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**1 - 5**of**5**### 1 PYTHAGOREAN POWERS or A CHALLENGE TO PLATONISM

"... I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot ..."

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I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects

### 1 Introduction

"... The basic question of ontology is “What exists?”. The basic question of metaontology is: are there objective answers to the basic question of ontology? Here ontological realists say yes, and ..."

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The basic question of ontology is “What exists?”. The basic question of metaontology is: are there objective answers to the basic question of ontology? Here ontological realists say yes, and

### 2. The Truth and Something but the Truth (5/9/12)

"... “What is only half true is untrue. Truth cannot tolerate a more or less ” (Frege [1956]). PURITANISM ABOUT TRUTH Insisting a thing is good, period, or bad, period, is silly—the pathology of black-white thinking. Insisting it is true, period, or false, period, seems forthright and healthy minded. Par ..."

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“What is only half true is untrue. Truth cannot tolerate a more or less ” (Frege [1956]). PURITANISM ABOUT TRUTH Insisting a thing is good, period, or bad, period, is silly—the pathology of black-white thinking. Insisting it is true, period, or false, period, seems forthright and healthy minded. Partial truth makes us uncomfortable. It seems sneaky, unclean... The notion HAS been used in some bad causes. To downgrade truths—part of the truth can be at most partly true (Bradley). Also, to upgrade falsehoods—they have to get something right to be in the reality-representing business at all (Joachim, inspired by Plato). They’re wrong, or it’s a different notion. Partial truth for us is truth of a part. For S to be less than the whole truth does not at all suggest that only a part of S can be true. Falsehoods need not have true parts. Everything is shot through with orgones is completely false. Something important happening in 1066 cannot be the truth in Columbus discovered America in 1066, because it isn’t part of it, or even implied by it. Truth-puritanism is wrong, but so what? Why utter falsehoods with true bits in them, rather than just the true bits? “[A] rule of thinking which would absolutely prevent me from acknowledging certain kinds of truth, if those kinds of truth were really there, would be an

### Field on the Contingency of Mathematical Objects

"... Any sceptic about abstract mathematical entities has somehow to come to terms with two facts: that many of the laws of physical science are formulated in ways which involve overt reference to mathematical entities, and that mathematical theory is pervasively applied in the practice of science. Hartr ..."

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Any sceptic about abstract mathematical entities has somehow to come to terms with two facts: that many of the laws of physical science are formulated in ways which involve overt reference to mathematical entities, and that mathematical theory is pervasively applied in the practice of science. Hartry Field's well-known variety of nominalism responds distinctively by finding no fault with the common-sense, platonist semantics of mathematical discourse, seeking to show mathematical entities, and that the justifiability of mathematics as a medium of inference within physical theory can be safeguarded without presupposing its truth. One purely philosophical difficulty for this approach concerns what modal sta-tus it should accord to the fundamental nominalist tenet that there are no mathe-matical entities. The problem is one to which we have, singly and jointly, called attention on several occasions. It was first noted in Hale (1987, Ch. 5) and receives its most detailed and considered formulation to date in our joint "Nom-inalism and the Contingency of Abstract Objects"(1992). ' Field (1993, pp. 285-99) has responded in his recent "The Conceptual Contingency of Mathematical Objects", and that response is the topic of this note. Unfortunately, much of Field's latest discussion is given to sniping at earlier formulations of the difficulty which were explicitly discarded in our (1992), and some confusion may thereby have been caused about the exact focus of the objec-tion we are lodging.2 While we recognise that clarity will likely be best served not by detailed skirmishing over previous formulations but by concentrating upon what Field has to offer by way of response to our most recent version, it 1

### By

"... ii The central question of this dissertation is whether we are justified in believing in the existence of abstract mathematical objects. In Part I, I provide an in-depth examination and criticism of the most popular argument for the justifiability of believing in the existence of mathematical object ..."

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ii The central question of this dissertation is whether we are justified in believing in the existence of abstract mathematical objects. In Part I, I provide an in-depth examination and criticism of the most popular argument for the justifiability of believing in the existence of mathematical objects, the Quine-Putnam Indispensability Argument. I argue that the naturalistic basis for the argument not only depends essentially upon an untenable form of radical confirmational holism, but is ultimately self-undermining. In Part II, I examine the most popular argument for the unjustifiability of believing in the existence of abstract mathematical objects, Field’s Inexplicability Argument. I argue that not only does the argument ignore contemporary epistemological theories of justified belief and knowledge, but that the justificatory constraint that it suggests is implausible and open to general counterexample. Thus, in Parts I and II, I show that the most popular mathematical objects rest upon untenable epistemological theories. In Part III, I develop