Last edited 3/27/2008 10:17:00 AM
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.
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. Its value can be computed at many different
numbers but it 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 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
apparent contradictions about 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).
Examples
¨
The variable x
is a variable real number. 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. It follows
that
a) 
The assertion “Either or ” is true.
b) The
assertion “ ” is false.
c)
The assertion “ ” is neither true nor false.
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.
¨ Suppose you wanted to prove that if n is divisible by 4 then it is
even. During the proof, you can use the
fact that n is divisible by 4. You cannot assume
it is 8 or 12, for example. In other
words, during
the proof the
statement “n is divisible by 4” is true, and “n = 3” is false, but the statement “n = 8” is neither true nor false.
¨
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.
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”).
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
dentist may tell you that he has a hole in his schedule at 3PM
next Monday; would you like to come then? That hole in his schedule is not a
physical object. It is an abstract object. But it 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.