Tag Archives: algebra

Mathematical Information I

Introduction

The January, 2016 meeting of the American Mathematical Society in Seattle included a special session on Mathe­matical Information in the Digital Age of Science. Here is a link to the list of talks in that session (you have to scroll down a ways to get to the list).

Several talks at that session were about communi­cating math, to other mathe­maticians and to the general public. Well, that’s what I have been about for the last 20 years. Mostly.

Overview

These posts discuss the ways we communi­cate math and (mostly in later posts) the revolution in math communication that the internet has caused. Parts of this discussion were inspired by the special session talks. When they are relevant, I include footnotes referring to the talks. Be warned that what I say about these ideas may not be the same as what the speakers had to say, but I feel I ought to give them credit for getting me to think about those concepts.

Some caveats

  • The distinctions between different kinds of math communi­cation are inevitably fuzzy.
  • Not all kinds of communication are mentioned.
  • Several types of communication normally occur in the same document.

Articles published in journals

Until recently, math journals were always published on paper. Now many journals exist only on the internet. What follows is a survey of the types of articles published in journals.

Refereed papers containing new results

These communications typically containing proofs of (usually new) theorems. Such papers are the main way that academic mathematicians get credit for their researchG for the purpose of getting tenure (at least in the USA), although some other types of credit are noted below.

Proofs published in refereed journals in the past were generally restricted to formal proofs, without very many comments intended to aid the reader’s under­standing. This restricted text was often enforced by the journal. In the olden days this would have been prompted by the expense of publishing on paper. I am not sure how much this restriction has relaxed in electronic journals.

I have been writing articles for abstractmath.org and Gyre&Gimble for many years, and it has taken me a very long time to get over unnecessarily restricting the space I use in what I write. If I introduce a diagram in an article and then want to refer to it later, I don’t have to link to it — I can copy it into the current location. If it makes sense for an informative paragraph to occur in two different articles, I can put it into both articles. And so on. Nowadays, that sort of thing doesn’t cost anything.

Survey articles and invited addresses

You may also get credit for an invited address to a prestigious organi­zation, or for a survey of your field, in for example the Bulletin of the AMS. Invited addresses and surveys may contain considerably more explanatory asides. This was quite noticeable in the invited talks at the AMS Seattle meeting.

Books

There is a whole spectrum of math books. The following list mentions some Fraunhofer lines on the spectrum, but the gamut really is as continuous as a large finite list of books could be. This list needs more examples. (This is a blog post, so it has the status of an alpha release.)

Research books that are concise and without much explanation.

The Bourbaki books that I have dipped into (mostly the algebra book and mostly in the 1970’s) are definitely concise and seem to strictly avoid explanation, diagrams, pictures, etc). I have heard people say they are unreadable, but I have not found them so.

Contain helpful explanations that will make sense to people in the field but probably would be formidable to someone in a substantially different area.

Toposes, triples and theories, by Michael Barr and Charles Wells. I am placing our book here in the spectrum because several non-category-theorists (some of them computer scientists) have remarked that it is “formidable” or other words like that.

Intended to introduce professional mathematicians to a particular field.

Categories for the working mathematician, by Saunders Mac Lane. I learned from this (the 1971 edition) in my early days as a category theorist, six years after getting my Ph.D. In fact, I think that this book belongs to the grad student level instead of here, but I have not heard any comments one way or another.

Intended to introduce math graduate students to a particular field.

There are lots of examples of good books in this area. Years ago (but well after I got my Ph.D.), I found Serge Lang’s Algebra quite useful and studied parts of it in detail.

But for grad students? It is still used for grad students, but perhaps Nathan Jacobson’s Basic Algebra would be a better choice for a first course in algebra for first-year grad students.

The post My early life as a mathematician discusses algebra texts in the olden days, among other things.

Intended to explain a part of math to a general audience.

Love and math: the heart of hidden reality. by Edward Frenkel, 2014. This is a wonderful book. After reading it, I felt that at last I had some clue as to what was going on with the Langlands Program. He assumes that the reader knows very little about math and gives hand-waving pictorial expla­nations for some of the ideas. Many of the concepts in the book were already familiar to me (not at an expert level). I doubt that someone who had had no college math courses that included some abstract math would get much out of it.

Symmetry: A Journey into the Patterns of Nature, by Marcus du Sautoy, 2009. He also produced a video on symmetry.

My post Explaining “higher” math to beginners, describes du Sautoy’s use of terminology (among others).

Secrets of creation: the mystery of the prime numbers (Volume 1) by Matthew Watkins (author) and Matt Tweed (Illustrator), 2015. This is the first book of a trilogy that explains the connection between the Riemann $\zeta$ function and the primes. He uses pictures and verbal descriptions, very little terminology or symbolic notation. This is the best attempt I know of at explaining deep math that might really work for non-mathe­maticians.

My post The mystery of the prime numbers: a review describes the first book.

Piper Harron’s Thesis

The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering, Ph.D. thesis by Piper Harron.

This is a remarkable departure from the usual dry, condensed, no-useful-asides Ph.D. thesis in math. Each chapter has three main parts, Layscape (explanations for nonspecialists — not (in my opinion) for nonmathe­maticians), Mathscape (most like what goes into the usual math paper but with much more explanation) and Weedscape (irrelevant stuff which she found helpful and perhaps the reader will too). The names of these three sections vary from chapter to chapter. This seems like a great idea, and the parts I have read are well-done.

These blog posts have useful comments about her thesis:

Types of explanations

Any explanation of math in any of the categories above will be of several different types. Some of them are considered here, and more will appear in Mathematical Information II.

The paper Varieties of Mathematical Prose, by Atish Bagchi and me, provides a more fine-grained description of certain types of math communication that includes some types of explanations and also other types of communication.

Images and metaphors

In abstractmath.org

I have written about images and metaphors in abstractmath.org:

Abstractmath.org is aimed at helping students who are beginning their study of abstract math, and so the examples are mostly simple and not at a high level of abstraction. In the general literature, the images and metaphors that are written about may be much more sophisticated.

The User’s GuideW

Luke Wolcott edits a new journal called Enchiridion: Mathematics User’s Guides (this link allows you to download the articles in the first issue). Each article in this journal is written by a mathematician who has published a research paper in a refereed journal. The author’s article in Enchiridion provides information intended to help the reader to understand the research paper. Enchiridion and its rationale is described in more detail in the paper The User’s Guide Project: Giving Experential Context to Research Papers.

The guidelines for writing a User’s Guide suggest writing them in four parts, and one of the parts is to introduce useful images and metaphors that helped the author. You can see how the authors’ user’s guides carry this out in the first issue of Enchiridion.

Piper Harron’s thesis

Piper Harron’s explanation of integrals in her thesis is a description of integrals and measures using creative metaphors that I think may raise some mathematicians’ consciousness and others’ hackles, but I doubt it would be informative to a non-mathematician. I love “funky-summing” (p. 116ff): it communicates how integration is related to real adding up a finite bunch of numbers in a liberal-artsy way, in other words via the connotations of the word “funky”, in contrast to rigorous math which depends on every word have an accumulation-of-properties definition.

The point about “funky-summing” (in my opinion, not necessarily Harron’s) is that when you take the limit of all the Riemann sums as all meshes go to zero, you get a number which

  • Is really and truly not a sum of numbers in any way
  • Smells like a sum of numbers

Connotations communicate metaphors. Metaphors are a major cause of grief for students beginning abstract math, but they are necessary for understanding math. Working around this paradox is probably the most important problem for math teachers.

Informal summaries of a proofW

The User’s Guide requires a “colloquial summary” of a paper as one of the four parts of the guide for that paper.

  • Wolcott’s colloquial summary of his paper keeps the level aimed at non-mathematicians, starting with a hand-waving explanation of what a ring is. He uses many metaphors in the process of explaining what his paper does.
  • The colloquial summary of another User’s Guide, by Cary Malkiewich, stays strictly at the general-public level. He uses a few metaphors. I liked his explanation of how mathematicians work first with examples, then finding patterns among the examples.
  • The colloquial summary of David White’s paper stays at the general-public level but uses some neat metaphors. He also has a perceptive paragraph discussing the role of category theory in math.

The summaries I just mentioned are interesting to read. But I wonder if informal summaries aimed at math majors or early grad students might be more useful.

Insights

The first of the four parts of the explanatory papers in Enchiridion is supposed to present the key insights and organizing principles that were useful in coming up with the proofs. Some of them do a good job with this. They are mostly very special to the work in question, but some are more general.

This suggests that when teaching a course in some math subject you make a point of explaining the basic techniques that have turned out very useful in the subject.

For example, a fundamental insight in group theory is:

Study the linear representations of a group.

That is an excellent example of a fundamental insight that applies everywhere in math:

Find a functor that maps the math objects you are studying to objects in a different branch of math.

The organizing principles listed in David White’s article has (naturally more specialized) insights like that.

Proof stories

“Proof stories” tell in sequence (more or less) how the author came up with a proof. This means describing the false starts, insights and how they came about. Piper Harron’s thesis does that all through her work.

Some authors do more than that: their proof stories intertwine the mathe­matical events of their progress with a recount of life events, which sometimes make a mathe­matical difference and sometimes just produces a pause to let the proof stew in their brain. Luke Wolcott wrote a User’s Guide for one of his own papers, and his proof story for that paper involves personal experiences. (I recommend his User’s Guide as a model to learn from.)

Reports of personal experiences in doing math seem to add to my grasp of the math, but I am not sure I understand why.

References

The talks in Seattle

  • List of all the talks.
  • W. Timothy Gowers, How should mathe­matical knowledge be organized? Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.
  • Colloquium notes. Gowers gave a series of invited addresses for which these are the notes. They have many instances of describing what sorts of problems obstruct a desirable step in the proof and what can be done about it.

  • Luke Wolcott, The User’s Guide. Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.

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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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Presenting binops as trees

Binary operations as trees

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. In some parts, I present various options that I have not decided between.

This post concerns the presen­ta­tion of binary operations as trees. The Mathematica code for the diagrams is in Substitution in algebra.nb

Binary operations as functions

A binary operation or binop $\Delta$ is a function of two variables whose value at $(a,b)$ is traditionally denoted by $a\Delta b$. Most commonly, the function is restricted to having inputs and outputs in the same set. In other words, a binary operation is a function $\Delta:S\times S\to S$ defined on some set $S$. $S$ is the underlying set of the operation. For now, this will be the definition, although binops may be generalized to multiple sets later in the book.

In AbAl:

  • Binops will be defined as functions in the way just described.
  • Algebraic expressions will be represented
    as trees, which exhibit more clearly the structure of the expressions that is encoded in algebraic notation.
  • They will also be represented using the usual infix expressions such as “$3\times 5$” and “$3-5$”,

Fine points

The definition of a binop as a function has termi­no­logical consequences. The correct point of view concerning a function is that it determines its domain and its codomain. In particular:

A binary operation determines its underlying set.

Thus if we talk about an arbitrary binop $\Delta$, we don’t have to give a name to its underlying set. We can just say “the underlying set of $\Delta$” or “$U(\Delta)$”.

Examples

“$+$” is not one binary operation.

  • $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a binary operation.
  • $+:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is another binary operation.

Mathematicians commonly refer to these particular binops as “addition on the integers” and “addition on the reals”.

Remark

You almost never see this attitude in textbooks on algebra. It is required by both category theory and type theory, two Waves flooding into math. Category theory is a middle-aged Wave and type theory, in the version of homo­topy type theory, is a brand new baby Wave. Both Waves have changed and will change our under­standing of math in deep ways.

Trees

An arbitrary binop $\Delta$ can be represented as a binary tree in this way:

generic binop

This tree represents the expression that in standard algebraic notation is “$a\Delta b$”.

In more detail, the tree is an ordered rooted binary tree. The “ordered” part means that the leaves (nodes with no descendants) are in a specific left to right order. In AbAl, I will define trees in some detail, with lots of pictures.

The root shows the operation and the two leaves show elements of the underlying set. I follow the custom in computing science to put the root at the top.

Metaphors should not dictate your life by being taken literally.

Remark

The Wikipedia treatment of trees is scat­tered over many articles and they almost always describe things mostly in words, not pictures. Describing math objects in words when you could use pictures is against my religion. Describing is not the same as defining, which usually requires words.

Some concrete examples:



    
    

3trees

These are represen­ta­tions of the expressions “$3+5$”, “$3\times5$”, and “$3-5$”.

Just as “$5+3$” is a different expression from “$3+5$”, the left tree in 3trees above is a different expression from this one:



    

switch

They have the same value, but they are distinct as expressions — otherwise, how could you state the commutative law?

Fine points

I regard an expression as an abstract math object that can have many repre­sentations. For example “$3+5$” and the left tree in 3trees are two different represen­ta­tions of the same (abstract) expression. This deviates from the usual idea that “expression” refers to a typographical construction.

In previous posts, when the operation is not commutative, I have sometimes labeled the legs like this:


I have thought about using this notation consistently in AbAl, but I suspect it would be awkward in places.

Evaluation and substitution

The two basic operations on algebraic expressions
are evaluation and substitution.

They and the Only Axiom of Algebra, which I will discuss in a later post, are all that is needed to express the true nature of algebra.

Evaluation

  • If you evaluate $3+5$ you get $8$.
  • If you evaluate $3\times 5$ you get $15$.
  • If you evaluate $3-5$ you get $-2$.

I will show evaluation on trees like this:




Evaluation with trace

A more elaborate version, valuation with trace, would look like this. This allows you to keep track of where the valuations come from.




You could also keep track of the operation used at each node. An interactive illustration of this is in the post Visible algebra I supplement. That illustration requires CDF Player to be installed on your computer. You can get it free from the Mathematica website.

Variables

In the tree above, the $a$ and $b$ are variables, just as they are in the equivalent expression $a\Delta b$. Algebra beginners have a hard time understanding variables.

  • You can’t evaluate an expression until you substitute numbers for the letters, which produces an instance of expression. (“Instance” is the preferable name for this, but I often refer to such a thing as an “example”.)
  • If a variable is repeated you have to substitute the same value for each occurrence. So $a\Delta b$ is a different expression from $a\Delta a$: $2+3$ is an instance of $a+b$ but it is not an instance of $a+a$. But $a\Delta a$ and $b\Delta b$ are the same expression: any instance of one is an instance of the other.
  • Substitute $a\Delta b$ for $a$ in $a\Delta b$ and you get $(a\Delta b)\Delta b$. You may have committed variable clash. You might have meant $(a\Delta b)\Delta c$. (Somebody please tell me a good link that describes variable clash.)
  • Later, you will deal with multiplication tables for algebraic structures. There the elements are denoted by letters of the alphabet. They can’t be substituted for.

Empty boxes

A straightforward way to denote variables would be to use empty boxes:

The idea is that a number (element of the underlying set) can be inserted in each box. If $3$ (left) and $5$ (right) are placed in the boxes, evaluation would place the value of $3\Delta5$ in the root. Each empty box represents a separate variable.

Empty boxes could also be used in the standard algebraic notation: $\Delta$ or $+$ or $-$.
I have seen that notation in texts explaining variables, but I don’t know a reference. I expect to use this notation with trees in AbAl.

To achieve the effect of one variable in two different places, as in

we can cause it to repeat, as below, where “$\text{id}$” denotes the identity function on the underlying set:

To evaluate at a number (member of the underlying set) you insert a number into the only empty box

which evaluates to

which of course evaluates to $3\Delta3$.

This way of treating repeated variables exhibits the nature of repeated variables explicitly and naturally, putting the values automatically in the correct places. This process, like everything in this section, comes from monad theory. It also reminds me of linear logic in that it shows that if you want to use a value more than once you have to copy it.

Substitution

Given two binary trees



      

you could attach the root of the first one to one of the leaves of the second one, in two different ways, to get these trees:



      


2trees

which in standard algebra notation would be written $(a-b)-c)$ and $a-(b-c)$ respectively. Note that this tree



would be represented in algebra as $(a-b)-b$.

In general, substituting a tree for an input (variable or empty box) consists of replacing the empty box by the whole tree, identifying the root of the new tree with the empty box. In graph theorem, “substitution” may be called “grafting”, which is a good metaphor.

You can evaluate the left tree in 2trees at particular numbers to evaluate it in two stages:



Of course, evaluating the right one at the same values would give you a different answer, since subtraction is not associative. Here is another example:


Binary trees in general

By repeated substitution, you can create general binary trees built up of individual trees of this form:

In AbAl I will give examples of such things and their counterparts in algebraic notation. This will include binary trees involving more than one binop, as well. I showed an example in the previous post, which example I repeat here:

It represents the precise unsimplified expression

\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]

Some of the operations in that tree are associative and commutative, which is why the expression can be simplified. The collection of all (finite) binary trees built out of a single binop with no assumption that it satisfies laws (associative, commutative and so on) is the free algebra on that binary operation. It is the mother of all binary operations, so it plays the same role for an arbitrary binop that the set of lists plays for associative operations, as described in Monads for High School III: Algebras. All this will be covered in later chapters of AbAl.

References

Preceding posts in this series

Other references

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Presenting binary operations

This is the first of a set of notes I am writing to help me develop my thoughts about how particular topics in my book Abstracting algebra should be organized. This article describes my plan for the book in some detail. The present post has some thoughts about presenting binary operations.

Before binary operations are introduced

Traditionally, an abstract algebra book assumes that the student is familiar with high school algebra and will then proceed with an observation that such operations as $+$ and $\times$ can be thought of as functions of two variables that take a number to another number. So the first abstract idea is typically the concept of binary operation, although in another post I will consider whether that really should be the first abstract concept.

The Abstracting Algebra book will have a chapter that presents concrete examples of algebraic operations and expressions on numbers as in elementary school and as in high school algebra. This section of the post outlines what should be presented there. Each subsection needs to be expanded with lots of examples.

In elementary school

In elementary school you see expressions such as

  • $3+4$
  • $3\times 4$
  • $3-4$

The student invariably thinks of these expressions as commands to calculate the value given by the expression.

They will also see expressions such as
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
\end{array}
\end{equation}\]
which they will take as a command to calculate the sum of the whole list:
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
474
\end{array}
\end{equation}\]

That uses the fact that addition is associative, and the format suggests using the standard school algorithm for adding up lists. You don’t usually see the same format with more than two numbers for multiplication, even though it is associative as well. In some elementary schools in recent years students are learning other ways of doing arithmetic and in particular are encouraged to figure out short cuts for problems that allow them. But the context is always “do it”, not “this represents a number”.

Algebra

In algebra you start using letters for numbers. In algebra, “$a\times b$” and “$a+b$” are expressions in the symbolic language of math, which means they are like noun phrases in English such as “My friend” and “The car I bought last week and immediately totaled” in that both are used semantically as names of objects. English and the symbolic language are both languages, but the symbolic language is not a natural language, nor is it a formal language.

Example

In beginning algebra, we say “$3+5=8$”, which is a (true) statement.

Basic facts about this equation:

The expressions “$3+5$” and “$8$”

  • are not the same expression
  • but in the standard semantics of algebra they have the same meaning
  • and therefore the equation communicates information that neither “$3+5$” nor “$8$” communicate.

Another example is “$3+5=6+2$”.

Facts like this example need to be communicated explicitly before binary operations are introduced formally. The students in a college abstract algebra class probably know the meaning of an equation operationally (subconsciously) but they have never seen it made explicit. See Algebra is a difficult foreign language.

Note

The equation “$3+5=6+2$” is an expression just as much as “$3+5$” and “$6+2$” are. It denotes an object of type “equation”, which is a mathematical object in the same way as numbers are. Most mathematicians do not talk this way, but they should.

Binary operations

Early examples

Consciousness-expanding examples should appear early and often after binary operations are introduced.

Common operations

  • The GCD is a binary operation on the natural numbers. This disturbs some students because it is not written in infix form. It is associative. The GCD can be defined conceptually, but for computation purposes needs (Euclid’s) algorithm. This gives you an early example of conceptual definitions and algorithms.
  • The maximum function is another example of this sort. This is a good place to point out that a binary operation with the “same” definition cen be defined on different sets. The max function on the natural numbers does not have quite the same conceptual definition as the max on the integers.

Extensional definitions

In order to emphasize the arbitrariness of definitions, some random operations on a small finite sets should be given by a multiplication table, on sets of numbers and sets represented by letters of the alphabet. This will elicit the common reaction, “What operation is it?” Hidden behind this question is the fact that you are giving an extensional definition instead of a formula — an algorithm or a combination of familiar operations.

Properties

The associative and commutative properties should be introduced early just for consciousness-raising. Subtraction is not associative or commutative. Rock paper scissors is commutative but not associative. Groups of symmetries are associative but not commutative.

Binary operation as function

The first definition of binary operation should be as a function. For example, “$+$” is a function that takes pairs of numbers to numbers. In other words, $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a function.

We then abstract from that example and others like it from specific operations to arbitrary functions $\Delta:S\times S\to S$ for arbitrary sets $S$.

This is abstraction twice.

  • First we replace the example operations by an arbitrary operation. such as multiplication, subtraction, GCD and MAX on $\mathbb{Z}$, or something complicated such as \[(x,y)\mapsto 3(xy-1)^2(x^2+xy^3)^3\].
  • Then we replace sets of numbers by arbitrary sets. An example would be the random multiplication on the set $\{1,2,5\}$ given by the table
    \[
    \begin{array}{c|ccc}
    \Delta& 1&2&5\\
    \hline
    1&2&2&1\\
    2&5&2&1\\
    5&2&1&5
    \end{array}
    \]
    This defines a function $\Delta:\{1,2,5\}\times\{1,2,5\}\to\{1,2,5\}$ for which for example $\Delta(2,1)=5$, or $2\Delta 1=5$. This example uses numbers as elements of the set and is good for eliciting the “What operation is it?” question.
  • I will use examples where the elements are letters of the alphabet, as well. That sort of example makes the students think the letters are variables they can substitute for, another confusion to be banished by the wise professor who know the right thing to say to make it clear. (Don’t ask me; I taught algebra for 35 years and I still don’t know the right thing to say.)

It is important to define prefix notation and infix notation right away and to use both of them in examples.

Other representations of binary operations.

The main way of representing binary operations in Abstracting Algebra will be as trees, which I will cover in later posts. Those posts will be much more interesting than this one.

Binary operations in high school and college algebra

  • Some binops are represented in infix notation: “$a+b$”, “$a-b$”, and “$a\times b$”.
  • “$a\times b$” is usually written “$ab$” for letters and with the “$\times$” symbol for numbers.
  • Some binops have idiosyncratic representation: “$a^b$”, “${a}\choose{b}$”.
  • A lot of binops such as GCD and MAX are given as functions of two variables (prefix notation) and their status as binary operations usually goes unmentioned. (That is not necessarily wrong.)
  • The symbol “$(a,b)$” is used to denote the GCD (a binop) and is also used to denote a point in the plane or an open interval, both of which are not strictly binops. They are binary operations in a multisorted algebra (a concept I expect to introduce later in the book.)
  • Some apparent binops are in infix notation but have flaws: In “$a/b$”, the second entry can’t be $0$, and the expression when $a$ and $b$ are integers is often treated as having good forms ($3/4$) and bad forms ($6/8$).

Trees

The chaotic nature of algebraic notation I just described is a stumbling block, but not the primary reason high school algebra is a stumbling block for many students. The big reason it is hard is that the notation requires students to create and hold complicated abstract structures in their head.

Example

This example is a teaser for future posts on using trees to represent binary operations. The tree below shows much more of the structure of a calculation of the area of a rectangle surmounted by a semicircle than the expression

\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]
does.

The tree explicitly embodies the thought process that leads to the formula:

  • You need to add the area of the rectangle and the area of the semicircle.
  • The area of the rectangle is width times height.
  • The area of the semicircle is $\frac{1}{2}(\pi r^2)$.
  • In this case, $r=\frac{1}{2}w$.

Any mathematician will extract the same abstract structure from the formula\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\] This is difficult for students beginning algebra.

References

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A mathematical saga

This post outlines some of the intellectual developments in the history of math. I call it a saga because it is like one:

  • It is episodic, telling one story after another.
  • It does not try to give an accurate history, but its episodes resemble what happened in math in the last 3000 years.
  • It tells about only a few of the things that happened.

Early techniques

We represented numbers by symbols.

Thousands of years ago, we figured out how to write down words and phrases in such a way that someone much later could read and understand them.

Naturally, we wanted to keep records of the number of horses the Queen owned, so we came up with various notations for numbers (number representing count). In some societies, these symbols were separate from the symbols used to represent words.

We invented algorithms

We discovered positional notation. We write $213$, which is based on a system: it means $2\times100+1\times10+3\times 1$. This notation encapsulates a particular computation of a number (its base-10 representation). (The expression $190+23$ is another piece of notation that encapsulates a computation that yields $213$.)

Compare that to the Roman notation $CCXIII$, which is an only partly organized jumble.
Try adding $CCXIII+CDXXIX$. (The answer is $DCXLII$.)

Positional notation allowed us to create the straightforward method of addition involving adding single digits and carrying:
\[\overset{\hspace{6pt}1\phantom{1}}
{\frac
{\overset{\displaystyle{213}}{429}}{642}
}
\]
Measuring land requires multiplication, which positional notation also allows us to perform easily.
The invention of such algorithms (methodically manipulating symbols) made it easy to calculate with numbers.

Geometry: Direct construction of mathematical objects

We discovered geometry in ancient times, in laying out plots of land and designing buildings. We had a bunch of names for different shapes and for some of them we knew how to calculate their area, perimeter and other things.

Euclid showed how to construct new geometric figures from given ones using specific methods (ruler and compasses) that preserve some properties.

Example

We can bisect a line segment (black) by drawing two circles (blue) centered at the endpoints with radius the length of the line segment. We then construct a line segment (red) between the points of intersection of the circle that intersects the given line segment at its midpoint. These constructions can be thought of as algorithms creating and acting on geometric figures rather than on symbols.



It is true that diagrams were drawn to represent line segments, triangles and so on.
But the diagrams are visualization helpers. The way we think about the process is that we are operating directly on the geometric objects to create new ones. We are thinking of the objects Platonically, although we don’t have to share Plato’s exact concept of their reality. It is enough to say we are thinking about the objects as if they were real.

Axioms and theorems

Euclid came up with the idea that we should write down axioms that are true of these figures and constructions, so that we can systematically use the constructions
to prove theorems about figures using axioms and previously proved theorems. This provided documented reasoning (in natural language, not in symbols) for building up a collection of true statements about math objects.

Example

After creating some tools for proving triangles are congruent, we can prove the the intersection of red and black lines in the figure really is the midpoint of the black line by constructing the four green line segments below and making appeals to congruences between the triangles that show up:



Note that the green lines have the same length as the black line.

Euclid thought about axioms and theorems as applying to geometry, but he also proved theorems about numbers by representing them as ratios of line segments.

Algebra

People in ancient India and Greece knew how to solve linear and quadratic equations using verbal descriptions of what you should do.
Later, we started using a symbolic language to express numerical problems and symbolic manipulation to solve (some of) them.

Example

The quadratic formula is an encapsulated computation that provides the roots of a quadratic equation. Newton’s method is a procedure for finding a root of an arbitrary polynomial. It is recursive in the loose sense (it does not always give an answer).

The symbolic language is a vast expansion of the symbolic notation for numbers. A major innovations was to introduce variables to represent unknowns and to state equations that are always true.

Logic

Aristotle developed an early form of logic (syllogisms) aimed at determining which arguments in rhetoric were sound. “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” This was written in sentences, not in symbols.

By explicit analogy with algebra, we introduced symbolism and manipulation rules for logical reasoning, with an eye toward making mathematical reasoning sound and to some extent computable. For example, in one dialect of logical notation, modus ponens (used in the Socrates syllogism) is expressed as $(P\rightarrow Q,\,P)\,\,\vdash\,\, Q$. This formula is an encapsulated algorithm: it says that if you know $P\rightarrow Q$ and $P$ are valid (are theorems) then $Q$ is valid as well.

Crises of understanding

We struggled with the notion of function as a result of dealing with infinite series. For example, the limit of a sequence of algebraic expressions may not be an algebraic expression. It would no longer do to think of a function as the same thing as an algebraic expression.

We realized that Euclid’s axioms for geometry lacked clarity. For example, as I understand it, the original version of his axioms didn’t imply that the two circles in the proof above had to intersect each other. There were other more subtle problems. Hilbert made a big effort to spell out the axioms in more detail.

We refined our understanding of logic by trying to deal with the mysteries of calculus, limits and spaces. An example is the difference between continuity and uniform continuity.
We also created infinitesimals, only to throw up our hands because we could not make a logic that fit them. Infinitesimals were temporarily replaced by the use of epsilon-delta methods.

We began to understand that there are different kinds of spaces. For example, there were other models of some of Euclid’s axioms than just Euclidean space, and some of those models showed that the parallel axiom is independent of the other axioms. And we became aware of many kinds of topological spaces and manifolds.

We started to investigate sets, in part because spaces have sets of points. Then we discovered that a perfectly innocent activity like considering the set of all sets resulted in an impossibility.

We were led to consider how to understand the Axiom of Choice from several upsetting discoveries. For example, the Banach-Tarski “paradox” implies that you can rearrange the points in a sphere of radius $1$ to make two spheres of radius $1$.

Mathematics adopts a new covenant… for awhile

These problems caused a kind of tightening up, or rigorizing.
For a period of time, less than a century, we settled into a standard way of practicing research mathematics called new math or modern math. Those names were used mostly by math educators. Research mathematicians might have called it axiomatic math based on set theory. Although I was around for the last part of that period I was not aware of any professional mathematicians calling it anything at all; it was just what we did.

First, we would come up with a new concept, type of math object, or a new theorem. In this creative process we would freely use intuition, metaphors, images and analogies.

Example

We might come up with the idea that a function reaches its maximum when its graph swoops up from the left, then goes horizontally for an infinitesimal amount of time, then swoops down to the right. The point at which it was going horizontally would obviously have to be the maximum.

But when we came to publish a paper about the subject, all these pictures would disappear. All our visual, metaphorical/conceptual and kinetic feelings that explain the phenomenon would have to be suppressed.

Rigorizing consisted of a set of practices, which I will hint at:

Orthodox behavior among mathematicians in 1950

Definition in terms of sets and axioms

Each mathematical object had to be defined in some way that started with a set and some other data defined in terms of the set. Axioms were imposed on these data. Everything had to be defined in terms of sets, including functions and relations. (Multiple sets were used occasionally.)

Definitions done in this way omit a lot of the intuition that we have concerning the object being defined.

Examples
  • The definition of group as a set with a binary operation satisfying some particular axioms does not tell you that groups constitute the essence of symmetry.
  • The definitions of equivalence relation and of partition do not even hint that they define the same concept.

Even so, definitions done in this way have an advantage: They tend to be close to minimal in the sense that to verify that something fits the definition requires checking no more (or not much more) than necessary.

Proofs had to be clearly translatable into symbolic logic

First order logic (and other sorts of logic) were well developed and proofs were written in a way that they could in principle be reduced to arguments written in the notation of symbolic logic and following the rules of inference of logic. This resulted in proofs which did not appeal to intuition, metaphors or pictures.

Example

In the case of the theorem that the maximum of a (differentiable) function occurs only where the derivative is zero, that meant epsilon-delta proofs in which the proof appeared as a thick string of symbols. Here, “thick” means it had superscripts, subscripts, and other things that gave the string a fractal dimension of about $1.2$ (just guessing!).

Example

When I was a student at Oberlin College in 1959, Fuzzy Vance (Elbridge P. Vance) would sometimes stop in the middle of an epsilon-delta proof and draw pictures and provide intuition. Before he started that he would say “Shut the door, don’t tell anyone”. (But he told us!)

Example

A more famous example of this is the story that Oscar Zariski, when presenting a proof in algebraic geometry at the board, would sometimes remind himself of a part of a proof by hunching over the board so the students couldn’t see what he was doing and drawing a diagram which he would immediately erase. (I fault him for not telling them about the diagram.)

It doesn’t matter whether this story is true or not. It is true in the sense that any good myth is true.

Commercial

I wrote about rigor in these articles:

Rigorous view in abstractmath.org.

Dry bones, post in this blog.

Logic and sets clarify but get in the way

The orthodox method of “define it by sets and axioms” and “makes proofs at least resemble first order logic” clarified a lot of suspect proofs. But it got in the way of intuition and excessive teaching by using proofs made it harder to students to learn.

  • The definition of a concept can make you think of things that are foreign to your intuition of the concept. A function is a mapping,. The ordered pairs are a secondary construction; you should not think of ordered pairs as essential to your intuition. Even so the definition of function in terms of ordered pairs got rid of a lot of cobwebs.
  • The cartesian product of sets is obviously an associative binary operation. Except that if you define the cartesian product of sets in terms of ordered pairs then it is not associative.
  • Not only that, but if you define the ordered pair $(a,b)$ as $\{\{a,b\},a\}$ the you have to say that $a$ is an element of $(a,b)$ but $b$ is not That is not merely an inconvenient definition of ordered pair, it is wrong. It is not bad way to show that the concept of ordered pair is consistent with ZF set theory, but that is a minor point mathematicians hardly ever worry about.

Mathematical methods applied to everything

The early methods described at the beginning of this post began to be used everywhere in math.

Algorithms on symbols

Algorithms, or methodical procedures, began with the addition and multiplication algorithms and Euclid’s ruler and compass constructions, but they began to be used everywhere.

They are applied to the symbols of math, for example to describe rules for calculating derivatives and integrals and for summing infinite series.

Algorithms are used on strings, arrays and diagrams of math symbols, for example concatenating lists, multiplying matrices, and calculating binary operations on trees.

Algorithms as definitions

Algorithms are used to define the strings that make up the notation of symbolic logic. Such definitions include something like: “If $E$ and $F$ are expressions than $(E)\land (F)$ and $(\forall x)(E)$ are expressions”. So if $E$ is “$x\geq 3$” then $(\forall x)(x\geq 3)$ is an expression. This had the effect of turning an expression in symbolic logic into a mathematical object. Deduction rules such as “$E\land F\vdash E$” also become mathematical objects in this way.

We can define the symbols and expressions of algebra, calculus, and other part of math using algorithms, too. This became a big deal when computer algebra programs such as Mathematica came in.

Example

You can define the set $RP$ of real polynomials this way:

  • $0\in RP$
  • If $p\in RP$ then $p+r x^n\in RP$, where $x$ is a variable and $n$ a nonnegative integer.

That is a recursive definition. You can also define polynomials by pattern recognition:

Let $n$ be a positive integer, $a_0,\,a_1\,\ldots a_n$ be real numbers and $k_0,\,k_1\,\ldots k_n$ be nonnegative integers. Then $a_0 x^{k_0}+a_1 x^{k_1}+\ldots+ a_n x^{k_n}$ is a polynomial.

The recursive version is a way of letting a compiler discover that a string of symbols is a polynomial. That sort of thing became a Big Deal when computers arrived in our world.

Algorithms on mathematical objects

I am using the word “algorithm” in a loose sense to mean any computation that may or may not give a result. Computer programs are algorithms, but so is the quadratic formula. You might not think of a formula as an algorithm, but that is because if you use it in a computer program you just type in the formula; the language compiler has a built-in algorithm to execute calculations given by formulas.

It has not been clearly understood that mathematicians apply algorithms not only to symbols, but also directly to mathematical objects. Socrates thought that way long ago, as I described in the construction of a midpoint above. The procedure says “draw circles with center at the endpoints of the line segment.” It doesn’t say “draw pictures of circles…”

In the last section and this one, I am talking about how we think of applying an algorithm. Socrates thought he was talking about ideal lines and circles that exist in some other universe that we can access by thought. We can think about them as real things without making a metaphysical claim like Socrates did about them. Our brains are wired to think of abstract ideas in some many of the same ways we think about physical objects.

Example

The unit circle (as a topological space at least) is the quotient space of the space $\mathbb{R}$ of real numbers mod the equivalence relation defined by: $x\sim y$ if and only if $x-y$ is an integer.

Mathematicians who understand that construction may have various images in their mind when they read this. One would be something like imagining the real line $\mathbb{R}$ and then gluing all the points together that are an integer apart. This is a distinctly dizzying thing to think about but mathematicians aren’t worried because they know that taking the quotient of a space is a well-understood construction that works. They might check that by imagining the unit circle as the real line wrapped around an infinite number of times, with points an integer apart corresponding to the same point on the unit circle. (When I did that check I hastily inserted the parenthetical remark saying “as a topological space” because I realized the construction doesn’t preserve the metric.) The point of this paragraph is that many mathematicians think of this construction as a construction on math objects, not a construction on symbols.

Everything is a mathematical object

A lot of concepts start out as semi-vague ideas and eventually get defined as mathematical objects.

Examples

  • A function was originally thought of as a formula, but then get formalized in the days of orthodoxy as a set of ordered pairs with the functional property.
  • The concept of equation has been turned into a math object many times, for example in universal algebra and in logic. I suspect that some practitioners in those fields might disagree with me. This requires further research.
  • Propositions are turned into math objects by Boolean Algebra.
  • Perhaps numbers were always thought of as math objects, but much later the set $\mathbb{N}$ of all natural numbers and the set $\mathbb{R}$ of all real numbers came to be thought of explicitly as math objects, causing some mathematicians to have hissy fits.
  • Definitions are math objects. This has been done in various ways. A particular theory is a mathematical object, and it is essentially a definition by definition (!): Its models are what the theory defines. A particular example of “theory” is first-order theory which was the gold standard in the orthodox era. A classifying topos is also a math object that is essentially a definition.

Category Theory

The introduction of categories broke the orthodoxy of everything-is-a-set. It has become widely used as a language in many branches of math. It started with problems in homological algebra arising in at least these two ways:

  • Homotopy classes of continuous functions are not functions in the set theory sense. So we axiomatized the concept of function as an arrow (morphism) in a category.
  • The concept of mathematical object is axiomatized as an object in a category. This forces all properties of an object to be expressed in terms of its external relations with other objects and arrows.
  • Categories capture the idea of “kind of math”. There is a category of groups and homomorphisms, a category of topological spaces and homeomorphisms, and so on. This is a new level of abstraction. Before, if someone said “I work in finite groups”, their field was a clear idea and people knew what they were talking about, but now the category of finite groups is a mathematical object.
  • Homology maps one kind of math (topology) into another kind (algebra). Since categories capture the general notion of “kind of math”, we invented the idea of functor to capture the idea of modeling or representing one branch of math in another one. So Homology became a mathematical object.
  • The concept of functor allowed the definition of natural transformation as a mathematical object. Before categories, naturality was only an informal idea.

Advantages of category theory

  • Categories, in the form of toposes, quickly became candidates to replace set theory as a foundation system for math. They are more flexible and allow the kind of logic you want to use (classical, intuitionistic and others) to be a parameter in your foundational system.
  • “Arrow” (morphism) replaced not only the concept of function but also the concept of “element of” (“$\in$”). It allows the concept of variable elements. (This link is to a draft of a section of abstractmath.org that has not been included in the table of contents yet.) It also requires that an “element” has to be an element of one single object; for example, the maps $1\to \mathbb{N}$ and $1\to \mathbb{R}$ that pick out the number $42$ are not the same maps, although of course they are related by the canonical inclusion map $\mathbb{N}\to\mathbb{R}$.
  • Diagrams are used in proofs and give much better immediate understanding than formulas written in strings, which compress a lot of things unnecessarily into thick strings that require understanding lots of conventions and holding things in your memory.
  • Categories-as-kinds-of-math makes it easy to turn an analogy, for example between products of groups and products of spaces, into two examples of the same kind of mathematical object: Namely, a product in a category.

Disadvantages of category theory

  • Category theory requires a new way of thinking. Some people think that is a disadvantage. But genuine innovation is always disruptive. New technology breaks old technology. Of course, the new technology has to turn out to be useful to win out.
  • Category theory has several notions of “equal”. Objects can be the same or isomorphic. Categories can be isomorphic or equivalent. When you are doing category theory, you should never worry about whether two objects are equal: that is considered evil. Category theorists generally ignored the fuzziness of this problem because you can generally get away with it. Still, it was an example of something that had not been turned into a mathematical definition. Two ways of accomplishing this are anafunctors and homotopy type theory.

I expect to write about homotopy type theory soon. It may be the Next Revolution.

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Explaining math

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

This post explains some basic distinctions that need to be made about the process of writing and explaining math.  Everyone who teaches math knows subconsciously what is happening here; I am trying to raise your consciousness.  For simplicity, I have chosen a technique used in elementary algebra, but much of what I suggest also applies to more abstract college level math.

An algebra problem

Solve the equation "$ax=b$" ($a\neq0$).

Understanding the statement of this problem requires a lot of Secret Knowledge (the language of ninth grade algebra) that most people don't have.

  • The expression "$ax$" means that $a$ and $x$ are numbers and $ax$ is their product. It is not the word "ax". You have to know that writing two symbols next to each other means multiply them, except when it doesn't mean multiply them as in "$\sin\,x$".

  • The whole expression "$ax=b$" ostensibly says that the number $ax$ is the same number as $b$.  In fact, it means more than that. The phrase "solve the equation" tells you that in fact you are supposed to find the value of $x$ that makes $ax$ the same number as $b$.

  • How do you know that "solve the equation" doesn't mean find the value of $a$ that makes $ax$ the same number as $b$? Answer: The word "solve" triggers a convention that $x$, $y$ and $z$ are numbers you are trying to find and $a$, $b$, $c$ stand for numbers that you are allowed to plug in to the equation.

  • The conventions of symbolic math require that you give a solution for any nonzero value of $a$ and any value of $b$.  You specifically are not allowed to pick $a=1$ and $b=33$ and find the value just for those numbers.  (Some college calculus students do this with problems involving literal coefficients.)

  • The little thingy "$(a\neq0)$" must be read as a constraint on $a$.  It does not mean that $a\neq0$ is a fact that you ought to know. ( I've seen college math students make this mistake, admittedly in more complex situations). Nor does it mean that you can't solve the problem if $a=0$ (you can if $b$ is also zero!).

So understanding what this problem asks, as given, requires (fairly sophisticated in some cases) pattern recognition both to understand the symbolic language it uses, and also to understand the special conventions of the mathematical English that it uses.

Explicit descriptions

This problem could be reworded so that it gives an explicit description of the problem, not requiring pattern recognition.  (Warning: "Not requiring pattern recognition" is a fuzzy concept.)  Something like this:  

You have two numbers $a$ and $b$.  Find a number $c$ for which if you multiply $a$ by $c$ you get $b$.

This version is not completely explicit.  It still requires understanding the idea of referring to a number by a letter, and it still requires pattern recognition to catch on that the two occurrences of each letter means that their meanings have to match. Also, I know from experience that some American first year college students have trouble with the syntax of the sentence ("for which…", "if…").

The following version is more explicit, but it cheats by creating an ad hoc way to distinguish the numbers.

Alice and Bob each give you a number.  How do you find a number with the property that Alice's number times your number is equal to Bob's number? 

If the problem had a couple more variables it would be so difficult to understand in an explicit form that most people would have to draw a picture of the relationships between them.  That is why algebraic notation was invented.

Visual descriptions

Algebra is a difficult foreign language.  Showing the problem visually makes it easier to understand for most people. Our brain's visual processing unit is the most powerful tool the brain has to understand things.  There are various ways to do this.  

Visualization can help someone understand algebraic notation better.  

You can state the problem by producing examples such as

  • $\boxed{3}\times\boxed{\text{??}}=\boxed{6}$ 
  • $\boxed{5}\times\boxed{\text{??}}=\boxed{2}$ 
  • $\boxed{42}\times\boxed{\text{??}}=\boxed{24}$

where the reader has to know the multiplication symbol and, one hopes, will recognize "$\boxed{\text{??}}$" as "What's the value?". But the reader does not have to understand what it means to use letters for numbers, or that "$x$ means you are suppose to discover what it is".  This way of writing an algebra problem is used in some software aimed at K-12 students.  Some of them use a blank box instead of "$\boxed{\text{??}}$".

Such software often shows the algorithm for solving the problem visually, using algebraic notation like this:

I have put in some buttons to show numbers as well as $a$ and $b$.  If you have access to Mathematica instead of just to CDF player, you can load SolvEq.nb and put in any numbers you want, but CDF's don't allow input data. 

You can also illustrate the algorithm using the tree notation for algebra I used in Monads for high school I  (and other posts). The demo below shows how to depict the value-preserving transformation given by the algorithm.  (In this case the value is the truth since the root operation is equals.)

This demo is not as visually satisfactory as the one illustrating the use of the associative law in Monads for high school I.  For one thing, I had to cheat by reversing the placement of $a$ and $x$.  Note that I put labels for the numerator and denominator legs, a practice I have been using in demos for a while for noncommutative operations.  I await a new inspiration for a better presentation of this and other equation-solving algorithms.

Another advantage of using pictures is that you can often avoid having to code things as letters which then has to be remembered.  In Monads for high school I, I used drawings of the four functions from a two-element set to itself instead of assigning them letters.  Even mnemonic letters such as $s$ for "switch" and $\text{id}$ for the identity element carry a burden that the picture dispenses with.

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Naming mathematical objects

Commonword names confuse

Many technical words and phrases in math are ordinary English words ("commonwords") that are assigned a different and precisely defined mathematical meaning.  

  • Group  This sounds to the "layman" as if it ought to mean the same things as "set".  You get no clue from the name that it involves a binary operation with certain properties.  
  • Formula  In some texts on logic, a formula is a precisely defined expression that becomes a true-or-false sentence (in the semantics) when all its variables are instantiated.  So $(\forall x)(x>0)$ is a formula.  The word "formula" in ordinary English makes you think of things like "$\textrm{H}_2\textrm{O}$", which has no semantics that makes it true or false — it is a symbolic expression for a name.
  • Simple group This has a technical meaning: a group with no nontrivial normal subgroup.  The Monster Group is "simple".  Yes, the technical meaning is motivated by the usual concept of "simple", but to say the Monster Group is simple causes cognitive dissonance.

Beginning students come with the (generally subconscious) expectation that they will pick up clues about the meanings of words from connotations they are already familiar with, plus things the teacher says using those words.  They think in terms of refining an understanding they already have.  This is more or less what happens in most non-math classes.  They need to be taught what definition means to a mathematician.

Names that don't confuse but may intimidate

Other technical names in math don't cause the problems that commonwords cause.

Named after somebody The phrase "Hausdorff space" leads a math student to understand that it has a technical meaning.  They may not even know it is named after a person, but it screams "geek word" and "you don't know what it means".  That is a signal that you can find out what it means.  You don't assume you know its meaning. 

New made-up words  Words such as "affine", "gerbe"  and "logarithm" are made up of words from other languages and don't have an ordinary English meaning.  Acronyms such as "QED", "RSA" and "FOIL" don't occur often.  I don't know of any math objects other than "RSA algorithm" that have an acronymic name.  (No doubt I will think of one the minute I click the Publish button.)  Whole-cloth words such as "googol" are also rare.  All these sorts of words would be good to name new things since they do not fool the readers into thinking they know what the words mean.

Both types of words avoid fooling the student into thinking they know what the words mean, but some students are intimidated by the use of words they haven't seen before.  They seem to come to class ready to be snowed.  A minority of my students over my 35 years of teaching were like that, but that attitude was a real problem for them.

Audience

You can write for several different audiences.

Math fans (non-mathematicians who are interested in math and read books about it occasionally) In my posts Explaining higher math to beginners and in Renaming technical conceptsI wrote about several books aimed at explaining some fairly deep math to interested people who are not mathematicians.  They renamed some things. For example, Mark Ronan in Symmetry and the Monster used the phrase "atom" for "simple group" presumably to get around the cognitive dissonance.  There are other examples in my posts.  

Math newbies  (math majors and other students who want to understand some aspect of mathematics).  These are the people abstractmath.org is aimed at. For such an audience you generally don't want to rename mathematical objects. In fact, you need to give them a glossary to explain the words and phrases used by people in the subject area.   

Postsecondary math students These people, especially the math majors, have many tasks:

  • Gain an intuitive understanding of the subject matter.
  • Understand in practice the logical role of definitions.
  • Learn how to come up with proofs.
  • Understand the ins and outs of mathematical English, particularly the presence of ordinary English words with technical definitions.
  • Understand and master the appropriate parts of the symbolic language of math — not just what the symbols mean but how to tell a statement from a symbolic name.

It is appropriate for books for math fans and math newbies to try to give an understanding of concepts without necessary proving theorems.  That is the aim of much of my work, which has more an emphasis on newbies than on fans. But math majors need as well the traditional emphasis on theorem and proof and clear correct explanations.

Lately, books such as Visual Group Theory have addressed beginning math majors, trying for much more effective ways to help the students develop good intuition, as well as getting into proofs and rigor. Visual Group Theory uses standard terminology.  You can contrast it with Symmetry and the Monster and The Mystery of the Prime Numbers (read the excellent reviews on Amazon) which are clearly aimed at math fans and use nonstandard terminology.  

Terminology for algebraic structures

I have been thinking about the section of Abstracting Algebra on binary operations.  Notice this terminology:

boptable

The "standard names" are those in Wikipedia.  They give little clue to the meaning, but at least most of them, except "magma" and "group", sound technical, cluing the reader in to the fact that they'd better learn the definition.

I came up with the names in the right column in an attempt to make some sense out of them.  The design is somewhat like the names of some chemical compounds.  This would be appropriate for a text aimed at math fans, but for them you probably wouldn't want to get into such an exhaustive list.

I wrote various pieces meant to be part of Abstracting Algebra using the terminology on the right, but thought better of it. I realized that I have been vacillating between thinking of AbAl as for math fans and thinking of it as for newbies. I guess I am plunking for newbies.

I will call groups groups, but for the other structures I will use the phrases in the middle column.  Since the book is for newbies I will include a table like the one above.  I also expect to use tree notation as I did in Visual Algebra II, and other graphical devices and interactive diagrams.

Magmas

In the sixties magmas were called groupoids or monoids, both of which now mean something else.  I was really irritated when the word "magma" started showing up all over Wikipedia. It was the name given by Bourbaki, but it is a bad name because it means something else that is irrelevant.  A magma is just any binary operation. Why not just call it that?  

Well, I will tell you why, based on my experience in Ancient Times (the sixties and seventies) in math. (I started as an assistant professor at Western Reserve University in 1965). In those days people made a distinction between a binary operation and a "set with a binary operation on it".  Nowadays, the concept of function carries with it an implied domain and codomain.  So a binary operation is a function $m:S\times S\to S$.  Thinking of a binary operation this way was just beginning to appear in the common mathematical culture in the late 60's, and at least one person remarked to me: "I really like this new idea of thinking of 'plus' and 'times' as functions."  I was startled and thought (but did not say), "Well of course it is a function".  But then, in the late sixties I was being indoctrinated/perverted into category theory by the likes of John Isbell and Peter Hilton, both of whom were briefly at Case Western Reserve University.  (Also Paul Dedecker, who gave me a glimpse of Grothendieck's ideas).

Now, the idea that a binary operation is a function comes with the fact that it has a domain and a codomain, and specifically that the domain is the Cartesian square of the codomain.  People who didn't think that a binary operation was a function had to introduce the idea of the universe (universal algebraists) or the underlying set (category theorists): you had to specify it separately and introduce terminology such as $(S,\times)$ to denote the structure.   Wikipedia still does it mostly this way, and I am not about to start a revolution to get it to change its ways.

Groups

In the olden days, people thought of groups in this way:

  • A group is a set $G$ with a binary operation denoted by juxtaposition that is closed on $G$, meaning that if $a$ and $b$ are any elements of $G$, then $ab$ is in $G$.
  • The operation is associative, meaning that if $a,\ b,\ c\in G$, then $(ab)c=a(bc)$.
  • The operation has a unity element, meaning an element $e$ for which for any element $a\in G$, $ae=ea=a$.
  • For each element $a\in G$, there is an element $b$ for which $ab=ba=e$.

This is a better way to describe a group:

  • A group consist of a nullary operation e, a unary operation inv,  and a binary operation denoted by juxtaposition, all with the same codomain $G$. (A nullary operation is a map from a singleton set to a set and a unary operation is a map from a set to itself.)
  • The value of e is denoted by $e$ and the value of inv$(a)$ is denoted by $a^{-1}$.
  • These operations are subject to the following equations, true for all $a,\ b,\ c\in G$:

     

    • $ae=ea=a$.
    • $aa^{-1}=a^{-1}a=e$.
    • $(ab)c=a(bc)$.

This definition makes it clear that a group is a structure consisting of a set and three operations whose axioms are all equations.  It was formulated by people in universal algebra but you still see the older form in texts.

The old form is not wrong, it is merely inelegant.  With the old form, you have to prove the unity and inverses are unique before you can introduce notation, and more important, by making it clear that groups satisfy equational logic you get a lot of theorems for free: you construct products on the cartesian power of the underlying set, quotients by congruence relations, and other things. (Of course, in AbAl those theorem will be stated later than when groups are defined because the book is for newbies and you want lots of examples before theorems.)

References

  1. Three kinds of mathematical thinkers (G&G post)
  2. Technical meanings clash with everyday meanings (G&G post)
  3. Commonword names for technical concepts (G&G post)
  4. Renaming technical concepts (G&G post)
  5. Explaining higher math to beginners (G&G post)
  6. Visual Algebra II (G&G post)
  7. Monads for high school II: Lists (G&G post)
  8. The mystery of the prime numbers: a review (G&G post)
  9. Hersh, R. (1997a), "Math lingo vs. plain English: Double entendre". American Mathematical Monthly, volume 104, pages 48–51.
  10. Names (in abmath)
  11. Cognitive dissonance (in abmath)
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Monads for high school I

 

Notes for viewing

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

Monads in Abstracting Algebra

I've been working on first drafts (topic posts) of several sections of my proposed book Abstracting algebra (AbAl), concentrating on the ideas leading up to monads.  This is going slowly because I want the book to be full of illustrations and interactive demos.  I am writing the demos in Mathematica simultaneously with writing the text, and designing demos is very s l o w work. It occurred to me that I should write an outline of the path leading up to monads, using some of the demos I have already produced. This is the first of probably two posts about the thread.

  • AbAl is intended to give people with a solid high school math background a mental picture of or way of thinking about the various levels of abstraction of high school algebra.
  • This outline is not a "Topic post" as described in the AbAl page. In particular, it is not aimed at high school students! It is a guided tour of my current thoughts about a particular thread through the book.
  • The AbAl page has a brief outline of the topics to be covered in the whole book.  Perhaps it should also have a list of threads like this post.

Associativity

AbAl will have sections introducing functions and binary operations using pictures and demos (not outlined in this thread).  The section on binary operations will introduce infix, prefix and postfix notation but will use trees (illustrated below) as the main display method.  Then it will introduce associativity, using demos such as this one: 

Using this computingscienceish tree notation makes it much easier to visualize what is happening (see Visible Algebra II), compared to, for example, \[(ab)(cd)=a(b(cd))=a((bc)d)=((ab)c)d=(a(bc))d\]  In this equation, the abstract structure is hidden.  You have to visualize doing the operation starting with the innermost parentheses and moving out.  With the trees you can see the computation going up the tree.

I will give examples of associative functions that are not commutative using $2\times2$ matrices and endofunctions on finite sets such as the one below, which gives all the functions from a two element set to itself. 


  • Note that each function is shown by a diagram, not by an arbitrary name such as "id" or "sw", which would add a burden to the memory for an example that occurs in one place in the book. (See structural notation in the Handbook.) 
  • The section on composition of functions will also look in some depth at permutations of a three-element set, anticipating a section on groups.

 By introducing a mechanism for transforming trees of associative binary operations, you can demonstrate (as in the demo below) that any associative binary operation is defined on any list of two or more elements of its domain.

For example, applying addition to three numbers $a$, $b$ and $c$ is uniquely defined. This sort of demo gives an understanding of why you get that unique definition but it is not a proof, which requires formal induction. AbAl is not concerned with showing the reader how to prove math statements.

In this section I will also introduce the oneidentity concept: the value of an associative binary operation on a an element $a$ is $a$.  Thus applying addition or multiplication to $a$ gives $a$.  (The reason for this is that you want an associative binary operation to be a unique quotient of the free associative binary operation.  That will come up after we talk about some examples of monads.)  

The oneidentity property also implies that for an associative binary operation with identity element, applying the operation to the empty set gives the identity element.  Now we can say:

An associative binary operation with identity element is uniquely defined on any finite list of elements of its domain.

Thus, in prefix notation,$+(2,3)=5$, $+(2,3,5)=10$, $+(2)=2$ and $+()=0$.  Similarly $\times(2)=2$ and $\times()=1$.

This fact suggests that the natural definition of addition, multiplication, and other associative binary operations is as functions from lists of elements of the domain to elements of the domain.   This fits with our early intuition of addition from grade school, not to mention from Excel:  Addition is something you do to lists.  That feeling (for me) is not so strong for multiplication; for many common business applications you generally multiply two things like price and number sold. That's because multiplication is usually for things of different data types, but you usually add things of the same data type (not apples and oranges?).   

That raises the question: Does every function taking lists to elements come from an associative binary operation?  I will give an example that says no.  But the next thing is to introduce joining lists (concatenation), where we discover that joining lists is an associative binary operation.  So it is really an operation on lists of lists.  This will turn out to give us a systematic way to define all associative binary operations by one mechanism, because it is an example of a monad.  That is for the second installment of this outline.

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Abstracting algebra

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