Produced by Charles Wells Revised 20150515 Introduction to this website website TOC website index blog Back to head of understanding math chapter
[Mathematics consists of] true facts about imaginary objects.  P. Davis and R. Hersh. 
Mathematical structures have an eerily real feel to them. Max Tegmark 
Math objects are not physical
objects,
but we think about them and talk about them as if they were.
That is one of the more important aspects of thinking about math objects. I will not go into what abstract or math objects "really are". For the purposes of doing math, what we need to understand is (among other things) how to think about them, not what they "are".
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 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". 
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. 
Remember: I am discussing how we think and talk about mathematical objects, not what they really are.

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."
Eudora: "That tree must be ten feet tall".
Rowena: "It can't be that short, it's taller than the twostory house next to it."
Eudora: "Oh yeah, good point."
We expect that there is always an
explanation
of an apparent contradiction in math,
even if we cannot find it. Kinds and propertiesMathematical 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 $40$").
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. ConstructorsMath objects can be constructed from other math objects by many different procedures.
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 relationshipsA math object is like a physical object in that you can do things to it.
You can't do most operations to every object, though.
You can perform more complicated operations involving several objects.
Math objects can have various relationships with each other.
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. ExampleThe number $42$ is an object about which you know a bunch of things:
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.
Other kinds of nonphysical objectsAbstract objectsThere 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!
Fictional objects
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 horsetype nonsplit 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. 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
Foundational aspectsSometimes, 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 $\{a\}$".
Specification for ordered pairsAll 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. ExampleUsually, 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.
The discussions in this chapter was influenced by these books and papers.
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