Last edited 5/12/2009 3:49:00 PM
Contents
Introduction.. 1
Examples
of math objects. 2
Thinking
about math objects. 2
Talking
about math objects. 2
Consistent
Experience. 3
Specific and variable mathematical objects. 3
Kinds
and properties. 4
Constructors. 4
Objects,
processes and relationships. 4
Appendices. 5
Other
kinds of non-physical objects. 5
Foundational aspects. 6
Answers
to questions. 8
Mathematical
Objects
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.
![Text Box: [Mathematics consists of] true facts about imaginary objects. -- P. Davis and R. Hersh](MMMathObj_files/image001.gif)
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.
Thinking about math objects
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.
These facts about
how we talk about math objects show how we think about them.
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.
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
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.
Merge with imgs & metaphors for variables
There are several ways to think about a
variable mathematical object x, and several names we can use to
refer to the concept of variable object.
Ways
of thinking:
- x is a math object but not all
statements about it have a truth value. This is the way
of thinking I was pushing in the section quoted. It comes from
category theory. During the process of proving something about it,
certain statements may acquire a truth value. At the end of
the proof you may or may not know everything about it. For example,
if you solve a linear equation in x then at the end you
know exactly what it is. If you solve a quadratic equation, you wind
up knowing only that x is one of two things (in general).
- x is a math object but we don't know everything about it.
Right after you say something like "Let x be a real
number" you know only that it is a real number, but if you then start
a mathematical argument you may put constraints on it. I have an
(over-) elaborate example of this in the section on context. At each step of the argument, you may know more about x than
you did before. This is what computer scientists call the
"context" of the argument, meaning everything known about the
values of all the variables at any given place in the
argument.
In some sense the m
and n
in the context section start out by being generic but in each
phrase the text may give additional information (m divides n
for example), so they no longer deserve to be called generic.
Category theorists would refer to x as
a "variable object", and that is the terminology I used. But
there are two things under discussion: How we think about it and what we call
it.
About variable real number vs generic:
If "generic" stops the students from thinking of the variable as
changing while they are looking at it, as you describe in your paper, then it
is a better choice of name. But there is something subtler going on
here. Right after you say something like "Let x be
a real number" you know only that it is a real number, but if you then
start a mathematical argument you may put restraints on it. The variable
doesn't "wiggle while you look at it" but your information about it
does change in stages as you read the mathematical discourse. Now that I
have said all this I don't know the right terminology to use. Maybe
Acknowledgments:
Susanna Epp.
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.
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!
Acknowledgment
Wesley Morris
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
(more about that function here).
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.
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.
§
When you say “i before e except after c…”
are you talking
about physical marks? It makes my head
hurt to think about this.
¨ 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.
Answers to questions
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.
Return.
You
can too multiply a triangle by 2. You
could define it to be the triangle that is similar but twice as big in area!
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.
Why did you call a unicorn an “object”?
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.