The branch of philosophy that relates to the foundation of mathematics, deals with the nature of mathematics, and ponders whether mathematics is discovered or invented is known as the philosophy of mathematics. Part 1 focuses upon the foundation of mathematics. The book One, Two, Three: Absolutely Elementary Mathematics by David Berlinski is a great source on this topic.
Whereas the techniques of absolutely elementary mathematics (AEM) are one thing, their explanations are quite another. For example, 1+1=2 is the simple addition of two natural numbers. However, it is not so simple to explain what addition means and why it is justified.
The foundational role of mathematics is to explain the meaning and to provide the justification. According to Berlinski there is one idea which predominates AEM. In his introduction he states, “that the calculations and concepts of absolutely elementary mathematics are controlled by the single act of counting by one.”
Berlinski also offers the following definitions: Geometry is the study of space as the mysterious stuff between points. Calculus is the study and analysis of time. Algebra is the study of symbols and structures. He also notes in his introduction, “Einstein required only high-school algebra in creating his theory of special relativity, but he required high-school algebra, and he would have been lost without it.”
Einstein’s theory of special relativity explains how speed affects mass, space, and time. His theory also introduced us to the most famous equation in science: E = mc². His equation means that energy (E) and mass (m) are interchangeable. The speed of light (c) squared is a large multiplier, so even a small amount of mass contains an enormous amount of energy.
From 1+1=2 toward E = mc², the foundational role of mathematics is to explain meaning and provide justification. AEM builds the foundation necessary to progress from arithmetic to geometry to algebra to calculus and beyond. Next in Part 2 we deal with the nature of mathematics.
~ Boethius ~