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Produced by Charles Wells     Revised  2017-04-18

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OTHER ASPECTS OF UNDERSTANDING MATH

Abstraction

An abstraction of a concept $C$ is a concept $C'$ with these properties:

$C'$ includes all instances of $C$ and
$C'$ is constructed by taking as a definition one or more assertions that are true of all instances of $C$.

The concept $C'$ may include instances not included in $C$.

In this article, I will give one example of abstraction in detail, describing the way that mathematicians have abstracted the concept of relation. This will be followed by some examples not given in detail, with references.

Example: Relations

Mathematicians have made use of relations between math objects since antiquity.

Examples of relations

For real numbers $r$ and $s$. "$r\lt x$" means that $r$ is less than $s$. So the statement "$5\lt 7$" is true, but the statement "$7\lt 5$" is false. We say that "$\lt$" is a relation on the real numbers. Other relations on real numbers denoted by symbols are "$=$" and "$\leq$".

Suppose $m$ and $n$ are positive integers. $m$ and $n$ are said to be relatively prime if the greatest common divisor of $m$ and $n$ is $1$. So $5$ and $7$ are relatively prime, but $15$ and $21$ are not relatively prime. Being relatively prime is a relation on positive integers. This is a relation that does not have a commonly used symbol.

The concept of congruence of triangles has been used for a couple of millenia. In recent centuries it has been denoted by the symbol "$\cong$". Congruence of triangles is a relation on triangles.

How to define "relation"

Let's consider how we could give a mathematical definition of "relation".

The logicians' definition of "relation" is to say that a relation is a true-or-false statement that can be made about a pair of math objects of a certain type. That defines "relation" in terms of the concept of "true-or-false statement".

In the late nineteenth century, mathematicians came up with a formal definition of "set" and began to use the theory of sets as a basis for all definitions in math. That resulted in this definition of "relation":

A relation on a set $S$ is a set $\alpha$ of ordered pairs of elements of $S$.

The usual notation for being related by a relation $\alpha$ is to write "$s\alpha t$" if and only if $s$ is related by $\alpha$ to $t$. ("$\alpha$" is the Greek letter alpha). This fits with the notations "$s\lt t$" for real numbers $s$ and $t$ and "$S\cong T$ " for triangles $S$ and $T$.
The idea is that if $\alpha$ is a relation, then "$s\alpha t$" means that $(s,t)\in\alpha$. As a result, the concept of relation is identified with the set of ordered pairs for which the relation holds.
This definition takes one assertion that is true of all relations used in practice (namely that it is a set of ordered pairs) and using it as the definition of "relation".
The definition abstracts the everyday concept of relationship by focusing on the property that a relation either holds or doesn’t hold between two given objects.

Consequences of this definition of "relation"

The definition of "relation" as any set of ordered pairs is an abstraction of the concept of relation generated by considering what examples of relations have in common. A consequence of this definition is that any set of ordered pairs is a relation.

Example: Let \[\xi:=\{(2,3),(2,9),(9,1),(9,2)\}\] Then by definition of relation, $\xi$ is a relation on the set $\{1,2,3,9\}$. Before this new definition of "relation" came up, no one would have called this a relation.

Your reaction may be: What relation IS it? Answer: just that set of ordered pairs. You know that $2\xi3$ and $2\xi9$, for example, but $1\xi9$ is false. There is no other definition of $\xi$.

Yes, the relation $\xi$ is weird. It is an arbitrary definition. It does not have any verbal description other than listing the pairs in $\xi$. It is probably useless. Live with it.

The symbol "$\xi$" is a Greek letter. It looks weird, so I used it to name a weird relation. Its upper case version is "$\Xi$", which is even weirder. I pronounce "$\xi$" as "ksee" but most mathematicians call it "si" or "zi" (rhyming with "pie").

Example: Functions

Mathematicians originally thought of a function as a formula containing a variable that gave a value when a number was substituted for the variable.
For example, the function $x^3-x$ has the value $6$ when you set $x=2$. The common way of describing this is: "Let $f(x)=x^3-x$. Then $f(2)=6$."
They also defined functions geometrically, such as the sine function shown below. The sine function and many other such functions can also be defined by infinite series, a kind of extension of the idea of "formula".

But as more and more weird functions were studied, such as the Dirichlet function and space-filling curves, a new approach to functions was needed.
That led to the modern definition discussed in detail in the abstractmath article Function: Specification and Definition, which defines a function as having a domain, a codomain, and a set of ordered pairs (called the graph of the function) with certain properties.
For example, the function whose formula is $x^3-x$ has domain and codomain the set of all real numbers, and the graph consists of an infinite set of pairs of real numbers, namely all pairs of the form $(r,r^3-r)$ for every real number $r$ -- including for example $(2,6)$, $(-1,0)$, $(\frac{1}{2},-\frac{3}{8})$, and $(42,74046)$.
In the old approach, given $t$ you find $f(t)$ by plugging $t$ into the formula. Now, by definition of function, $f(t)$ is the second coordinate of the pair in the graph whose first coordinate is $t$.
That definition is an abstraction of the concept of function which includes all the example already studied and many more.

Axiomatic method

The axiomatic method is a technique for studying math objects of some kind by formulating them as a type of math structure.

You take some basic properties of the kind of structure you are interested in and set them down as axioms
Then you deduce other properties (that you may or may not have already known) as theorems.
The point of doing this is to make your reasoning and all your assumptions completely explicit.
Nowadays research papers typically state and prove their theorems in terms of math structures defined by axioms, although a particular paper may not mention the axioms but merely refer to other papers or texts where the axioms are given.
For some common structures such as the real numbers and sets, the axioms are not only not referenced, but the authors clearly don’t even think about them in terms of axioms: they use commonly-known properties (of real numbers or sets, for example) without reference.

The axiomatic method in practice

Typically when using the axiomatic method some of these things may happen:

You discover that there are other examples of this system that you hadn’t previously known about. This makes the axioms more broadly applicable.
You discover that some properties that your original examples had don’t hold for some of the new examples. Depending on your research goals, you may then add some of those properties to the axioms, so that the new examples are not examples any more.
You may discover that some of your axioms follow from others, so that you can omit them from the system.

Example: Continuity

A continuous function (from the set of real numbers to the set of real numbers) is sometimes described as a function whose graph you can draw without lifting your chalk from the board. This is a physical description, not a mathematical definition.

In the nineteenth century, mathematicians talked about continuous functions but became aware that they needed a rigorous definition. One possibility was functions given by formulas, but that didn’t work: some formulas give discontinuous functions and they couldn’t think of formulas for some continuous functions.

This description of nineteenth century math is an oversimplification.

Cauchy produced the definition we now use (the epsilon-delta definition) which is a rigorous mathematical version of the no-lifting-chalk idea and which included the functions they thought of as continuous.

To their surprise, some clever mathematicians produced examples of some weird continuous functions that you can’t draw, for example the sine blur function. In the terminology in the discussion of abstraction above, the abstraction $C'$ (epsilon-delta continuous functions) had functions in it that were not in $C$ (no-chalk-lifting functions.) On the other hand, their definition now applied to functions between some spaces besides the real numbers, for example the complex numbers, for which drawing the graph without lifting the chalk doesn’t even make sense.

Example: Rings

Suppose you are studying the algebraic properties of numbers. You know that addition and multiplication are both associative operations and that they are related by the distributive law: $x(y+z)=xy+xz$. Both addition and multiplication have identity elements ($0$ and $1$) and satisfy some other properties as well: addition forms a commutative group for example, and if $x$ is any number, then $0\cdot x=0$.

One way to approach this problem is to write down some of these laws as axioms on a set with two binary operations without assuming that the elements are numbers. In doing this, you are abstracting some of the properties of numbers.

Certain properties such as those in the first paragraph of this example were chosen to define a type of math structure called a ring. (The precise set of axioms for rings is given in the Wikipedia article.)

You may then prove theorems about rings strictly by logical deduction from the axioms without calling on your familiarity with numbers.

When mathematicians did this, the following events occurred:

They discovered systems such as matrices whose elements are not numbers but which obey most of the axioms for rings.
Although multiplication of numbers is commutative, multiplication of matrices is not commutative.
Now they had to decide whether to require commutative of multiplication as an axioms for rings or not. In this example, historically, mathematicians decided not to require multiplication to be commutative, so (for example) the set of all $2\times 2$ matrices with real entries is a ring.
They then defined a commutative ring to be a ring in which multiplication is commutative.

So the name "commutative ring" means the multiplication is commutative, because addition in rings is always commutative. Mathematical names are not always transparent.

You can prove from the axioms for a ring that in any ring, $0 x=0$ for all $x$, so you don’t need to include it as an axiom.

Nowadays, all math structures are defined by axioms.

Other examples

Historically, the first example of something like the axiomatic method is Euclid’s axiomatization of geometry. The axiomatic method began to take off in the late nineteenth century and now is a standard tool in math. For more about the axiomatic method see the Wikipedia article.
Partitions and equivalence relations are two other concepts that have been axiomatized. Remarkably, although the axioms for the two types of structures are quite different, every partition is in fact an equivalence relation in exactly one way, and any equivalence relation is a partition in exactly one way.

Remark

Many articles on the web about the axiomatic method emphasize the representation of the axiom system as a formal logical theory (formal system). Mathematicians in practice create and use a particular axiom system as a tool for research and understanding, and state and prove theorems of the system in semi-formal narrative form rather than in formal logic.

Generalization

To generalize a mathematical concept $C$ is to find a concept $C'$ with the property that all instances of $C$ are also instances of $C'$.

Examples

a) $\mathbb{R}^n$, for arbitrary positive integer $n$, is a generalization of $\mathbb{R}^2$ (the set of all ordered pairs of real numbers). One replaces the ordered pairs in $\mathbb{R}^2$ by ordered $n$-tuples, and much of the arithmetic and spatial structure (except for the representation of $\mathbb{R}^2$ using complex numbers) and even some of our intuitions carry over to the more general case.

b) The concept of abstract vector space is a generalization of ${\mathbb{R}}^n$. To get it you forget that the elements of ${\mathbb{R}^n}$ are $n$-tuples of real numbers and your produce a different, more abstract definition obtained by finding properties of $\mathbb{R}^n$ such as the existence of addition, multiplication, scalar multiplication, and so on, and using these properties as axioms for the new concept of vector space.

Expansive generalization

Example (a) is an example of an expansive generalization, obtained by changing a datum in the definition of a concept (the dimension $2$ in this case) into a parameter.

Reconstructive generalization

Example (b) is obtained taking the concept and reconstructing it, producing a very different-looking definition that includes all the original examples and others as well. This is reconstructive generalization.

Both expansive and reconstructive generalizations, if done correctly, take a concept and introduce another concept that includes all the examples of the original concepts and (in general) others as well. These are legitimate generalizations.

Generalization from examples

It can happen that all the examples you know of a specific concept have a certain property $P$ not required by the definition. If you conclude from this that all the examples of the concept have property $P$ you are engaging in generalization from examples.

Generalization from examples
is an illegitimate mathematical method.
It does not always give true statements about math objects.

Example: The statement "An infinite sequence gets close to its limit but never reaches it" is a MYTH. Many newcomers to abstract math believe it because in the typical examples you see of limits of sequences the elements of the sequence in fact are never equal to the limit. For example

\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}=0\]

but 1/$n$ is never equal to $0$.

On the other hand,

\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{\sin \left(\frac{\pi n}{4}\right)}{n}=0\]

but every fourth term is zero. The sequence starts out like this:

\[\frac{1}{\sqrt{2}},\,\frac{1}{2},\,\frac{1}{3 \sqrt{2}}\, ,\,0,-\frac{1}{5 \sqrt{2}}\, ,\,-\frac{1}{6},\,-\frac{1}{7 \sqrt{2}}\, ,\,0,\,\frac{1}{9 \sqrt{2}}\, ,\,\frac{1}{10},\,\frac{1}{11 \sqrt{2}}\, ,\,0,\,-\frac{1}{13 \sqrt{2}}\, ,\,-\frac{1}{14},\,-\frac{1}{15 \sqrt{2}}\, ,\,0\]

The names "expansive" and "reconstructive" are due to Harel and Tall.

Note: The usual meaning of the word "generalization" in colloquial American English is generalization from examples, often used in a derogatory way. When mathematicians refer to a generalization it usually means an expansive or reconstructive generalization and has no derogatory intent. See cognitive dissonance.

Ratchet effect

Once you acquire an insight, you may not be able to understand how someone else can't understand it. It becomes obvious, or trivial to prove. That is the ratchet effect.

Example

“How did you know that \[{{\left(\frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\] is never negative?”

“Because it is a sixth power, so it is a square of a number, and the square of a number is never negative.”

(Smiting forehead) “Of course, why didn’t I see that!”

This observation stays in your head. You can’t forget it. Your neurons have been permanently changed and you can’t go back and regain your former state of non-understanding!

This is an example of pattern recognition – of ignoring the details and seeing \[{{\left(\frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\]as\[\left(\left(\text{some stuff}\right)^3\right)^2\] See also chunking. There are other examples like this here, here and here

Bad behavior

Mathematicians can be quite cruel once they know how to do something and they enjoy seeing you squirm as you struggle to catch on. –Marcus du Sautoy

“How could you NOT see that \[{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1}\right)}^{6}}\] is never negative?! It’s OBVIOUS that it’s never negative. It’s a SIXTH POWER YOU DUMMY!”

It is distressingly common that a mathematician for whom a concept has become obvious because of the ratchet effect will then tell someone else that the concept is obvious or trivial, leaving them to feel put down. It is one thing for s to do this to each other; that is a kind of normal competitiveness among members of a tribe. I believe behaving this way to non-mathematicians is a major contribution to the dislike and fear of math that many people have.

When you are explaining abstract math to non-mathematicians
DO NOT DUMP on those who are baffled
because they haven’t been RATCHETED UP yet.