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"Mathematics consists of true facts about imaginary objects." -- P. Davis and R. Hersh.

"Mathematical structures have an eerily real feel to them."--Max Tegmark

MATHEMATICAL OBJECTS

CONTENTS

Mathematical objects

Examples of math objects

Talking about math objects

Consistent experience

Kinds and properties

Constructors

Objects, processes and relationships

What IS a mathematical object?

Abstract objects

Foundational Aspects

Mathematical objects

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

Numbers, functions, triangles, matrices, groups and more complicated things such as vector spaces and infinite series are all examples of mathematical objects. See Examples.
This chapter discusses how we think about math objects in general.
Other parts of abstractmath.org discuss how we think about particular kinds of math objects, such as sets and numbers.

Math objects are abstract objects. They are not physical objects, but we think about them and talk about them as if they actually existed.
Math objects have certain properties that other kinds of abstract objects may not have. In particular, unlike other kinds of abstract objects, math objects are inert. This is discussed in more detail in the chapter on Images and metaphors

Math objects don't move or change over time.
Math objects don't interact with other objects or with the real world.

I will not go into what math objects "really are". For the purposes of doing math, what we need to understand is (among other things) how mathematicians think about them, not what they "are".

Examples of math objects

The number $42$ is a math object. (More about $42$ here.)
The rectangle with sides of length $2$,$3$,$2$,$3$ is a math object. This rectangle is pictured below. The picture is not a math object, it is a bunch of pixels (unless you printed this out in which case it is a bunch of inkspots.)
The set $\{3,5,6,7,8\}$ is a math object. It is not five different numbers, it is one object and its defining property is that the numbers $3$, $5$, $6$, $7$ and $8$ are elements of the set and nothing else is an element of the set.
The matrix \[\left[ \begin{matrix} -1 & 5 & -3.2 \\ 7.1 & 0 & 0 \\ 1 & 99 & 7 \\\end{matrix} \right]\] is one single math object with $9$ entries.
The set $\mathbb{R}$ of real numbers is a single math object with unimaginably many elements. It is still one single abstract thing. (Note that "the set of real numbers" means the set containing every real number and nothing else. See "the".)

The function $f(x):=2{{\sin }^{2}}x-1$ is a math object. You can compute its value at many different numbers. But the function $f(x)=2{{\sin }^{2}}x-1$ is a single, static math object. More.
You can take the derivative of $f(x)$. It is another function, namely $f'(x):=4\sin x\cos x$. Differentiation itself is a math object, an operator, which is a function from a function space to another function space.
In fact a function space is a math object. For example the space of all differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ is a math object. Each point in a function space is a whole function (not individual values of it) considered as a single math object, and on top of that the whole function space is a single math object, conceived of as in one chunk.
Certain branches of math regard things like equations, diagrams, theorems and proofs as math objects. These branches of math include mathematical logic, category theory, and type theory, which includes the hot new approach called HoTT.
I'm not trying to blow your mind. Well, yes I am a little. But this list gives a clue to how basic a concept "mathematical object" is in math.

Talking about math objects

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

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.

We say "This rectangle has area $6$ square units" in the same way we say "This house has $2000$ square feet." So we refer to rectangles using demonstratives such as "this".
A rectangle can be the subject or object in a sentence. "This rectangle has area $6$ square units", or "Rotate the rectangle $180$ degrees."
A rectangle can be described as having properties in the same way that we would describe a physical object having properties. "A rectangle has four corners."

We use the same grammatical constructs for the number $42$ as for names.

We say "$42$ is an even number" in the same way we say "Mars is a planet". In particular, we do not use "the" to refer to $42$, just as we don't (in English) say "the Latvia".
We do say "the Bronx" and "the Hague". We used to say "the Ukraine" but the government of Ukraine objected. On the other hand, the people in the Bronx demand that you use "the".
We could also give our rectangle a name, for example "$R$". Then "$R$" would be treated like a proper noun, as in for example "$R$ is half again as long as it is high", in the same way as we say "Latvia is smaller than Russia".

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 ten 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."

Examples:

If we ask two people to find the zeroes of a polynomial and they give different answers, we expect to find their mistake. We don't think the polynomial is a physical object but our experience with it is similarly repeatable and consistent.
In the case of a polynomial we probably will find the mistake. When we are faced with contradictory mathematical claims in more complicated situations it may be quite difficult to find the mistake, and we may have to leave it a mystery.
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.

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 $\pi$ is a real number but not an integer. A kind of object can be named by a noun or 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.

Your mental image of "kind" may be that it is a more permanent and intrinsic thing than "property". (Being human is a kind but being red-headed is a property.) This distinction is not important when thinking rigorously about math because in rigorous mode, math objects are thought of as inert and unchanging.

Constructors

Math objects can be constructed from other math objects by many different procedures.

The integers $1$, $2$ and $4$ are math objects, and from them you can construct the set $\{1, 2, 4\}$.
You can construct the ordered pair $(1, 2)$ from the numbers $1$ and $2$.
From those same numbers you can construct another ordered pair $(2,1)$ and a bunch of ordered triples such as $(1,2,1)$.
For any given set you can construct its set of subsets. For example, the set $\{\emptyset,\{1\},\{2\},\{1,2\}\}$ is the set of subsets of $\{1,2\}$.

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

In abmath, we don't (usually) talk about the set of pairs of people consisting of husbands and wives. We don't talk about the list of American states ordered alphabetically. 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!

Acknowledgments to 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 add $77$ to itself, getting $154$.
You can take the derivative of the function $x^3$, getting $3x^2$.

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 add $77$ to itself but you can't add a triangle to itself.
You can't take the derivative of the function defined by: \[F(x)=\left\{\begin{align} &1\,\,\,\,\,\,\text{if }x\,\,\text{is rational} \\ &0\,\,\,\,\,\,\text{if }x\,\,\text{is irrational} \\ \end{align} \right.\] (more about that function here).

You can perform more complicated operations involving several objects.

You can nail a birdhouse to a tree.
You can add $42$ and $63$.
You can calculate \[\int_{3}^{5}{3{{x}^{2}}\,dx=98}\] which means applying the process of calculating the definite integral to three objects: two numbers, $3$ and $5$, and one function $x\mapsto 3{{x}^{2}}$.

Math objects can have various relationships with each other.

The Washington Monument is taller than the White House.
$42$ is less than $63$.
Two triangles may or may not be congruent. If they are not congruent, they may or may not be similar

What IS a mathematical object?

I am myself and my circumstances. José Ortega y Gasset.

As I said at the top of this chapter, I am not going to talk about what math objects are, but how you should think about them.

Example

The number $42$ is an object about which you know a bunch of things:

It is called "42" in decimal notation and "101010" in binary notation.
It is an integer.
It is divisible by $2$.
It is one more than $41$.
You can add, subtract and multiply 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. That is all you ever need to know about a math object. "What it is" is irrelevant.

All you can know about a math object is its properties,
the processes you can apply to it,
and its relationships with other math objects.
There is nothing else to know.

Abstract objects

Interactive 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 too much 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.

Fictional objects

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

A unicorn is a fictional 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 $\pi $ to be equal to $3$: The meaning of the symbol $\pi $ and the concept of unicorn are part of our common culture (at least the common culture of the intersection of fantasy fans and mathematicians) and we violate our expectations and confuse people if we use these symbols and words with other meanings.

The preceding paragraph is controversial. Many mathematicians would say you certainly may define "unicorn" to be anything you want. Some of them redefine common terminology in mathematics books they write. Those books are very confusing to read.

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 math objects, abstract objects and fictional objects have 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 use proper nouns to refer to them. We say:

"$42$ is an even number."
"September has thirty days."
"Darth Vader wears a ventilator."

We also refer to them using common nouns:

"An integer is even if it is divisible by $2$."
"A month has at least 28 days."
"A unicorn has cloven hooves."

You can experiment with coming up with sentences in which each type of object is referred to as the object of a verb, the object or a preposition, and so on.
This suggests that these concepts are all stored and manipulated in the brain in the same way words denoting physical objects are.

References

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.

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

It is common in texts on foundations to define each kind of mathematical object as a special kinds of set. For example, the ordered pair $(a,b)$ may be defined to be the set $\{\{a\},\{a,b\}\}$. This has technical importance in foundations, but it is totally misleading to say things such as "The first coordinate of $(a,b)$ is $\{a\}$".

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

Specification for ordered pairs

All we need to know about the ordered pair $(a,b)$ is that if two ordered pairs are equal, then their first coordinates are equal and their second coordinates are equal. So for example, if $(a,b)=(c,d)$, then $a=c$ and $b=d$. For example, $(1,3)\neq(2,3)$ because $1\neq 2$. That is essentially the definition of "ordered pair" in homotopy type theory, a proposed system for providing a foundation for math other than set theory.

Example

Usually, a mathematician will simply talk about the number $2$ 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.

For purposes of foundations, the natural numbers may be defined by the Peano axioms. It is correct to say that in that axiom system, $2$ may be defined by saying that it is something like "the successor of the successor of $0$." Other axiom systems define $2$ in other ways. It is wrong to say that what $2$ really is (its "true inner essence") is the successor of the successor of $0$.

Two controversial statements:

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.

Advice to math majors: Observe and emulate what successful mathematicians say and do when they do math.

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

Acknowledgments

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

Azzouni, J. Metaphysical Myths, Mathematical Practice. Cambridge University Press, 1994.
Dubinsky, E. and McDonald, M. (2002). APOS: A constructivist theory of learning in undergraduate mathematics education researCH.

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.
David Tall, Michael Thomas, Garry Davis, Eddie Gray, Adrian Simpson, What is the object of the encapsulation of a process?
Wikipedia, Mathematical object.

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