- Finitely Supported Mathematics
- Philosophy of Mathematics
- INTRODUCTION
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At least some of the basic principles of logic are, or seem to be, absolutely necessary and a priori knowable. If one doubts the basic principles of logic, then, perhaps by definition, she cannot go on to think coherently at all. Prima facie, to think coherently just is to think logically. Like mathematics, logic has also been a central focus of philosophy, almost from the very beginning.

## Finitely Supported Mathematics

Aristotle is still listed among the four or five most influential logicians ever, and logic received attention throughout the ancient and medieval intellectual worlds. Today, of course, logic is a thriving branch of both mathematics and philosophy. It is incumbent on any complete philosophy of mathematics and any complete philosophy of logic to account for their at least apparent necessity and apriority. Broadly speaking, there are two options. The straightforward way to show that a given discipline appears a certain way is to demonstrate that it is that way.

Alternatively, the philosopher can p. The conflict between rationalism and empiricism reflects some tension in the traditional views concerning mathematics, if not logic. Mathematics seems necessary and a priori, and yet it has something to do with the physical world.

How is this possible? How can we learn something important about the physical world by a priori reflection in our comfortable armchairs? As noted above, mathematics is essential to any scientific understanding of the world, and science is empirical if anything is—rationalism notwithstanding.

Immanuel Kant's thesis that arithmetic and geometry are synthetic a priori was a heroic attempt to reconcile these features of mathematics. According to Kant, mathematics relates to the forms of ordinary perception in space and time. On this view, mathematics applies to the physical world because it concerns the ways that we perceive the physical world. Mathematics concerns the underlying structure and presuppositions of the natural sciences. Mathematical knowledge is a priori because we can uncover these presuppositions without any particular experience chapter 2 of this volume.

This set the stage for over two centuries of fruitful philosophy. For any field of study X , the main purposes of the philosophy of X are to interpret X and to illuminate the place of X in the overall intellectual enterprise. The philosopher of mathematics immediately encounters sweeping issues, typically concerning all of mathematics.

Most of these questions come from general philosophy: matters of ontology, epistemology, and logic. What, if anything, is mathematics about? How is mathematics pursued? Do we know mathematics and, if so, how do we know mathematics? What is the methodology of mathematics, and to what extent is this methodology reliable? What is the proper logic for mathematics? To what extent are the principles of mathematics objective and independent of the mind, language, and social structure of mathematicians?

Some problems and issues on the agenda of contemporary philosophy have remarkably clean formulations when applied to mathematics. Examples include matters of ontology, logic, objectivity, knowledge, and mind. The philosopher of logic encounters a similar range of issues, with perhaps less emphasis on ontology. Given the role of deduction in mathematics, the philosophy of mathematics and the philosophy of logic are intertwined, to the point that there is not much use in separating them out. A mathematician who adopts a philosophy of mathematics should gain something by this: an orientation toward the work, some insight into the role of mathematics, and at least a tentative guide to the direction of mathematics—What sorts of problems are important?

What questions should be posed?

## Philosophy of Mathematics

What methodologies are reasonable? What is likely to succeed? And so on? One global issue concerns whether mathematical objects —numbers, points, functions, sets—exist and, if they do, whether they are independent of the mathematician, her mind, her language, and so on. Define realism in ontology to be the view that at least some mathematical objects exist objectively. According to ontological realism, mathematical objects are prima facie abstract, acausal, indestructible, eternal, and not part of space and time.

Realism in ontology does account for, or at least recapitulate, the necessity of mathematics. If the subject matter of mathematics is as these realists say it is, then the truths of mathematics are independent of anything contingent about the physical universe and anything contingent about the human mind, the community of mathematicians, and so on.

What of apriority? However, such a connection is denied by most contemporary philosophers. If mathematical objects are in fact abstract, and thus causally isolated from the mathematician, then how is it possible for this mathematician to gain knowledge of them? It is close to a piece of incorrigible data that we do have at least some mathematical knowledge. If the realist in ontology is correct, how is this possible? Georg Kreisel is often credited with shifting attention from the existence of mathematical objects to the objectivity of mathematical truth.

Accordingly, mathematics has the objectivity of a science.

### INTRODUCTION

Mathematical and everyday discourse has variables that range over numbers, and numerals are singular terms. Realism in ontology is just the view that this discourse is to be taken at face value.

Singular terms denote objects, and thus numerals denote numbers. According to our two realisms, mathematicians mean what they say, and most of what they say is true.

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Nevertheless, a survey of the recent literature reveals that there is no consensus on the logical connections between the two realist theses or their negations. Each of the four possible positions is articulated and defended by established philosophers of mathematics. A closely related matter concerns the relationship between philosophy of mathematics and the practice of mathematics. In recent history, there have been disputes concerning some principles and inferences within mathematics.

One example is the law of excluded middle, the principle that for every sentence, either it or its negation is true.

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For a second example, a definition is impredicative if it refers to a class that contains the object being defined. For example, if mathematical objects are mental constructions or creations, then impredicative definitions are circular. One cannot create or construct an object by referring to a class of objects that already contains the item being created or constructed.

Realists defended the principles. On that view, a definition does not represent a recipe for creating or constructing a mathematical object. Rather, a definition is a characterization or description of an object that already exists. As far as contemporary mathematics is concerned, the aforementioned disputes are over, for the most part.

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The law of excluded middle and impredicative definitions are central items in the mathematician's toolbox—to the extent that many practitioners are not aware when these items have been invoked. But this battle was not fought and won on philosophical grounds. Mathematicians did not temporarily don philosophical hats and decide that numbers, say, really do p.

If anything, the dialectic went in the opposite direction, from mathematics to philosophy. The practices in question were found to be conducive to the practice of mathematics, as mathematics—and thus to the sciences but see chapters 9 , 10 , and 19 in this volume. There is nevertheless a rich and growing research program to see just how much mathematics can be obtained if the restrictions are enforced chapter 19 in this volume. The research is valuable in its own right, as a study of the logical power of the various once questionable principles.