[1]
J. Franklin, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, Palgrave Macmillan UK, 2014
Preface Introduction
In [1], the author defines a purely structural property as one that “can be defined wholly in terms of the concepts same and different, and part and whole (along with purely logical concepts).” This definition and its reference to parts and wholes calls to mind the history of the word algebra itself, which comes from the Arabic al-jabr, literally meaning “the reunion of broken parts”. One of the concepts fundamental to the historical development of algebra is the notion of factorization; closely related questions that have driven the development of algebra over the centuries are: when does a polynomial equation have solutions in a particular number system, and is there a systematic way to find them?
The goal of this book is to explore the idea of factorization from an abstract perspective. In Chapter 1, we develop our foundations in the integers. Much of this chapter will be familiar to students who have had a first course in number theory, but we are especially concerned with results that preview the structural questions we’ll investigate in more abstract settings.
In Chapter 2, we begin the process that so defines modern mathematics: abstraction. We ask: from the point of view of algebra, what properties of the integers are really important? And then we study those. We find that they are also held by several other familiar collections of numbers and algebraic objects, and then study those objects in increasing depth.
Chapter 3 begins with an exploration of factorization properties of polynomials in particular. We then precisely describe what we mean by “unique factorization” before demonstrating that every Euclidean domain possesses the unique factorization property. We conclude with a brief exploration of the ways in which systems of numbers and polynomials can fail to possess the unique factorization property.
Finally, in Chapter 4, we explore the concept of ideals in general, and use them to build new rings and study properties of homomorphisms.
Throughout this book, we will walk in the realms of abstraction, and catch glimpses of the beauty and incredible power of this perspective on mathematics.
