1. In the immediately previous post, I observed that I now find myself enjoying the leisure that will allow me “to engage in some serious thinking in the philosophy of mathematics.” More specifically, I announced my intention “to begin the preparation necessary for the rewriting, the much needed rewriting, of Quantity and Reality. The Bases of an Aristotelian Philosophy of Mathematics,” the doctoral dissertation I wrote and defended nearly forty years age, but which, however, left me then and leaves me now quite dissatisfied. I also let it be known that, given that which I have begun to understand, the rewritten document would be so restricted as to warrant the title the present series of posts bears, Number and Reality, along with the sub-title, The Bases of an Aristotelian Philosophy of Arithmetic.

I further announced that I would be using Stewart Shapiro’s accessible and lucid introduction to the philosophy of mathematics, Thinking about Mathematics. The Philosophy of Mathematics,* as an initial guide for my reflections. And, as I closed the post, I invited any and all interested parties to join with me in those reflections.

It is my intention, then, using Shapiro’s Thinking about Mathematics as an initial guide, to begin the preparation necessary for the rewriting, the much needed rewriting, of Quantity and Reality and to record the progress that I hope to make in that preparation in the form of posts to this blog. And it is my hope that others will join with me in discussions of those reflections.

2. The aim specific to the present post is that of coming to an initial judgment about the logically first step that the philosophy of arithmetic motivating “Number and Reality” needs to take. I’ll have the coming to that initial judgment arise out of a reading of the first paragraph of the preface to Thinking about Mathematics. It reads:

This is a philosophy book about mathematics. There are, first of all, matters of metaphysics. What is mathematics all about? Does it have a subject matter? What is this subject matter? What are numbers, sets, points, functions, and so on? Then there are semantics matters. What do mathematical statements mean? What is the nature of mathematical truth? And epistemology. How is mathematics known? What is its methodology? Is observation involved, or is it a purely mental exercise? How are disputes among mathematicians adjudicated? What is a proof? Are proofs absolutely certain, immune from rational doubt? What is the logic of mathematics? Are there unknowable mathematical truths?

3. The opening two sentences raise several significant questions. The one is:

What is philosophy?

The second is:

What is mathematics?

The third is:

What is metaphysics?

A fourth is, or a few more questions are:

What are the relations between and among philosophy, mathematics, and metaphysics?

These are all questions that we will have to answer, at some future point in this series of posts. For now, however, let’s content ourselves with a possible answer to the question of what philosophy is, suggested by the rest of the paragraph, that philosophy is an intellectual activity that asks questions like the ones the paragraph poses.

3. The paragraph itself contains three sets of questions. The first set contains, as Shapiro tells us, questions about the metaphysics of mathematics, the second set contains questions about the semantics of mathematics, and the third set contains questions about the epistemology of mathematics (two more questions raise their head, about semantics and epistemology). In the present post we’ll do some initial thinking focused on the first set alone of the questions Shapiro himself asks, though that initial thinking will itself include yet more questions.

The first set of questions contains four questions. Now the first question, “What is mathematics all about?” is a complex question, complex in that it presupposes an affirmative answer to the logically prior question, “Is there something that mathematics is all about?” (I am, of course, not saying that this complex question is an example of the fallacious argument, or fallacy, of the complex question. Ditto for the further complex questions that we will take note of presently.)

The latter question, “Is there something that mathematics is all about?” is equivalent to Shapiro’s second question, “Does it have a subject matter?” This is so at least if that which mathematics is “all about” is identical with its “subject matter.” And, if that is the case, then the third question, “What is this subject matter?” is equivalent to the first, “What is mathematics all about?” It, the third question, is then also a complex question, one presupposing an affirmative answer to the second.

4. The fourth of Shapiro’s questions, “What are numbers, sets, points, functions, and so on?” gives us, if not a complete answer to the question of what the subject matter of mathematics is, at least a partial answer to the question of what it includes; it includes numbers, sets, points, functions, and so on. But we have also to take note that the question, quite evidently, is a compound question, the component questions of which are:

What are numbers?

What are sets?

What are points?

What are functions?

And so on.

I expect to be expected to offer answers to these and other questions as we proceed. But the questions just posed are also complex questions, presupposing affirmative answers to the questions:

Do numbers exist?

Do sets exist?

Do points exist?

Do functions exist?

And so on.

Thus, the affirmative answers:

Numbers exist.

Sets exist.

Points exist.

Functions exist.

And so on.

It is, I think, only if such entities are existents that they are capable of being whatever they are.

This would seem to be the place where we should notice that the title of Shapiro’s eighth chapter is, in fact, “Numbers exist” and that his preface tells us (pp. viii-ix):

Chapter 8 is about views that take mathematical language literally, at face value, and hold that the bulk of the assertions of mathematicians are true. These philosophers hold that numbers, functions, points, and so on exist independent of the mathematician. They then try to show how we can have knowledge about such items, and how mathematics, so interpreted, relates to the physical world.

5. I noted above that the question, “What are numbers?” and the similar questions about sets, points, functions, and so on are complex questions, presupposing affirmative answers to the questions, “Do numbers exist?” and the similar questions about sets, points, functions, and so on. Now, in the post immediately preceding this one, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I also took note of the defining thesis of what I there nearly, though not quite, called realism in the ontology of mathematical objects. This is the thesis that Shapiro, in his Thinking about Mathematics* (p. 25), identified as “realism in ontology” and wanted us to define as “the view that at least some mathematical objects exist objectively, independent of the mathematician.” Let us, for present purposes, follow him, though using terminology that are in part mine and not fully his, and define realism in the ontology of numbers as “the view that at least some numbers exist objectively, independent of the mathematician.” Thus too, of course, for the realisms that are possible in the ontologies of sets, points, functions, and so on.

In, again, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I also noted that I incline towards one of the two versions of realism that present themselves for our attention, that of an Aristotelian, or of a something like an Aristotelian realism, as opposed to a Platonist realism. About this difference there is more, much more, to come in future posts.

5. First, however, negative answers to those questions are also logically possible. And so let us define anti-realism in the ontology of numbers as the view that no numbers exist objectively, independent of the mathematician.” Thus too, of course, for the anti-realisms that are possible in the ontologies of sets, points, functions, and so on:

No numbers exist.

No sets exist.

No points exist.

No functions exist.

And so on.

This, of course, would seem to be the place where we should notice that the title of Shapiro’s ninth chapter counters that of the eighth chapter, announcing that, “No they don’t” and that his preface tells us (p. ix):

Chapter 9 concerns philosophers who deny the existence of specifically mathematical objects. The authors covered here either reinterpret mathematical assertions so that they come out true without presupposing the existence of mathematical objects, or else they delimit a serious role for mathematics other than asserting truths and denying falsehoods.

6. In the light of the above, I will close by presenting the provisional statement that:

There is a theory of numbers, arithmetic, and so of, first, whether any numbers exist and only then, if they do, of just what numbers are.

The aim specific to the present post, that of coming to an initial judgment about the logically first step that the series of posts, “Number and Reality,” needs to take, has been reached. It is that of answering the question of whether any numbers exist.

But wait. I described the statement given just above as provisional. And so it is. More needs to be said, some of which will be said in the next post.

There we go.

Until next time,

Richard

*Stewart Shapiro, Thinking about Mathematics. The Philosophy of Mathematics (Oxford and New York: Oxford University Press, 2000). Thinking about Mathematics is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

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