Tag Archives: commutative

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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Dysfunctions in doing math I

I am in the middle of revising the article in abstractmath.org on dysfunctional attitudes and behaviors in doing math. Here are three of the sections I have finished.

Misuse of analogy

When William Rowan Hamilton was trying to understand the new type of number called quaternions (MW, Wik) that he invented, he assumed by analogy that like other numbers, quaternion multiplication was commutative. It was a major revelation to him that they were not commutative.

Analogy may suggest new theorems or ways of doing things. But it is fallible. What happens particularly often in abstract math is applying a rule to a situation where it is not appropriate. This is an easy trap to fall into when the notation in two different cases has the same form; that is an example of formal analogy.

Matrix multiplication

Matrix multiplication is not commutative

If $r$ and $s$ are real numbers then the products $rs$ and $sr$ are always the same number. In other words, multiplication of real numbers is commutative : $rs = sr$ for all real numbers $r$ and $s$.

The product of two matrices $M $and $N$ is written $MN$, just as for numbers. But matrix multiplication is not commutative. For example,
\[\left(
\begin{array}{cc}
1 & 2 \\
3 & 4\\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & 1 \\
3 &2\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
9 & 5\\
21 & 11 \\
\end{array}
\right)\]
but
\[\left(
\begin{array}{cc}
3 & 1 \\
3 & 2\\
\end{array}
\right)
\left(\begin{array}{cc}
1 & 2 \\
3 & 4\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
6 & 10\\
91 & 14 \\
\end{array}
\right)\]
Because $rs = sr$ for numbers, the formal similarity of the notation suggests $MN$ = $NM$, which is wrong.

This means you can’t blindly manipulate $MNM$ to become $M^2N$. More generally, a law such as $(MN)^n=M^nN^n$ is not correct when $M$ and $N$ are matrices.

You must understand the meanings
of the symbols you manipulate.

The product of two nonzero matrices can be 0

If the product of two numbers is 0, then one or both of the numbers is zero. But that is not true for matrix multiplication:
\[\left(
\begin{array}{cc}
-2 & 2 \\
-1 & 1\\
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 1 \\
1 &1\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
0 &0\\
0 & 0 \\
\end{array}
\right)\]

Canceling sine

  • Beginning calculus students have already learned algebra.
  • They have learned that an expression such as $xy$ means $x$ times $y$.
  • They have learned to cancel like terms in a quotient, so that for example \[\frac{3x}{3y}=\frac{x}{y}\]
  • They have learned to write the value of a function $f$ at the input $x$ by $f(x)$.
  • They have seen people write $\sin x$ instead of $\sin(x)$ but have never really thought about it.
  • So they write \[\frac{\sin x}{\sin y}=\frac{x}{y}\]

This happens fairly often in freshman calculus classes. But you wouldn’t do that, would you?

Boundary values of definitions

Definitions are usually inclusive

Definitions of math concepts usually include the special cases they generalize.

Examples

  • A square is a special case of rectangle. As far as I know texts that define “rectangle” include squares in the definition. Thus a square is a rectangle.
  • A straight line is a curve.
  • A group is a semigroup.
  • An integer is a real number. (But not always in computing languages — see here.)

But not always

  • The axioms of a field include a bunch of axioms that a one-element set satisfies, plus a special axiom that does nothing but exclude the one-element set. So a field has to have at least two elements, and that fact does not follow from the other axioms.
  • Boolean algebras are usually defined that way, too, but not always. MathWorld gives several definitions of Boolean algebra that disagree on this point.

When boundary values are not special cases

Definitions may or may not include other types of boundary values.

Examples

  • If $S$ is a set, it is a subset of itself. The empty set is also a subset of $S$.
  • Similarly the divisors of $6$ are $-6$, $-3$, $-2$, $-1$, $1$, $2$, $3$ and $6$, not just $2$ and $3$ and not just $1$, $2$, $3$ and $6$ (there are two different boundaries here).

But …

  • The positive real numbers include everything bigger than $0$, but not $0$. ( Note).

Blunders

A definition that includes such special cases may be called inclusive; otherwise it is exclusive. People new to abstract math very commonly use words defined inclusively as if their definition was exclusive.

  • They say things such as “That’s not a rectangle, it is a square!” and “Is that a group or a semigroup?”
  • They object if you say “Consider the complex number $\pi $.”

This appears to be natural linguistic behavior. Even so, math is picky-picky: a square is a rectangle, a group is a semigroup and $\pi$ is a complex number (of course, it is also a real number).

Co-intimidator

  • You attend a math lecture and the speaker starts talking about things you never heard of.
  • Your fellow students babble at you about manifolds and tensors and you thought they were car parts and lamps.
  • You suspect your professor is deliberately talking over your head to put you down.
  • You suspect your friends are trying to make you believe they are much smarter than you are.
  • You suspect your friends are smarter than you are.

There are two possibilities:

  • They are not trying to intimidate you (most common).
  • They are deliberately setting out to intimidate you with their arcane knowledge so you will know what a worm you are. (There are people like that.)

Another possibility, which can overlap with the two above, is:

  • You expect to be intimidated. You may be what might be called a co-intimidator, Similar to the way someone who is codependent wants some other person to be dependent on them. (This is not like the “co” in category theory: “product” and “coproduct” have a symmetric relationship with each other, but the co-intimidator relation is asymmetric.)

There are many ways to get around being intimidated.

  • Ask “What the heck is a manifold?”
  • (In a lecture where it might be imprudent or impractical to ask) Write down what they say, then later ask a friend or look it up.
  • Most teachers like to be asked to explain something. Yes, I know some professors repeatedly put down people. Change sections! If you can’t, live with it! Not knowing something says nothing bad about you.

And remember:

If you don’t know something
probably many other students don’t know it either.

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Monads for High School III: Algebras

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 MonadAlg.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This is a continuation of Monads for high school I and Monads for High School II: Lists. This post covers the concept of algebras for the monad for lists.

Lists

$\textrm{Lists}(S)$ is the set of all lists of finite length whose entries are elements of $S$.

  • $\boxed{2\; 2\; 4}$ is the way I denote the list of length $3$ whose first and second entries are each $2$ and whose third entry is $4$.
  • A list with only one entry, such as $\boxed{2}$, is called a singleton list.
  • The empty list $\boxed{\phantom{2}}$ has no entries.
  • $\textrm{Lists}^*(S)$ is the set of all nonempty lists of finite length whose entries are elements of $S$.
  • $\textrm{Lists}(\textrm{Lists}(S))$ is the list whose entries are lists with entries from $S$.
  • For example, $\boxed{\boxed{5\; 7}\; \boxed{2\; 12\; 7}}$ and $\boxed{\boxed{5\; 7\; 2\; 12\; 7}}$ are both entries in $\textrm{Lists}^*(\textrm{Lists}^*(\mathbb{Z}))$. The second one is a singleton list!
  • $\boxed{\boxed{\phantom{3}}\; \boxed{2}}
    $ and $\boxed{\boxed{\phantom{3}}}$ are entries in $\textrm{Lists}^*(\textrm{Lists}(\mathbb{Z}))$.
  • The empty list $\boxed{\phantom{2}}$ is an entry in $\textrm{Lists}(\mathbb{Z})$, in $\textrm{Lists}(\textrm{Lists}^*(\mathbb{Z}))$ and in $\textrm{Lists}(\textrm{Lists}(\mathbb{Z}))$. If you have stared at this for more than ten minutes, do something else and come back to it later.

The star notation is used widely in math and computing science to imply that you are including everything except some insignificant shrimp of a thing such as the empty list, the empty set, or $0$. For example, $\mathbb{R}^*$ denotes the set of all nonzero real numbers.

More details about lists are in Monads for High School II: Lists.

Join

The function join (or concatenation) takes two lists and creates a third list. For example, if you join $\boxed{5\; 7}$ to $\boxed{2\; 12\; 7 }$ in that order you get $\boxed{5\; 7\; 2\; 12\; 7}$.

  • I will use this notation: join$\boxed{\boxed{5\; 7}\; \boxed{2\; 12\; 7}}=\boxed{5\; 7\; 2\; 12\; 7}$.
  • This notation means that I am regarding join as a function that takes a two-element list in $\textrm{Lists}(\textrm{Lists}(S))$ to an element of $\textrm{Lists}(S)$.
  • join removes one level of lists
  • join is not commutative: join$\boxed{\boxed{2\; 12\; 7}\; \boxed{5\; 7}}=\boxed{2\; 12\; 7\; 5\; 7}$
  • Join is associative, and as for any associative binary operation, join is defined on any finite list of lists of elements of $S$. So for example, join$\boxed{\boxed{5\; 7}\; \boxed{2\; 12\; 7}\; \boxed{1}}=\boxed{5\; 7\; 2\; 12\; 7\; 1}$.
  • For any single list $\boxed{a\; b\; c}$, join$\boxed{\boxed{a\; b\; c}}=\boxed{a\; b\; c}$. This is required to make the theory work. It is called the oneidentity property.
  • If the empty list $\boxed{\phantom{2}}$ occurs in a list of lists, it disappears when join is applied: join $\boxed{\boxed{2\; 3}\; \boxed{\phantom{2}}\; \boxed{4\; 5\; 6}}=\boxed{2\; 3\; 4\; 5\; 6}$.

More details about join in Monads for High School II: Lists.

The main monad diagram

When you have a list of lists of lists, join can be applied in two different ways, "inside" and "outside" as illustrated in the diagram below. It gives you several different inputs to try out as a way to understand what is happening.

This is the special case of the main diagram for all monads as it applies to the List monad.

As you can see, after doing either of "inside" and "outside", if you then apply join, you get the same list. That list is simply the list of entries in the beginning list (and the two intermediate ones) in the same order, disregarding groupings.

From what I have just written, you must depend on your pattern recognition abilities to learn what inside and outside mean. But both can also be described in words.

  • The lists outlined in black are lists of elements of $\mathbb{Z}$. In other words, they are elements of $\textrm{Lists}(\mathbb{Z})$.
  • The lists outlined in blue are lists of elements of $\textrm{Lists}(\mathbb{Z})$. In other words, they are list of lists of elements of $\mathbb{Z}$. Those are the kinds of things you can apply join to.
  • The leftmost list in the diagram, outlined in green, is a list in $\textrm{Lists}(\textrm{Lists}(\mathbb{Z}))$. This means you can apply join in two different ways:
  • Each list boxed in blue is a list of lists of integers (two of the are singletons!) so you can apply join to each of them. This is joining inside first.
  • You can apply join directly to the leftmost list, which is a list of lists (of lists, but forget that for the moment), so you can apply join to the blue lists. This is join outside first.

To understand this diagram, staring at the diagram (for most people) uses the visual pattern recognition part of your brain (which uses over a fifth of the energy used by your brain) to understand what inside and outside mean, and then check your understanding by reading the verbal description. Starting by reading the verbal description first does not work as well for most people.

The unit monad diagram

There is a second unitary diagram for all monads:

The two right hand entries are always the same. Again, I am asking you to use your pattern recognition abilities to learn what singleton list and singleton each mean.

The main and unit monad diagrams will be used as axioms to give the general definition of monad. To give those axioms, we also need the concepts of functor and natural transformation, which I will define later after I have finished the monad algebra diagrams for Lists and several other examples.

Algebras for the List monad

If you have any associative binary operation on a set $S$, its definition can be extended to any nonempty list of elements (see Monads for High School I.)

Plus and Times are like that:

  • $(3+2)+4$ and $3+(2+4)$ have the same value $9$, so you can write $3+2+4$ and it means $9$ no matter how you calculate it.
  • I will be using the notation Plus$\boxed{3\; 2\; 4}$ instead of $3+2+4$.
  • Times is also associative, so for example we can write Times$\boxed{3\; 2\; 4}=24$.
  • Like join, we require that these operations satisfy oneidentity, so we know Plus$\boxed{3}=3$ and Times$\boxed{3}=3$.
  • When the associative binary operation has an identity element, you can also define its value on the empty list as the identity element: Plus$\boxed{\phantom{3}}=0$ and Times$\boxed{\phantom{3}}=1$. I recommend that you experiment with examples to see why it works.

An algebra for the List monad is a function algop:$\textrm{Lists}(S)\to S$ with certain properties: It must satisfy the Main Monad Algebra Diagram and the Unit Monad Algebra Diagram, discussed below.

The main monad algebra diagram

Example using Plus and Times

The following interactive diagram allows you to see what happens with Plus and Times. Afterwards, I will give the general definition.

Plus insides replaces each inside list with the result of applying Plus to it, and the other operation Join is the same operation I have used before.

Another example

The main monad algebra diagram requires that if you have a list of lists of numbers such as the one below, you can add up each list (Plus insides) and then add up the list of totals (top list in diagram), you must get the same answer that you get when you join all the lists of numbers together into one list (bottom list in the diagram) and then add up that list.

This is illustrated by this special case of the main monad algebra diagram for Plus:

General statement of the main monad algebra diagram

Suppose we have any function $\blacksquare$ $:\textrm{Lists}(S)\to S$ for any set $S$.
If we want to give the main monad algebra diagram for $\blacksquare$ we have a problem. We know for example that Plus$\boxed{1\; 2}=3$. But for some elements $a $ and $b$ of $S$, we don’t know what $\blacksquare\boxed{a\; b}$ is. One way to write it is simply to write $\blacksquare\boxed{a\; b}$ (the usual way we write a function). Or we could use tree notation and write

newalopdouble.

I will use tree notation mostly, but it is a good exercise to redraw the diagrams with functional notation.

Main monad diagram in prose

Below is a presentation of the general main monad algebra diagram using (gasp!) English phrases to describe the nodes.

genalgdiag

The unit monad algebra diagram

Suppose $\blacksquare$ is any function from $\textrm{Lists}(S)$ to $S$ for any set $S$. Then the diagram is

UnitMAdiag

This says that if you apply $\blacksquare$ to a singleton you get the unique entry of the singleton. This is not surprising: I defined above what it means when you apply an operation to a singleton just so this would happen!

A particular example

These are specific examples of the general main monad algebra diagram for an arbitrary operation $\blacksquare$:

stalgdiagleft

staldiagright

These examples show that if $\blacksquare$ is any function from $\textrm{Lists}(S)$ to $S$ for any set $S$, then

newalopleft

equals

newaloptriple

and

newalopright

equals

newaloptriple

Well, according to some ancient Greek guy, that means

newalopleft

equals

newalopright

which says that
newalopdouble
is an associative binary operation!

The mother of all associative operations

We also know that any associative binary $\blacksquare$ on any set $S$ can be extended to a function on all finite nonempty lists of elements of $S$. This is the general associative law and was discussed (without using that name) in Monads fo High School I.

Let’s put what we’ve done together into one statement:

Every associative binary operation $\blacksquare$ on a set $S$ can be extended uniquely to a function $\blacksquare:\textrm{Lists}^*(S)\to S$ that satisfies both the main monad algebra diagram and the unit monad algebra diagram. Furthermore, any function $\blacksquare:\textrm{Lists}^*(S)\to S$ that satisfies both the main monad algebra diagram and the unit monad algebra diagram is an asssociative binary operation when applied to lists of length $2$ of elements of $S$.

That is why I claim that the NonemptyList monad is the mother of all associative binary operations.

I have not proved this, but the work in this and preceding posts provide (I think) a good intuitive understanding of this fundamental relationship between lists and associative binary operations.

Things to do in upcoming posts

  • I have to give a proper definition of monads using the concepts of functor and natural transformation. I expect to do this just for set functors, not mentioning categories.
  • Every type of binary operation that is defined by equations corresponds to a monad which is the mother of all binary operations of that type. I will give examples, but not prove the general case.

Other examples of monads

  • Associative binary operations on $S$ with identity element (monoids) corresponds to all lists, including the empty list, with entries from $S$.
  • Commutative, associative and idempotent binary operations, like and and or in Boolean algebra, correspond to the set monad: $\text{Sets}(S)$ is the set of all finite and countably infinite sets of elements of $S$. (You can change the cardinality restrictions, but you have to have some cardinality restrictions.) Join is simply union.
  • Commutative and associative binary operations corresponds to the multiset monad (with a proper definition of join) and appropriate cardinality restrictions. You have to fuss about identity elements here, too.
  • Various kinds of nonassociative operations get much more complicated, involving tree structures with equivalence relations on them. I expect to work out a few of them.
  • There are lots of monads in computing science that you never heard of (unless you are a computing scientist). I will mention a few of them.

  • Every type of binary operation defined by equations corresponds to a monad. But some of them are unsolvable, meaning you cannot describe the monad precisely.

There will probably be long delay before I get back to this project. There are too many other things I want to do!

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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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Visible algebra II

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 Wolfram website. The code for the demos is in the Mathematica notebook algebra2.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.

More about visible algebra

I have written about visible algebra in previous posts (see References). My ideas about the interface are constantly changing. Some new ideas are described here.

In the first place I want to make it clear that what I am showing in these posts is a simulation of a possible visual algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

Commutative and associative operations

Times and Plus are commutative and associative operations.  They are usually defined as binary operations.  A binary operation $*$ is said to be commutative if for all $x$ and $y$ in the underlying set of the operation, $x*y=y*x$, and it is associative if for all $x$,$y$ and $z$ in the underlying set of the operation, $(x*y)*z=x*(y*z)$. 

It is far better to define a commutative and associative operation $*$ on some underlying set $S$ as an operation on any multiset of elements of $S$.  A multiset is like a set, in particular elements can be rearranged in any way, but it is not like a set in that elements can be repeated and a different number of repetitions of an element makes a different multiset.  So for any particular multiset, the number of repetitions of each element is fixed.  Thus $\{a,a,b,b,c\} = \{c,b,a,b,a\}$ but $\{a,a,b,b,c\}\neq\{c,b,a,b,c\}$. This means that the function (operation) Plus, for example, is defined on any multiset of numbers, and \[\mathbf{Plus}\{a,a,b,b,c\}=\mathbf{Plus} \{c,b,a,b,a\}\] but $\mathbf{Plus}\{a,a,b,b,c\}$ might not be equal to $\mathbf{Plus} \{c,b,a,b,c\}$.

This way of defining (any) associative and commutative operation comes from the theory of monads.  An operation defined on all the multisets drawn from a particular set is necessarily commutative and associative if it satisfies some basic monad identities, the main one being it commutes with union of multisets (which is defined in the way you would expect, and if this irritates you, read the Wikipedia article on multisets.). You don't have to impose any conditions specifically referring to commutativity or associativity.  I expect to write further about monads in a later post. 

The input process for a visible algebra system should allow the full strength of this fact. You can attach as many inputs as you want to Times or Plus and you can move them around.  For example, you can click on any input and move it to a different place in the following demo.

Other input notations might be suitable for different purposes.  The example below shows how the inputs can be placed randomly in two dimensions (but preserving multiplicity).  I experimented with making it show the variables slowly moving around inside the circle the way the fish do in that screensaver (which mesmerizes small children, by the way — never mind what it does to me), but I haven't yet made it work.

A visible algebra system might well allow directly input tables to be added up (or multiplied), like the one below. Spreadsheets have such an operation In particular, the spreadsheet operation does not insist that you apply it only as a binary operation to columns with two entries.  By far the most natural way to define addition of numbers is as an operation on multisets of numbers.

Other operations

Operations that are associative but not commutative, such as matrix multiplication, can be defined the monad way as operations on finite lists (or tuples or vectors) of numbers.  The operation is automatically associative if you require it to preserve concatenation of lists and some other monad requirements.

Some binary operations are neither commutative nor associative.  Two such operations on numbers are Subtract and Power.  Such operations are truly binary operations; there is no obvious way to apply them to other structures.  They are only binary because the two inputs have different roles.  This suggests that the inputs be given names, as in the examples below.

Later, I will write more about simplifying trees, solving the max area problem for rectangles surmounted by semicircles, and other things concerning this system of doing algebra.

References

Previous posts about visible algebra

Other references

 

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