y)|x^2+y^2\lt1\}\]
  • But the “interior” metaphor is imperfect: The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere. 
  • This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

Real number line

  • You may think of the real
    numbers
    as lying along a straight line (the real line) that extends infinitely far in both directions.  This is both visual and a metaphor (a real number “is” a place on the real line).
  • This metaphor is imperfect because you can’t draw the whole real line, but only part of it. But you can’t draw the whole graph of the curve $y=25-(t-5)^2$, either.

Continuous functions

No gaps

“Continuous functions don’t have gaps in the graph”.    This is a visual image, and it is usually OK.

  • But consider the curve defined by $y=25-(t-5)^2$ for every real $x$ except $x=1$. It is not defined at $x=1$ (and so the function is discontinuous there) but its graph looks exactly like the graph in the figure above because no matter how much you magnify it you can’t see the gap.
  • This is a typi­cal math example that teachers make up to raise your consciousness.

  • So is there a gap or not?
No lifting

“Continuous functions can be drawn without lifting the chalk.” This is true in most familiar cases (provided you draw the graph only on a finite interval). But consider the graph of the function defined by $f(0)=0$ and \[f(t)=t\sin\frac{1}{t}\ \ \ \ \ \ \ \ \ \ (0\lt t\lt 0.16)\]
(see Split Definition). This curve is continuous and is infinitely long even though it is defined on a finite interval, so you can’t draw it with a chalk at all, picking up the chalk or not. Note that it has no gaps.

Keeping concepts separate by using mental “space”

I personally use visual images to remember relationships between abstract objects, as well.  For example, if I think of three groups, two of which are isomorphic (for example $\mathbb{Z}_{3}$ and $\text{Alt}_3$), I picture them as in three different places in my head with a connection between the two isomorphic ones.

Notation

Here I give some examples of thinking of math objects in terms of the notation used to name them. There is much more about notation as mathe­matical represen­tation in these sections of abmath:

Notation is both something you visualize in your head and also a physical represen­tation of the object.  In fact notation can also be thought of as a mathe­matical object in itself (common in mathe­matical logic and in theoretical computing science.)   If you think about what notation “really is” a lot,  you can easily get a headache…

Symbols

  • When I think of the square root of $2$, I visualize the symbol “$\sqrt{2}$”. That is both a typographical object and a mathe­matically defined symbolic represen­tation of the square root of $2$.
  • Another symbolic represen­tation of the square root of $2$ is “$2^{1/2}$”. I personally don’t visualize that when I think of the square root of $2$, but there is nothing wrong with visualizing it that way.
  • What is dangerous is thinking that the square root of $2$ is the symbol “$\sqrt{2}$” (or “$2^{1/2}$” for that matter). The square root of $2$ is an abstract mathe­matical object given by a precise mathe­matical definition.
  • One precise defi­nition of the square root of $2$ is “the positive real number $x$ for which $x^2=2$”. Another definition is that $\sqrt{2}=\frac{1}{2}\log2$.

Integers

  • If I mention the number “two thousand, six hundred forty six” you may visualize it as “$2646$”. That is its decimal represen­tation.
  • But $2646$ also has a prime factorization, namely $2\times3^3\times7^2$.
  • It is wrong to think of this number as being the notation “$2646$”. Different notations have different values, and there is no mathe­matical reason to make “$2646$” the “genuine” represen­tation. See represen­tations and Models.
  • For example, the prime factor­ization of $2646$ tells you imme­diately that it is divisible by $49$.

When I was in high school in the 1950’s, I was taught that it was incorrect to say “two thousand, six hundred and forty six”. Being naturally rebellious I used that extra “and” in the early 1960’s in dictating some number in a telegraph mes­sage. The Western Union operator corrected me. Of course, the “and” added to the cost. (In case you are wondering, I was in the middle of a postal Diplomacy game in Graustark.)

Formulas

Set notation

You can think of the set containing $1$, $3$ and $5$ and nothing else as represented by its common list notation $\{1, 3, 5\}$.  But remember that $\{5, 1,3\}$ is another notation for the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.


Kinetic

Shoot a ball straight up

  • The arch function could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere.
  • The ball starts upward at time $t=0$ at elevation $0$, reaches an elevation of (for example) $16$ units at time $t=2$, and lands at $t=10$.
  • The parabola is not the path of the ball. The ball goes up and down along the $x$-axis. A point on the parabola shows it locaion on the $x$ axis at time $t$.
  • When you think about this event, you may imagine a physical event continuing over time, not just as a picture but as a feeling of going up and down.
  • This feeling of the ball going up and down is created in your mind presumably using mirror neuron. It is connected in your mind by a physical connection to the understanding of the function that has been created as connections among some of your neurons.
  • Although $h(t)$ models the height of the ball, it is not the same thing as the height of the ball.  A mathe­matical object may have a relationship in our mind to physical processes or situations, but it is distinct from them.

Remarks

  1. This example involves a picture (graph of a function).  According to this report, kinetic
    understanding can also help with learning math that does not involve pictures. 
    For example, when I think of evaluating the function ${{x}^{2}}+1$ at 3, I visualize
    3 moving into the x slot and then the formula $9^2+1$ transforming
    itself into $10$. (Not all mathematicians visualize it this way.)
  2. I make the point of emphasizing the physical existence in your brain of kinetic feelings (and all other metaphors and images) to make it clear that this whole section on images and metaphors is about objects that have a physical existence; they are not abstract ideals in some imaginary ideal space not in our world. See Thinking about thought.

I remember visualizing algebra I this way even before I had ever heard of the Transformers.

Process 

It is common to think of a function as a process: you put in a number (or other object) and the process produces another number or other object. There are examples in Images and metaphors for functions.

Long division

Let’s divide $66$ by $7$ using long division. The process consists of writing down the decimal places one by one.

  1. You guess at or count on your fingers to find the largest integer $n$ for which $7n\lt66$. That integer is $9$.
  2. Write down $9.$ ($9$ followed by a decimal point).
  3. $66-9\times7=3$, so find the largest integer $n$ for which $7n\lt3\times10$, which is $4$.
  4. Adjoin $4$ to your answer, getting $9.4$
  5. $3\times10-7\times4=2$, so find the largest integer $n$ for which $7n\lt2\times10$, which is $2$.
  6. Adjoin $2$ to your answer, getting $9.42$.
  7. $2\times10-7\times2=6$, so find the largest integer for which $7n\lt6\times10$, which is $8$.
  8. Adjoin $8$ to your answer, getting $9.428$.
  9. $6\times10-7\times8=4$, so find the largest integer for which $7n\lt4\times10$, which is $5$.
  10. Adjoin $5$ to your answer, getting $9.4285$.

You can continue with the procedure to get as many decimal places as you wish of $\frac{66}{7}$.

Remark

The sequence of actions just listed is quite difficult to follow. What is difficult is not understanding what they say to do, but where did they get the numbers? So do this exercise!


Exercise worth doing:

Check that the procedure above is exactly what you do to divide $66$ by $7$ by the usual method taught in grammar school:




Remarks
  • The long division process produces as many decimal places as you have stamina for. It is likely for most readers that when you do long division by hand you have done it so much that you know what to do next without having to consult a list of instructions.
  • It is a process or procedure but not what you might want to call a function. The process recursively constructs the successive integers occurring in the decimal expansion of $\frac{66}{7}$.
  • When you carry out the grammar school procedure above, you know at each step what to do next. That is why is it a process. But do you have the procedure in your head all at once?
  • Well, instructions (5) through (10) could be written in a programming language as a while loop, grouping the instructions in pairs of commands ((5) and (6), (7) and (8), and so on). However many times you go through the while loop determines the number of decimal places you get.
  • It can also be described as a formally defined recursive function $F$ for which $F(n)$ is the $n$th digit in the answer.
  • Each of the program and the recursive definition mentioned in the last two bullets are exercises worth doing.
  • Each of the answers to the exercises is then a mathematical object, and that brings us to the next type of metaphor…

Object

A particular kind of metaphor or image for a mathematical concept is that of a mathematical object that represents the concept.

Examples

  • The number $10$ is a mathematical object. The expression “$3^2+1$” is also a mathematical object. It encapsulates the process of squaring $3$ and adding $1$, and so its value is $10$.
  • The long division process above finds the successive decimal places of a fraction of integers. A program that carries out the algorithm encapsulates the process of long division as an algorithm. The result is a mathematical object.
  • The expression “$1958$” is a mathematical object, namely the decimal represen­tation of the number $1958$. The expression
    “$7A6$” is the hexadecimal represen­tation of $1958$. Both represen­tations are mathematical objects with precise definitions.

Represen­tations as math objects is discussed primarily in represen­tations and Models. The difference between represen­tations as math objects and other kinds of mental represen­tations (images and metaphors) is primarily that a math object has a precise mathematical definition. Even so, they are also mental represen­tations.

Uses of mental represen­tations

Mental represen­tations of a concept make up what is arguably the most important part of the mathe­matician’s understanding of the concept.

  • Mental represen­tations of mathe­matical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us) .
  • They are necessary for seeing how the theory can be applied.
  • They are useful for coming up with proofs. (See example below.) 

Many represen­tations

 Different mental represen­tations of the same kind of object
help you understand different aspects of the object. 


Every important mathe­matical object
has many different kinds of represen­tations
and mathe­maticians typically keep
more that one of them in mind at once.

But images and metaphors are also dangerous (see below).

New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us .  But if we work with it for awhile, finding lots of examples, and
eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness.

Then, when someone asks us about this concept that we are now experts with, we
trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathe­maticians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept. They are wrong to do this. That behavior encourages the attitude of many people that

  • Mathe­maticians can’t explain things.
  • Math concepts are incomprehensible or bizarre.
  • You have to have a mathe­matical mind to understand math.

In my opinion the third statement is only about 10 percent true.

All three of these statements are half-truths. There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors. 

Images and metaphors on this website

This website has many examples of useful mental represen­tations.  Usually, when a chapter discusses a particular type of mathe­matical object, say rational numbers, there will be a subhead entitled “Images and metaphors for rational numbers”.  This will suggest ways of thinking about them that many have found useful. 

Two levels of images and metaphors

Images and metaphors have to be used at two different levels, depending on your purpose. 

  • You should expect to use rich view for understanding, applications, and coming up with proofs.
  • You must limit yourself to the rigorous view when constructing and checking proofs.

Math teachers and texts typically do not make an explicit distinction between these views, and you have to learn about it by osmosis. In practice, teachers and texts do make the distinction implicitly.  They will say things
like, “You can think about this theorem as …” and later saying, “Now we give a rigorous proof of the theorem.”  Abstractmath.org makes this distinction explicit in many places throughout the site.

The
rich view

The kind of metaphors and images discussed in the #mentalrepresen­tations>mental represen­tations section above make math rich, colorful and intriguing to think about.  This is the rich view of math.  The rich view is vitally important.  

  • It is what makes math useful and interesting.
  • It helps us to understand the math we are working with.
  • It suggests applications.
  • It suggests approaches to proofs.
Example

You expect the ball whose trajectory is modeled by the function h(t) above  to slow down as it rises, so the derivative of h must be smaller at t
= 4
 than it is at t = 2.  A mathe­matician might even say that that is an “informal proof” that $h'(4)<h'(2)$.  A rigorous proof is given below.

The rigorous view: inertness

When we are constructing a definition or proof we cannot
trust all those wonderful images and metaphors. 

  • Definitions must
    not use metaphors.
  • Proofs must use only logical reasoning based on definitions and
    previously proved theorems.

For the point of view of doing proofs, math
objects must be thought of as inert (or static),
like your pet rock. This means they

  • don’t move or change over time, and
  • don’t interact with other objects, even other mathe­matical objects.

(See also abstract object).

  • When
    mathe­maticians say things like, “Now we give a rigorous proof…”, part of what they mean is that they have to forget about all the color
    and excitement of the rich view and think of math objects as totally
    inert. Like, put the object under an anesthetic
    when you are proving something about it.
  • As I wrote previously, when you are trying to understand arch function $h(t)=25-{{(t-5)}^{2}}$, it helps to think of it as representing a ball thrown directly upward, or as a graph describing the height of the ball at time $t$ which bends over like an arch at the time when the ball stops going upward and begins to fall down.
  • When you proving something about it, you must be in the frame of mind that says the function (or the graph) is all laid out in front of you, unmoving. That is what the rigorous mode requires. Note that the rigorous mode is a way of thinking, not a claim about what the arch function “really is”.
  • When in rigorous mode,  a mathe­matician will
    think of $h$ as a complete mathe­matical object all at once,
    not changing over time. The
    function is the total relationship of the input values of the input parameter
    $t$ to the output values $h(t)$.  It consists of a bunch of interrelated information, but it doesn’t do anything and it doesn’t change.

Formal proof that $h'(4)<h'(2)$

Above, I gave an informal argument for this.   The rigorous way to see that $h'(4)\lt h'(2)$ for the arch function is to calculate the derivative \[h'(t)=10-2t\] and plug in 4 and 2 to get \[h'(4)=10-8=2\] which is less than $h'(2)=10-4=6$.

Note the embedded
phrases
.

This argument picks out particular data about the function that
prove the statement.  It says nothing about anything slowing down as $t$
increases.  It says nothing about anything at all changing.

Other examples

  • The rigorous way to say that “Integers go to infinity in both directions” is something like this:  “For every integer n there is an integer k such that k < n  and an integer m such that n < m.”
  • The rigorous way to say that continuous functions don’t have gaps in their graph is to use the $\varepsilon-\delta $ definition of continuity.
  • Conditional assertions are one important aspect of mathe­matical reasoning in which this concept of unchanging inertness clears up a lot of misunderstanding.   “If… then…” in our intuition contains an idea of causation and of one thing happening before another (see also here).  But if objects are inert they don’t cause anything and if they are unchanging then “when” is meaningless.

The rigorous view does not apply to all abstract objects, but only to mathe­matical objects.  See abstract objects for examples.

Metaphors and images are dangerous

The price of metaphor is eternal vigilance.–Norbert Wiener

Every
mental represen­tation has flaws. Each oneprovides a way of thinking about an $A$ as a kind of $B$ in some respects. But the represen­tation can have irrelevant features.  People new to the subject will be tempted to think  about $A$ as a kind of $B$ in inappropriate respects as well.  This is a form of cognitive dissonance.

 It may be that most difficulties students have with abstract math are based on not knowing which aspects of a given represen­tation are applicable in a given situation.  Indeed, on not being consciously aware that in general you must restrict the applicability of the mental pictures that come with a represen­tation.

In abstractmath.org you will sometimes see this statement:  “What is wrong with this metaphor:”  (or image, or represen­tation) to warn you about the flaws of that particular represen­tation.

Example

The graph of the arch function $h(t)$ makes it look like the two arms going downward become so nearly vertical that the curve has vertical asymptotes
But it does not have asymptotes.  The arms going down are underneath every point of the $x$-axis. For example, there is a point on the curve underneath the point $(999,0)$, namely $(999, -988011)$.

Example

A set is sometimes described as analogous to A container. But consider:  the integer 3 is “in” the set of all odd integers, and it is also “in” the set $\left\{ 1,\,2,\,3 \right\}$.  How could something be in two containers at once?  (More about this here.)

An analogy can be help­ful, but it isn’t the same thing as the same thing. – The Economist

Example

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. See density of the real line.

Example

We commonly think of functions as machines that turn one number into another.  But this does not mean that, given any such function, we can construct a machine (or a program) that can calculate it.  For many functions, it is not only impractical to do, it is theoretically
impossible to do it.
They are not href=”http://en.wikipedia.org/wiki/Recursive_function_theory#Turing_computability”>computable. In other words, the machine picture of a function does not apply to all functions.

Summary


The images and metaphors you use
to think about a mathe­matical object
are limited in how they apply.


The images and metaphors you use to think about the subject
cannot be directly used in a proof.
Only definitions and previously proved theorems can be used in a proof.

Final remarks

Mental represen­tations are physical represen­tations

It seems likely that cognitive phenomena such as images and metaphors are physically represented in the brain as collec­tions of neurons connected in specific ways.  Research on this topic is pro­ceeding rapidly.  Perhaps someday we will learn things about how we think physi­cally that actually help us learn things about math.

In any case, thinking about mathe­matical objects as physi­cally represented in your brain (not neces­sarily completely or correctly!) wipes out a lot of the dualistic talk about ideas and physical objects as
separate kinds of things.  Ideas, in partic­ular math objects, are emergent constructs in the
physical brain. 

About metaphors

The language that nature speaks is mathe­matics. The language that ordinary human beings speak is metaphor. Freeman Dyson

“Metaphor” is used in abstractmath.org to describe a type of thought configuration.  It is an implicit conceptual identification
of part of one type of situation with part of another. 

Metaphors are a fundamental way we understand the world. In particular,they are a fundamental way we understand math.

The word “metaphor”

The word “metaphor” is also used in rhetoric as the name of a type of figure of speech.  Authors often refer to metaphor in the meaning of  thought configuration as a conceptual metaphor.  Other figures of speech, such as simile and synecdoche, correspond to conceptual metaphors as well.

References for metaphors in general cognition:

Fauconnier, G. and Turner, M., The Way We Think: Conceptual Blending And The Mind’s Hidden Complexities . Basic Books, 2008.

Lakoff, G., Women, Fire, and Dangerous Things. The University of Chicago Press, 1986.

Lakoff, G. and Mark Johnson, Metaphors We Live By
The University of Chicago Press, 1980.

References for metaphors and images in math:

Byers, W., How mathe­maticians Think.  Princeton University Press, 2007.

Lakoff, G. and R. E. Núñez, Where mathe­matics Comes
From
. Basic Books, 2000.

Math Stack Exchange list of explanatory images in math.

Núñez, R. E., “Do Real Numbers Really Move?”  Chapter
in 18 Unconventional Essays on the Nature of mathe­matics, Reuben Hersh,
Ed. Springer, 2006.

Charles Wells,
Handbook of mathe­matical Discourse.

Charles Wells, Conceptual blending. Post in Gyre&Gimble.

Other articles in abstractmath.org

Conceptual and computational

Functions: images and metaphors

Real numbers: images and metaphors

represen­tations and models

Sets: metaphors and images

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Pattern recognition in understanding math

Abstract patterns

This post is a revision of the article on pattern recognition in abstractmath.org.

When you do math, you must recognize abstract patterns that occur in

  • Symbolic expressions
  • Geometric figures
  • Relations between different kinds of math structures.
  • Your own mental representations of mathematical objects

This happens in high school algebra and in calculus, not just in the higher levels of abstract math.

Examples

Most of these examples are revisited in the section called Laws and Constraints.

At most

For real numbers $x$ and $y$, the phrase “$x$ is at most $y$” means by definition $x\le y$. To understand this definition requires recognizing the pattern “$x$ is at most $y$” no matter what expressions occur in place of $x$ and $y$, as long as they evaluate to real numbers.

Examples

  • “$\sin x$ is at most $1$” means that $\sin x\le 1$. This happens to be true for all real $x$.
  • “$3$ is at most $7$” means that $3\leq7$. You may think that “$3$ is at most $7$” is a silly thing to say, but it nevertheless means that $3\leq7$ and so is a correct statement.
  • “$x^2+(y-1)^2$ is at most $5$” means that
    $x^2+(y-1)^2\leq5$. This is true for some pairs $(x,y)$ and false for others, so it is a constraint. It defines the disk below:

The product rule for derivatives

The product rule for differentiable functions $f$ and $g$ tells you that the derivative of $f(x)g(x)$ is \[f'(x)\,g(x)+f(x)\,g'(x)\]

Example

You recognize that the expression ${{x}^{2}}\sin x$ fits the pattern $f(x)g(x)$ with $f(x)={{x}^{2}}$ and $g(x)=\sin x$. Therefore you know that the derivative of ${{x}^{2}}\,\sin x$ is \[2x\sin x+{{x}^{2}}\cos x\]

The quadratic formula

The quadratic formula for the solutions of an equation of the form $a{{x}^{2}}+bx+c=0$ is usually given as\[r=\frac{-b\pm
\sqrt{{{b}^{2}}-4ac}}{2a}\]

Example

If you are asked for the roots of $3{{x}^{2}}-2×-1=0$, you recognize that the polynomial on the left fits the pattern $a{{x}^{2}}+bx+c$ with

  • $a\leftarrow3$ (“$a$ replaced by $3$”)
  • $b\leftarrow-2$
  • and $c\leftarrow-1$.

Then
substituting those values in the quadratic formula gives you the roots $-1/3$ and $1$.

Difficulties with the quadratic formula

A little problem

The quadratic formula is easy to use but it can still cause pattern recognition problems. Suppose you are asked to find the solutions of $3{{x}^{2}}-7=0$. Of course you can do this by simple algebra — but pretend that the first thing you thought of was using the quadratic formula.

  • Then you got upset because you have to apply it to $a{{x}^{2}}+bx+c$
  • and $3{{x}^{2}}-7$ has only two terms
  • but $a{{x}^{2}}+bx+c$ has three terms…
  • (Help!)
  • Do Not Be Anguished:
  • Write
    $3{{x}^{2}}-7$ as $3{{x}^{2}}+0\cdot x-7$, so $a=3$, $b=0$ and $c=-7$.
  • Then put those values into the quadratic formula and you get $x=\pm \sqrt{\frac{7}{3}}$.   
  • This is an example of the following useful principle:


    Write zero cleverly.

    I suspect that most people reading this would not have had the problem with $3{{x}^{2}}-7$ that I have just described. But before you get all insulted, remember:


    The thing about really easy examples is that they give you the point without getting you lost in some complicated stuff you don’t understand very well.

    A fiendisher problem

      Even college students may have trouble with the following problem (I know because I have tried it on them):

    What are the solutions of the equation $a+bx+c{{x}^{2}}=0$?

    The answer

             

    \[r=\frac{-b\pm
    \sqrt{{{b}^{2}}-4ac}}{2a}\]

    is wrong. The correct answer is

                                     \[r=\frac{-b\pm
    \sqrt{{{b}^{2}}-4ac}}{2c}\]


    When you remember a pattern with particular letters in it and an example has some of the same letters in it, make sure they match the pattern!

    The substitution rule for integration

    The chain rule says that the derivative of a function of the form $f(g(x))$ is $f'(g(x))g'(x)$. From this you get the substitution rule for finding indefinite integrals:

                                      \[\int{f'(g(x))g'(x)\,dx}=f(g(x))+C\]

    Example

    To find $\int{2x\,\cos
    ({{x}^{2}})\,dx}$, you recognize that you can take $f(x)=\sin x$and $g(x)={{x}^{2}}$ in the formula, getting \[\int{2x\,\cos ({{x}^{2}})\,dx}=\sin ({{x}^{2}})\]    Note that in the way I wrote the integral, the functions occur in the opposite order from the pattern. That kind of thing happens a lot.

    Laws and constraints

    • The statement “$(x+1)^2=x^2+2x+1$” is a pattern that is true for all numbers $x$. $3^2=2^2+2\times2+1$ and $(-2)^2=(-1)^2+2\times(-1)+1$, and so on. Such a pattern is a universal assertion, so it is a theorem. When the statement is an equation, as in this case, it is also called a law.
    • The statement “$\sin x\leq 1$” is also true for all $x$, and so is a theorem.
    • The statement “$x^2+(y-1)^2$ is at most $5$” is true for some real numbers and not others, so it is not a theorem, although it is a constraint.
    • The quadratic formula says that:
      The solutions of an equation
      of the form $a{{x}^{2}}+bx+c=0$ is
      given by\[r=\frac{-b\pm
      \sqrt{{{b}^{2}}-4ac}}{2a}\]

      This is true for all complex numbers $a$, $b$, $c$.
      The $x$ in the equation is not a free variable, but a “variable to be solved for” and does not appear in the quadratic formula. Theorems like the quadratic formula are usually called “formulas” rather than “laws”.

    • The product rule for derivatives

      The derivative of $f(x)g(x)$ is $f'(x)\,g(x)+f(x)\,g'(x)$

      is true for all differentiable functions $f$ and $g$. That means it is true for both of these choices of $f$ and $g$:

      • $f(x)=x$ and $g(x)=x\sin x$
      • $f(x)=x^2$ and $g(x)=\sin x$

      But both choices of $f$ and $g$ refer to the same function $x^2\sin x$, so if you apply the product rule in either case you should get the same answer. (Try it).

    Some bothersome types of pattern recognition

    Dependence on conventions

    Definition: A quadratic polynomial in $x$is an expression of the form $a{{x}^{2}}+bx+c$.   

    Examples

    • $-5{{x}^{2}}+32×-5$ is a quadratic polynomial: You have to recognize that it fits the pattern in the definition by writing it as $(-5){{x}^{2}}+32x+(-5)$
    • So is ${{x}^{2}}-1$: You have to recognize that it fits the definition by writing it as ${{x}^{2}}+0\cdot x+(-1)$ (I wrote zero cleverly).

    Some authors would just say, “A quadratic polynomial is an expression of the form $a{{x}^{2}}+bx+c$” leaving you to deduce from conventions on variables that it is a polynomial in $x$ instead of in $a$ (for example).

    Note also that I have deliberately not mentioned what sorts of numbers $a$, $b$, $c$ and $x$ are. The authors may assume that you know they are using real numbers.

    An expression as an instance of substitution

    One particular type of pattern recognition that comes up all the time in math is recognizing that a given expression is an instance of a substitution into a known expression.

    Example

    Students are sometimes baffled when a proof uses the fact that ${{2}^{n}}+{{2}^{n}}={{2}^{n+1}}$ for positive integers $n$. This requires the recognition of the patterns $x+x=2x$ and $2\cdot
    \,{{2}^{n}}={{2}^{n+1}}$.

    Similarly ${{3}^{n}}+{{3}^{n}}+{{3}^{n}}={{3}^{n+1}}$.

    Example

    The assertion

    \[{{x}^{2}}+{{y}^{2}}\ge 0\ \ \ \ \ \text{(1)}\]

    has as a special case

    \[(-x^2-y^2)^2+(y^2-x^2)^2\ge
    0\ \ \ \ \ \text{(2)}\]

    which involves the substitutions $x\leftarrow -{{x}^{2}}-{{y}^{2}}$ and $y\leftarrow
    {{y}^{2}}-{{x}^{2}}$.

    Remarks
    • If you see (2) in a text and the author blithely says it is “never negative”, that is because it is of the form \[{{x}^{2}}+{{y}^{2}}\ge 0\] with certain expressions substituted for $x$ and $y$. (See substitution and The only axiom for algebra.)
    • The fact that there are minus signs in (2) and that $x$ and $y$ play different roles in (1) and in (2) are red herrings. See ratchet effect and variable clash.
    • Most people with some experience in algebra would see quickly that (2) is correct by using chunking. They would visualize (2) as

      \[(\text{something})^2+(\text{anothersomething})^2\ge0\]
      This shows that in many cases


      chunking is a psychological inverse to substitution

    • Note that when you make these substitutions you have to insert appropriate parentheses (more here). After you make the substitution, the expression of course can be simplified a whole bunch, to

      \[2({{x}^{4}}+{{y}^{4}})\ge0\]

    • A common cause of error in doing this (a mistake I make sometimes) is to try to substitute and simplify at the same time. If the situation is complicated, it is best to

      substitute as literally as possible and then simplify

    Integration by Parts

    The rule for integration by parts says that

                             \[\int{f(x)\,g'(x)\,dx=f(x)\,g(x)-\int{f'(x)\,g(x)\,dx}}\]

    Suppose you need to find $\int{\log x\,dx}$.(In abstractmath.org, “log” means ${{\log }_{e}}$).  Then we can recognize this integral as having the pattern for the left side of the parts formula with $f(x)=1$ and $g(x)=\log \,x$. Therefore

    \[\int{\log x\,dx=x\log x-\int{\frac{1}{x}dx=x\log \,x-x+c}}\]

    How on earth did I think to recognize $\log x$ as $1\cdot \log x$??  
    Well, to tell the truth because some nerdy guy (perhaps I should say some other nerdy guy) clued me in when I was taking freshman calculus. Since then I have used this device lots of times without someone telling me — but not the first time.

    This is an example of another really useful principle:


    Write $1$ cleverly.

    Two different substitutions give the same expression

    Some proofs involve recognizing that a symbolic expression or figure fits a pattern in two different ways. This is illustrated by the next two examples. (See also the remark about the product rule above.) I have seen students flummoxed by Example ID, and Example ISO is a proof that is supposed to have flummoxed medieval geometry students.

    Example ID

    Definition: In a set with an associative binary operation and an identity element $e$, an element $y$ is the inverse of an element $x$ if

    \[xy=e\ \ \ \ \text{and}\ \ \ \ yx=e \ \ \ \ (1)\]

    In this situation, it is easy to see that $x$ has only one inverse: If $xy=e$ and $xz=e$ and $yx=e$ and $zx=e$, then \[y=ey=(zx)y=z(xy)=ze=z\]

    Theorem: ${{({{x}^{-1}})}^{-1}}=x$.

    Proof: I am given that ${{x}^{-1}}$ is the inverse of $x$, By definition, this means that

    \[x{{x}^{-1}}=e\ \ \ \text{and}\ \ \ {{x}^{-1}}x=e \ \ \ \ (2)\]

    To prove the theorem, I must show that $x$ is the inverse of ${{x}^{-1}}$. Because $x^{-1}$ has only one inverse, all we have to do is prove that

    \[{{x}^{-1}}x=e\ \ \ \text{and}\ \ \ x{{x}^{-1}}=e\ \ \ \ (3)\]  

    But (2) and (3) are equivalent! (“And” is commutative.)

    Example ISO

    This sort of double substitution occurs in geometry, too.

    Theorem: If a triangle has two equal angles, then it has two equal sides.

    Proof: In the figure, assume $\angle ABC=\angle ACB$. Then triangle $ABC$ is congruent to triangle $ACB$ since the sides $BC$ and $CB$ are equal (they are the same line segment!) and the adjoining angles are equal by hypothesis.

    The point is that although triangles $ABC$ and $ACB$ are the same triangle, and sides $BC$ and $CB$ are the same line segment, the proof involves recognizing them as geometric figures in two different ways.

    This proof (not Euclid’s origi­nal proof) is hundreds of years old and is called the pons asinorum (bridge of donkeys). It became famous as the first theorem in Euclid’s books that many medi­eval stu­dents could not under­stand. I con­jecture that the name comes from the fact that the triangle as drawn here resembles an ancient arched bridge. These days, isos­ce­les tri­angles are usually drawn taller than they are wide.

    Technical problems in carrying out pattern matching

    Parentheses

    In matching a pattern you may have to insert parentheses. For example, if you substitute $x+1$ for $a$, $2y$ for
    $b$ and $4$ for $c$ in the expression \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] you get \[{{(x+1)}^{2}}+4{{y}^{2}}=16\]
    If you did the substitution literally without editing the expression so that it had the correct meaning, you would get \[x+{{1}^{2}}+2{{y}^{2}}={{4}^{2}}\] which is not the result of performing the substitution in the expression ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$.   

    Order switching

    You can easily get confused if the patterns involve a switch in the order of the variables.

    Notation for integer division

    • For integers $m$ and $n$, the phrase “$m$ divides $n$” means there is an integer $q$ for which $n=qm$.
    • In number theory (which in spite of its name means the theory of positive integers) the vertical bar is used to denote integer division. So $3|6$ because $6=2\times 3$ ($q$ is $2$ in this case). But “$3|7$” is false because there is no integer $q$ for which $7=q\times 3$.
    • An equivalent definition of division says that $m|n$ if and only if $n/m$ is an integer. Note that $6/3=2$, an integer, but $7/3$ is not an integer.
    • Now look at those expressions:
    • “$m|n$” means that there is an integer $q$ for which $n=qm$.In these two expressions, $m$ and $n$ occur in opposite order.
    • “$m|n$” is true only if $n/m$ is an integer. Again, they are in opposite order. Another way of writing $n/m$ is $\frac{n}{m}$. When math people pronounce “$\frac{n}{m}$” they usually say, “$n$ over $m$” using the same order.
  • I taught these notation in courses for computer engineering and math majors for years. Some of the students stayed hopelessly confused through several lectures and lost points repeatedly on homework and exams by getting these symbols wrong.
  • The problem was not helped by the fact that “$|$” and “$/$” are similar but have very different syntax:

    Math notation gives you no clue which symbols are operators (used to form expressions) and which are verbs (used to form assertions).

  • A majority of the students didn’t have so much trouble with this kind of syntax. I have noticed that many people have no sense of syntax and other people have good intuitive understanding of syntax. I suspect the second type of people find learning foreign languages easy.
  • Many of the articles in the references below concern syntax.
  • References

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    The only axiom of algebra

    This is one of a series of posts I am writing to help me develop my thoughts about how particular topics in my book Abstracting Algebra (“AbAl“) should be organized. This post concerns the relation between substitution and evaluation that essentially constitutes the definition of algebra. The Mathematica code for the diagrams is in Subs Eval.nb.

    Substitution and evaluation

    This post depends heavily on your understanding of the ideas in the post Presenting binary operations as trees.

    Notation for evaluation

    I have been denoting evaluation of an expression represented as a tree like this:



    In standard algebra notation this would be written:\[(6-4)-1=2-1=1\]

    Comments

    This treatment of evaluation is intended to give you an intuition about evaluation that is divorced from the usual one-dimensional (well, nearly) notation of standard algebra. So it is sloppy. It omits fine points that will have to be included in AbAl.

    • The evaluation goes from bottom up until it reaches a single value.
    • If you reach an expression with an empty box, evaluation stops. Thus $(6-3)-a$ evaluates only to $3-a$.
    • $(6-a)-1$ doesn’t evaluate further at all, although you can use properties peculiar to “minus” to change it to $5-a$.
    • I used the boxed “1” to show that the value is represented as a trivial tree, not a number. That’s so it can be substituted into another tree.

    Notation for substitution

    I will use a configuration like this

    to indicate the data needed to substitute the lower tree into the upper one at the variable (blank box). The result of the substitution is the tree

    In standard algebra one would say, “Substitute $3\times 4$ for $a$ in the expression $a+5$.” Note that in doing this you have to name the variable.

    Example

    “If you substitute $12$ for $a$ in $a+5$ you get $12+5$”:

    results in

    Example

    “If you substitute $3\times 4$ for $a$ in $a+b$ you get $3\times4+b$”:

    results in

    Comments

    Like evaluation, this treatment of substitution omits details that will have to be included in AbAl.

    • You can also substitute on the right side.
    • Substitution in standard algebraic notation often requires sudden syntactic changes because the standard notation is essentially two-dimensional. Example: “If you substitute $3+ 4$ for $a$ in $a\times b$ you get $(3+4)\times b$”.
    • The allowed renaming of free variables except when there is a clash causes students much trouble. This has to be illustrated and contrasted with the “binop is tree” treatment which is context-free. Example: The variable $b$ in the expression $(3\times 4)+b$ by itself could be changed to $a$ or $c$, but in the sentence “If you substitute $3+ 4$ for $a$ in $a\times b$ you get $(3+4)\times b$”, the $b$ is bound. It is going to be difficult to decide how much of this needs explaining.

    The axiom

    The Axiom for Algebra says that the operations of substitution and evaluation commute: if you apply them in either order, you get the same resulting tree. That says that for the current example, this diagram commutes:

    The Only Axiom for Algebra

    In standard algebra notation, this becomes:

    • Substitute, then evaluate: If $a=3\times 4$, then $a+5=3\times 4+5=12+5$.
    • Evaluate, then substitute: If $a=3\times 4$, then $a=12$, so $a+5=12+5$.

    Well, how underwhelming. In ordinary algebra notation my so-called Only Axiom amounts to a mere rewording. But that’s the point:


    The Only Axiom of Algebra is what makes algebraic manipulation work.

    Miscellaneous comments

    • In functional notation, the Only Axiom says precisely that $\text{eval}∘\text{subst}=\text{subst}∘(\text{eval},\text{id})$.
    • The Only Axiom has a symmetric form: $\text{eval}∘\text{subst}=\text{subst}∘(\text{id},\text{eval})$ for the right branch.
    • You may expostulate: “What about associativity and commutativity. They are axioms of algebra.” But they are axioms of particular parts of algebra. That’s why I include examples using operations such as subtraction. The Only Axiom is the (ahem) only one that applies to all algebraic expressions.
    • You may further expostulate: Using monads requires the unitary or oneidentity axiom. Here that means that a binary operation $\Delta$ can be applied to one element $a$, and the result is $a$. My post Monads for high school III. shows how it is used for associative operations. The unitary axiom is necessary for representing arbitrary binary operations as a monad, which is a useful way to give a theoretical treatment of algebra. I don’t know if anyone has investigated monads-without-the-unitary-axiom. It sounds icky.
    • The Only Axiom applies to things such as single valued functions, which are unary operations, and ternary and higher operations. They also apply to algebraic expressions involving many different operations of different arities. In that sense, my presentation of the Only Axiom only gives a special case.
    • In the case of unary operations, evaluation is what we usually call evaluation. If you think about sets the way I do (as a special kind of category), evaluation is the same as composition. See “Rethinking Set Theory”, by Tom Leinster, American Mathematical Monthly, May, 2014.
    • Calculus functions such as sine and the exponential are unary operations. But not all of calculus is algebra, because substitution in the differential and integral operators is context-sensitive.

    References

    Preceding posts in this series

    Remarks concerning these posts
    • Each of the posts in this series discusses how I will present a small part of AbAl.
    • The wording of some parts of the posts may look like a first draft, and such wording may indeed appear in the text.
    • In many places I will talk about how I should present the topic, since I am not certain about it.

    Other references

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