## Representations of mathematical objects

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This is a long post. Notes on viewing.

A mathematical object, or a type of math object, is represented in practice in a great variety of ways, including some that mathematicians rarely think of as "representations".

In this post you will find examples and comments about many different types of representations as well as references to the literature. I am not aware that anyone has considered all these different ideas of representation in one place before. Reading through this post should raise your consciousness about what is going on when you do math.

This is also an experiment in exposition.  The examples are discussed in a style similar to the way a Mathematica command is discussed in the Documentation Center, using mostly nonhierarchical bulleted lists. I find it easy to discover what I want to know when it is written in that way.  (What is hard is discovering the name of a command that will do what I want.)

## Types of representations

### Using language

• Language can be used to define a type of object.
• A definition is intended to be precise enough to determine all the properties that objects of that type all have.  (Pay attention to the two uses of the word "all" in that sentence; they are both significant, in very different ways.)
• Language can be used to describe an object, exhibiting properties without determining all properties.
• It can also provide metaphors, making use of one of the basic tools of our brain to understand the world.
• The language used is most commonly mathematical English, a special dialect of English.
• The symbolic language of mathematics (distinct from mathematical English) is used widely in calculations. Phrases from the symbolic language are often embedded in a statement in math English. The symbolic language includes among others algebraic notation and logical notation.
• The language may also be a formal language, a language that is mathematically defined and is thus itself a mathematical object. Logic texts generally present the first order predicate calculus as a formal language.
• Neither mathematical English nor the symbolic language is a formal language. Both allow irregularities and ambiguities.

### Mathematical objects

The representation itself may be a mathematical object, such as:

• A linear representation of a group. Not only are the groups mathematical objects, so is the representation.
• An embedding of a manifold into Euclidean space. A definition given in a formal language of the first order predicate calculus of the property of commutativity of binary operations. (Thus a property can be represented as a math object.)

### Visual representations

A math object can be represented visually using a physical object such as a picture, graph (in several senses), or diagram.

• The visual processing of our brain is our major source of knowledge of the world and takes about a fifth of the brain's processing power.  We can learn many things using our vision that would take much longer to learn using verbal descriptions.  (Proofs are a different matter.)
• When you look at a graph (for example) your brain creates a mental representation of the graph (see below).

### Mental representations

If you are a mathematician, a math object such as "$42$", "the real numbers" or "continuity" has a mental representation in your brain.

• In the math ed literature, such a representation is called "mental image", "concept image", "procept", or "schema".   (The word "image" in these names is not thought of as necessarily visual.)
• The procept or schema describe all the things that come to mind when you think about a particular math object: The definition, important theorems, visual images, important examples, and various metaphors that help you understand it.
• The visual images occuring in a mental schema for an object may themselves be mental representations of physical objects. The examples and theorems may be mental representations of ideas you learned from language or pictures, and so on.  The relationships between different kinds of representations get quite convoluted.

### Metaphors

Conceptual metaphors are a particular kind of mental representation of an object which involve mentally associating some aspects of the objects with some aspects of something else — a physical object, an image, an action or another abstract object.

• A conceptual metaphor may give you new insight into the object.
• It may also mislead you because you think of properties of the other object that the math object doesn't have.
• A graph of a function is a conceptual metaphor.
• When you say that a point on a graph "rises as it goes from left to right" your metaphor is an action.
• When you say that the cosets of a normal subgroup of a group "get along" with the group multiplication, your metaphor identifies a property they have with an aspect of human behavior.

## Properties of representations

A representation of a math object may or may not

• determine it completely
• exhibit some of its properties
• suggest easy proofs of some theorems
• provide a useful way of thinking about it

## Examples of representations

This list shows many of the possibilities of representation.  In each case I discuss the example in terms of the two bulleted lists above. Some of the examples are reused from my previous publications.

### Functions

Example (F1) "Let $f(x)$ be the function defined by $f(x)=x^3-x$."

• This is an expression in mathematical English that a fluent reader of mathematical English will recognize gives a definition of a specific function.
• (F1) is therefore a representation of that function.
• The word "representation" is not usually used in this way in math.  My intention is that it should be recognized as the same kind of object as many other representations.
• The expression contains the formula $x^3-x$.  This is an encapsulated computation in the symbolic language of math. It allows someone who knows basic algebra and calculus to perform calculations that find the roots, extrema and inflection points of the function $f$.
• The word "let" suggests to the fluent reader of mathematical English that (F1) is a definition which is probably going to hold for the next chunk of text, but probably not for the whole article or book.
• Statements in mathematical English are generally subject to conventions.  In a calculus text (F1) would automatically mean that the function had the real numbers as domain and codomain.
• The last two remarks show that a beginner has to learn to read mathematical English.
• Another convention is discussed in the following diatribe.

#### Diatribe

You would expect $f(x)$ by itself to mean the value of $f$ at $x$, but in (F1) the $x$ has the property of a bound variable.  In mathematical English, "let" binds variables. However, after the definition, in the text the "$x$" in the expression "$f(x)$" will be free, but the $f$ will be bound to the specific meaning.  It is reasonable to say that the term "$f(x)$" represents the expression "$x^3-x$" and that $f$ is the (temporary) name of the function. Nevertheless, it is very common to say "the function $f(x)$" to mean $f$.

A fluent reader of mathematical English knows all this, but probably no one has ever said it explicitly to them.  Mathematical English and the symbolic language should be taught explicitly, including its peculiarities such as "the function $f(x)$".  (You may want to deprecate this usage when you teach it, but students deserve to understand its meaning.)

### The positive integers

You have a mental representation of the positive integers $1,2,3,\ldots$.  In this discussion I will assume that "you" know a certain amount of math.  Non-mathematicians may have very different mental representations of the integers.

• You have a concept of "an integer" in some operational way as an abstract object.
• "Abstract object" needs a post of its own. Meanwhile see Mathematical Objects (abstractmath) and the Wikipedia articles on Mathematical objects and Abstract objects.
• You have a connection in your brain between the concept of integer and the concept of listing things in order, numbering them by $1,2,3,\ldots$.
• You have a connection in your brain between the concept of an integer and the concept of counting a finite number of objects.  But then you need zero!
• You understand how to represent an integer using the decimal representation, and perhaps representations to other bases as well.
• Your mental image has the integer "$42"$ connected to but not the same as the decimal representation "42". This is not true of many students.
• The decimal rep has a picture of the string "42" associated to it, and of course the picture of the string may come up when you think of the integer $42$ as well (it does for me — it is a an icon for the number $42$.)
• You have a concept of the set of integers.
• Students need to be told that by convention "the set of integers" means the set of all integers.  This particularly applies to students whose native language does not have articles, but American students have trouble with this, too.
• Your concept of  "the set of integers" may have the icon "$\mathbb{N}$" associated with it.  If you are a mathematician, the icon and the concept of the set of integers are associated with each other but not identified with each other.
• For me, at least, the concept "set of integers" is mentally connected to each integer by the "element of" relation. (See third bullet below.)
• You have a mental representation of the fact that the set of integers is infinite.
• This does not mean that your brain contains an infinite number of objects, but that you have a representation of infinity as a concept, it is brain-connected to the concept of the set of integers, and also perhaps to a proof of the fact that $\mathbb{N}$ is infinite.
• In particular, the idea that the set of integers is mentally connected to each integer does not mean that the whole infinite number of integers is attached in your brain to the concept of the set of integers.  Rather, the idea is a predicate in your brain.  When it is connected to "$42$", it says "yes".  To "$\pi$" it says "No".
• Philosophers worry about the concept of completed infinity.  It exists as a concept in your brain that interacts as a meme with concepts in other mathematicians' brains. In that way, and in that way only (as far as I am concerned) it is a physical object, in particular an object that exists in scattered physical form in a social network.

### Graph of a function

This is a graph of the function $y=x^3-x$:

• The graph is a physical object, either on a screen or on paper
• It is processed by your visual system, the most powerful sensory management system in your brain
• It also represents the graph in the mathematical sense (set of ordered pairs) of the function $y=x^3-x$
• Both the mathematical graph and the physical graph are represented by modules in your brain, which associates the two of them with each other by a conceptual metaphor
• The graph shows some properties of the function: inflection point, going off to infinity in a specific way, and so on.
• These properties are made apparent (if you are knowledgeable) by means of the powerful pattern recognition system in your brain. You see them much more quickly than you can discover them by calculation.
• These properties are not proved by the graph. Nevertheless, the graph communicates information: for example, it suggests that you can prove that there is an inflection point near $(0,0)$.
• The graph does not determine or define the function: It is inaccurate and it does not (cannot) show all of the graph.

### Continuity

Example (C1) The $\epsilon-\delta$ definition of the continuity of a function $f:\mathbb{R}\to\mathbb{R}$ may be given in the symbolic language of math:

A function $f$ is continuous at a number $c$ if $\forall\epsilon(\epsilon\gt0\implies(\forall x(\exists\delta(|x-c|\lt\delta\implies|f(x)-f(c)|\lt\epsilon)))$

• To understand (C1), you must be familiar with the notation of first order logic.  For most students, getting the notation right is quite a bit of work.
• You must also understand  the concepts, rules and semantics of first order logic.
• Even if you are familiar with all that, continuity is still a difficult concept to understand.
• This statement does show that the concept is logically complicated. I don't see how it gives any other intuition about the concept.

Example (C2) The definition of continuity can also be represented in mathematical English like this:

A function $f$ is continuous at a number $c$ if for any $\epsilon\gt0$ and for any $x$ there is a $\delta$ such that if $|x-c|\lt\delta$, then $|f(x)-f(c)|\lt\epsilon$.

• This definition doesn't give any more intuition that (C1) does.
• It is easier to read that (C1) for most math students, but it still requires intimate familiarity with the quirks of math English.
• The fact that "continuous" is in boldface signals that this is a definition.  This is a convention.
• The phrase "For any $\epsilon\gt0$" contains an unmarked parenthetic insertion that makes it grammatically incoherent.  It could be translated as: "For any $\epsilon$ that is greater than $0$".  Most math majors eventually understand such things subconsciously.  This usage is very common.
• Unless it is explicitly pointed out, most students won't notice that  if you change the phrase "for any $x$ there is a $\delta$"  to "there is a $\delta$ for any $x$" the result means something quite different.  Cauchy never caught onto this.
• In both (C1) and (C2), the "if" in the phrase "A function $f$ is continuous at a number $c$ if…" means "if and only if" because it is in a definition.  Students rarely see this pointed out explicitly.

Example (C3) The definition of continuity can be given in a formally defined first order logical theory

• The theory would have to contain function symbols and axioms expressing the algebra of real numbers as an ordered field.
• I don't know that such a definition has ever been given, but there are various semi-automated and automated theorem-proving systems (which I know little about) that might be able to state such a definition.  I would appreciate information about this.
• Such a definition would make the property of continuity a mathematical object.
• An automated theorem-proving system might be able to prove that $x^3-x$ is continuous, but I wonder if the resulting proof would aid your intuition much.

Example (C4) A function from one topological space to another is continuous if the inverse of every open set in the codomain is an open set in the domain.

• This definition is stated in mathematical English.
• In definitions (C1) – (C3), the primitive data are real numbers and the statement uses properties of an ordered field.
• In (C4), the data are real numbers and the arithmetic operations of a topological field, along with the open sets of the field. The ordering is not mentioned.
• This shows that a definition need not mention some important aspects of the structure.
• One marvelous example of this is that  a partition of a set and an equivalence relation on a set are based on essentially disjoint sets of data, but they define exactly the same type of structure.

Example (C4) "The graph of a continuous function can be drawn without picking up the chalk".

• This is a metaphor that associates an action with the graph.
• It is incorrect: The graphs of some continuous functions cannot be drawn.  For example, the function $x\mapsto x^2\sin(1/x)$ is continuous on the interval $[-1,1]$ but cannot be drawn at $x=0$.
• Generally speaking, if the function can be drawn then it can be drawn without picking up the chalk, so the metaphor provides a useful insight, and it provides an entry into consciousness-raising examples like the one in the preceding bullet.

## References

1. 1.000… and .999… (post)
2. Conceptual blending (post)
3. Conceptual blending (Wikipedia)
4. Conceptual metaphors (Wikipedia)
5. Convention (abstractmath)
6. Definitions (abstractmath)
7. Embodied cognition (Wikipedia)
8. Handbook of mathematical discourse (see articles on conceptual blendmental representationrepresentationmetaphor, parenthetic assertion)
9. Images and Metaphors (abstractmath).
10. The interplay of text, symbols and graphics in math education, Lin Hammill
11. Math and the modules of the mind (post)
12. Mathematical discourse: Language, symbolism and visual images, K. L. O’Halloran.
13. Mathematical objects (abmath)
14. Mathematical objects (Wikipedia)
15. Mathematical objects are “out there?” (post)
16. Metaphors in computing science ​(post)
17. Procept (Wikipedia)
18. Representations 2 (post)
19. Representations and models (abstractmath)
20. Representations II: dry bones (post)
21. Representation theorems (Wikipedia) Concrete representations of abstractly defined objects.
22. Representation theory (Wikipedia) Linear representations of algebraic structures.
23. Semiotics, symbols and mathematical visualization, Norma Presmeg, 2006.
24. The transition to formal thinking in mathematics, David Tall, 2010
25. Theory in mathematical logic (Wikipedia)
26. What is the object of the encapsulation of a process? Tall et al., 2000.
27. Where mathematics comes from, by George Lakoff and Rafael Núñez, Basic Books, 2000.
28. Where mathematics comes from (Wikipedia) This is a review of the preceding book.  It is a permanent link to the version of 04:23, 25 October 2012.  The review is opinionated, partly wrong, not well written and does not fit the requirements of a Wikipedia entry.  I recommend it anyway; it is well worth reading.  It contains links to three other reviews.

### Notes on Viewing

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

## Visible algebra II

### MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } }); Notes on viewing:

• This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
• To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb.

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.

## MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } });

Note: This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

In the post Algebra is a difficult foreign language  I listed some of the difficulties of the syntax of the symbolic language of math (which includes high school algebra and precalculus).  The semantics causes difficulties as well.  Again I will list some examples without any attempt at completeness.

## The status of the symbolic language as a language

There is a sharp distinction between the symbolic language of math and mathematical English, which I have written about in The languages of math and in the Handbook of mathematical discourse. Other authors do not make this sharp distinction (see the list of references at the end of this post). The symbolic language occurs embedded in mathematical English and the embedding has its own semantics which may cause great difficulty for students.

The symbolic language of math can be described as a natural formal language. Pieces of it were invented by mathematicians and others over the course of the last several hundred years. Individual pieces (notation such as "$3x+1=2y$") can be given a strictly formal syntax, but the whole system is ambiguous, inconsistent, and context-sensitive.  When you get to the research level, it has many dialects: Research mathematicians in one field may not be able to read research articles in a very different field.

## Examples

I think the examples below will make these claims plausible.  This should be the subject of deep research.

### Superscripts and functions

• A superscript, as in $5^2$ or $x^3$, has a pretty standard meaning denoting a power, at least until you get to higher level stuff such as tensors.
• A function can be denoted by a letter, symbol, or string, and the notation $f(x)$ refers to the value of the function at input $x$.

For functions defined on numbers, it is common in precalculus and higher to write $f^2(x)$ to denoted $(f(x))^2=f(x)\,f(x)$.  Since the value of certain multiletter functions are commonly written without the parentheses (for example, $\sin\,x$), one writes $\sin^2x$ to mean $(\sin\,x)^2$.

The notation $f^n$ is also widely used to mean the $n$th iterate of $f$ (if it exists), so $f^3(x)=f(f(f(x)))$ and so on.  This leads naturally to writing $f^{-1}(x)$ for the inverse function of $f$; this is common notation whether the function $f$ is bijective or not (in which case $f^{-1}$ is set-valued).  Thus $\sin^{-1}x$ means $\arcsin\,x$.

It is notorious that words in mathematical English have different meanings in different texts.  This is an example in the symbolic language (and not just at the research level) of a systematic construction that can give expressions that have ambiguous meanings.

This phenomenon is an example of why I say the symbolic language of math is a natural formal language: I have described a natural extension of notation used with multiplication of values that has been extended to being used for the binary operation of composition.  And that leads to students thinking that $\sin^{-1}x$ means $\frac{1}{\sin\,x}$.

History can overtake notation, too: Mathematicians probably took to writing $\sin\,x$ instead of $\sin(x)$ because it saves writing.  That was not very misleading in the old days when mathematical variables were always single symbols.  But students see multiletter variable names all the time these days (in programming languages, Excel and elsewhere), so of course some of them think $\sin\,x$ means $\sin$ times $x$. People who do this are not idiots.

### Juxtaposition

Juxtaposition of two symbols means many different things.

• If $m$ and $n$ are numbers, $mn$ denotes the product of the two numbers.
• Multiplication is commutative, so $mn$ and $nm$ denote the same number, but they correspond to different calculations.
• If $M$ and $N$ are matrices, $MN$ denotes the matrix product of the two matrices.
• This is a binary operation but it is not the same operation denoted by juxtaposition of numbers. (In fact it involves both addition and multiplication of numbers.)
• Now $MN$ may not be the same matrix as $NM$.
• If $A$ and $B$ are points in a geometric drawing, $AB$ denotes the line segment from $A$ to $B$.
• This is a function of two variables denoting points whose value is a line segment.
• It is not what is usually called a binary operation, although as an opinionated category theorist I would call it a multisorted binary operation.
• It is commutative, but it doesn't make sense to ask if it is associative.

This phenomenon is called overloaded notation.

• In order to understand the meaning of the juxtaposition of symbols, you have to know the type of the variables.
• The surrounding text may tell you specifically the variables denote matrices or whatever. So this is an instance of context-sensitive semantics.
• Students tend to expect that they know what any formula means in isolation from the text.  It may make them very sad to discover that this doesn't work — once they believe it, which can take quite a while.
• In many cases the problem is alleviated by the use of convention.
• Matrices are usually denoted by capital letters, numbers by lower case letters.
• But points in geometry are usually denoted by capital letters too.  So you have to know that referring to a geometric diagram is significant to understanding the notation. This is an indirect form of context-sensitivity.  Did any teacher every point this out to students?  Does it appear anywhere in print?

The earlier example of $\sin^{-1}x$ is a case which is not context-sensitive.  Knowing the types of the variables won't help.  Of course, if the author explains which meaning is meant, that explanation is within the context of the book!  That is not a lot of help for grasshoppers like me that look back and forth at different parts of a math book instead of reading it straight through..

### Equations

Consider the expressions

1. $x^2-5x+4=0$
2. $x^2+y^2=1$
3. $x^2+2x+1=(x+1)^2$

They are assertions that two expressions have the same value. A strictly logical view of an equation containing variables is that it puts a constraint on the variables.  It is true of some numbers (or pairs of numbers) and false of others.  That is the defining property of an equation. Equation 1 requires that $x=1$ or $x=4$.  Equation 2 imposes a constraint which is satisfied by uncountably many pairs of real numbers, and is also not true of uncountably many pairs. But equation 3 puts no constraint on the variable.  It is true of every number $x$.

A strictly logical view of symbolic notation does math a disservice.  Here, the notion that an equation is by definition a symbolic statement that has a truth set and a falsity set may be correct but it is not the important thing about any particular equation. When we read and do math we have many different metaphors and images about a concept.  The definition of a kind of object is often in terms of things that may not be the most important things to know about it.  (One of the most important fact about groups is that it is an abstraction of symmetries, which the axioms don't mention at all.)

Equation 1. is something that would make most people set out to discover the truth set.  Equation 2. calls out for drawing its graph.  Equation 3. being an identity means that is useful in algebraic reasoning.  The images they call up are different and what you do with them is different.  The images and metaphors that cluster around a concept are an important part of the semantics of the symbolic language.

I expect to post separately about the semantics of variables and about the semantics of symbolic language embedded in mathematical English.

## Algebra is a difficult foreign language

Note: This post uses MathJax.  If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

## Algebra

In a previous post, I said that the symbolic language of mathematics is difficult to learn and that we don't teach it well. (The symbolic language includes as a subset the notation used in high school algebra, precalculus, and calculus.) I gave some examples in that post but now I want to go into more detail.  This discussion is an incomplete sketch of some aspects of the syntax of the symbolic language.  I will write one or more posts about the semantics later.

### The languages of math

First, let's distinguish between mathematical English and the symbolic language of math.

• Mathematical English is a special register or jargon of English. It has not only its special vocabulary, like any jargon, but also used ordinary English words such as "If…then", "definition" and "let" in special ways.
• The symbolic language of math is a distinct, special-purpose written language which is not a dialect of the English language and can in fact be read by mathematicians with little knowledge of English.
• It has its own symbols and rules that are quite different from spoken languages.
• Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
• A mathematical article or book is typically written using mathematical English interspersed with expressions in the symbolic language of math.

### Symbolic expressions

A symbolic noun (logicians call it a term) is an expression in the symbolic language that names a number or other mathematical object, and may carry other information as well.

• "3" is a noun denoting the number 3.
• "$\text{Sym}_3$" is a noun denoting the symmetric group of order 3.
• "$2+1$" is a noun denoting the number 3.  But it contains more information than that: it describes a way of calculating 3 as a sum.
• "$\sin^2\frac{\pi}{4}$" is a noun denoting the number $\frac{1}{2}$, and it also describes a computation that yields the number $\frac{1}{2}$.  If you understand the symbolic language and know that $\sin$ is a numerical function, you can recognize "$\sin^2\frac{\pi}{4}$" as a symbolic noun representing a number even if you don't know how to calculate it.
• "$2+1$" and "$\sin^2\frac{\pi}{4}$" are said to be encapsulated computations.
• The word "encapsulated" refers to the fact that to understand what the expressions mean, you must think of the computation not as a process but as an object.
• Note that a computer program is also an object, not a process.
• "$a+1$" and "$\sin^2\frac{\pi x}{4}$" are encapsulated computations containing variables that represent numbers. In these cases you can calculate the value of these computations if you give values to the variables.

symbolic statement is a symbolic expression that represents a statement that is either true or false or free, meaning that it contains variables and is true or false depending on the values assigned to the variables.

• $\pi\gt0$ is a symbolic assertion that is true.
• $\pi\lt0$ is a symbolic assertion that it is false.  The fact that it is false does not stop it from being a symbolic assertion.
• $x^2-5x+4\gt0$ is an assertion that is true for $x=5$ and false for $x=1$.
• $x^2-5x+4=0$ is an assertion that is true for $x=1$ and $x=4$ and false for all other numbers $x$.
• $x^2+2x+1=(x+1)^2$ is an assertion that is true for all numbers $x$.

### Properties of the symbolic language

The constituents of a symbolic expression are symbols for numbers, variables and other mathematical objects. In a particular expression, the symbols are arranged according to conventions that must be understood by the reader. These conventions form the syntax or grammar of symbolic expressions.

The symbolic language has been invented piecemeal by mathematicians over the past several centuries. It is thus a natural language and like all natural languages it has irregularities and often results in ambiguous expressions. It is therefore difficult to learn and requires much practice to learn to use it well. Students learn the grammar in school and are often expected to understand it by osmosis instead of by being taught specifically.  However, it is not as difficult to learn well as a foreign language is.

In the basic symbolic language, expressions are written as strings of symbols.

• The symbolic language gives (sometimes ambiguous) meaning to symbols placed above or below the line of symbols, so the strings are in some sense more than one dimensional but less than two-dimensional.
• Integral notation, limit notation, and others, are two-dimensional enough to have two or three levels of symbols.
• Matrices are fully two-dimensional symbols, and so are commutative diagrams.
• I will not consider graphs (in both senses) and geometric drawings in this post because I am not sure what I want to write about them.

## Syntax of the language

One of the basic methods of the symbolic language is the use of constructors.  These can usually be analyzed as functions or operators, but I am thinking of "constructor" as a linguistic device for producing an expression denoting a mathematical object or assertion. Ordinary languages have constructors, too; for example "-ness" makes a noun out of a verb ("good" to "goodness") and "and" forms a grouping ("men and women").

### Special symbols

The language uses special symbols both as names of specific objects and as constructors.

• The digits "0", "1", "2" are named by special symbols.  So are some other objects: "$\emptyset$", "$\infty$".
• Certain verbs are represented by special symbols: "$=$", "$\lt$", "$\in$", "$\subseteq$".
• Some constructors are infixes: "$2+3$" denotes the sum of 2 and 3 and "$2-3$" denotes the difference between them.
• Others are placed before, after, above or even below the name of an object.  Examples: $a'$, which can mean the derivative of $a$ or the name of another variable; $n!$ denotes $n$ factorial; $a^\star$ is the dual of $a$ in some contexts; $\vec{v}$ constructs a vector whose name is "$v$".
• Letters from other alphabets may be used as names of objects, either defined in the context of a particular article, or with more nearly global meaning such as "$\pi$" (but "$\pi$" can denote a projection, too).

This is a lot of stuff for students to learn. Each symbol has its own rules of use (where you put it, which sort of expression you may it with, etc.)  And the meaning is often determined by context. For example $\pi x$ usually means $\pi$ multiplied by $x$, but in some books it can mean the function $\pi$ evaluated at $x$. (But this is a remark about semantics — more in another post.)

### "Systematic" notation

• The form "$f(x)$" is systematically used to denote the value of a function $f$ at the input $x$.  But this usage has variations that confuse beginning students:
• "$\sin\,x$" is more common than "$\sin(x)$".
• When the function has just been named as a letter, "$f(x)$" is more common that "$fx$" but many authors do use the latter.
• Raising a symbol after another symbol commonly denotes exponentiation: "$x^2$" denotes $x$ times $x$.  But it is used in a different meaning in the case of tensors (and elsewhere).
• Lowering a symbol after another symbol, as in "$x_i$"  may denote an item in a sequence.  But "$f_x$" is more likely to denote a partial derivative.
• The integral notation is quite complicated.  The expression $\int_a^b f(x)\,dx$ has three parameters, $a$, $b$ and $f$, and a bound variable $x$ that specifies the variable used in the formula for $f$.  Students gradually learn the significance of these facts as they work with integrals.

### Variables

Variables have deep problems concerned with their meaning (semantics). But substitution for variables causes syntactic problems that students have difficulty with as well.

• Substituting $4$ for $x$ in the expression $3+x$ results in $3+4$.
• Substituting $4$ for $x$ in the expression $3x$ results in $12$, not $34$.
• Substituting "$y+z$" in the expression $3x$ results in $3(y+z)$, not $3y+z$.  Some of my calculus students in preforming this substitution would write $3\,\,y+z$, using a space to separate.  The rules don't allow that, but I think it is a perfectly natural mistake.

### Using expressions and writing about them

• If I write "If $x$ is an odd integer, then $3+x$ is odd", then I am using $3+x$ in a sentence. It is a noun denoting an unspecified number which can be constructed in a specified way.
• When I mention substituting $4$ for $x$ in "$3+x$", I am talking about the expression $3+x$.  I am not writing about a number, I am writing about a string of symbols.  This distinction causes students major difficulties and teacher hardly ever talk about it.
• In the section on variables, I wrote "the expression $3+x$", which shows more explicitly that I am talking about it as an expression.
• Note that quotes in novels don't mean you are talking about the expression inside the quotes, it means you are describing the act of a person saying something.
• It is very common to write something like, "If I substitute $4$ for $x$ in $3x$ I get $3 \times 4=12$".  This is called a parenthetic assertion, and it is literally nonsense (it says I get an equation).

## Bugs in the Mathematical Dialect of English

The mathematical dialect of English is what I call Mathematical English in the abstractmath website.  It is a different language from the symbolic language, which is not a dialect of English.

I have written about the problems with Mathematical English in a ridiculous number of places.  (See references in The Handbook of Mathematical Discourse).  It is normal for a dialect of a language to use words and grammatical structures that in the original language mean different things.  (See Dialects below).

### Words with different meanings

• A set is a group in standard English, but not in math English.
• The number 2+3i is a real number in standard English, but not in math English.
• And so on.

### Use of adjectives and prefixes

• A "noncommutative ring" has commutative addition.
• A "semigroup" has a fully defined binary operation.

### If, then

The bug that grabs math newbies by the throat and won't let go is the meaning of "If P, then Q".

• "If a number is divisible by 4, then it is even" in math dialect means a number not divisible by 4 might be even anyway.
• "If you eat your broccoli you will get your dessert" in standard American Parental English does not mean you might get your dessert if you don't eat your broccoli.

And then there is the phenomenon of Vacuous Implication, which leaves students gasping and writhing.

Most Americans are not familiar with dialects in the sense I am using the word here, since the only really different dialects we have are Gullah and Hawaiian Pidgin, both of which are very hard to understand; although for example Appalachian English and African-American urban vernacular [1] are dialects of a milder sort.  I grew up in Savannah and heard diluted Gullah sometimes on the street (didn't understand much).  I am also rather familiar with Züritüütsch since we lived in Zürich for a year.

What the rest of the world call dialects have many distinctive properties:

• They have nonstandard pronunciation to the point where they are difficult to understand.
• They have differences in grammar.  (Both Gullah and especially Hawaiian Creole have differences in grammar from Standard English.)
• They have differences in vocabulary, enough sometimes to cause misunderstanding.

I grew up speaking an Atlanta dialect, which really did have differences in all those parameters.  But what people today call a Southern accent is really just an accent (minor variations in pronunciation), not a dialect.

Hawaiian Creole, and possibly Gullah, but not the other dialects I mentioned, are singled out by linguists as creoles because they been modified heavy influence from another language.  Züritüütsch is not a creole, but it is quite difficult for native German-speakers to understand.  The Swiss situation particularly emphasizes the distinction between "dialect" and "accent".  The typical native of Zürich speaks Züritüütsch and also speaks standard German with a Swiss accent.

### Reference

[1] What Language Is (And What It Isn't and What It Could Be) by John H. McWhorter. Gotham, 2011.

## The languages of mathematics

Conjecture: Mathematical English (ME) and the symbolic language of math (SL) are two distinct languages, not dialects of the same language.

I have asserted this in several places (Handbook, abstractmath.org) but I am not a linguist and it could be that linguists would disagree with this conjecture, or that the study of a mathematical corpus would reveal that another theoretical take on the situation would be more appropriate.

Some relevant points are listed below. I intend to expand on them in later posts.

1) Is ME a dialect of English or a register of English? Or does it have some other relationship to English?

2) ME appears to have several dialects or registers. One register is that used for what mathematicians call “formal proofs”. These are not formal in the sense of first order predicate logic, but their language is constrained, with the intent of making it easier to see the logical structure of the argument. Another register is that of “intuitive [or informal] explanations”. This is more like standard English.

3) The SL is clearly not a spoken language. It is a two-dimensional written language using symbols from English and other languages and some symbols native only to math. People do try to speak formulas aloud occasionally but this is well known to be difficult and can be done successfully only for fairly simple expressions.

4) There are other non-spoken languages such as ASL for example. I don’t know whether there are other non-spoken languages that are written. I don’t think dead languages count.

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