abstractmath.org

help with abstract math

`Produced by Charles Wells.  Home    Website TOC    Website Index   Blog `
`Back to top of Understanding Math chapter`

Last edited 5/12/2009 3:49:00 PM

### Contents

Introduction

Mathematical Objects

# Introduction

Mathematical objects are what we talk and write about when we do math.

Numbers, functions, triangles, matrices, and more complicated things such as vector spaces and infinite series are all examples of math objects.  See Examples.

Those new to abstract math often complain that they don’t understand some type of math object.   What they really mean in many cases is that they don’t know how to think about it.  Here I discuss how we think about math objects in general.   The sections on special types of math objects, such as sets and numbers, talk about how we think specifically about those objects.

Math objects are not physical objects, but we think about them and talk about them as if they were.

This is discussed here.

There are other types of non-physical objects.  Math objects are a type of abstract object, as is for example the month of September or an organizational tree.  More about abstract objects.  They are similar to fictional objects, such as a unicorn or Darth Vader.  More about fictional objects.

I will not go into what abstract or math objects "really are", which is a question that belongs to the philosophy of mathematics.   For the purposes of doing math, what we need to understand is how to think about them, not what they “are”.

Math objects on this site

When this site talks about math objects it (usually) refers only to math objects constructed from other math objects.  We don’t talk about “the set of American Presidents” for example.  More about this.

# Examples of math objects

¨  The number 42 is a math object.  The decimal notation ‘42’ and the binary notation ‘101010’ are two different representations of the same math object.  But what is the number 42?

¨  A rectangle is a math object.  A rectangle with sides of length 2, 3, 2, and 3 is a specific math object.  (Some would say, “The rectangle with sides of length 2, 3, 2, and 3”.  See isomorphism and identity.)

¨  The matrix

is one single math object with 9 entries.

¨  The set  is a math object.  It is not five different numbers, it is one object and its defining property is that only the numbers 3, 5 ,6, 7 and 8 are elements of the set.  The set  of all real numbers is a single math object with unimaginably many elements.   It is still one single abstract thing.  More.

¨  The function  is a math object.  You can compute its value at many different numbers.  You can take its derivative, which is another function.   But the function  is a single, static math object.  More.

Mathematicians talk about math objects as if they were physical objects.

We say

“This rectangle has area 6 square units.”

in the same way we say

“This house has 2000 square feet.”

We say

42 is an even number.”

in the same way we say

“Arnold Schwarzenegger is a Republican.”

Of course we know that a rectangle is not a physical object like a house and that the number 42 is not a person.  Nevertheless:

¨  We use the same grammatical constructs for rectangles as for objects such as houses.  A rectangle can be referred to by demonstratives such as “this”, can be the subject or object in a sentence, and can be described as having properties in the same way that we would describe a physical object having properties.

¨  We use the same grammatical constructs for the number 42 as for proper names. In particular, we do not use “the” to refer to 42, just as we don’t (in English) say “the Latvia”.   We could also give our rectangle a name, for example “R”.  Then “R” would be treated like a proper noun.

We talk about math objects in these ways because we think of them as things, although not as physical things.

Mathematicians think of math objects as things.

To understand abstract math, you need to think that way too.

Remember: I am discussing how we think and talk about mathematical objects, not what they really are.

## Consistent Experience

Mathematical objects are like physical objects in that our experience with them is repeatable.

¨  If you ask some mathematicians about a property of some particular mathematical object that is not too hard to verify, they will generally agree on what they say about it.

¨  When there is disagreement they commonly discover that someone has made a mistake or has misunderstood the problem. This is analogous to the way we deal with physical objects.

#### Example

Eudora:  “Take a prime number as an example, such as 111.”

Rowena:  “111 is not a prime.  It is divisible by 37.”

Eudora:  “Well my stars, you’re right.  I never noticed that.”

Compare this to

Eudora:  “That tree must be 10 feet tall”.

Rowena:  “It can’t be that short, it’s taller than the two-story house next to it.”

Eudora:  “Oh yeah, good point.”

#### Example:If we ask two people to find the area of a rectangle and they give different answers, we expect to find their mistake.  We don’t think the rectangle is a physical object but our experience with it is similarly repeatable and consistent.

In the case of a simple rectangle we probably will find the mistake.  When we are faced with contradictory mathematical claims in complicated situations it may be quite difficult to find the mistake, and we may have to leave it a mystery.  This can also happen with a physical experiment that gives a deviant reading.  But we operate as if we believe that mathematics is consistent:

We expect that there is always an explanation

of an apparent contradiction in math

even if we cannot find it.

## Specific and variable mathematical objects

It is useful to distinguish between specific mathematical objects and variable mathematical objects.  The number 42 (the mathematical object called “42”, not the representation “42”) is a specific mathematical object. So is the sine function (once you decide whether you are using radians or degrees).  But math books are full of references to mathematical objects, typically named by a letter, that are not completely specified.  I call these variable objects (not standard terminology).

Example

If an author or lecturer says “Let x be a real variable”, you can then think of  x as a variable real number.   If they say, “Let G be a group” you can think of G as a variable group.

Example

If you are going to prove a theorem about functions, you might begin, "Let  f  be a continuous function", and in the proof refer to  f  and various objects connected to  f.    This makes  f  a variable mathematical object. When you are proving things about it you may use the fact that it is continuous.  But you cannot assume that it is (for example) the sine function or any other particular function.

### How to think about variable objects

The remarks in the section on variables and substitution apply to more general math objects.  If someone says “Let x be a real number, you can think of x as playing the role of a real number, or as a slot in which you can put any real number.

These metaphors refer to the name x.  The idea about x as a variable object means thinking of x as a genuine mathematical object, but with limitations about what you can say or think about it.  There are two related points of view:

1. Some statements about the object are neither true nor false.

This means x is a genuine mathematical object and you can make assertions about it, but some of the assertions might have no truth value (but there are other ways to think about it that you may like better).  From “Let x be a real number” you know these things:

¨  The assertion “Either  or  ” is true.

¨  The assertion “  ” is false.

¨  The assertion “  ” is neither true nor false.

The assertion “x is a real number” is true is in a certain sense ithe most general true statement you can make about x.   In other words, x is a mathematical object given by an incomplete specification, so you are limited in what you can say about it or in what conclusions you can draw about it.

If you say, “Let n be an integer divisible by 4, you cannot assume it is 8 or 12, for example.  In other words, the statement “n is divisible by 4” is true, and “n = 3” is false, but the statement “n = 8” is neither true nor false, and you can’t derive any conclusions from n being 8.

2. The object is fixed but some things are not known about it.

If you say x is a real number, you know x is a real number (duh) and:

¨  You know x is either positive or nonnegative.

¨  You know  is not equal to any negative number.

¨  You don’t know whether x is positive or not.

This way of looking at it involves thinking of x as a particular real number.  During the process of solving the equation  you are thinking of x as a particular number, but note that when you are finished, all you know is x = 2 or x = 3:  you still don’t know which it is.  This point of view causes me cognitive dissonance, but the point of view that some statements are neither true nor false upsets other people.  I like point of view (1) better than (2) but I have to admit that it is much less familiar to most mathematicians.

## Kinds and properties

Mathematical objects come in different kinds and have various properties.

One kind of mathematical object is “integer”.  Another is “real number”.  The number 42 is both an integer and a real number.  The number  is a real number but not an integer.  A kind of object can be named by a noun phrase.

The number 42 has the property of being even.  The number 43 is an integer but not an even integer.  Both numbers have the property that they are greater than 40.   Thus properties can be named by adjectives (“even”) or phrases (“greater than 40”).

 Your mental image of “kind” may be of a more permanent and intrinsic thing that “property”.  (Being human is a kind but being blond is a property.)  This distinction is not important when thinking rigorously about math because math objects are inert and unchanging.

The ideas of kind and property are not really different.

You could define a kind of mathematical object called ent to denote an even integer and another kind ont to denote odd integers.  Then you could say “42 is an ent” using the same grammar as when you say “42 is an integer”.   So although we think of “integer” as a kind of thing and “even” as a property of things, calling even integers ents allows us to think of “even integer” as a kind.

## Constructors

If you are given some math objects you can construct the set that contains just those objects.  For example, the integers 1, 2 and 4 are math objects, and they are the only elements of the set {1, 2, 4}.   Similarly you can construct the ordered pair (1, 2).  These are examples of  constructors.

In abstractmath.org, we construct math objects only out of math objects.

We don’t (usually) talk about pairs of people consisting of husbands and wives.  We don’t talk about the set of American Presidents.   This is to avoid certain philosophical complications that would be distracting and largely irrelevant.

Other websites and texts do  make constructions using math constructors on physical objects.    They are not wrong to do this.  Here I am merely wimping out on getting into the complexities they involve!

## Objects, processes and relationships

A math object is like a physical object in that you can do things to it.

¨  You can burn a piece of wood.

¨  You can differentiate the function f defined by the formula , getting the function h defined by , usually denoted by .  You input a function to the operator and out comes its derivative (if it has one).  More about that here.

You can’t do most operations to every object, though.

¨  You can burn a piece of wood but you can’t burn a molecule of helium.

¨  You can multiply 42 by 2 but you can’t multiply a triangle by 2.  (Objection).

¨  You can’t differentiate the function defined by:

for all real x

You can perform more complicated operations involving several math objects.

¨  For example, you can add 42 and 63.

¨  You can calculate

which means applying the process of calculating the definite integral to three objects: two numbers, 3 and 5, and one function .

Math objects can have various relationships with each other.

¨  42 is less than 63.

¨  Two triangles may or may not be congruent.  If they are not congruent, they may or may not be similar.

# Appendices

## Other kinds of non-physical objects

### Abstract objects

There are other kinds of abstract objects besides math objects.  I will give some examples here.  But note:  you can get carried away and write books about abstract objects and classify them in also sorts of ways.  You may think I have got carried away a bit here!

¨  “September” is an abstract object with a proper name.  It certainly is not a physical object.   Its properties change over time (sometimes “this month is September” is true and sometimes it is false) and it affects what people do (some of us have to go back to school).   Neither of these is true of mathematical objects (see rigorous).

¨  A schedule is an abstract object.  When we think about our schedule for Wednesday afternoon, it may be represented as a physical piece of handwriting, or as a bunch of pits on our hard disk (which if we push the right buttons becomes characters for us to read on a screen), or it may be only in our mind.   The schedule is an abstract object with possible physical representations, and we refer to it as we refer to physical objects.  We say “my schedule” and “I need to make a schedule”.  A schedule is not a mathematical object; it affects what people do and it changes over time.

¨  A letter of the alphabet, such as “c”, is an abstract object.   However the letter “c” is also associated with a certain physical shape of a mark on paper or computer screen.  Sometimes we say “c” to refer to an actual mark of that shape and sometimes we are talking about the letter in a more abstract sense.

§  When you say “This is a ‘c’: C” you are talking about a particular mark.

§  When you say, “Let c denote the speed of light”, you are not talking about the particular physical mark you see in that sentence.. What you mean is that anywhere in your discourse when you say or write ‘c’ you mean the speed of light.

¨  A variable, say Height, in a running computer program is an abstract object, but it is not a mathematical object. At different times when the program is running, it may have different values (quite possibly stored at different locations in the computer’s memory), so it is not inert. It may be in a subroutine, in which case it may not exist except when the subroutine is running, so it does not have an independent existence. And it can certainly interact (in a sense that would not be easy to explicate) with physical objects, for example if it keeps track of the height of a missile in order to send it a signal to explode if its height becomes less than 100 meters.

None of these examples are mathematical objects but they are abstract objects.

The idea of distinguishing between abstract objects and math objects and in particular the schedule example are from What is Mathematics, Really? by Reuben Hersh.

You can find out what philosophers say about abstract objects starting here.

### Fictional objects

A unicorn is a fictional object.  (Why “object”?)  We may talk about them as if they are real physical objects, and we think of them as having many of the properties of physical objects.  A unicorn has a horn in the middle of its forehead, allows only virgins to ride it, and has cloven hooves.  If you show a picture of a supposed unicorn that shows horse-type non-split hooves to certain types of fantasy fans, they will object strongly and say, “That’s wrong!  Unicorns have cloven hooves!”   But what does that mean?  Why not define unicorns any way you want, since they are not real?  Answer: For the same reason we must not define  to be equal to 3:  The meaning of the symbol  and the concept of unicorn are part of our common culture (at least the common culture of the intersection of fantasy fans and math students) and we violate our expectations and confuse people if we use these symbols and words with other meanings.

Another type of fictional object is a character in a book, for example Sherlock Holmes.  The source of information we have about Sherlock Holmes is the set of stories Arthur Conan Doyle wrote about him.

### What they have in common

Every abstract object, every fictional object, and every real physical object have something in common.

¨  Each one corresponds to a physical arrangement in our brain that connects the concept to certain properties, relationships with other objects, and expectations of behaviors.

¨  Each one is talked about using many of the same grammatical constructions as for physical objects.     We say “September has thirty days” and “Darth Vader wears a ventilator” (proper nouns), and “A month has at least 28 days” and “A unicorn has cloven hooves” (common nouns).  This suggests that these concepts are all stored in similar ways in the brain.

## Foundational aspects

Sometimes, when asked about how they think about an object, a mathematician will give a description of the object based on some particular construction used in the study of the foundations of mathematics.  Such constructions are useful for showing that some part of math is consistent if some other more primitive part is.  However, in my opinion they are not usually much help in studying the particular object.

Example  In many treatments of foundations, it appears that there are many copies of the natural numbers that can be constructed, and if pressed a mathematician might say something like: “For definiteness let’s pick a copy of the natural numbers.”  If the subject of foundations does not come up, the mathematician will simply talk about 42 as if it were a specific individual that we are all familiar with.  That is because that is the way a mathematician thinks about it.

Example  It is common in texts on foundations to define ordered pairs as special kinds of sets.  For example, the ordered pair  may be defined to be the set .  This or something like it is worth doing.  It shows that if set theory is consistent, then so is the theory of ordered pairs.

 Wikipedia has an illuminating discussion of how  ordered pairs have been defined.

As a result, some mathematicians (erroneously in my opinion) say that  is .  Nevertheless, these mathematicians do not think  of  as .   Suppose out of the blue (not when you have just been discussing foundations) you ask:  “Is  an element of the pair ?”  The  typical mathematician will correct you to say, no, it is not an element of , it is the singleton set containing the first coordinate of .

I claim:

When they are actively doing mathematics,

as opposed to philosophizing about math,

most mathematicians think about mathematical objects

in the ways suggested by how they talk about them.

That is a controversial statement!

My advice to math majors is:

Observe and emulate

what successful mathematicians say and do

when they do math.

What they say about it when they philosophize about it is (in my opinion) not necessarily trustworthy.

What is  the number 42, if it is not the string “42” of numerals?

As I said at the top of the page, I am not going to talk about what math objects are, but how you should think about them.  The number 42 is an object about which you know a bunch of things:

¨  It is called “42” in decimal notation.

¨  It is called “101010” in binary notation.

¨  It is an integer.

¨  It is divisible by 2.

¨  It is one more than 41.

¨  You can add, subtract, multiply and divide it by other numbers.

In other words, you know some of its properties,  some processes you can apply to it, and some relationships it has with other objects.

In the same way, you know

¨  The Queen of England is a human being.

¨  She may be called “Queen Elizabeth II”.

¨  She can walk and talk.

¨  You can take her photograph.

¨  She opens Parliament every year.

So we know some properties she has and some actions and relationships she takes part in.  So what does it mean to say what IS she?  Do I need to define “human being”, say by giving her DNA?   You can see that this type of reasoning gets into a philosophical mess and is not the way we usually think about objects, physical or otherwise.

You could also define it to be the similar triangle whose sides were twice as long.  You could define it to be two copies of the triangle.

A written text exists in a relationship with its readers of sharing meanings of words and phrases (and other concepts).  It is very likely that you the reader thinks “mutiplying 42 by 2” means the same thing I think it means.  There is no standard definition of multiplying a triangle by 2 that most readers of this text share.

Return.

I am using the word “object” in the philosophical sense of object of thought.  Such a thing can be animate or inanimate.  Sherlock Holmes is a fictional object in this sense.

Return

### Acknowledgments

The discussions in this section was influenced by these books and papers.

¨  Azzouni, J. Metaphysical Myths, Mathematical Practice. Cambridge University Press, 1994.

¨  Hersh, R. What is Mathematics, Really?  Oxford University Press, 1997

¨  Sfard. A.  ‘Symbolizing mathematical reality into being  Or how mathematical discourse and mathematical objects create each other.’  In the book Symbolizing and communicating in mathematics classrooms:  Perspectives on discourse, tools and instructional design.  Paul Cobb, Erna Yackel and Kay McClain, eds.  Lawrence Erlbaum, 2000.

¨  Shapiro, S.  Thinking about mathematics: The philosophy of mathematics.  Oxford University Press, 2000.