A philosophical approach to relational thinking in mathematics

The basis of this work is to lay the groundwork for relational thinking in mathematics by giving a general mathematical definition of relational thinking in mathematics that builds on the theory of relational thinking in arithmetic and then extends that theory to include all other mathematics subjects, especially algebra and geometry. The necessity to include all other mathematics subjects in relational thinking is predicated on the need for students at all levels to be able to think relationally. In an effort to further establish relational thinking in mathematics, this work attempts to merge mathematics and philosophy by examining Plato's Meno and Wittgenstein's Philosophical Investigations to show the importance of deductive reasoning, logic, and language in the use of relational thinking in mathematics. Further, this work also sets out to establish relations in a mathematical sense as a unifying concept in algebra and geometry. I therefore define relational thinking in mathematics as the skill and propensity to use deductive reasoning and logic in order to make connections between and among abstract mathematical concepts and specific instances thereof. This definition stems from mathematics being built on two pillars--that is, deductive reasoning and logic--and being of two different branches--that is, abstract mathematics and applied mathematics