What do mathematicians do? If we have to describe it in ten minutes, the first thing coming to mind is that mathematicians are engaged in abstraction.
Real or not so real known objects and their properties are investigated by mathematicians and new purely artificial objects are constructed by them which, in some way, are simplified versions of those real and not so real known prototypes. These simplified abstract objects concentrate the quintessential properties of their prototypes common with many other similar objects without small details that make the prototypes different.
Symbolically, if one prototype object has properties B, C and F while another one has A, B, C, D and E, we can abstract properties B and C (common for both prototypes) and consider an abstract object with only these two properties. This new abstract object can be studied and every statement like "abstract object with properties B and C has also properties X, Y, Z" can be extended to both prototypes since B and C are their properties as well. For instance, rectangularly shaped football field and rectangularly shaped table can be represented by an abstract object "rectangle". All properties of mathematical rectangles (like a square of a diagonal equals to a sum of squares of two sides) will be shared by both prototypes (since they are both "rectangular") and mathematicians do not have to be concerned with the type of grass growing on the football field or the material the table is made of.
Why is this process of abstraction so important?
First of all, because abstract objects are simpler and well defined. Since they are artificially created, they have exactly the properties mathematicians want them to have (which, in theory, are those properties shared by many prototypes) and they do not have minor peculiarities of their prototypes that make these prototypes so difficult to research.
Secondly, once studied, the properties of an abstract object can be extended to all prototypes represented by an abstraction. Proven once for an abstract object, the theorem can be considered proven for all prototypes of this object.
Here is an example of this second point.
Consider all the integer numbers with an operation of addition as a prototype for an abstract object called "group" in mathematics. Group is a set of elements (all integers, as a prototype, are a set of numbers, so we abstracted "integer number" into "element of a group" and a set of all integer numbers into "group"). Assume that there is some binary operation defined for any pair of elements of a group which results in an element from the same group (our prototype, set of all integers, has operation of addition defined for any pair of integers and a sum of any two integers is an integer,
i.e. an element of the same set).
Let's further assume that there is a special element of a group called "unit element" that, if participated in a defined binary operation paired with any other element, results in the same element (in our prototype of integer numbers and an operation of addition the role of "unit element" is played by a number 0 because 0 + X = X and X + 0 = X for any integer X).
A couple of more assumptions about groups based on properties of a set of integer numbers as a prototype are:
- for every element in a group there is a so called "inverse" element which, if paired with original element in a defined operation, results in "unit element" (this is based on a concept of negative numbers in a set of integers because, for any integer X, the number (-X), added to X, results in 0: X + (-X) = 0 for any X);
- associative law of a group operation which is based on associativity of addition for integers, i.e. X+(Y+Z)=(X+Y)+Z.
Now, when we've built our abstract "group" which generalizes properties of a set of all integer numbers, we can start studying some properties of this group. Once proven, they not only will apply to a set of integers but also can be extended on any other mathematical object with properties of a group. For example, consider a set of all rotations of a geometrical figure on a plane around some fixed point with a group operation defined on a pair of elements (i.e. rotations) as a consecutive application of both rotations. The "unit element" is a rotation by zero degree. The "inverse" rotation is a rotation in an opposite direction. Associativity is also in place. So, all these rotations represent a group.
Let's prove some simple theorem about groups and extend it towards all its prototypes. For instance, let's prove that there is only one unit element in a group. That is, if there are two unit elements, they are equal to each other. Here is the proof.
Let U and V be two unit elements. From a definition of a unit element, for any group element X the following is true (we use ^ as a group operation):
(1) U ^ X = X
(2) X ^ U = X
(3) V ^ X = X
(4) X ^ V = X
Since X can be any group element, substitute V instead of X in (1) and U instead of X
(1) U ^ V = V
(4) U ^ V = U
From these two equations with equal left parts we conclude that right parts are equal: U = V
Now recall that this proof is made for any group. It means that we can extend it to any group realization: a set of only even integers with addition as an operation and number 0 as a unit element, a set of rotations of a figure on a plane with consecutive application of two rotations as an operation and rotation by 0 degree as a unit element, a set of rational numbers with multiplication as an operation and number 1 as a unit element, a set of quadratic polynomials with addition as an operation and polynom with all coefficients equal to zero as a unit element and thousands others.
That is what makes abstraction such a universal and powerful tool and that is why abstraction is in a foundation of mathematics.
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