Forms and Algebra

                When studying mathematics at any level, one might find themselves asking where did this stuff come from? , or why is this useful?.  The answer to these questions can be summed up in one word , Algebra.    Why algebra? Why is it so important?  Well, let’s look at what algebra actually means.  Algebra, throughout history, has developed many rules, relations, and symbols to answer some of the everyday problems we encounter.  The history of algebra and how it was formed, in my opinion, has little to do with what it actually means. I will instead explain algebra by use of philosophy. We can call it the philosophical theory of algebraic reasoning.

The Theory of forms and Algebra

      The theory of Forms is basically that real-world objects have forms. For example, when you look at your table, how do you know it’s a table?  In fact, your table could be extremely different from all other tables, but you know it’s a table because it has the form of a table. The form of a table is that it has a flat surface at the top, and a support structure at the bottom.  The Form of the table can be restated as the abstraction of a table.  By having the abstract model or form of a table, we can then manipulate or customize our table to look how we want so long as it stays within the basic bounds of the form.  We can also now take our base form and use it to define specific types of tables.

    Algebra works in the same way as the theory of forms. In algebra we use letters to symbolize or initialize the form of a number. We call the letters variables because they can change their value. Let’s look at the statement 1+1=2. Even though we have numbers and not variables this is still an abstraction. This is a perfect abstraction for general purposes it gives us a specific form unity. The 1 is the form or abstraction of a real world object. It can be any object as long as it is a single object. So to translate this statement we would say that a single object and a single object together has the form of two objects.  For most the meaning of this statement is obvious, however; the statement a + a = b is not as obvious. Also, this statement does not mean the same as 1+1=2.   Table + Table = 2 tables this statement is equivalent to 1+1=2 only because we are assuming the Knowledge of what the symbols 1, 2, + , =, represent. So we build our philosophical theory of algebra off of the assumption that we don’t need to define the symbols for numbers and what they mean.

     The symbols 1 and 2 have a very specific meaning, but their properties, and interactions have a more complicated philosophical meaning. So we have a gap in our ability to abstract numbers and their properties, which is why we use the letters or variables.  The variables are in fact a form of a form. Sounds redundant, but it is necessary to provide abstraction from something that is already an abstraction. This concept is where it gets a little tricky.  We can say a variable is the form of a number but not all numbers only numbers that have the specific properties we provide with the rest of the statement. Take our a + a = b statement. This statement gives us a relationship between any two numbers that can be represented in this form a + a = b which has many numerical equivalences.                            It can mean 1+1 =2, 2+2 = 4, 3+3 =6, 4+4 = 8…etc. In fact this statement provides an abstract definition of even numbers.  Statements like this should then be called definitions of forms.  We shall go back to our table example again. The basic form of a table is flat surface at the top and support structure at the bottom. What if we said table + wood? We are giving an abstract definition of tables that are made out of wood. So table + wood is the same as a + a =b they are both abstract definitions of forms.   The only difference is abstract definitions of numerical forms take a little more interpretation. However both assume an understanding of something else. The table + wood example assumes an understanding of the form of wood and the form of table. The a+a=b example assumes an understanding that 2,4,6,8,10…etc are called even numbers. a+a=b just gives us a definition of even numbers without trying to list all of the even numbers up to infinity. So abstraction is actually necessary to define specific numbers and number relationships, because we cannot realistically list every even number to infinity.

Why is algebra the heart of mathematics?

                The abstraction of algebra gives us strong foundation by which to build many more complex definitions with a wide range of uses in everyday life. A lot of these concepts we use every day without even realizing it. In fact nearly every branch of mathematics would not exist without the foundation laid down by algebra. For example statistics, uses many definitions that would not exist without the framework of algebra. We even use algebraic concepts in many other subjects. For example in composition classes we use abstraction as a tool to write papers we start with an outline, which is simply an abstraction. Our outline gives us a simple abstract definition of our paper which makes writing our paper much easier.  Language itself is an abstract definition of our thoughts and feelings.