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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

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

There are other types of non-physical objects.

Examples of math objects

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

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.

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:

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 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 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.

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

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 $(1,2,1)$.
  • Not to mention the set $\{1, 2\}$ and the set of sets $\{\emptyset,\{1\},\{2\},\{1,2\}\}$, which is the set of all subsets of $\{1,2\}$. It has four elements.

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: \[F(x)=\left\{\begin{align} &1\,\,\,\,\,\,\text{if }x\,\,\text{is rational} \\ &0\,\,\,\,\,\,\text{if }x\,\,\text{is irrational} \\ \end{align} \right.\] for all real $x$ (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?

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 also a real number.
  • It is divisible by $2$.
  • It is one more than $41$.
  • You can add, subtract, multiply and divide it by other numbers.

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

In other words, you know some of its properties, some processes you can apply to it, and some relationships it has with other objects. This is all you ever need to know about a math object. "What it is" is irrelevant. In my opinion asking "what is it?" is a meaningless question about any object, physical or abstract.

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.

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 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.

  • 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 write a "C" on the board, point to it, and say, "This is a C", you are referring to a particular mark.
  • 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.

    • 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.

Fictional objects

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.

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!"

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

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.

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 abstract 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 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 and manipulated in similar ways by 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

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

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 $\{a\}$".

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 they say about it 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.


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