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 ...... 51 Metamathematics ...... 52 Part Four: Platonic Intuitionism ...... 53 Origins of Intuitionism ...... 53 The Totality of Integers ...... 54 Real Numbers ...... 56 Part Five: Platonic Ideas in Modern Set Theory ...... 57 Cantor’s Theorem ...... 57 R is Uncountable ...... 57

vi Axioms of Set Theory ...... 60 R and Decimal Numbers ...... 62 Godel’s Platonism ...... 62

Conclusions ...... 63 Bibliography ...... 77

vii 1

CHAPTER ONE: PLATONISM DEFINED

In this chapter we will define and articulate Plato’s theory of ideas as a contribution to the philosophy of mathematics and as an origin of logic. We identify key concepts central to the notion of Platonism and show how each of these concepts shows up in the logic of modern mathematics through symbolic notation. In this manner we aim to show how, in Plato’s works, the notion of logos sets the foundation of modern mathematics. Included also is a discussion of classical Platonism as viewed by ancient critics and by modern Platonists. The central claim is that there is little agreement as to what actually constitutes Platonism. Each key criterion of Platonism is identified, numerated, and commented upon. We then explore the relationships between the themes of logos, knowledge, and forms.

Part One: Basic Key Concepts

Defining Platonism (related to modern math theory) means tracing the development through time and space of Platonic notions of idea, forms (universal

knowledge and its negation, existential knowledge), maieutics (induction, inference, and

dialectics), logos, knowledge, science, mathematics, truth, philosophy, sense, intelligence, and categorical methods for defining philosophical concepts, as well as illustrating how these concepts have come to shape what we now know constitutes mathematical logic.

Ideas and Universal Quantifiers

To begin, let us consider the connection of Plato’s original ‘idea’ and the

‘universal quantifier’ used in mathematical logic. The term ‘idea’ is synonymous with 2

form. According to the original notion articulated by Plato, an idea (or form) is a changeless object of knowledge; form involves problems and relationships between questions of knowledge, science, happiness, and politics, and distinguishes between knowledge and opinion. From Plato’s original theory of forms, the idea is a universal form. In modern logic, Plato’s ‘idea’ is synonymous with the ‘universal quantifier.’

In modern mathematics and logic, both universal and existential quantifiers are logico-mathematical operators used to bind variables to quantify some condition in a given universe of discourse. The universal quantifier in symbolic notation is expressed as an upside down letter A, ( ∀) which means ‘every’ or ‘for all.’ We use the universal quantifier to bind a variable – call it x – to quantify a conditional sentence – call it p(x) – that translates literally, ‘Every x has some property p(x).’

So ()∀x ∈ S (p(x))is a universally quantified conditional statement, that says for

every x in the universe of discourse S, the open sentence p(x) holds.

To represent the opposite of the universal quantifier ∀ is the specific, or

existential, form of knowledge, quantified by ∃. This existential quantifier is symbolized

in notation as a backwards capital E, and ∃ means, ‘there exists’ or ‘there is.’ So the

existential binds a variable x to a conditional statement – call it p(x) – to say, ‘There is an

x that has a property p(x).’

So ()∃x ∈ S (p(x)) says there is an x in the universe of discourse S where p(x)

holds. Notice that in both the universal and existential casex is the bounded dependent

variable and the independent variable y is equivalent to the open sentence p(x) (namely

that y = p(x)). 3

The universal and existential forms are categories of being known also as genera and species, ‘idea and things,’ ‘logoi and pragmata.’

Negation is a logical operation, symbolized by ‘ ⌐ ’ in front of a quantifier that switches the value of the quantifier to its opposite value. For example, by negating a universal statement of mathematical knowledge, we get an existential statement of mathematics and vice versa. So the universal quantifier is equivalent to (or defined as) the negation of the existential quantifier. Symbolically, we say that

¬∀[]()x ∈ S ()p(x) ≡∃( x ∈ S)(¬p(x)).

Maieutics

The object of philosophy is to define concepts. The object of logic is the

formation of general concepts, since truth is found in general concepts. Classically,

concepts were defined by maieutics (the art of delivering truth) via induction, procedure,

and dialectics (types of generalization of inference, along with irony and induction), thus leading to general definition.

Induction is considered one of humanity’s oldest methods of scientific analysis.

Even today, as is admitted by many detractors, induction is ever present in logic and the methodology of science. Related to induction are the terms irony, inference, and maieutics. Naturally, induction is a tool of the mind and is fundamental to the purpose of philosophy. When we consider logic, among other things, as the “rules of correct thinking”1 and explore Plato’s claim that “the mind is knowledge,”2 the mind

1 Kant, Logic (1801: translated by Hartman and Schwartz), p.13. 2 Dumitiru, A History of Logic, vol. 1, p.25. 4

(inductively) knows and is, hence, the foundation of knowledge. The mind implies to know.

How does this relate to mathematics and logic? First and foremost, induction has been delineated as the most basic type of logic, evolving later into more refined logic types resulting into inference, dialectics, and more formalized proof methods. Nowadays, just as at the dawn of logic and philosophy (since logic is considered a sub-discipline of philosophy), the sole objective mathematical logic is the formation of general concepts; it is in the general concepts where ‘truth’ is found. In this process, a distinction between ideas (logoi) and things (pragmata) occur.

Prior to the symbolism of modern logic, the aim and goal of philosophy, and logic in particular, was articulated “by the ancients with the analogy of delivering a baby

(maieutics)”3. Hence, the art of delivering truth was symbolically represented by the art

of delivering a baby. Truth was found in general definition from an induction procedure.

Maieutics eventually took shape through the method of asking questions (irony)

combined with the more complex procedure of dialectics.

Dialectics was developed after induction became inference. Godel’s math

theorems on the incompleteness of formal systems rely heavily on induction procedures,

leading to the inference that given a formal system Φ that is complex enough to produce

whole numbers, there are logical statements φ that are true but unprovable within the

formal system Φ.

3 Ibid., p.26. 5

Induction

As a specific example of induction, consider the most general case of induction used in today’s formal methods, the so called mathematical induction. Define P(n): = An open mathematical sentence. Define P(n) as the math equation n+2n = 3n, for every natural number n. We must prove P(n) is true for all N.

Using the principle of mathematical induction, let n=1, then P(1): = (1) + 2(1) =3(1).

So P(1) is equal to 3, for n =1.

So P(n) holds for n=1. Now we assume P(n) = n + 2n holds for some particular natural number n. Now we need to show that P(n+1) holds for n+1.

So P(n) is n + 2n = 3n.

Now add 3 to both sides of the equation to obtain P(n+1). So P(n+1) holds for all n+1.

Thus, by the principle of mathematical induction, our original statement n +2n = 3n is true for all natural numbers n.

Notice that we did not simply use arithmetic to show that n + 2n = 3n, but that we used the principle of induction; if the special case for unity (n = 1) was true, then we needed to show if induction worked for some natural number, then (n+1) + 2(n+1) = 3(n+1). This method is valid since the set of all natural numbers is an inductive set of numbers, the first natural number is 1 and all other natural numbers can be found by repeatedly adding

1 to the first natural number. This made our proof become a general statement. Even though this illustration is of a modern type, the ideas and elements of this math induction go back through time to around the origins of philosophy, logic, and math itself. 6

Prior to modern math symbolism, philosophical induction served the same purpose of generalizing from a special basis. Once a specific case was known to be true, one could generalize (assume the case more generally) by adding (or conjoining) a specific to a general; a new special general was developed (similar to Cantor’s countable infinites ∞, ∞ + 1, ∞ – 1, etc.); and if the existing elements, 1, n, n + 1 in N could establish the ‘consistency’ or ‘equivalence’ of the math statements, then a mathematical truth was established.

Dialectics

Before modern mathematics symbolically distinguished the ideas of 1, n, n + 1 in

N, this distribution of objects into general and specifics was known as ‘dialectics.’

Initially, dialectics began as a verbal ‘art of argumentation,’ where, specifically through induction and inference, objects were distributed into general (ideas) and specific

(things). Such objects were defined and concepts were constructed in discussions with adversaries, compelling them to contradict themselves.

Dialectics, as a method of maieutics, took place mainly through the art of asking questions (irony). The actual dialogue, or argumentation, taking place between two orators, was known as physical dialectics, where “material implied a factual position of driving adversaries into contradictions; thereby compelling one to accept the victor’s solution”4 (similar to legal court proceedings between prosecutor and defendant).

Metaphysical dialectics referred to the more general method of finding the intelligible

upon the existent, which formed into a theory of being. It was from metaphysical

4 Ibid., p.18. 7

dialectics that we arrived at modern mathematics, via logic in the form of scientific method.

Here we explore the epistemological and ontological foundations of metaphysical dialectics, rooted in Plato’s original philosophy and how they relate to Godel’s later math philosophy. How does a proof predicate of Peano Arithmetic (PA) equate to an epistemological forming of ontology?

Recall that the results of the seven step skeleton of Godel’s proof predicate of

Peano Arithmetic states the following: ‘Every proof of a theorem of N is not shown by a well formed formula of N.’

To establish ‘provability’ of a well formed formula φ; y must be a free variable and x, the actual proof of φ, must be bounded by an existential quantifier. Choosing a value for y will yield a truth value of the proof predicate of the formal system (in our case

Peano Arithmetic). The negation of such a proof predicate will result in a statement that does not yield a true/false value, but nonetheless holds, ultimately, as a true statement about natural numbers.

In terms of epistemology, when we view mathematics as a theory of knowledge, in the words of Klein, mathematics has lost its certainty; because there exist mathematical truths that cannot be proved within the given theory, you must go outside of the theory to prove the theorems. It is like needing to be in three dimensions to totally understand the two dimensional world. Throwing out Platonic forms means limiting math in such a way that all truths are not provable. 8

Ontologically, in asking the question, “What is mathematics?” (or “What ‘be’ mathematics?”) math is not simply formal methods. To say that math is also informal methods arriving to truth opens logic to the original metaphysical enterprise that modernists tried to avoid. (At this stage, math is whatever we want it to be, except, of course, formal.)

General Definition

In philosophy as a method, initial elements are general notions with a well defined existence. This leads to constructed thinking: a formation of general concepts. As a study of ideas (logoi) and not things (pragmata), philosophy arrives at truth (the essence of a concept) as the principle of a thing. As an objective science, philosophy determines concepts. There are two types of science, the indirect science (dianoia), which includes dialectics and the direct science (intuition) which includes intelligence (essence). Thus, science is knowledge of the intelligible world, the universe, if you will, of discourse used in mathematical logic.

In defining objects, the method of philosophy expresses essence, divides genus into species, and species into subspecies, stopping when division is no longer possible and the individual is reached. This methodology is similar to the original arithmetical sequence binary code as prototype for modern computer programming.

We have identified six key concepts that constitutes Platonism: namely, ideas, quantifiers, maieutics, induction, dialectics, and general definition. These concepts are used in virtually every branch of mathematics. However, there remains the question of how these key concepts are forms of knowledge and logos, and how Plato’s articulations 9

around these forms constitute the basis for mathematical logic. In the next section, we will pay close attention to the relationship between knowledge, form, and logos to gain clarity around how they help us to delineate between the different interpretations of modern mathematics.

Part Two: Forms and Logos

First, we will examine closely what Plato says about logos. We will also show the

modern equivalence of Plato’s dictums using precise mathematical notation. We also intend to show modern Platonic views of, as well as ancient critiques of, ‘logos’ and

‘ideas.’ We will also cite two examples from Meno and the Symposium of what Plato says about the logos and the ideas. We conclude by showing that there is much disagreement among many classical philosophers, as well as among modern Platonists, as to what actually constitutes Platonism.

Traditionally, knowledge was considered the combination of belief and logos.

Given A, A is knowable. So, there is no logos, only names in the first elements of which all things (matter) are composed. That is, things composed of elements are defined as being complex. Elements are named and the combined names of elements are in a logos.

Thus, logoi (being unknowable) are not members of elements. Instead, such elements

imply perception. So complexes relate to the knowable (that which is statable), so that

one can have a true notion of them.

Concerning logos, Plato says, “Whenever then anyone gets hold of the true notion

of everything without a logos, his soul thinks truly of it; but he does not know it, for if one cannot give and receive a logos of anything, one has no knowledge of that thing; but 10

when he has acquired a logos, then all things are realized and he is fully equipped for knowledge.”5 Coupled with this quote, consider the conditional statement in modern

mathematical logic. For elements p and q, where p is the subject and q is the predicate,

the condition ‘if p then q’ is logically equivalent (in set theory) to A < B (‘A is a subset of

B’ or ‘for every x in A, A is a subset of B’). Note that ‘if’ is really a universal quantifier

and p, the logical subject, is an element, and q:= the logical predicate. Note that the

universe of discourse U has an element x, such that every x in some subset A in U (we

can call the subset A the subject or hypothesis) is moved, through the implication

connective of an open sentence p(x), to set B in U (we can call B the predicate or conclusion).

‘Logos’ is classically discussed by Plato but barely understood by the modern

Platonists. One modern theory of Platonism states that logos is just the composite (name)

of names of knowable elements. A second theory is that logos is more than just the nouns out of which knowable elements are composed. Logos is a new linguistic unit that conveys something more than what is conveyed by bare enumeration of individual names in it. The problem is that both of these theories hold that logos would not convey

knowledge. Another theory between the problem and criticism of Socrates’ dream in

Thaetetus is that there is a relation with logical atomist theories of words and sentences

found in Russell’s early writing.

Even though considered a weak claim, forms imply verbs, adjectives, and nouns,

and constitute the names of simple nameables. Hence, if form implies the logical subject

5 Theaetetus: (202), cited in Allen, Studies in Plato’s Metaphysics, p.14. 11

of predication, then verbal expression of form is equivalent to name, not sentence. Thus, it is not false, but nonsense of knowledge, belief, etc.

Refuting the old critique of the theory of ideas would say that it is related to perception, which is Plato’s concern. For unknowable elements, which are perfected simples, there is a relation between argument (that holds every simple nameable as either objects of thought-perception) and the theory of forms. Plato knew of arguments damaging to the theory of forms but kept them anyway. No one is prepared to say that

Plato abandoned forms. Logos does not entail knowledge; there is alogo implying knowledge if there is all knowledge. Plato’s favorite doctrine was ‘knowledge entails logos.’ Consider the classical entailment argument,

Γ ={roses are red, violets are blue}, S1 = “roses are red and violets are blue,” S2 =

“violets are blue.”

Γ entails S2 since “S1” and ⌐ “S1” are logically inconsistent.

Plato simply stated that Γ ={knowledge}, S1 = “logos,” S2 = “a name.”

Γ entails S2 since “S1” and ⌐“S1” are logically inconsistent.6

Citing Cross’ and Ryles’ views of Platonism shows that most of these arguments of what constitutes Platonism have not been settled.7 If Plato still holds a doctrine after

writing an unanswered critique, then we would doubt Plato’s credentials. Form as a

doctrine implies logical atomical difficulties. Ryles, a modern Platonist, holds an

interpretation of Platonism that says that verbs, adjectives, and nouns are themselves

6 Another way to say this is that Γ = {knowledge, logos}, S1 = "logos and knowledge," S2 = "logos" Γ entails S2 since "S1" and ⌐"S1" are logically inconsistent. 7 For Cross and Ryles’s views, see Allen, Studies in Plato’s Metaphysics. 12

names of simple nameables, a tenet of the doctrine of forms. And since we are not sure if other interpretations are possible, we reject Ryles.

Ryles’ interpretation of the theory of ideas states that proper names imply particulars, and that other substantiative adjectives, verbs, prepositions are names of forms. So there are a class of entities called ‘universals,’ entirely different from sensible things. The language use implies such entities, like existence. So every word is not a proper name, which implies that the name of something is one or many instances of something. This also implies the form of immediate apprehension such that we are directly acquainted with the term, which in turn implies a special faculty of knowledge characterized by direct contact of subject and object. Such an interpretation could be corroborated by Plato, but this would make the theory of forms less illuminating, since such claims are orthodox to Platonism. But Plato had other things in mind. In Meno, a specific definition is applied. Given a term, what is it? Is it a term? Notice that we use a

‘word’ as a ‘general term.’ Every one group of particulars is called by one name—‘virtue’ or ‘bee,’ for example—even though there are many different kinds of ‘virtues’ and ‘bees.’

But this is not the only interest of Meno or Socrates.

Meno’s worry is about a special kind of ‘virtue’; he has difficulty that Socrates is trying to discover what ‘virtue’ this one is. Socrates provides Meno with an example (a standard) regarding how to proceed to answer the question “What is a figure?” by answering “the only thing which follows a color.” So, “A man can’t understand the name of a thing when he does not know what the thing is; he gives an illustration of the sort of 13

answer he wants....”8 Meno later says, about mathematical roots: “just as you found a

single character to embrace these many roots, so now try to find a single logos that

applies to many kinds of knowledge.”9 So back to the question, of “What is x?” Unless

the question is put into context, (specific) different answers could appear to be legitimate.

The real business for Meno is Plato trying to tell him that ‘ideas’ are ‘universals’ via the

‘logoi.’ This is an important element in the theory of ideas.

In the Symposium, Plato says, “After long training, the soul sees beauty in itself...there is no ‘logos’ of it and no knowledge of it...it is higher knowledge in the ordinary sense.”10 Thus, with knowledge in its ordinary senses there exists a logos. And the ‘knowledge of acquaintance’ (‘mathema auto to kalon’, ‘μάθημα αύτό τό κάλον’)

tells us that knowledge (‘mathema’, ‘μάθημα’) as ‘scientific knowledge, from direct

experience’ (‘episteme’, ‘έπιστήμη’) belongs to the moment of ‘one admitted into the

higher mysteries’ (‘epopteis’, ‘έπόπτης’); it goes beyond ordinary norms of knowledge.

So usually knowledge and logos go hand in hand.11

The knowledge that Plato is usually concerned with is not ordinary. In his non-

enthusiastic moments, Plato is concerned with the knowledge that logos is involved in.

This knowledge is discussed in the forms, not knowledge by acquaintance (which is

Cross’ opinion).12

8 Allen. Studies in Plato’s Metaphysics (1965), p.20. 9 Ibid., p.21. 10 Plato’s Symposium, Jowett edition, (210e) and (212a). 11 Allen, Studies in Plato’s Metaphysics, p.23. 12 Ibid., p.25. 14

We claim that ‘knowledge by acquaintance’ is modern mathematics and logic

(from an intuitive, or intuitionist perspective). Knowledge and logos go together in

Plato’s philosophy except in cases like ‘knowledge by acquaintance.’ Plato’s dialogues speak of knowledge, forms, logos, the proper duties of a dialectician, and discussing knowledge. So that for every beginning of (intellectual) exercise, exercise must not be directed to visibles but forms.

Knowledge is defined as a product of belief and logos. So belief is not necessarily knowledge, since it can lack logos. Logos has also been shown to be synonymous with naming or defining an object. We also looked at various interpretations of logos, from various Platonic thinkers including Plato himself. Some theories state that logos is a composite name of knowable elements. Others state that logos is a new linguistic unit, or that logos is a relationship between words and sentences. We formalized Plato’s statement that knowledge entails logos and showed how two leading Platonic thinkers of our era,

Cross and Ryles, both differ in their interpretations of Plato’s claim. These fundamentally different views of forms are as follows. Cross thinks that formal knowledge is not knowledge by acquaintance and Ryles thinks that forms are simply verbs, adjectives, and nouns, especially names. The main point is that even today there is no consensus, outside of logos, on a strict definition of Platonism. This point justifies some thinkers in distinguishing between philosophical and mathematical Platonism. In this study we do not agree with distinguishing between the two. 15

Part Three: Knowledge and Logos

In the previous section, we established that knowledge is a form. From this premise we can now delve into the origin of the term ‘logos’ to provide a commentary on its etymology. Then we show a common theme in modern scholarship of how recent translations of the term are not clean, which confuses the actual meaning of the term. We show through such analysis how the question of naming arrives in the implicit meaning of the term ‘logos,’ and how ‘idea’ leads analytically to the notion and utility of a formula, thus showing that knowledge is a formula.

Etymologically ‘logos’ comes from the verb ‘legein,’ which means ‘to tell, state, say, discourse, a statement, or hypothesis.’ As a linguistic operation, ‘hypothesis’ keeps logos in the domain of language. Plato, in defining or describing logos, limits it to the linguistic domain. Aristotle refers to Platonists as “thinkers who occupy themselves with

verbal discussions, tending to rank universals as substances (for they tend to describe as

principles or substances, owing to the abstract nature of their inquiry).”13 Aristotle makes the distinction between thinkers who get down colloquially to brass tacks of things and Platonists who interest themselves in talk.

‘Logikos’ is not equal to ‘rationally.’ Moreover, bad translations of ‘logos’ as

‘reason’ or ‘thought’ make difficult the clarification of Platonic conceptions. Consider the following three translations.

(i) “I have better have discourse to the world of the mind and seek there the truth of

existence” to “I had better have recourse to statements, etc.”

13 Ibid., p.26. A better translation by Cross is “through pursuing their inquiry by means of logoi.” 16

(ii) “In reference to thought and what may be called ideas” to: “In reference to those

things which are essentially grasped by ‘statement’ (or ‘discourse’).”

(iii)“For immaterial things, which are the noblest and greatest, are shown only in

‘thought and idea’ and in no other way” to: “Are shown only in ‘discourse’ or

‘statement’ and in no other way.”14

Recall that the idea is an object (nota) of intelligence (good, beautiful, etc.). Objects of perception are ‘aestheta.’ Mental experiences (‘pathemata en ta psyche’) are high (gnosis and dianoia) intelligible—the two modes of operation of the reasoning part of the soul, and the low mental experiences are perception and belief. The mathematical sciences are equivalent to ‘gnosis,’ which is equivalent to ‘dianoia,’ which is equivalent to ‘logismos,’ that is, reflection or intelligence or abstract thinking. Thus, gnosis and dianoia are distinct according to dialectics. Mathematics, dialectics in objects, methods of procedure, movements of thoughts, and states of mind are grounds of distinction between gnosis and dianoia.

Objects imply a division within intelligibles, not a class of mathematical numbers and figures intermediate between ideas and sensible things. In the intelligible world, education (schema) implies figures in moral and mathematical ideas and their truths.

Thus, moral ideas are not in a higher class reached by gnosis and math ideas are not in a lower class reached by dianoia, since math objects can be also objects of gnosis when seen in connection with a first principle.

14 Ibid., p.27. 17

Plato is given credit for discovering the method of analysis. Heath tells us,

But analysis and synthesis following each other are related in the same way as upward and downward progressions in the dialectician’s intellectual method. Plato observed the importance, from the point of view of logical rigor, of the confirmatory synthesis following analysis. The mind must possess the power of taking a step or leap upwards from the conclusion to the premise implied in it. Prior truth can’t be deduced or proved from conclusion; it must be grasped by an act of analytical penetration, such an act is involved in the solution, by way of hypothesis.15

Thus, Proclus associated Plato’s method of dialectical ascent to genuine principles with

method of analysis in geometry.

New interpretations of Platonism include the basis that forms are simple

nameables known ultimately by acquaintance. For example in Theaetetus, when the

question “What is a figure?” was asked, Plato asks for the idea of a figure and expects as

a proper answer to the question, to be given a statement. Thus, the idea of a figure is

displayed in the logos, and the form is displayed in the predicate of the logos. Theaetetus

found the idea of the mathematical roots, and Socrates replied, “just as you found a single

character to embrace all that multitude try to find a single formula that applies to many

kinds of knowledge.”16 Thus, the idea is equivalent to the logos, that is, giving an idea

implies giving logos, which embodies a formula. So a form is like a formula, the logical

predicate in a logos, not the logical subject: what is said of something, not something

about which something else is said.

15 Proclus, "Commentary on the First Book of the Elements," in Heath, Greek Mathematics, p.211-218. 16 Allen, Studies in Plato’s Metaphysics, pp.28-29. 18

It is incorrect when ‘we talk about ideas (form),’ but it is correct to say that when

‘we talk with ideas (form), logoi (pieces of talk) are necessary to display ideas (form)’ to us. Thus, ‘idea’ is a logical predicate. So when asking “What is a figure?” the answer is to produce a logos (in predicate form of which ‘idea’ is displayed). The idea of ‘figure’ is expressed in the predicate of the statement ‘figure is...[the] boundary of [a] solid.’ So what is the statement about? The logos is not about the form (displayed in the predicate)

‘figure.’ The whole process is of an unreal definition becoming a nominal definition

(defining a word). Thus, we arrive at necessary statements; that is, logically necessary (or self-contradictory) to deny, not truth about things, but logical truths of how we talk about things. Plato knows that idea is a ‘thing’ question, and to reach certainty you must pay the price.

The main point is that forms are logical predicates displayed in logoi and not simple nameables known by acquaintance. This is not to deny that there exist things:

Plato says this can be construed to fit the ‘simple nameables’ view. Nor do we pretend that the view of forms as logical predicates displayed in logoi is to be found explicitly formulated in Plato. (But it does allow greater flexibility in more open interpretations, such as views). Especially before exploring ‘logos’ and its notions of subject and predicate, the view is there implicitly in how Plato develops the theory of forms. So we must stay focused on the relationship and prominence of the logos/knowledge/forms combination in Plato’s dialogues.

Cross, a Platonist, put the theory of ideas in the context of language and logic.

Most puzzles of Plato’s time were logical puzzles of language. The Socratic method of 19

question/answer to attain knowledge (‘elenchus’) requires ‘logoi’ to proceed. The mathematical interest of ‘logoi’ and deductive procedures—not the simple entities known by acquaintance—are the goal of Plato. So forms are not simple entities: how could they either be premises or conclusions of any argument? Plato liked the material mode of speech and existing propositions.17

After examining the etymology of the term logos as meaning ‘to say,’ logos can

be interpreted as a kind of verbal activity of argumentation. Aristotle’s view also supports

this. Not equating ‘logikos’ with ‘logos’ shows how bad translations have led to muddled

meanings of ‘logos.’ Accordingly, as shown by Plato, even amidst the confusion

translators create around our understanding of ‘logos,’ ‘knowledge’ is the result of the

‘high intelligible.’ This shows ‘logos’ as the basis of the dialectics of abstract thinking, or

so-called ‘mathematical reasoning.’ In the light of this inquiry, we will next look at

‘mathematical Platonism’ as a transition into our investigation of identifying Platonic

ideas in the modern schools of mathematical thought.

In this chapter, we have defined Platonism by looking at key concepts Plato

articulated in his original theory of ideas and identified them in contemporary

mathematical symbolism. We have also looked into the intimate relationship between

logos, forms, and knowledge and used this relationship as a basis to show not only the

interpretative disagreements that exist among Platonic thinkers, but also to show how

these notions provide a basis for our understanding of what constitutes mathematical

17 Ibid., p.31. 20

thinking, or abstract reasoning. Next we will continue to explore the presence and evolution of such ideas in the various branches of the mathematics foundations.

CHAPTER TWO: THE FIVE SCHOOLS OF MATHEMATICS

This chapter explores the five schools of mathematical thought: logicism, nominalism, formalism, intuitionism, and set theory. We distinguish nominalism as its own school even though it developed out of logicism, the section under which it will be discussed.

Part One: The Origin and Development of Logicism

Logicism is the school of mathematical thought that maintains that mathematics is

reducible to logic. In the early 1900’s, the laws of logic were accepted by almost all

mathematicians as a body of truths. Hence, the logicists contended that mathematics must

also be a body of truth. “And since truth is consistent, so they claimed, mathematics must be.”18 This school of thought began with Leibniz and Dedekind, was developed by Frege,

and improved by Russell and Whitehead’s Principia Mathematica.

According to Leibniz,

A truth is necessary when the opposite implies a contradiction, and when it is not necessary then it is called contingent. That God exists, that all right angles are equal, etc., are necessary truths; but that I myself exist and there are bodies in nature which possess an angle of exactly 90 degrees are contingent truth. These could be true or false; for the whole universe might be otherwise.19

18 Kline, Mathematics: The Loss of Certainty, p.216. 19 Ibid., p.217. 21

Klein continues, “Dedekind affirmed flatly that number is not derived from intuitions of space and time, but is ‘an immediate emanation from the pure laws of thought. From number we gain precise concepts of space and time.’ He started to develop this but did not pursue it,”20 attempting instead to build up (natural) numbers from logic.

Frege developed the symbolic notation used today in modern logic and is credited

with transforming logic from a rhetorical art to a deductive science by the means of his

symbolic notation. Arguments therefore could be constructed according to a logical calculus, bringing into fruition the goal of Leibniz’ philosophy as a universal science.

Frege’s work was then carried forward by Russell and Whitehead in their work

Principia Mathematica. Russell found a flaw in Frege’s definition of a set. “Frege…in

1902 received a letter from Russell informing him that his work involved a concept; the

set of all sets; that can lead to a contradiction.” This flaw, known as Russell’s Paradox, provided a fatal blow to Frege’s logical system, since his axiomatic definition of what

constitutes a set did not hold.

Acknowledging Frege’s contribution to the foundations of mathematics, Russell

(and Whitehead) attempted to resolve Frege’s flawed Axiom 5 via the Principia

Mathematica. This full attempt to reduce all of mathematics, numbers systems, algebra,

geometry, etc., to logical statements, was essentially based on Frege’s logical symbolism

and notation. The only addition was a ‘theory of types’ added to resolve Frege’s systemic

flaw. This theory of types arranged sets in a form of hierarchical classes, to avoid the

possibility of ‘a set of all sets that contains itself.’

20 Ibid., p.218. 22

According to Klein,

Russell did know of course that Peano had derived the real numbers from axioms about the whole numbers, and he was also aware that Hilbert had given a set of axioms for the entire real numbers. However he remarked in 1919, in his Introduction to Mathematical Philosophy, ‘The method of postulating what we want has many advantages; they are the same as the advantages of theft over honest toil.’21

Russell’s real concern was that postulation of, say, 10 or 15 axioms about number

does not ensure the consistency and truth of the axioms. As he put it, it gives unnecessary

hostages to fortune. Whereas in the early 1900’s Russell was sure that principles of logic

were truths and therefore consistent, Whitehead cautioned in 1907, “There can be no

formal proofs of the consistency of the logical premises themselves.”22

In his search for mathematical truth, Russell could not distinguish between

Euclidean and non-Euclidean concepts. That is, he could not affirm which was the truth.

But in his Essay in the Foundations of Geometry (1898), he did manage to find some

mathematical laws, such as that as physical space must be homogeneous (possess the

same properties everywhere); hence, he believed that mathematical laws were physical

truths.

This logistic theory of types requires that statements be carefully distinguished by

type. But, if one attempts to build mathematics in accordance with type theory, the development becomes exceedingly complex. For example, in Principia Mathematica

(hereafter referred to as “PM”), the logical foundation for mathematics established by

Russell and Whitehead, two objects a and b are equal if every proposition and

21 Ibid., p.220. 22 Ibid., p.221. 23

propositional function that applies to or holds for a also holds for b, and conversely.

Hence, type theory introduces a complication also in the concept of the least upper bound of a bounded set of real numbers.23

Consider that the least upper bound (l.u.b.) is defined as the smallest of all upper

bounds, so the l.u.b. is defined in terms of the set of real numbers, R. Hence the l.u.b.

must be of higher type than R and so is not itself a real number.

Also, through the axiom of reducibility, in PM, every property p of higher type; p

is equivalent to one of first order. Through the axiom of reducibility, math induction is

supported in PM.

The PM system was then refuted and shown to be contradictory by Godel’s

incompleteness theorems. Godel showed that any time one attempted to formalize

arithmetic, according to PM or any axiomatic system, there will be certain true arithmetic

statements that cannot be proven in that logical system. Godel qualifies as a logicist since

his methods in proving the incompleteness theorem are wholly logical.

Critics of Logicism

The axiom of reducibility seems arbitrary, but no proof of its falsity has been

shown. It is a happy accident, but not a logical necessity. That is, “the axiom of infinity,

although it is couched in logical terms, thus seems to pose the problem of whether the

universe is composed of a finite or infinite number of ultimate particles; a question

answered by physics, not math or logic.”24 In short, if mathematics was to be ‘reduced’ to

logic, then logic seemingly would have to include the axiom of infinity.

23 Ibid., p.222. See also Russell and Whitehead, Principia Mathematica, Chapter IX. 24 Ibid., p.225. 24

A geometric criticism of logicism made by the logician Hemphill is that

by using analytic geometry one could do so (develop geometry in PM). Nevertheless, it is sometimes argued that PM, by reducing to logic a set of axioms for N, thereby reduced number theory, algebra, and analysis to logic, but did not reduce to logic the non-arithmetical parts of math; such as geometry, topology, and abstract algebra.25

A philosophical critique of logicism by Weyl says that “if the logistic view is

correct, then all math is a purely formal, logico-deductive science whose theorems follow

from laws of thought. Just how deductive elaboration of the laws of thought can represent wide varieties of natural phenomenon; the uses of N, the geometry of space acoustics, electromagnetism and mechanics seem unexplained.”26 Logistic programs affirm that in

the creation of math, perceptual or imaginative intuition must supply new concepts

whether or not derived from experience. Else, how could new knowledge arise? But in

the PM all concepts reduce to logical ones.

Formalization apparently does not represent math in any real sense. It is the husk,

not the corn. Russell’s own statement, made in another context, that mathematics is the

subject in which we never know what we are talking about, nor whether what we are

saying is true, can be found against logicism.

Nominalism

Nominalism is a reaction to the applications of Platonism to logicism and set

theory and results from the discovery of the paradoxes. Nominalists oppose the view of

certain interpretations of math objects and are concerned with what object or objects are

25 Ibid., p.227. 26 Ibid., pp.227-28. 25

denoted by a single term. In set theory, for example, the nominalists criticized the revised axiom of comprehension (that each mathematical entity is the element of a set). They attempt to construct a system for the foundations of mathematics and logic by reinterpreting the axiom of comprehension in terms of nominalism.

Quine’s Semantics

For example, Quine has a semantic approach to the natural deduction of formal systems. His theory of quantification is a restricted form of elementary logic. He introduced “a logical system which enables philosophy to a totality of pure math without appeal to any special axioms”27. Quine’s New Foundation (NF) and Mathematical Logic

(ML) are distinct from Fraenkel Skolem’s set theory (FS). His system (NF) is weaker than (FS). Thus, consistency can be established by more elementary methods; and (FS) provides proofs of certain theorems which are unprovable in (NF).

Nominalistic logic denies that a universal term can be used to denote a multitude

of concrete objects. Nominalists feel that the collective usage of universal terms need not

justify the existence of some abstract entity; instead, universal terms of concrete objects

(‘good,’ ‘bad,’ ‘natural number,’ etc.) correspond to their original sense. Quine postpones

in his logical theory, as much as possible, the supposition of the existence of abstract entities. Nominalism is similar to intuitionism with removal of negation, the existential quantifier, and disjunction.

The nominalists’ current mathematical logic is distinct from the nominalism of classical philosophy. Today nominalism is a spontaneous reaction to Platonism in Frege’s

27 Beth, Foundations of Mathematics, p.507. 26

and Cantor’s system as well as paradoxes. It is a critique of Russell-Zermelo’s logico-set theory system (RZ) and Platonic background; it constructs a system of logic and math based on nominalist views; it nominalistically reinterprets the RZ system.

One nominalist objection is the negation of the compression of multitudes into unity. Beth tells us, “admission of this unity as a value of certain bound variables; implies attributing a substance to compressed multitude.” Quine calls this objection a ‘liberal policy’ used in establishing pure math as a branch of logic. Even though he attempts to avoid this axiom of comprehension, his logical systems New Foundations (NF) and

Mathematical Logic (ML) avoids Russell’s and Cantor’s paradox, but NF still falls victim to the incompleteness paradox of denotation.28 Rosser proved that ML was vulnerable to

the Burali-Forti paradox. Quine retried ML to account for the Burali-Forti paradox, then

Rosser and Wang showed that NF cannot handle standard models and, thus, NF and ML

are hardly consistent, failing the initial program of logicism. So nominalism is a tendency

of universalism, with the philosophical implications that Godel’s result that ‘Zermelo-

Fraenkel’s axioms of set theory including the axiom of choice are consistent (Con(ZFC))’

provides for different interpretations of abstract set theory.

For distinctions of nominalism, consider the theory of syllogism. Both the

Platonist and the nominalist accept the same modes. They just justify them differently.

This debate is restricted to the foundations of logic; no repercussion in the field of

application. The new nominalism involves itself with logic theory and applications. The

axiom of comprehension of Frege and Cantor states that sets are multitudes of

28 Ibid., p.508. 27

mathematical entities sharing a certain property, a unity capable of appearing as a member of another multitude. Comprehension is a transformation from multiple to unity; it is the source of many paradoxes of logic and set theory; it was also a source for logicians and mathematicians turning away from and criticizing Platonism. Quine attempted to revise the comprehension axiom.

Russell’s axiom of reducibility is not logically necessary. No one is sure if the axiom of infinity is an axiom of logic, and the use of the theory of types is for propositional functions. For example: (∀x ∈ S)(p(x)) where ∀= quantifier, p(x) =

proposition, p(x) = 2+3, for S⊆N. Nominalists include Quine, Tarski, and Henkin,

because they oppose the reducibility view to interpretations of math objects. But the nominalists have a methodology and goal that are Platonic in essence; the only difference between them and traditional Platonists is in their interpretations of mathematical objects.

Techniques of nominalist logic are used by logicians Godel, Quine, and Tarski in their quest to ground logic in mathematics. These methods are consistent with the original notion of ‘idea’ and the claim that the ‘logos’ constitutes the logical predicate of the idea

(or form).

Part Two: Intuitionism

Descartes defined ‘intuitio’ as ‘genuine knowledge, clarifying, intuitive

knowledge.’29 For Kant, math intuition implies space and time; hence intuition is a theory

of reason. For Leibniz, intuition was ‘intuitive knowledge,’ ‘clear distinct and adequate.’

Godel’s intuition was a composite of Leibniz’s ‘intuitive knowledge’ and the modern

29 Descartes, Rules for the Direction of the Mind, in Philosophical Works of Descartes, vol. 1, p.7. 28

math intuition, an object relational, propositional attitude known as realism (which is synonymous with mathematical Platonism). Here the math objects we talk about exist independently of our thoughts and knowledge about them.

Godel’s intuitionism, which is a realism of abstract math objects extended to set theory, type theory, and the idea of concepts as classes or categories, is equivalent to

Descartes’ ‘intuitio’ (intuitive knowledge), as well as the relations of objects. For

Brouwer, mathematics is equivalent to synthetic, a priori truths where all numbers and mathematics derive from intuition in time. So a math theorem constitutes empirical fact construction (2+2 = 3+1) and is a source of logical principles. Thus, “it is in vain to hope complete formalization of math, excluding the law of the excluded middle.”30

In intuitionist set theory, there is no law of the excluded middle, identity, non-

contradiction, and syllogism. Thus, all math cannot be solved, and there is no such thing

as disjunction like ‘p or ~p.’ So in general the properties of intuitionism are not the basis

for possible knowledge of the strongest math axioms. They are restricted to demand in its

application, equivalent to perception and yielding knowledge of propositions involving

abstract concepts in a central way, distinct from intuitive evidence.31 Thus, all intuition is

knowledge, especially in propositions, and Kantian relations of objects are ‘an attitude of

intuitive knowledge.’32 Intuition is not always used in a mode of knowledge.

Philosophical intuition is what one thinks is true, and is sometimes fallible.

30 Klein, Mathematics: The Loss of Certainty, pp.243-44. 31 See Charles Parsons, “Platonism and Mathematical Intuition in Kurt Godel’s Thought” in The Bulletin of Symbolic Logic, vol.1, no. 1 (March 1995), pp.44-74. 32 Ibid., p.47. 29

Brouwer conceived of mathematics as a thinking process: a mental construction that builds its own universe, independent of experience and restricted only insofar as it must be based upon the fundamental mathematical intuition. This fundamental intuitive concept must not be thought of as in the nature of an undefined idea, such as what occurs in axiomatic theories, but rather as something in terms of which all undefined ideas are to be intuitively conceived, if they are indeed to serve some mathematical thinking.

Moreover, mathematics is synthetic. It composes truths rather than derives implications of logic.33

Mathematical ideas are more deeply imbedded in the human mind than in

language since the world of mathematical intuition is opposed to the world of

perceptions. To the latter world belongs language, where it serves the understanding of

common dealings. Language evokes copies of ideas in man’s mind, symbols and sounds.

The distinction is similar to that between climbing a mountain and describing it in words.

According to Klein,

Mathematics is not bound to respect the rules of logic. Knowing mathematics does not require knowing formal proofs, and for this reason the paradoxes are unimportant even if we were to accept the mathematical concepts and constructions they involve. Paradoxes are a defect of logic but not of mathematics. Thus, consistency is a hobglobin.34

In the realm of logic, there are some clear intuitively acceptable logical principles

or procedures that can be used to assert new theorems from old ones. These principles are

part of the fundamental mathematical intuition. Intuitionist Herman Weyl adds, “The

33 Klein, Mathematics: Loss of Certainty, p.234. 34 Ibid., p.236. 30

principle of excluded middle may be valid for God who surveys the infinite sequence of actual numbers, as it were; with a glance; but not for human logic.”35

Weyl observed that non-constructive existence proofs inform the world that a

treasure exists without disclosing its location. Such proofs cannot replace construction

without loss of significance and/or value. He also pointed out that adherence to the

intuitionist philosophy means the abandonment of basic existence theorems in classical

analysis. Cantor’s hierarchy of transfinite numbers was described by Weyl as a fog on a

fog. “Analysis,” he wrote in Das Kontinuum (1918), “is a house built on sand. One can be

certain only of what is established by ‘intuitionistic methods.’”36

So the different types of intuitionists are those that:

1. Eliminate all set theory and form only constructed concepts.

2. For every real number in the math universe, p(r) holds if R is a class.

3. For every integer in the math universe, p(z) is a function.

Here’s a typical mathematical example to show the problem with construction.

()−1 p Define N = 1+ , N = number. 10 p

()−1 3 If p =3, then N = 1 + = 1+(-.001) = 1-.001 = .999. 103

If p = 2, then N = 1+(.01) = 1.01.

If p = first digit in pi, whose sequence 123456789 is an expansion,

then if there is no p, then N = 1.

If p does exist, then N = 1.0000....0 (0 p times, if p is even) or N – 0.99999....(if p is odd).

35 Ibid., p.237. 36 Ibid., p.239. 31

So the conclusion is that we are not sure if N exists. If N does not exist, N = 1. If N does exists, N cannot be written. Even if N is defined, is N constructed?

Existence proofs, axiom of choice and the continuum hypothesis are nonconstructible and therefore unacceptable to intuitionists. Constructivists rebuilt a good deal of mathematics. But in practice, intuitionists, as all other mathematicians, depend on normal methods of creation and even on classical logic, even though intuitionists seek to reconstruct proofs to accord with their own principles.

Thus, to demand ultimate reliance upon logical deduction from axioms, Brouwer says that a system of axioms must be proven consistent by using interpretations or models which are already known to be consistent. Hilbert and Bernays argue in their work on the foundations of mathematics that the great weakness of intuitionist philosophy is the inability to answer the questions: (1) What concepts and reasoning may we rely upon if correctness were to mean self-evidence to the human mind? And (2) where is truth objectively valid for all human beings? Brouwer admits that intuitionistic mathematics is useless for practical applications. He also rejects human domination over nature.

Regardless, Weyl says in 1951, “I think everybody has to accept Brouwer’s critique who wants to hold to the belief that math propositions tell the sheer truth, truth based on evidence.”37

Hence, “If mathematicians abstained from talking of infinity in the manner of

Cantor, they would not have to face Cantor’s peculiar puzzles, and he might perhaps have

argued that it was useless to spend time on the detailed discussion of an assumption

37 Ibid., p.242. 32

whose absurdity could be seen without references to paradoxes.”38 The research program

of mathematical infinity was avoided not only due to the contradictions inherent in its

problems, but also because it would disturb the hierarchy of power amongst the leading mathematician of the 19th century. For example, “At the beginning of 19th century, Gauss said there was no place for talk of infinity, so Kroenecker conducted a campaign against

Cantor’s program.”39

From Brouwer’s statement that “neo-intuitionism considers the falling apart of the

moments of life into qualitatively different parts, to be reunited only while remaining

separated by time, as the fundamental phenomena of the human intellect,”40 we get the

intuition (of sets as being) of the betweenness which is not exhaustible by interposition of

new units and therefore cannot be thought of as a mere collection of units. Thus, there is

continuity, even though geometry is dependent upon the same intuition. There is no set

o). Defineא) ,except denumerable; there are no transfinite cardinals except aleph-nought

o, as the cardinal of a set whose members can be correlated one to oneא ,aleph-nought

with the sequence of N. Thus, there is no meaning to the set notation statement: ‘{For

every real number, 0 < r < 1}.’

In a Platonic sense, Brouwer rejects all connection to actual infinity and real

numbers as an open interval. N is an open manifold always in growth, unfinished and

comprehended only by the law of number construction. The intuition of which they speak

is just the mind’s clear apprehension of what it has itself constructed. They assert that in

38 Kneale & Kneale, Development of Logic, p.673. 39 See Gauss’ Works, vol. 8, p.216. 40 Kneale & Kneale, Development of Logic, p.674. 33

mathematics all satisfactory proofs are constructive. They deny the dependence of math on all special language, and also distrust formal techniques. Hence, constructive proofs imply a performance of an experiment in imagination. Modern intuitionists, following the tradition of Kant, think that all math theorems are objective (valid for all intelligent beings), and there is no existential quantifier in math only if it is demonstrated by an instant production. Thus, there is no use of excluded middle, implying that math is derived from intuition. It does not imply a system of logic; instead math is equivalent to a source of logical principles.

Intuitionistic Logic

Intutionistic logic is basically logic without the law of the excluded middle. Every statement is true. Consider the following example of Intuitionistic Logic:

Let A be the formal statement: ¬(∃x)[Fx.¬Gx] ⇒ (x)¬[Fx.Gx]. Let B be another

formal math statement: ¬x¬[Fx.Gx] ⇒ (∃x)[Fx.¬Gx].

Statement A holds in intuitionistic logic. It says, ‘there is not an x such that F(x)

and not G(x) implies that x and not F(x) and not G(x) if x and not F(x) or G(x).’

Statement B does not hold in intuitionistic logic, since it says that ‘not x, not (F

and G)(x) implies that there is an x such that F(x) and not G(x).’

In statement A, we see that refuting a universal statement(S) by showing it

involves a contradiction is equivalent to saying that statement S is equivalent to the

statement (FS and ~GS). Thus, the existential quantifier is unnecessary.

Statement B is non-constructive, since the existential quantification of the

conclusion of a proposition poses intellectual danger. Abandoning statement B implies 34

abandoning the law of the excluded middle, eliminating double negation. Thus, to

Brouwer every math problem is solvable.

From Intuition to Construction

Intuitionism shares a core part with most other forms of constructivism.

Constructivism in general is concerned with constructive mathematical objects and reasoning. From constructive proofs one can, at least in principle, extract algorithms that compute the elements and simulate the constructions whose existence is established in the proof. Most forms of constructivism are compatible with classical mathematics, as they are in general based on a stricter interpretation of the quantifiers and the connectives and the constructions that are allowed, while no additional assumptions are made.

The logic accepted by almost all constructive communities is the same, namely intuitionistic logic. Many existential theorems in classical mathematics have a constructive analogue in which the existential statement is replaced by a statement about approximations. We saw an example of this for the intermediate value theorem in the section on weak counterexamples above. Large parts of mathematics can be recovered constructively in a similar way. The reason not to treat them any further here is that the focus in this section is on those aspects of intuitionism that set it apart from other constructive branches of mathematics.41

Part Three: Formalism

For Hilbert and the formalists, one cannot deduce mathematics from logic because

math is its own autonomous discipline, not a consequence of logic. Moreover

41 See Beth, Foundations of Mathematics, p.644. 35

mathematics must be treated not as factual knowledge but as a formal discipline, abstract, symbolic, and without reference to meaning. In this manner deductions are to be manipulations of symbols according to logical principles. Thus, to avoid ambiguities of language, the unconscious use of intuitive knowledge (the main causes of mathematical paradoxes), to eliminate other paradoxes, and to achieve precision of proof and objectivity, Hilbert decided that all statements in logic and math must be expressed in symbolic form.42

The formalist believes that in the real world only a finite number of objects exists

and that matter is composed of a finite number of elements. Using an analogy from

number theory, the irrational number has no intuitive meaning as number. Even though

we can introduce lengths whose measures are irrational, the lengths themselves do not

furnish any intuitive meaning for irrational numbers. Yet the complex number ‘i’ as ideal

element is necessary even for elementary mathematics; this is why math used it even

without logical basis until the 1870’s when Hilbert (and the formalists) made some point

with respect to the set of complex numbers C, defining i = square root of -1. These have

no real immediate counterparts. Yet they make possible general theorems, such as that

exactly of n roots, as well as an entire theory of functions of complex variables, which

proved immensely useful even in physical investigations.

Formalist Logic

Formalist logic is Plato’s logic via Aristotle’s logic. To the formalist then,

mathematics proper is a collection of formal systems, each building its own logic along

42 Klein, Mathematics: The Loss of Certainty, p.247. 36

with its mathematics, each having its own concepts, its own axioms, its own rules for deducing theorems, and its own theorems. The development of each of these deductive systems is the task of math. The formalist school, including Hilbert and his students,

Wilhelm Ackermann, Paul Bernays, and John von Neumann, gradually evolved into what is known as Hilbert’s proof theory or metamathematics. They pioneered a method of establishing the consistency of every formal system.

Hilbert proposed that a special logic be used, free from all objections. The logical principles would be so obviously true all would accept them. Actually, they were very close to the intuitionistic principles. Controversial reasoning—such as proof of existence by contradiction, transfinite induction, actually infinite sets, and impredicative definitions and axioms of choice—was not to be used.43 Metamathematical concepts and proof

methods he called finitary. For example, ‘if p = prime, then there exists prime larger than

p’ is a non-finitary math statement since it asserts about every integer m and n, where m,

n > p. But,‘if p = prime, then there is a prime between p and p+1’ is a finitary math

statement since every prime number p can be checked if p exists among the finite

numbers between p and p+1.

Like the logicist school of mathematics, the intuitionists objected to formalist

concepts of existence. Hilbert maintained that “the existence of every entity was

guaranteed by the consistency of the branch of math in which it is introduced.”44 For

Brouwer and the intuitionists, math rigor was found in the human intellect; to Hilbert and the formalists, math rigor was found on paper. For the formalist, intuitionism is treason

43 Ibid., p.250. 44 Ibid., p.252. 37

against science. But in formalist metamathematics, as Weyl pointed out, “essentially intuitionistic principles were used.”45 And by 1930, Russell and logicists agreed that

axioms of logic were not sound truths and consistency was not assumed. Intuitionists maintained only soundness of their guaranteed consistency. The formalists had a well thought-out, thorough procedure for establishing consistency; success with simple systems made them confident they would succeed with showing natural numbers to be consistent, as well as all math.

Hilbert produced a set of axioms for Euclidean geometry on foundations of logic and math, suggesting a new program for eliminating newly discovered paradoxes by axiomatizing logic, number theory, analysis, and set theory. Thus, to call a formal mathematical system consistent is to say that the application of the rules of inference to the axioms can never lead to a pair of consequences, one of which is the negation of the other. Apart from the problem of consistency, formalist techniques are appropriate for questions about deductive systems, such as the completeness of axiom systems devising decision procedures and solution of problems in the various branches of math.46

When undertaking to axiomatize all math, the formalist assumes that his axioms are postulates that determine the undefined symbols that he uses. Thus, a mathematical entity to the formalist is just what his axioms allow, neither one or less. All math statements must have same status as a pure geometry statement.

Formalist math is not open to intuitionistic objections. Anyone talking of real

numbers as a formalist will say ‘real number’ is any object satisfying certain conditions,

45 Ibid., p.253. 46 Kneale & Kneale, Development of Logic, pp.684-85. 38

assertions if you will. These axioms have certain consequences, so by virtue of their form they will be truths of logic. Are formalists content with this conclusion? We are not sure.

Nor do we know of their view of the ultimate relation of axiomatized math to constructive math, for “After a proof of consistency of his axioms by constructive mathematics he would be content to forget the difference and treat non-constructive math as a continuation of constructive math.”47

Constructive mathematics is beside, not interior to, classical math, according to the intuitionists. Hence, the formalist program can never be carried out in full. But the formalists did stimulate the study of axiom sets that led to some interesting discoveries of the properties of logic and other deductive systems.

Part Four: Set Theory

Though set theory is incorporated in the logicist approach to math, set theory

prefers to approach mathematics directly through axioms. The axiomatization of set

theory was first undertaken by Zermelo in 1908. He believed paradoxes arose because

Cantor did not restrict the concept of a set. His axiom system contained undefined formal concepts of a set and the relation of one set being included in another. These and defined

concepts were to satisfy statements in the axioms.

In the axioms, there are infinite sets and operations like the union of sets and

formation of subsets as well as the axiom of choice. Fraenkel improved Zermelo’s axioms

since Zermelo failed to distinguish the property of a set and the set itself. ZF, named after

Zermelo/Fraenkel, is the most commonly used axiom system of set theory. Thus, for set

47 Ibid., pp.687-88. 39

theorists, mathematical logic was used but not specialized. Logical principles, to them, were outside of math.

Zermelo-Fraenkel Axioms

The ZF axioms are as follows:

a. Two sets are identical if they have the same members.

b. The empty set exists.

c. If x,y are sets, then {x,y} is a set.

d. x ∪ y is a set (union).

e. Infinite sets exist.

f. Every property formalized in theory language can determine a set.

g. Power set of any set.

h. Axiom of choice.

i. x does not belong to x.

ZF axioms for set theory can build all mathematics. They are most general and fundamental to build analysis and geometry. The Bourbakists, a radical and influential school of mathematics, expressed their position on logic: “In other words, logic, insofar as we mathematicians are concerned, is no more and no less than grammar of a language which we use, a language which had to exist before the grammar could be constructed.”48

The Bourbaki School

The Bourbaki school renounces logicism, formalism, and intuitionism. Their philosophical position is best summed up in the following quotation: “For 25 centuries

48 Klein, Mathematics: Loss of Certainty, p.256. 40

mathematicians have been correcting their errors and seeing their science enriched and not impoverished in consequence; and this gives them the right to contemplate their future with equanimity.”49 The Bourbakists have put forth about thirty volumes in their development of the set theoretic approach.

Cantor’s Set Theory

Cantor’s original theory of sets was a logical theory of natural numbers prior to

Frege. The totality of the real numbers, R in the interval (0,1) is not denumerable

(countably infinite, or countable). To deal with this situation, Cantor elaborated set

theory. A set is defined as collection into one whole of definite distinct objects of our

perception or our thought, which are called elements of the set. The term ‘set’ is

equivalent to the terms ‘manifold,’ ‘ensemble,’ ‘totality,’ ‘aggregate,’ ‘class,’ etc. So the

sets S and T are equivalent if there is a one-to-one correspondence between them. The

cardinal set, S*, is defined as the general concept with the aid of our active intelligence

and results from a set when we abstract from the nature of its various elements and from

the order of their being given. For a finite S, S* is an element of N. S is infinite if and

only if there is a one-to-one correspondence with a proper subset. Thus, the set of Z+ can

have one-to-one correspondence with a set of squares of Z+.

The original set operations outlined by Cantor are as follows:

S +T – logical sum

ST – inner product

S× T – cross product, ordered pairs

49 Ibid., p.257. 41

S T – insertion of S into T

U S – power set ={in, out} S

n (oא) = 0 אn =n + א = 0 א

Cantor thought his work was a discovery of laws not made by man. He was just a scribe

with no claim to merit except for the style and economy of his exposition.50

Following the development of logicism by Russell, the Vienna Circle, and Godel,

its many contributions to mathematics included the introduction of precise symbolic

notation to modern mathematics and type theory. Nominalism tries to revise the

comprehensive axiom through a semantic approach to natural deductions of formal

systems; quantification is restricted to elementary logic. The school of intuitionism traces

its development back from Descartes then splits accordingly to Kant through Brouwer on

one path and Leibniz to Godel on another path. For the formalist, the consistency of a

postulate system may be established without production of a model. When undertaking to

axiomatize all mathematics, the formalist assumes that his axioms are postulates that

determine the undefined symbols that he uses. In establishing set theory, Zermelo hoped

that clear and explicit axioms would clarify the meaning of a set and its properties.

Zermelo aimed to limit the size of possible sets. He had no philosophical basis, but only

sought to avoid contradictions.

We have shown that logicism attempts to build all mathematics from rules of

logic. Nominalism, as a spin-off of logicism, avoids naming objects as universals.

Intuitionists view mathematics as not bound to the rules of logic, but being a creative,

50 Kneale & Kneale. Development of Logic, p.689. 42

man-made enterprise. Formalists are concerned with using finite procedures to establish proof methods, leading to a rich theory of metamathematics, analyzing the syntax of math proofs. And set theorists express numbers as sets of objects and build modern mathematics from such axioms and numbers. In the next chapter, we will look more closely into these five schools to identify Platonic ideas.

CHAPTER THREE: MATHEMATICAL PLATONISM IN EACH SCHOOL

In this last chapter, we will explore logic from the classical viewpoint of Plato’s view of logic as logos. Then we will see logos in relation to forms and knowledge. Then we will look at some of the competing views of modern Platonists’ interpretation of what

Plato means by logos. We will examine what is known as mathematical Platonism, its methodology, and show its existence in each school of mathematical thought.

Mathematical Platonism is a form of realism that, it is argued here, is inherent in all of the schools of modern mathematical thought. Its origins lie in Plato’s beliefs in a

‘World of Ideas’ and that the everyday world imperfectly approximates an unchanging, ultimate reality, as well as with the ancient Pythagoreans who believed that the world was generated by numbers. According to Plato’s theories of ideas (or forms), the objects we talk about do exist independently of our thoughts and knowledge of them. Such interpretations are held for mathematical objects, to varying degrees. Throughout the history of foundations of mathematics there has been debate among philosophers as to what actually constitutes Platonism. In this thesis, we assume that the main school of modern logical thought has a common Platonic kernel but appears distinct only on the 43

surface. This Platonic interpretation of the foundations of mathematics offers a fresh perspective in viewing mathematics proper.

The questions of Mathematical Platonism include: where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one that is occupied by the mathematical entities?

How can we gain access to this separate world and discover truths about the entities?

Mathematical Platonism is ‘realism’ since it holds the view that mathematical entities exist independently of the human mind that humans do not invent mathematics, but rather discover it. So there is really one sort of mathematics that can be discovered.

Mathematical entities are abstract, have no spatio-temporal or causal properties, and are eternal and unchanging. Most people have a Platonic conception of numbers.

Platonism also postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. And mathematicians see themselves as discoverers of naturally occurring objects. Mathematical Platonists, such as Godel, believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles of set theory could be directly seen to be true, but some conjectures, like the continuum hypothesis, might be unprovable just on the basis of such principles.

Mathematical Platonism suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.

Within mathematical Platonism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them. In so- 44

called Platonism, dialogues are the source of the theory of ideas (eternal prototypes), and application to natural phenomena (fact perceptible by senses). The things that we see

(phenomena of nature) are in the earthly realm, merely copies (idols) of the prototypes

(paradigms). Ideas and noumena exist in the heavenly realm; real and perfect, while the phenomena is the unreal and imperfect. Thus, the purpose of philosophy is to enable the mind to rise above contemplation of visible copies of ideas (idols) and advance to knowledge of ideas.

Part One: Logicism as Platonism

Logicism in its development has elements of Platonism in it, especially in its aims

to situate the rules of clear thinking with number theory, as well as in Russell’s theory of

types. Godel considered Russell’s type theory as being ‘Platonic’ because it claimed the

existence of ‘classes’without a logical proof. Godel considered himself a Platonist

because the results of his incompleteness theorem led to those ubiquitous—true but

unprovable—statements about numbers, showing him their Platonic essences.

Logicism

Logicism claims that mathematics is reducible to and a part of logic, and that

mathematics can be known a priori, but that our knowledge of mathematics is just part of

our knowledge of logic in general. Mathematics is thus analytic, not requiring any special

faculty of mathematical intuition. So logic is the proper foundation of mathematics and

all mathematical statements are necessary logical truths. Carnap (1931) presents the

logicist thesis in two parts: (1) the concepts of mathematics can be derived from logical 45

concepts through explicit definitions; (2) the theorems of mathematics can be derived from logical axioms through purely logical deduction.

Logicists use Platonic notions when they add new axioms to logic to reformulate old ‘unworkable ones’ such as Russell’s axiom of reducibility that replaced Frege’s basic law V and the axiom of comprehension.51 Logicists use abstraction principles such as

Hume’s principle that the number of objects falling under the concept F equals the

number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence. If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense,

logicism can be seen as shifting questions about the philosophy of mathematics to

questions about logic without fully answering them.

Logicism’s main business is to define the notions of pure mathematics, especially

those considered as fundamental and irreducible, to pure logic in order to prove theorems of pure mathematics, starting with postulates, nothing but the basic principles of logic, and applying only logical methods of proof. This tradition along with Frege holds the view of “the possibility of deriving, the basic notions and the fundamental principles of pure mathematics.”52 He emphasizes that “logical justification does not depend on

psychological data, and that a logical justification of pure mathematics does not

necessarily need an intuitive construction of mathematical entities.”53 Also,

51 See Frege’s Foundations of Arithmetic, p.45. 52 Kneale & Kneale, Development of Logic, p.403. 53 Ibid., p.413. 46

“mathematical symbols have a definite meaning of their own and need not be devoid of meaning.”54 Logicism stems from Plato, to Neo-Platonists such as Proclus, down to

Leibniz, who, influencing Dedekind, influenced Frege, down to Russell, who refined

Frege’s logic by using Peano’s symbolism to describe such mathematics.

Type Theory

In the logicist PM-system, Russell and Whitehead uses a theory of types to avoid

the contradictions of Frege by using a Platonic notion known as an axiom of reducibility.

Accordingly, the definition of a number is a class, and PM builds the natural numbers N,

the real numbers R, the complex numbers C, related functions for such numbers systems

and analysis.

The axiom of reducibility and the theory of types in general are Platonic notions

because for propositions P: ‘Every P of a higher type is equivalent to one of 1st order.’

And for propositional functions f: ‘Every f such that f(x) or f (x,y) is coextensive with

f(x) and the same number or variables whatever the variable type’ (propositional

functions are used to justify mathematical induction in PM). Logicism is Platonic because

it corrects and adds new axioms (such as the axiom of reducibility). According to the type

theory of PM, “statements must be clearly distinguished by type”55; for example, in

solving the liar paradox, Russell states “The statement I am lying means ‘∃ proposition p

such that I am affirming p and it is false.’ So if p is of the nth order, then the assertion

about p itself is of a higher order.”56

54 Ibid., p.426. 55 Ibid., p.527. 56 Ibid., p.530. 47

So the (imperfect) phenomena falls short of ideas; it partakes in an imitated participation of ideas. Thus, the nous is the world soul that acts as the mediator between ideas and the natural phenomena; it is the cause of life, motion, order, and knowledge in the universe. The idea retains its unity, unchangeableness, and perfections. It is the element of reality in a thing, the concept by which a thing is known, thus implying the element of reality in a thing, implying that the concrete thing itself in unreal. Thus, the equivalence of the real/unreal and the ideas/natural phenomena is simply an application of the doctrine of opposites, also stated as heavenly realm/this world = being/not being.

Part Two: Platonic Nominalism

Nominalism

In mathematics the term ‘nominal’ is synonymous with the term ‘data.’ Data is the

observations gathered from an experiment, survey, or observational study. Often the data

are a randomly selected sample from an underlying population. Numerical data are

discrete if the underlying population is finite or countably infinite and are continuous if

the underlying interval forms an interval, finite of infinite.57 Data are descriptive and

have no natural order. Data specifying country of origin, type of vehicle, or subject studied, for example, are nominal.

The nominalists have a methodology and goal that are Platonic in essence; the only difference between them and traditional Platonists resides in their interpretations of mathematical objects. So Platonic Nominalism is concerned with the problem of finding out which objects are denoted by universal terms. Thus, each singular term denotes a

57 Oxford’s Concise Dictionary of Mathematics. 48

certain substance or substantial unity. And all mathematical proof methods are valid and sound. In this case mathematics is viewed as an application of logic; Platonic nominalism does not make much distinction between pure and applied logic. Techniques of nominalist logic have been used and developed by such logicians such as Godel, Quine, and Tarski.

Naming Universals

The basis for Platonic nominalism is a different interpretation of Frege and

Cantor’s logical construction of an axiom of comprehension (AC), used to describe a number. ‘AC: The mathematical entities which share a certain property constitute a set of which they are the elements and which is uniquely determined by a characteristic property; Each set is a mathematical entity and thus may appear in its turn as an element of a set; Two sets which contain the same elements are identical.’

AC is Platonic because sets appear first as multitudes of mathematical entities sharing a certain property. Thus, turning a multitude into a unity is a compression. This compression has been the source of many paradoxes in logic and set theory.

To remedy such contradictions, nominalists have explored the historically Platonic distinction of extension and intension, with extension referring to denotation and intension referring to sense. “The distinction is indispensable as certain qualities may happen to inhere ‘by accident’ in one and the same individual object.”58 For example,

‘the set of all morning stars’ and ‘the set of all evening stars’ have the same extension but

58 Beth, Foundations of Mathematics, p.466. 49

different intensions. The same could be said for mathematical expressions like Ex(x=2+2) and Ex(x =2×2).

Nominalists have also attempted to revise the comprehension axiom in the following manner: ‘Revised AC: (i) The mathematical entities which share a certain property constitute a class of which they are the elements and which is uniquely determined by the characteristic property; (ii) Classes satisfying such and such conditions can be composed and are called sets; each set is a mathematical entity and hence may appear as an element of a class; (iii) Two classes which contain the same elements are identical.’

Various systems of constructed logic are characterized by choosing conditions of compression by (ii). But regardless of how the axiom of comprehension is revised, the same impredicative definitions (in the original axiom) occur in all subsequent revisions.

According to Quine, realism, conceptualism and nominalism are found in all mathematics. By ‘realism’ he means apparently a willingness to assume the existence of classes and classes of classes whenever our symbolism suggests the possibility of such assumptions, by ‘conceptualism’ a policy of admitting the existence of classes only when we are in a position to establish their constitution, and by ‘nominalism’ an attempt to dispense with any assumption about the existence of classes. In practice the three attitudes are supposed to show themselves in greater or lesser willingness to apply quantifiers to class variables.59

59 Kneale & Kneale, Development of Logic, p.626. 50

Platonic Nominalism is sometimes noted in modern literature as empiricism, or naturalism. It denies that mathematics can be known a priori at all. It says that we discover mathematical facts by empirical research, just like facts in any of the other sciences, because it makes statements like ‘2 + 2 = 4’ come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet. For the empiricist, physics becomes the outlet to show and prove the existence of mathematrical entities. Such Platonic Nominalism is primarily supported by the argument that mathematics is indispensable to all empirical sciences. Thus, to believe in the phenomena of reality described by the sciences, we ought also to believe in the reality of those entities required for this description.

Since nature needs to talk about numbers in offering any of its explanations, numbers must exist. In keeping with this view, Quine naturalistically argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of some of its distinctness from the other sciences.

If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Quine’s case, the empirical justification comes indirectly through the coherence of our scientific theory as a whole.

Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.60

60 Ibid., p.672. 51

Part Three: Platonic Formalism

Hilbert’s Program

Formalism holds that mathematical statements may be thought of as statements

about the consequences of certain string manipulation rules. For example, in the ‘game’

of Euclidean geometry (which is seen as consisting of some strings called ‘axioms,’ and

some ‘rules of inference’ to generate new strings from given ones), one can prove that the

Pythagorean theorem holds (that is, one can generate the string corresponding to the

Pythagorean theorem). According to formalism, mathematical truths are not about

numbers and sets and triangles and the like—in fact, they are not ‘about’ anything at all.

Platonic formalism is often known as deductivism. In deductivism, the

Pythagorean theorem is not an absolute truth, but a relative one: if one assigns meaning to

the strings in such a way that the rules of the game become true (i.e., true statements are

assigned to the axioms and the rules of inference are truth-preserving), then one has to

accept the theorem, or, rather, the interpretation one has given it must be a true statement.

The same is held to be true for all other mathematical statements. Thus, Platonic

formalism means that there exists some interpretation in which the rules of the game

hold. But it does allow the working mathematician to continue in his or her work and

leave such problems to the philosopher or scientist. Many formalists would say that in

practice, the axiom systems to be studied will be suggested by the demands of science or

other areas of mathematics.

A major early proponent of Platonic formalism was David Hilbert, whose

program was intended to establish correspondence between semantic truth and syntactic 52

provability in first order logic and have no contradictions derived from the system axiomatization of all of mathematics. Such consistency of mathematical systems could be reached from the assumption that ‘finitary arithmetic’ (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent.

Metamathematics

The Platonic forms of ‘model’ and ‘interpretation’ are attributed to Hilbert’s formalism. To metamathematically distinguish between a model and an interpretation, we begin with key concepts of:

(i) Inclusive or: (φ ∨ψ) is a formula assigned truth value T under an interpretation of ℑ if and only if φ is assigned a truth value T under ℑ of φ is assigned the truth value T under

ℑ.

(ii) Let S be a symbol set, composed of S-terms, S-formulas or S-interpretations.

ℑ = (U,β), ∀t, ℑ(t) from domain A. ℑ ╞ t 1=t 2 iff ℑ(t1 ) = ℑ(t2).

(iii) Relation: Let ℑ = (U,β),be all interpretations such that ℑ╞ ∃xφ if and only if there is

a an a∈A such that ℑ b ╞ φ.

Thus, to make precise the notion of a formula being true under an interpretation we use

the satisfaction relation. Fix a symbol set S, a simple formal system. For a set Ф of S-

formulas ℑ╞ Ф. ‘ℑ is a model of Ф.’61

Since Hilbert’s program of formalism deals mainly with the syntax of the formal system of first order logic, we list the basic components of a first order formal system.

61 Ebbinghaus, Flum, and Thomas, Mathematical Logic, pp.32-35. 53

A = alphabet (symbols)

= words = finite sequences of symbols, strings.

A* = the set of all strings (words) over A terms = formulas.

S-Terms are composed of variables, constants. If a string are S-terms and f is a n-ary

function symbol in S, then f(t 1...t n ) is also a S-term.

T S = set of all S-terms.

Thus, an interpretation is the metamathematical way of expressing a form, or formula,

which is a model of a theory. Model theory is a rich area of study in metamathematics.

Part Four: Platonic Intuitionism

Origins of Intuitionism

Intuitionism is the mathematical philosophy which states that it is not possible to

penetrate the foundations of mathematics without paying attention to conditions under

which mental activity takes place. Mathematics should be developed independently of

preconceived ideas of the nature of mathematical entities or of mathematical activity, as

well as mathematical research. Brouwer first articulated this intuitionist formulation,

noting that logic is treated as a special branch of mathematical investigation.62

Intuitionism is inherited from Kant; specifically, the notion that math intuition implies

space and time as forms of our sensibilities. Intuitionists treat logic and physics as

synonymous, since physics’ tasks of interpreting nature had been used as analogy for the

business of logic to obtain ‘truths’ by many philosophers, including Wittgenstein, Godel,

62 Beth, Foundations of Mathematics, p.410. 54

and Kant. Intuitionists say of math that ‘there are no non-experienced mathematical truths.’ They seek “to reconstruct the foggy portions of mathematics in accordance with

Platonic-Kantian concepts of being, becoming, intuition, and knowledge. Mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.”63 Kroenecker said: “The natural numbers come from God, everything

else is man’s work.” These Platonists rejected the usefulness of formalized logic of any

sort for mathematics, and postulated an intuitionistic logic, different from the classical

Aristotelian logic; that logic does not contain the law of the excluded middle and

therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in

most intuitionistic set theories, though in some versions it is accepted.

The Totality of Integers

Weak arithmetical Platonism can illustrate intuitionistic mathematics. Consider the totality of integers (Z). If P is a predicate of Z, then either P is true of every integer or there is at least one exception. Thus, disjunction (implying the law of the excluded middle) is continually applied in analysis. For example, in the real numbers R: ‘for every real number a and b given by the convergent series, either a = b, or a ≤ b, or b ≤ a.’ That

is, for some ordered integers p,n, where p ≤ n; f(n) is defined for very integer n where p ≤

n ≤ q:64

(a) ∑f(n) ↔ f(p) +f(p +1)+ . . . +f(q), where n is a dummy index that can be placed by

any other symbol without change in meaning.

(b) Given a sequence{xn}, the expression ∑xn = x1 +,x2+ . . . +xn+ . . . is an infinite

63 Parsons, “Platonism and Mathematical Intuition,” p.542. 64 Rudin, Principles of Mathematical Analysis, p.8. 55

series with the terms x1, x2, .., xn, if the sequence s1=x, s2=x1+x2.., sn = x1+x2+ . . .

+xn +. . ., of partial sums of terms of ∑xn = x1 +, x2+ . . . +xn+ . . . , then ∑xn = x1 +, x2+ . . . +xn+ . . . is convergent with sum s. Thus, for two real numbers a and b; ∑xn ↔ a

= b, ∑xn ↔ a ≤ b, where the sequence is increasing, or ∑xn ↔ b ≤ a, the sequence is decreasing.

A sequence of Q+ either comes as close as you please to zero or E a positive rational such the number is smaller than all members of the sequence. {xn} → ∑xn = x1

+ x2+ . . . +xn+ . . .

So ∀a > 0∈R, ∃b > 0 such that |xn-A| < a iff 0 < |x-A| < b, or (∀ε > 0)(a∈R)(b > a) so |xn -a| = |xn| = 1/a < ε, ∀a ≥ ai.

As a disjunction, (i) (∃a, b∈R)(lim∑xn = c), either a = b or a < b or b < a

(ii) {xn}∈Q+, either {xn} → 0 or ∃q ≤ {x1 + x2 + . . . + xn}, q < xi, q < x2 . . . q ≤ xn.

Trivial disjunctions are subtle assumptions. Analysis is not content with such

Platonism; it reflects more with sets of numbers, sequences of numbers, and functions.

Analysis also abstracts from the possibility of giving definitions of sets, sequences, and functions. Thus, in a quasi-combinatorial manner from infinite, to finite, and vice versa, given a function (in N), where F(a) = b, there exists nⁿ functions, each obtained by n determinations.

In the infinite case, f ∞ (n) are functions engendered by ∞ of independent

determinations which assign to each m ∈ Z an integer n ∈ Z, such that we reason on totality of such functions. Z is the set of the rest of infinitely many independent sets deciding if every natural number should be included, and the totality of these sets. 56

Sequences and sets of R have the same analogy as Z, the constructive definition of specific functions, sequences, and sets is the only way to pick out objects that exist independently of, and prior to, construction.

The axiom of choice is also an application of ‘quasi-combinatorial concepts’ (read

Platonic) in questions used in real analysis. Let M1, M2,..., be a sequence of nonempty sets of R. Then there exists sequence m1,m2,..., such that for every natural number n, an∈Mn. Thus, principles imply demanding, effective construction of sequence.

Poincare’s impredicative definition of R, ‘(∀r ∈ R)(Have property P or ∃r ∈ R such that r has property P)’ depends on the assumption of the existence of ‘totality of sequences of integers’ since r ∈ R is represented by decimal functions (a special kind of sequence of integers).

Real Numbers

The real numbers defined as ‘the union of rationals and irrationals’ are used to show the fundamental theorem of real numbers that ‘a bounded set of S⊄R always has a least upper bound.’65 That is, for S ordered, E ⊄ S, E bounded above, there is a β ∈ S such that x ≤ β, ∀x ∈ E, E is bounded above, β = upper bound of E.

Fundamental Theorem (1.11)66: Suppose S is an ordered set with the least upper

bound property, B ⊄ S, B ≠ 0 and B is bounded below. Let L = {all lower bounds of B},

then ∃α = supL ∈ S, α = inf B, inf B = S. Proof: Since B is bounded below, this implies

that L ≠∅; Since L includes all y ∈ S such that y ≤ x, ∀x ∈B, this implies that ∀x ∈ B is

65 Bernays, “Platonism in Mathematics,” p.5. 66 See Rudin, Principles of Mathematical Analysis, p.11. 57

an upper bound of L. Thus, L is bounded above. Assumption of S implies that L has a supremum in S, call it α. If γ < α, then γ is not upper bound of L, thus γ ∈ B. Therefore,

α.≤ x ∀x ∈ B, so α ∈L. If α < β, then β ∉ L since α is an upper bound of L. Thus, α ∈ L but β ∉ L if β > α. (That is, α is a lower bound of B, but β is not if β > α. This means that

α = inf B.)

Part Five: Platonic Ideas in Modern Set Theory

Set theory, as initially elaborated by Cantor, was considered by critics at the time

as Platonic and was heavily criticized by mathematicians of the day.

Cantor’s Theorem

In Cantor’s theorem, Platonic conceptions extend beyond real analysis by

iterating use of a quasi-combinatorial concept of function and adding methods of

collection, the well-known method of set theory. Thus, Platonic conception of analysis and set theory is applied to algebra and topology.

Let A = set of all sequences whose elements are 0 and 1. Then A is uncountable. A

= [1,0,0,1,0,1,1,1,0,111,0,...].

Proof: Let E be a countable subset of A, Let E include sequences s1,s2,s3,...

Construct s: if nth digit in sn =1, let nth digit of s =1, vice versa. Then sequences differ

from all members in E in at least one place. Thus, S ∉ E. But S ∈ A, so E ⊄ A. Thus,

every countable subset E ∉ A is proper, thus A is uncountable (or A< A, which is false).

R is Uncountable

Cantor showed that there are infinite sets that cannot be put into one to one

correspondence with the infinite set of natural numbers. (This logic was later used by 58

Godel in showing the incompleteness formal systems.) Proof: Suppose R is countable: R infinite implies R and N have the same cardinality. (R~N) Let there exist a bijective f from N onto R (f: N → R). We will show a contradiction that there is a real number x that is not in the image of f. Define a real number x (x ∈ R) such that x differs from f(1) in the first decimal place;

f(2) in the second decimal place,

f(n) in at least one decimal place, x ≠ f(n),∀ n∈N.

So, using Cantor’s diagonalization construction, let f(1) = a0.a1a2a3 . . . f(2) = b0.b1b2b3. . . f(3) = c0.c1c2c3 . . .

. . . . . * . . .

...... * . .

...... * . f(n)= z0.z1z2z3 . . . zn where f(n) ∈ R; (a0,b0,c0,...,z0) ∈ Z and ai,bi,ci,...,zi represent digits of decimal numbers; we form the decimal a0.b1c2… zn to have same nth place as f(n). So x is the anti-diagonal element, and it differs from diagonal decimal in every decimal place. Thus, x differs from every number of the form f(n).

About Platonism in mathematics, recall that Bernays says, “The application is so widespread that it is not an exaggeration to say that Platonism reigns today in 59

mathematics.”67 Platonism as a principle has been criticized, reinforced by paradoxes in

set theory, which refutes ‘extreme’ Platonism. Recall, restricted Platonism implies the

ideal projection of a domain of thought. As conceptual realism, restricted Platonism

posits the existence of a world of ideal objects, contrary for all objects and relations of math. This form of conceptual realism was shown untenable by Russell-Zermelo’s paradox.

According to Russell’s paradox, T = {S ∈U | S ∈ S}, where U = ‘universe set,’ T

= ‘truth set’ of ‘S ∉ S.’ Is T ∈ S? IF T ∈ T, then ‘T ∉ T.’ Russell’s paradox shows the impossibility of extreme Platonism in set theory (objects and relations in math). The paradox shows the impossibility of (i) the idea of totality of math objects and general concept of sets and functions [totality = domain of elements for sets and functions] (ii) totality = domain of arguments and values for functions.

To escape paradoxes by eliminating Platonism, various thinkers have told us to replace ‘quasi-combinatorial concepts’ with ‘constructive concepts’ of set, sequence, and function and to reject the idea of ‘infinity of independent determination.’ So the infinite sequence of a decimal fraction can be given only by arithmetic laws. The continuum is a set of elements defined by arithmetic laws.

The continuum hypothesis asks the question, are there any cardinalities between Q and R? That is: ‘(∃T ∈ R)(Q < T and T < R)’ and ‘¬(∃T ∈ R)Q < T and T < R).’68 So the questions are raised, are infinite sequence and decimal functions given only by N-laws?

67 Bernays, “Platonism in Mathematics,” p.5. 68 ‘<’ = strict ordering. Fendel & Resek, Foundations of Mathematics, p.310. 60

Is the continuum, as a set of elements defined by N-laws? The complete arithmetization of analysis not by usual methods, is not completely reducible to Z and logic.

Thus, If (∀x∈R)(x: = defined by N-law), then [the idea of totality of R not indispensable] (the axiom of choice is not evident) if ⌐∃ (auxiliary assumptions) then certain conclusions are unfounded. For example, algebraic numbers are defined, but transcendental numbers are not algebraic.

Axioms of Set Theory

The Zermelo Fraenkel Axioms of Set Theory are as follows:

a. Extension A=B iff x∈A and x∈B

b. Empty ∃φ such that x ∉ φ

c. Pairing (∀X, Y⊄S)(∃ T such that X ∩ Y ∈ T)

d. Union (∀X ∈ S)( T = S ) U n e. Separation (∀X ⊄ S)(∀ Y∈X)(P(Y) → {∃Y∈ X | P(Y)}

f. Replacement (∀X ∈S)(∀ f: X → Im(f), Im(f) ⊄ S)

g. Infinity (∃X ⊄S)(∅∈ X)

h. Regularity (∀X ≠ ∅, X ⊄ S)(∃ Y ⊄ X)(Y ∩ X =∅)

Objects of analysis and set theory are viewed as elements of a totality such that, for each property expressible using notions of theory, it is objectively determinate whether some set E or ~E is an element of the totality which possesses this property.

When an object is viewed as itself, this is a form of mathematical Platonism. “The value of Plato inspired mathematical concepts is that they furnish models of abstract 61

imagination. These stand out by their simplicity and logical strength. They form representations which extrapolate from certain regions of experience and intuition.”69

Since it is possible to arithmetize geometry and physics, Cantor’s and Bernays’ focus on

Platonism in the arithmetic of analysis and set theory.

In Cantor’s set theory, especially in light of the philosophical implication,

Platonism shows itself in two ways. First, sets can be considered as domains or multitudes (in Fraenkel-Skolem’s system and in von Neumann-Bernays’ systems of set theory). Secondly, these sets can be considered as arbitrary multitudes, to answer the questions of whether and how they are defined. In the first case, we must accept that there exist sets that we can never hope to find definitions for, in terms of the specific properties of their elements. Even if a definition is found for such a set, it may involve impredicative properties. And this second Platonic interpretation of set theory, involving impredicative properties (that there are definitions that do not define a set) is a necessary interpretation of set theory. Set theory is considered as a mathematical school of actual infinity, as opposed to the potential infinity attributed to Aristotle, from Eudoxus’ theory of proportions. In the modern era, Cantor first offered a satisfactory positive solution to the countability of the infinite by two methods: counting by cardinal numbers and counting by ordinal numbers.

69 Bernays, “Platonism in Mathematics.” 62

R and Decimal Numbers

Let x be a positive real number and let n0 be the largest integer where n0 ≤ x (the existence of n0 in Z depends on the Archimedian property of R). Choose n0, n1, . . ., nk-

n1 n k 1, let nk be the largest integer such that n0 + + . . . = ≤ x (k = 0,1,2,...). Then x = 10 10k

sup E.

The decimal expansion of x is n0.n1n2n3... Thus, every infinite decimal has the form

n0.n1n2n3...,

n1 n k the set E of numbers = n0 + + . . . = is bounded above and n0.n1n2n3... is the 10 10k

decimal expansion of supE.

Godel’s Platonism

Godel was a self-proclaimed Platonist who made contributions to virtually every

school of mathematical thought. His Platonism represents a unique combination of all the

Platonic schools of mathematics. His completeness theorems, incompleteness theorems,

and work in set theory and Platonism, provide the best examples of how Platonism is

inherent in modern logico-mathematical thought. Although both Godel and Brouwer

inherit intuitionism from Kant, Godel’s philosophy articulates more clearly to Kant’s

notion of ‘a theory of reason.’ We will show how Godel innovated Kantian-Brouwerian

intuitionism to develop a more robust realism, taking into account certain Leibnizian and

Cartesian notions of intuitionism as well as taking into account elements of modern

physics. 63

Given a formal language Φ, where φ1, φ2, ..., φn are formulas constructed from elementary signs that form fundamental vocabulary, Φ has primitive formulas, called axioms, that can be used to construct new formulas φi. Godel showed that it is possible to assign a unique number to each elementary sign, formula and proof of Φ.

One of the limitations of a formal theory Φ is that ‘if Φ is consistent, then Φ is incomplete.’ Godel constructed a demonstrable formula θ that says, ‘there is at least one formula for which no proposed sequence of formulas with Godel’s number of θ constitutes a proof in Φ.’70

CONCLUSIONS

Platonism Defined

We have shown that key Platonic forms and ideas are expressed in modern logic

as universal and existential quantifiers.To properly identify Platonic notions in modern

mathematics means tracing the development through time and space of such concepts as

knowledge, maieutics induction, inference, dialectics, logos, science, mathematics, truth,

philosophy, sense, intelligence, and categories. From such analysis we show how these

concepts are used to construct a method to define and shape such philosophical concepts

in what we now call mathematical logic, by showing the connection of Plato’s ‘idea’ and

the ‘universal quantifier’ ∀ used in mathematical logic. This operator, along with its

negation, the existential quantifier, is used to bind variables to quantify some condition in

a given universe of discourse. These universal and existential forms are categories of

70 For more details see Nagel, Godel’s Proof, p.76. 64

being known also as genera and species, ‘idea and things,’ or ‘logoi and pragmata.’

Negation is a logical operation, symbolized by ‘⌐’ in front of a quantifiers that switches the value of the quantifier to its opposite value.

Maieutics

Since one of the objects of philosophy is to find truth, the definition and formation of general concepts via logic helps to clarify philosophy. As we recall,

induction is considered one of humanity’s oldest methods of scientific analysis. How

does this relate to mathematics and logic? First and foremost, induction has been

delineated as the most basic type of logic, evolving later into more refined logic types

resulting in inference, dialectics, and more formalized proof methods. Prior to the

symbolism of modern logic, the aim and goal of philosophy, and logic in particular, was

compared by the ancients to the analogy of delivering a baby (maieutics).

Dialectics

Dialectics took place after induction became inference. As a specific example of induction, consider the most general case of induction used in today’s formal methods: that is, so-called mathematical induction. Prior to the symbolism of modern mathematics, philosophical induction served the same purpose of generalizing from a special basis.

Before modern math symbolically distinguished induction (1, n, n+1) in the natural numbers (N), this distribution of objects into general and specifics was known as

‘dialectics.’ Dialectics, as a method of maieutics, took place mainly through the art of asking questions (irony). Here we explore the epistemological and ontological 65

foundations of metaphysical dialectics, rooted in Plato’s original philosophy and how they relate to Godel’s later math philosophy.

Provability

To establish ‘provability’ of a well formed formula (y), y must be a free variable, and x, the actual proof of the formula, must be bounded by an existential quantifier.

Epistemologically, when we view mathematics as a theory of knowledge; in the words of

Klein, mathematics has lost its certainty. Ontologically, in asking the question, “What is mathematics?” or “What ‘be’ mathematics?” we can certainly state that math is not simply formal methods. In philosophy as a method, initial elements are general notions with a well-defined existence. This leads to constructed thinking, a formation of general concepts. In defining objects, the method of philosophy expresses essence, divides genus into species, and species into subspecies, stopping when division is no longer possible and the individual is reached.

Logicism

Logicism is the school of mathematical thought that maintains that mathematics is reducible to logic. According to Leibniz, “A truth is necessary when the opposite implies a contradiction, and when it is not necessary then it is called contingent.”71 Frege developed the symbolic notation used today in modern logic and is credited with transforming logic from a rhetorical art to a deductive science by the means of his symbolic notation. Acknowledging Frege’s contribution to the foundations of mathematics, Russell (and Whitehead) attempted to resolve Frege’s flawed Axiom 5 via

71 Klein, Mathematics: Loss of Certainty, p.60. 66

the Principia Mathematica (PM). Russell did know, of course, that Peano had derived the

real numbers from axioms about the whole numbers, and he was also aware that Hilbert

had given a set of axioms for the entire real numbers. Russell’s real concern was that

postulation of, say, 10 or 15 axioms about number does not ensure the consistency and

truth of the axioms. Russell could not distinguish between Euclidean and non-Euclidean

or affirm which was the truth. But in his Essay in the Foundations of Geometry (1898),

he did manage to find some mathematical laws, such as that physical space must be

homogeneous (possess the same properties everywhere), which he then believed were

physical truths.

The logistic theory of types requires that statements be carefully distinguished by

type. Consider that the least upper bound (l.u.b.) is defined as the smallest of all upper

bounds, so the l.u.b. is defined in terms of the set of real numbers, R. Also, through the

axiom of reducibility in PM, in every property p of higher type, p is equivalent to one of

first order. The PM system was then refuted and shown to be contradictory by Godel’s

incompleteness theorems. Logicism in its development has elements of Platonism in it,

especially in its aims to situate the rules of clear thinking with number theory, as well as

in Russell’s theory of types.

Critics of logicism: The axiom of reducibility seems arbitrary, but no proof of its

falsity has been shown. It is a happy accident, but not a logical necessity. A geometric

criticism of logicism is that, ‘by using analytic geometry one could do so’ (develop

geometry in PM). A philosophical critique of logicism says that ‘if the logistic view is

correct, then all math is a purely formal, logico-deductive science whose theorems follow 67

from laws of thought.’ Formalization apparently does not represent math in any real sense. Nominalism is a spin off of logicism, and set theory interpreting and supposing every subset of a mathematical universe is definable in constructive, liberal means.

Intuitionism

The school of intuitionism traces its development back from Descartes to Kant through Brouwer and also to a lesser extent from Descartes to Leibniz to Godel. Godel’s intuitionism, which is a realism of abstract math objects extended to set theory, type theory, and the idea of concepts as classes or categories, is equivalent to Descartes’

‘intuitio’ (intuitive knowledge), as well as the relations of objects. In intuitionist set theory, there is no law of the excluded middle, identity, non-contradiction, or syllogism.

Brouwer conceived of mathematics as a thinking process, a mental construction that builds its own universe, independent of experience and restricted only insofar as it must be based upon the fundamental mathematical intuition.

Mathematical ideas are more deeply imbedded in the human mind than in language since the world of mathematical intuition is opposed to the world of perceptions. Mathematics is not bound to respect the rules of logic. In the realm of logic, there are some clear intuitively acceptable logical principles or procedures that can be used to assert new theorems from old ones. The intuitionist Herman Weyl said that non- constructive existence proofs inform the world that a treasure exists without disclosing its location. So the different types of intuitionists are those that: (1) Eliminate all set theory and form only constructed concepts. (2) Hold that: ‘For every real number in the math universe, p(r) holds if R is a class.’ That is, for every integer, z, in the math universe, p(z) 68

is a function. Existence proofs, axiom of choice, and the continuum hypothesis are non- constructible and therefore unacceptable to intuitionists. Thus, to demand ultimate reliance upon logical deduction from axioms, Brouwer says that a system of axioms must be proven consistent by using interpretations or models which are already known to be consistent. Regardless, Weyl says in 1951, “I think everybody has to accept Brouwer’s critique who wants to hold to the belief that math propositions tell the sheer truth, truth based on evidence...”72 Hence, mathematicians should abstain from talking of infinity in the manner of Cantor so that they can avoid Cantor’s peculiar puzzles. Then they might

argue and see that it was useless to detailed discussion of an assumption whose absurdity

could be seen without even referring to paradoxes. From Brouwer’s statement that “neo-

intuitionism considers the falling apart of the moments of life into qualitatively different parts, to be reunited only while remaining separated by time, as the fundamental phenomena of the human intellect,”73 we get the intuition of mathematically defined sets

as being of the betweenness which is not exhaustible by interposition of new units and therefore can not be thought of as a mere collection of units. In a Platonic sense, Brouwer rejects all connection to actual infinity and real numbers as an open interval. N is an open manifold always in growth, unfinished, and comprehended only by the law of number construction. Intuitionism shares a core part with most other forms of constructivism. The logic accepted by almost all constructive communities is the same, namely intuitionistic

logic.

72 Ibid., p.319. 73 Kneale & Kneale, Development of Logic, p.673. 69

Formalism

This school begins with David Hilbert in 1900. For Hilbert and the formalists, one cannot deduce math from logic because math is its own autonomous discipline, not a consequence of logic. The formalist believes that in the real world only a finite number of objects exist and that matter is composed of a finite number of elements. Formalist logic is Aristotle’s logic. Hilbert proposed that a special logic be used free from all objections.

Like the logicist school of mathematics, the intuitionists objected to the formalist concept of existence. Hilbert produced a set of axioms for Euclidean geometry and, on foundations of logic and math, suggested a new program for eliminating newly discovered paradoxes by axiomatizing logic, theory, analysis, and set theory. For the formalist, the consistency of a postulate system may be established without production of a model. Formalist math is not open to intuitionistic objections. Constructive mathematics is beside, not interior to, classical math, according to the intuitionists.

Set Theory

Set theory gradually gained adherents to their explicit program. Though set theory is incorporated in the logicist approach to math, set theory prefers to approach math directly through axioms. In the axioms, there are infinite sets and operations like the union of sets and formation of subsets, as well as the axiom of choice. ZF axioms for set theory can build all mathematics. The Bourbaki school renounces logicism, formalism, and intuitionism. Cantor’s original theory of sets was a logical theory of natural numbers prior to Frege. Cantor thought his work was a discovery of laws not made by man. 70

Knowledge

Traditionally, knowledge was considered the combination of belief and logos.

Concerning logos, Plato says,

Whenever then anyone gets hold of the true notion of everything without a logos, his soul thinks truly of it; but he does not know it, for if one cannot give and receive a logos of anything, one has no knowledge of that thing; but when he has acquired a logos, then all things are realized and he is fully equipped for knowledge.74

‘Logos’ is classically discussed by Plato but barely understood by the modern Platonists.

Even though considered a weak claim, forms imply verbs, adjectives, nouns, and

constitute names of simple nameables. Refuting the old critique of the theory of ideas

would say that knowledge is related to perception, which is Plato’s concern. Citing the

interpretations by two Platonists provides evidence that most of these arguments of what

constitutes Platonism are never settled. Ryles’ interpretation of theory of ideas states that

proper names imply particulars, and that other substantiatives—adjectives, verbs,

prepositions—are names of forms. Meno’s worry is to a special kind of ‘virtue.’ In the

Symposium Plato says, “After long training, the soul sees beauty in itself.”75 Knowledge

Plato is usually concerned with is not ordinary. It seems apparent that ‘knowledge by acquaintance’ is modern mathematics and logic.

Logos

Etymologically, ‘logos’ comes from the verb ‘legein,’ which means ‘to tell, state,

say, discourse, a statement, or hypothesis.’ ‘Logikos’ is not equal to ‘rationally.’ New interpretations of Platonism include the basis that forms are simple nameables known

74 Theaetetus: (202), cited in Allen’s Studies in Plato’s Metaphysics, p.62. 75 Symposium: (210e -212a) and Ryles cited in Allen’s Studies in Plato’s Metaphysics, p.65. 71

ultimately by acquaintance. It is incorrect when we talk about idea (form), but it is correct when we talk with idea (form), and logoi (pieces of talk) are necessary to display idea

(form) to us. The main point is that forms are logical predicates displayed in logoi and

not simple nameables known by acquaintance, and that Platonists put the theory of ideas

in context of language and logic.

Mathematical Platonism

Mathematical Platonism is a form of realism that is inherent in all of the schools

of modern logico-mathematical thought. The questions of mathematical Platonism

include: where and how do the mathematical entities exist, and how do we know about

them? Mathematical entities are abstract, have no spatio-temporal or causal properties,

and are eternal and unchanging. Mathematical Platonism suggests that quasi-empirical

methodology could be used to provide sufficient evidence to be able to reasonably

assume such a conjecture. Logicism is the school of mathematics that associates with the

thesis that mathematics is reducible to, and a part of, logic.

Logicists use Platonic notions when they correct and add new axioms in order to

reformulate old unworkable ones, such as the axiom of reducibility that replaced Frege’s

basic law V and the axiom of comprehension. Logicism’s main business is to define the

notions of pure mathematics, especially those considered as fundamental and irreducible,

to pure logic, to prove theorems of pure mathematics, starting with postulates, nothing

but the basic principles of logic, and applying only logical methods of proof. In the

logicist PM-system, Russell and Whitehead use a theory of types to avoid the

contradictions of Frege by using a Platonic notion known as an axiom of reducibility. The 72

axiom of reducibility and the theory of types in general are Platonic notions because for propositions, ‘every p of a higher type is equivalent to one of 1st order.’ So the (imperfect)

phenomena falls short of ideas; it partakes in an imitated participation of ideas.

A Mathematical Way of Naming the Universal

Of the differing interpretations of Platonism in modern thinking, many of the

internal criticisms include the question of naming universals. For example, Quine has a

semantic approach to natural deduction of formal systems. Nominalistic logic, according

to Quine’s interpretation of it, states the existence of logical and mathematical entities; various views have been defended by specialists in math foundations. One nominalist objection is the negation of the compression of multitudes into unity, but ‘admission of this unity as a value of certain bound variables implies attributing a substance to

compressed multitude.’ According to the logician Beth, “We are compelled to admit the

existence of sets for which we can’t hope ever to find a definition in terms of specific

properties of their elements, where for a set such a definition can be found, it may turn

out to involve impredicative properties.”76 For distinctions of nominalism, consider the

theory of syllogism. The axiom of reducibility of Russell is not logically necessary. No

one is sure if the axiom of infinity is an axiom of logic, and the use of the theory of types

is for propositional functions.

76 Beth, Foundations of Mathematics, p.408. 73

Platonic Formalism

Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. Another version of Platonic formalism is often known as deductivism. A major early proponent of Platonic formalism was David Hilbert, whose program was intended to establish correspondence between semantic truth and syntactic provability in first order logic and have no contradictions derived from the system axiomatization of all of mathematics. Platonic formalism considers certain metamathematical methods to yield intrinsically meaningful results, especially with respect to finitary arithmetic. The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above.

Objects of analysis and set theory are viewed as elements of a totality, such that for each property expressible using notions of theory, it is objectively determinate whether E or ~E is an element of the totality which possesses this property. When an object is viewed as itself, this is a form of mathematical Platonism. Weak N-Platonism includes (i) the totality of integers.

A sequence of Q+ either comes as close as possible to zero or E, a positive rational, such that the number is smaller than all members of the sequence. In the infinite case, f ∞ (n) are functions engendered by ∞ of independent determinations which assign to each m ∈ Z an n ∉ Z, such that we reason on the totality of such functions. The axiom of choice is an application of quasi-combinatorial concepts in questions used in real 74

analysis. Fundamental Theorem (1.11)77: Suppose S is an ordered set with the least upper

bound property, B ⊄ S, B ≠ 0 and B is bounded below.

Platonic Intuition

Intuition is the mathematical philosophy which states that it is not possible to

penetrate the foundations of mathematics without paying attention to conditions under

which mental activity takes place. The weak N-Platonism used to illustrate formal

mathematics can also be used to illustrate intuitionistic mathematics. A sequence of Q+

either comes as close as possible to zero or E, a positive rational, such the number is

smaller than all members of the sequence. In the infinite case, f ∞ (n) are functions

engendered by ∞ of independent determinations which assign to each m ∈ Z an n ∈ Z, such that we reason on the totality of such functions. The axiom of choice is an application of quasi-combinatorial concepts in questions used in real analysis. The real

numbers defined as the union of rationals and irrationals are used to show the fundamental theorem of real numbers that in ‘a bounded set of S ⊄ R always has a least

upper bound.’ According to intuitionism, the term ‘explicit construction’ is not cleanly

defined, and that has led to criticisms. Recall that the idea is an object (nota) of

intelligence (good, beautiful, etc.). Objects imply a division within intelligibles, not a

class of mathematical numbers and figures intermediate between ideas and sensible

things. Plato is given credit for discovering the method of analysis. Godel was a self-

proclaimed Platonist who made contributions to virtually every school of mathematical

thought. Godel is not a modest intuitionist such as Brouwer, since he inherits and

77 Rudin, Principles of Mathematical Analysis, p.13. 75

incorporates Leibniz’ notion of ‘intuitive knowledge’ (knowledge is clear, distinct, and adequate) into his mathematical intuition.

Cantor’s Theorem

In Cantor’s theorem, Platonic conceptions extend beyond real analysis by iterating use of quasi-combinatorial concepts of function and adding methods of collection, the well-known method of set theory. On contradictions in applications to

Platonism in mathematics, Bernays says that, “The application is so widespread that it is not an exaggeration to say that Platonism reigns today in mathematics.”78 According to

Russell’s paradox, T = {S ∈ U | S ∈ S}, where U = ‘universe set,’ T = ‘truth set’ of ‘S ∉

S.’ Is T ∈ S? If T ∈ T, then ‘T ∉ T.’ A critique of this analysis is how to escape paradoxes

by eliminating Platonism, even though arithmetization of analysis is not a usual method.

So the questions are raised, are infinite sequence and decimal functions given only by N-

laws? When an object is viewed as itself, this is a form of mathematical Platonism. In

Cantor’s set theory, especially in light of the philosophical implication, Platonism shows

itself in two ways. For positive real numbers, r ∈ R, when constructing decimal numbers,

we let x be a positive real number and let n0 be the largest integer where n0 ≤ x. The

existence of n0 in Z depends on the Archimedian property of R, which is itself Platonic.

Platonism is inherent in all of mathematics in the same manner that all the schools

of mathematics share their techniques and strategies with each other in order to articulate

and solve more mathematical problems. For example, even though Brouwer is an

intuitionist, his fixed point theorem was used by the formalist von Neumann to develop

78 Bernays, “Platonism in Mathematics,” p.17. 76

game theory. Years later, Nash got a Nobel Prize in Economics for his work exploiting

Brouwer’s fixed point theorem to develop Nash’s equilibrium in game theory. The application of mathematics in various fields and subdisciplines begins first with the foundational concepts and ideas to be symbolized, articulated, and constructed. These ideas exist in the realm of the intelligible, and only through intellect via logos are such

ideas reached. 77

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Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics (1953). Oxford: Blackwell, 1978.

Wittgenstein, Ludwig. Philosophical Investigations. New Jersey: Prentice Hall, 1958.

Mathematics References

Beth, E.W. The Foundations of Mathematics. Amsterdam: North Holland Publishing, 1965.

Borowski, E. J. & Borwein, J.M. Collins Web-linked Dictionary of Mathematics. New York: Harper Collins, 2002.

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Davis, Martin. Ed. The Undecidable. New York: Raven Press, 1965.

Ebbinghaus, H. D., Flum, J. and Thomas, W. Mathematical Logic. New York: Springer, 1994.

Fendel, D. and Resek, D. Foundations of Higher Mathematics: Exploration and Proof. Boston: Addison-Wesley, 1999.

Kuttler, Kenneth. Basic Analysis. New Jersey: Rinton Press, Inc., 2001.

Lawvere, F. W. and Schanuel, S.H. Conceptual Mathematics. Cambridge: Cambridge University Press, 1997.

Lay, Steven. Analysis with an Introduction to Proof. New Jersey: Prentice Hall, 2001.

Rudin, Walter. Principles of Mathematical Analysis. Singapore: McGraw Hill, 1976.

Silverman, Richard. Essential Calculations with Applications. New York: Dover, 1989.

Tymoczo, Tom and Henle, Jim. Sweet Reason: A Field Guide to Modern Logic. New York: Freeman and Co., 1995.

Wade, W. Introduction to Analysis. New Jersey: Prentice Hall, 2000.

Wallace, E. and West, S. Roads to Geometry. New Jersey: Prentice Hall 1998.

Papers

Bays, T. “On Floyd and Putnam on Wittgenstein on Godel.” The Journal of Philosophy CI.4 (April 2004): 197–210.

Bernays, P. “Platonism in Mathematics.” Accessed on 08/01/12 at http://www.phil.cmu.edu/projects/bernays/Pdf/platonism.pdf.

Findlay, J.N. “Godelian Sentences: A Non-Numerical Approach.” Mind, New Series, Vol.51, No 203 (July 1942), pp.259-265.

Floyd, J. and Putnam, H. “A Note on Wittgenstein’s ‘Notorious Paragraph’ about Godel’s Theorem.” The Journal of Philosophy 97 (2000): 624-632.

Floyd, J. “Wittgenstein 2,2,2...: The Opening Remarks on the Foundations of Mathematics.” Synthese, Vol.87, No.1 Wittgenstein, Part 1(April 1991), pp. 143- 180.

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Floyd, J. “Wittgensteins’ ‘Notorious’ Paragraph about the Gödel Theorem: Recent Discussions” in Hilary Putnam, Philosophy in an Age of Science: Physics, Mathematics, and Skepticism. Eds. Mario De Caro and David Macarthur. Cambridge, MA: Harvard University Press, 2012, pp. 458-81.

Parsons, Charles “Platonism and Mathematical Intuition in Kurt Gödel’s Thought.” The Bulletin of Symbolic Logic, Vol. 1, No. 1 (Mar., 1995), pp. 44-74.

Quine, W.V.O. “Two Dogmas of Empiricism.” The Philosophical Review 60 (1951): 20- 43.

Dissertation

Abrams, Russell Z. “Meaning Postulates.” Ph.D. dissertation, Yale University, 1974.