NOEMAYA

Mathematics is a view to the logical structure of the world. The mathematical principles are a priori propositions whose truth-values are independent of empirical evidence, and hence the a priori.

Mathematics involves two types of things: Sets and Relations. Sets are the collections of objects, the collection being defined provisionally. For instance, a mathematician may define his set to be the set of all triangles, while another one may work with the set of all imaginary arrows in space or the set of all even numbers. One can make up any collection of objects and consider them to be members of a set. Objects in a set are usually called members of that set. Relations are the relationships that exist between the objects of the set. These relations too are defined provisionally; we define a relation between the objects of a set. For instance, if we define our set to be the set of all triangles and number them, then we can impose the relation that the odd-numbered triangles are twice as big as the even-numbered triangles. This relation created a relationship between every two triangles: They are either the same size or one is twice as the other, or vice versa. So relations are such kinds of things. Notice that how relations can be used as a way of creating sets or collections. We can use a relation as a criterion for creating a set. For instance, I can say that I want a collection whose members are numbers such that they are all divisible by 2. Using this condition I collect all such numbers and define the set; obviously this is the set of all even numbers, and each two members are related to one another such that they are both divisible by 2. An important point is that relations themselves can be treated as objects so that we may create collections whose members are relations. For instance, I can define a collection, a set, whose members are all algebraic operations, say plus, minus, multiplications, etc. These objects are relations; addition is something that exists between two or more numbers. If we define a set whose members are pairs of numbers, any numbers we wish, like (a,b), then we can say addition is a relation between any pair of numbers and a third number c such that (a,b) is related to c in the form a+b equals c or a+b=c. Addition is an instance of a relation between some objects.

Therefore, the most general structures in mathematics are sets and relations. Everything else is a specializations and variations of these two concepts. We defined relations in its most general sense. We can specialize it further and refer to all relations such that each object of the set is related to exactly one object and not two. For instance, if a is related to be, then a is not related to c unless b=c. All such specialized relations are known as functions. The another kinds of relations that I mentioned above as operations, the operations between a pair of numbers and a third number, are known as binary operations. Binary operations such as +, -, *, / are special kinds of relations.

We can play more and define new objects: The new object is a set of numbers equipped with a binary operation. Consider the set S with members a, b, and c. We write it as: S={a,b,c}. The name of the set and its members. Call our binary operation this symbol *. We define the set A equipped with * and write it as <S , *> which is a set and relation together. We define the relation to be the following:

a*b=c    a*c=b    a*a=a    b*a=c    b*c=a    b*b=b    c*a=b    c*b=a    c*c=c

See that ever two members of S are related to one another by the binary operation *. Notice that I have not mentioned anything about the nature of a,b, and c and also the nature of the binary operation which can be addition, multiplication, and anything else. But I have defined a structure which can exist regardless of how we fill in the unknown symbols. The object we defined above, which is a set equipped with an operation <S,*>, has a special property: Notice that all members are related to one another; there is no member that is not related to another; and no matter how many times we apply the operation between members we never go out of S; the operation between any two members of S always gives another elements which is still a member of S. We never go outside S in order to perform the operation *. This property is called closure; we say that the set S is closed under the operation *, that is: Operation of * on the members of S always gives another member of S. Now the new object that we define to be a set equipped with an operation under which S is closed has a special name in algebra: It is called a group. To have a group we need some more restrictions: The operation must be associative; it must permit an identity and an inverse. We need not go into these now. The point is to say that everything in mathematics is composed of a set and a relation. In fact, everything in the world can be viewed in the same fashion.

Now we can play more and extend our definitions: We define a group to be a set equipped with a binary operation which relates the members of the set with one another. Now we can define a set equipped with two binary operations. We can call these two operations multiplication and addition. In fact the set of real numbers of which we are familiar and use on a daily basis is such a mathematical object; we multiply and add them and get another object which is still a real number. This new object which is a set equipped with two binary operations, under some group constraints, is called a Ring in mathematics. If we add more relations to this ring we can make a more interesting object called a Field. In fact all of us know a field and work with it everyday: The set of real numbers. Field is a special kind of ring. If the operation * that relates two elements a and b is such that a*b=b*a we say that * is commutative. The real numbers that we know is a commutative ring because 2*3=3*2=6. We can now say: Field is a commutative division ring.

All of the above are created by playing with set and relations:

Set is a collection of object.

Relations are the relationships existing between members of a set.

Function is a monogamous relation.

Binary operation is a function.

Group is a set with a binary operation.

Ring is a group with a binary operation.

Field is a commutative division ring.

Division ring is a ring whose non-zero elements are all units.

A unit is an element of a set that has an inverse.

Inverse of an element a is another element b such that a*b is the identity element. We write: a*b=e.

Identity element e is an element of a set such that a*e=e*a=a for all elements a of the set.

[The purpose of this post was to review the subject for myself and prepare for a lecture on the concepts of mathematics.]