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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

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.  Its value can be computed at many different numbers but it is a single, static math object.  More.

Thinking about math objects

Talking 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. 

¨  Text Box: Mathematical structures have an eerily real feel to them. –Max TegmarkWe 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.

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 apparent contradictions about math

even if we cannot find it.

Specific and variable mathematical objects

Text Box: A logician would refer to the symbol  f  as a variable, but mathe¬maticians in general would not use the word “variable” in that situation. 

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)     Text Box: The idea of a variable mathe¬mati¬cal object is not often taught as such in undergraduate classes but it is worth pondering. It has certainly clarified my thinking about expressions with variables.  

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”). 

Text Box: Your mental image of “kind” may be of a more permanent and intrinsic thing that “prop-erty”.  (Being human is a kind but being blond is a property.)  This distinc¬tion 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!   

Acknowledgment

Wesley Morris

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  

(more about that function here).

You can perform more complicated operations involving several math objects. 

¨  For example, you can add 42 and 63.

¨  Text Box: read about paren¬thetic assertionsYou 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 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.