PLATONISM IN MODERN MATHEMATICS A University Thesis Presented to the Faculty of California State University, East Bay In Partial Fulfillment ofthe Requirements for the Degree Master of Science in Interdisciplinary Studies By Ramal Lamar October, 2012 PLATONISM IN MODERN MATHEMATICS By Ramal Lamar Date: INTRODUCTION In this thesis we investigate the existence of Platonic ideas in the concepts and foundations of modern mathematics. Platonism refers to the viewpoint that the objects and entities constructed and defined in the work of mathematics actually exist independent of our sense preception. In the first chapter, we look at how Platonism itself has been interpreted differently by modern thinkers and how this leads to confusion at times as to what is actually Platonism. Therefore by articulating specific ideas constructed by Plato and showing their analogs in today’s mathematical notation, we can speak of ‘mathematical Platonism.’ For example, numbers are not just figments of the imagination; there is a symbol known as one to refer to the concept of ‘oneness.’ Of course, philosophers disagree as to what constitutes Platonism, whether Platonism, also known as realism, is necessary in order to obtain truth and to solve mathematical problems with implications in the applied areas of science and technology. Also in this thesis, we describe the various schools of mathematical thought. Each school tackles its own philosophical problem and we highlight some of the major contributions associated with each school. In the second chapter, we survey a recent history of mathematics by exploring the five foundations of mathematics. The philosophical program of the logicist school is to ground mathematics proper in the rules of logic. Frege tried to deduce arithmetic from the rules of logic. Whitehead and Russell remedied the flaw of Frege’s system with type theory. For the nominalists, mathematical objects exist in space and time and logic’s business is to approximate reality. Nominalists such as Carnap and Quine equate logic ii with physics. The formalist school holds the view that every mathematical problem is solvable by finite proof methods, thus a new field of studying the structure, or syntax and mathematical arguments and proof, was developed in this school known as mathematics. Thus, the meaning of mathematical symbols are removed, and a math proof is analyzed by deriving such a proof from a series of formal axioms and rules of inference performed in a series of logical steps. The intuitonist school of mathematics simply assumes that mathematics is a man-made activity stemming from imagination and creativity. Thus logic is not math, but logic itself is an area of mathematical study, and in no way do they rely on each other. Finally, set theorists describe all mathematical objects as sets. Cantor developed this school with his studies in the infinite. When he showed that there are certain infinite sets that can not map to infinite numbers, this led to the analysis of infinity in modern mathematics. All the schools of mathematics at times share and refine each other’s techniques, and set theory itself has been revised on numerous occasions to work within a given school. In the third chapter, we look more closely at each of the philosophical questions posed by each school and determine how Platonism is used in articulating or solving the problem. From the offset we see in logicism the axiom of comprehension and impredicative definitions as Platonic concepts. We even see mathematicians of other schools criticizing logicists as being too Platonic. ‘Type theory,’ the lasting contribution of the logicists, is Platonic since it leads to a class of hierarchies associated with mathematical objects, to avoid loops and paradoxes. The empirical and naturalistic focus iii of nominalism as a way of avoiding naming ‘universals’ has in its approach inherent Platonism, especially in its use of set theory and logicists’ notation to construct new formal systems to achieve such a goal. Quine’s initial critique of claiming there was essentially no distinction between subject and object in logic sounds very Platonic. The intuitionist constructs universal mathematical objects from some of the same Platonic tools used by the nominalists. The formalists with their whole metamathematics program introducing an entirely new idea to the study of mathematics, however finitary they want it to be, are very Platonic, with their focus on the structure of math sentences. Set theory, with its emphasis on the infinite, axioms of choice, and ability to describe all of mathematics, is the most Platonic of them all. Important to note is how set theory is used in all branches of math as well as all foundations of mathematics. In the conclusions of the thesis, we enumerate the key Platonic ideas inherent in modern mathematics. We trace these ideas through time and see how they ended up as basic mathematical objects. Then we show how maieutics and dialectics lead to the notion of provability, especially in foundational schools such as logicism and formalism. Logos and knowledge, from the world of the intelligible, fall in line with intuitionists’ prescription of mathematics as being primarily a mental activity and with the nominalists’ requirement that logic describe natural reality. Thus talking about infinity does indeed complicate the tasks of the nominalists, but there are many ways to describe reality. Finally, we cite Godel as an example of a Platonist who made contributions to all areas of mathematics. He used logicist methodology in his incompleteness theorems to show that it is impossible to prove all the truths of a theory within the given theory. Logic iv must be somehow extended to capture all truths, and this extension cannot be just by merely adding new axioms. Godel constructed a true but unprovable mathematical sentence, using the diagonalization method that Cantor used to show the uncountability of real numbers. The intuitionist Brouwer constructed mathematical objects that were not only used by formalists, but were actually applied to solve real world phenomena. All too often mathematics follows the vicious-circle principle. And this vicious-circle principle is the Platonism that is inherent in all mathematics. v TABLE OF CONTENTS Chapter One: Platonism Defined ................................................................................... 1 Part One: Basic Key Concepts ........................................................................... 1 Ideas and Universal Quantifiers ................................................................ 1 Maiuetics ................................................................................................... 3 Induction ................................................................................................... 5 Dialectics ................................................................................................... 6 General Definition .................................................................................... 8 Part Two: Forms and Logos ............................................................................... 9 Part Three: Knowledge and Logos .................................................................. 15 Chapter Two: The Five Philosophical Schools of Mathematics .................................. 20 Part One: The Origin and Development of Logicism 20 Critics of Logicism ................................................................................. 23 Nominalism . ............................................................................................ 24 Quine’s Semantics ................................................................................... 25 Part Two: Intuitionism ..................................................................................... 27 Intuitionistic Logic .................................................................................. 33 From Intuition to Construction ............................................................... 34 Part Three: Formalism ..................................................................................... 34 Formalist Logic ....................................................................................... 35 Part Four: Set Theory ....................................................................................... 38 Zermelo-Fraenkel Axioms ...................................................................... 39 The Bourbaki School .............................................................................. 39 Cantor’s Set Theory ................................................................................ 40 Chapter Three: Mathematical Platonism in Each School ............................................ 42 Part One: Logicism as Platonism ..................................................................... 44 Logicism ................................................................................................. 44 Type Theory ............................................................................................ 46 Part Two: Platonic Nominalism ....................................................................... 47 Nominalism ............................................................................................. 47 Naming Universals .................................................................................. 48 Part Three: Platonic Formalism ....................................................................... 51 Hilbert’s Program ...................................................................................