Eric’s Philosophy of Mathematics

Philosophical Papers --- 10-31-2019

Two aspects of my Philosophy of Mathematics center on abstract relations and unitization.

The Philosophy of Mathematics is tightly connected with the Foundations of Mathematics, as the Philosopher Bertrand Russell demonstrated by reducing all of mathematics to set theory and logic.

I will here present some aspects of my own philosophy of mathematics.  It should be noted that many philosophers have talked about mathematics in their philosophy, as for example Kant with his theory of mathematics as containing both a priori analytical propositions and a priori synthetic propositions.  Many philosophers were also mathematicians as DesCartes who developed analytic geometry and Leibniz who developed calculus.  The attraction to philosophers to mathematics is that philosophy is the study of knowledge and mathematics in some sense is the only/an ? area where there is perfect knowledge (universal truths). 

My own theory of mathematics is incomplete at this time, but what I have developed so far has to do with two notions which I will explain:  abstract relations, and unitization.

When I say that mathematics is about relations, what do I mean?  First of all, I am not talking about the classical definition of relations in mathematics which has to do with ordered pairs.  I am using the term in a much more broad sense.  First, math is about quantitative or numerical relations.  It is true that some mathematical structures such as graphs can be semantic and contain non-quantitative information.  However, the graph itself and its structure can be described purely mathematical or as a set of mathematical relations.

Math Qua a System of Abstract Relations

Let’s start with a simple example of a mathematical relation.  The Pythagorean theorem that says that for any right triangle the sides are RELATED in the following way:  the hypotenuse is the square root of the two other sides squared (verify if you remember your Pythagorean theorem).  This is true of any right triangle regardless of size of the sides.  I.e., there is an abstract relation between the three sides of the triangle.  How far can this notion of mathematical relations be extended into the field of mathematics?  We shall see.  Logic, which underlies all mathematics, itself is about abstract relations.  A logical proposition logically equivalent to another logical proposition is equivalent by logical form i.e., as the relations between the variables as given by the connectives.  Again, as in the case of the triangle, it doesn’t matter what specific terms evaluate the variables, the abstract relations between variables as given by the connectives will always make such a logical equivalence true.  In mathematical proofs, identities are often used which are universal relations between variables which allow steps in a proof to proceed from one mathematical form to an equivalent but different mathematical form.  Certainly, my time doing Boolean algebraic proofs indicated this is how you proceed in those proofs (there are other logical methods used in non-Boolean mathematical proofs).  What you are doing largely in mathematics is specifying universal abstract relations which hold between variables and proofs, which themselves proceed by these abstract relations, are used to discover more universal abstract relations which are mathematical theorems.  It seems that mathematical operations can be viewed as abstract relations too.  In operations (be they unary, binary, ternary, or whatever) stated abstractly the operation holds for any evaluation of the variables involved in the operation, it is the abstract relation between variables specified by the operation which holds universally.  Graphs, one of my favorite areas, related to combinatorics, are also a collection of abstract relations.  The nodes, or variables, or evaluated variables, are connected to other nodes by virtue of a certain topology.  In a graph where there is the mere connectivity and connections (or edges) are not evaluated (as in a weighted graph) or semantically instantiated (as in a graph like one of human relations sister-of, mother-of etc), the connectivity gives the relations between the nodes (like what the shortest path is between two nodes).  In a semantically instantiated graph the relations show the relations between concepts or as in my example family members in an analytic fashion (if a is the mother of b and c is the brother of b, c is the son of a), i.e., the family tree (a tree is just a special case of a graph) or family graph.  Anyway, this is the beginning of one part of my philosophy of mathematics and I need to keep thinking about how far this notion of abstract relations can be extended into the various areas of mathematics which I hope to do by continue to read my math foundations book.

Unitization

Another aspect of my philosophy of mathematics has to do with what I call “unitization.”  This in some sense is related to the set theory aspect of mathematics.  Let’s consider unitization and operations.  A sum, in some sense (or sum sense), is a unitization of several numbers.  In the same way, a product is the unitization of a set of sums n times.  Exponentiation is the unitization of a set of products n times, and so forth.  A difference is a de-unitization of a sum.  A divisor is a de-unitization of a product, and so forth (a log is a de-unitization of an exponentiation, verify).  I need to think about how far this notion of unitization can be extended.  Can it be extended to geometry?  Is a shape a unitization of a set of coordinates? Etc  implications on topology

Copyright Eric Wasiolek 10-31-2019