What is abstraction and generalization ?

Question

Mathematicians does something when they mean by doing "abstraction" or "generalization". I can't exactly decipher what they mean, but I suppose

i. Abstraction is of finding patterns from some objects(and conctretizing something is probably adding structure to the pattern to produce some examples)

Ex: From Sun, Blood, Traffic light stopping color etc the pattern red could be found (which is an abstraction). Reversly, adding structure to the red will produce Blood/Sun (Concrete)

ii. Generalization is probably making some constant variable.

Ex: The binomial theorem can be thought of a good generalization of $(1+x)^3, (x+p)^{13}, (a-b)^{50}$ (Random)

Is my notions of abstraction and generalization right ? If not, then please explicitly state what it is with a lot of example. (An example) What can be treated as a generalization and (separately) abstraction of $1^2 + 2^2 + 3^2 + 4^4 + ... + 66^2 $ (The last term is quite arbitrary)?

Answer 1

By abstraction, we mean that we step back from concrete objects to consider a number of objects with identical properties simultaneously.

For instance, consider the following three objects:

1. The set of functions $A,B,C$ defined on the set $\{1,2,3\}$ by $A(1)=1,\;A(2)=2,\;A(3)=3$,
   $B(1)=2,\;B(2)=3,\;B(3)=1$,
   $C(1)=3,\;C(2)=1,\;C(3)=2$,

2. The set of complex numbers $a = 1, b = e^{i2\pi/3},c = e^{i4\pi/3}$.

3. The set of matrices $ \alpha =\left( \begin{array}{c c}1 & 0\\ 0& 1 \end{array}\right), \beta = \left( \begin{array}{c c}0 & -1\\ -1& -1 \end{array}\right), \gamma = \left( \begin{array}{c c}-1 & -1\\ -1& 0 \end{array}\right)$

Consider these notations: AB means A(B(x)), $ab$ is ordinary multiplication of complex numbers, and $\alpha \beta$ means ordinary matrix multiplication. Verify the following "multiplication" tables:

$$\begin{array}{c | c c c}& A & B & C\\ \hline A & A & B & C \\ B & B & C & A\\C & C & A & B \end{array}\;\;\;\;\;\begin{array}{c | c c c}& a & b & c\\ \hline a & a &b & c \\ b & b & c & a\\c & c & a & b \end{array}\;\;\;\;\;\begin{array}{c | c c c}& \alpha & \beta & \gamma \\ \hline \alpha & \alpha & \beta & \gamma \\ \beta & \beta & \gamma & \alpha\\\gamma & \gamma & \alpha & \beta \end{array}$$

Notice that these tables are identical. Then let us by abstraction define an abstract object which is the set of three elements $\{e,g,g^{-1}\}$ paired with a binary operator $\cdot$ such that set acts on itself in the following manner with respect to the operator:

\begin{array}{c | c c c} \cdot & e & g & g^{-1}\\ \hline e & e & g & g^{-1} \\ g & g & g^{-1} & e\\g^{-1} & g^{-1} & e & g \end{array}

In Group Theory an object with such structure is called the cyclic group of order three. Then the examples above are representations of this abstract object. It is an abstract object because while we have now given it a definition, notice that it is itself a stand-in for a variety of objects that have the properties that it demonstrates. You might even consider the abstract object to be more of a set of properties or a structure than an object itself, since its elements can refer to many (indeed, in this case an infinite number of) actual sets. This abstraction is powerful, since proving any property of this abstract object (like the fact that $x^3$ = e for any element $x$) demonstrates this property for each and every realization, in the same way that observing the properties of an artist's posable figurine allows the artist to determine how to sketch all sorts of different people in many different configurations because it is an abstraction of a human figure.

Generalization, on the other hand, is carried out when we consider an object under inspection in the context of a larger domain for the parameters involved.

For instance, we might define the sine function as the ratio of the side opposite to the acute base angle to the side adjacent to said angle in a right triangle. With this definition, we can only consider the sine of an acute angle.

By considering reference angles, we generalize our definition of the sine of an angle to "the height of a point on the unit circle intersected by a ray at the terminal side of the angle in standard position". With this more general definition, all of our previous results still hold and we can consider more values of the sine function- any real number.

Generalizing further, we find that defining sine as a power series admits us to evaluate the sine of a complex number. We see that our original definition was very limited in scope, and by generalizing we obtained new and broader domains which agree with our more narrow previous domains, but in which we find richer or more widely applicable discussion of our query.

Here are a few more examples of generalization:

• The improper Riemann integral generalizes the definite Riemann integral, since the definite Riemann integral is only defined for a continuous function on a closed interval.

• The Lebesgue integral is a generalization of the Riemann integral. For a function defined over a Lebesgue measurable set, if the function is Riemann integrable then the values of the two integrals agree. But the Lebesgue integral provides meaningful results in some cases that the Riemann integral is meaningless.

• The norm is a generalization of distance. The $l^p$ norms are generalizations of Euclidean distance, and there are some interesting norms that allow us to define a "distance" between objects that are not geometrical, like the Hamming distance. It turns out that all norms are equivalent up to a scalar and so one can consider a single abstract norm for many purposes.

• The inner product allows one to generalize the notion of an angle. Using the inner product, one can, for example, find the angle between two vectors or two matrices with complex entries or describe which polynomials are "orthogonal". Orthogonal functions are important in the solution of partial differential equations.

• Algorithms are generalizations of solutions to particular problems. The Euclidean algorithm, for instance, allows one to find the greatest common divisor of any two integers, and it is always a solution.

Answer 2

Andrew's answer is very good. However, I'm going to describe abstraction slightly differently.

When we're doing mathematics, we often notice similarities in different subfields. Abstraction can be thought of ignoring the differences. We define a new concept in terms of the similarities only and give it a name so that we can talk about it independently of any particular instance.

The hard part is that it's sometimes not clear what the similarities are or how we can express them independently. It took half a century, for example, to come up with category theory, which is probably the most powerful abstraction language yet found.

This is my excuse for answering this question. Since you're self-learning, I wanted to give you just a little picture of just how powerful abstraction can be.

First, let's consider sets.

Suppose that $A$ and $B$ are finite sets. Then the number of functions with domain $A$ and codomain $B$ is $|B|^{|A|}$.

Now let's think about the empty set $\emptyset$ for a moment.

For any non-empty finite set $A$, how many functions are there with domain $A$ and codomain $\emptyset$? Well, it's just $0^{|A|}$, which is $0$. There are no functions $f$ such that $f : A \rightarrow \emptyset$ if $A$ is nonempty.

What about the other way? How many functions are there with domain $\emptyset$ and codomain $A$? Well, it's $|A|^0$, which is $1$. There is exactly one function $f$ such that $f : \emptyset \rightarrow A$.

Note that the above argument also applies to infinite set, however we also need to consider the case $f : \emptyset \rightarrow \emptyset$. By convention, when doing combinatorics, $0^0 = 1$, and it turns out (if, for example, you consider the graph of the function) it makes sense to do this too.

The empty set is the only set with the following property:

For any set $A$, there is exactly one function $f$ such that $f : \emptyset \rightarrow A$.

Now, let's think about logic.

For logic, I'm going to use some notation that may be unfamiliar at the high school level. Given two propositions $A$ and $B$, we will denote $A \vdash B$ (pronounced "$A$ entails $B$", or "$A$ yields $B$") to mean that given the premise $A$, you can conclude $B$.

The proposition $\hbox{False}$ has the interesting property that from that premise, you can prove anything; this is known as ex falso quodlibet, or the principle of explosion.

But more than that, if you formalise the notion of what constitutes a "distinct proof" very carefully, you can show that $\hbox{False}$ is the only proposition with this property:

For any proposition $A$, there is exactly one proof of $\hbox{False} \vdash A$.

There's an abstraction to be discovered here.

In category theory, an object with this property is known as an "initial object". There is also the dual notion of a "terminal object", that is, an object $T$ such that for any other object $A$, there is exactly one "arrow" such that $A \rightarrow T$. It won't surprise you to learn that $\hbox{True}$ is the unique terminal object in logic.

There is also a terminal object in sets, and it's actually the "singleton set", that is, the set with only one element in it. Of course, I say "the" singleton set as if there's only one, when you can probably think of lots.

In category theory terms, we say that all singleton sets are isomorphic. They all have the same "shape", much like in Andrew's group theory example, where all the groups have the same "shape". You can think of them as different "physical" groups, or you can think of them as the same abstract group.

Informally, you can't tell the difference between any two singleton sets without looking inside the set to see what the element is. That's the price you pay for abstraction: the actual element inside the set is an irrelevant detail that you're abstracting away.

But what you gain is that it's possible to prove a theorem which is true of both sets-and-functions and propositions-and-proofs. An example of such a theorem is: "All initial objects are identical up to isomorphism." That's a very abstract statement, but you should now have an idea just how powerful that statement is.

Incidentally, since group theory has already come up, the "trivial group" (i.e. the group with one element) is both an initial object and a terminal object, which makes it very interesting indeed. That's a topic for another time.

Answer 3

I'm not sure what you mean by abstraction, but you can make a universal generalizations from a specific case in one of two ways:

Direct proof:

Start by assuming for that sake of argument that $P(x)$ is true, where $P$ is a logical predicate (or truth function) and $x$ is a newly introduced free variable (not previously used in your proof). Suppose that, by using whatever rules and axioms you may have at your disposal, you infer $Q(x)$ is also true, where $Q$ is another logical predicate and $x$ is the same object introduced the initial premise above, and $Q(x)$ has no free variables introduced after that initial premises. Then you can conclude that, for all $x$, $P(x)$ implies $Q(x)$.

Proof by contradiction:

Start by assuming for that sake of argument that $P(x)$ is true, where $P$ is a logical predicate (or truth function) and $x$ is newly introduced free variable (not previously used in your proof). Suppose that, by using whatever rules and axioms you may have at your disposal, you infer a contradiction of the form $Q$ and not $Q$, or $Q$ if and only if $\neg Q$, where $Q$ is any logical proposition. Then you can conclude that, there does not exist an $x$ such $P(x)$ is true.