1. At long last I find myself enjoying a set of circumstances that will allow me to engage in some serious thinking in the philosophy of mathematics; the two September posts, “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics,” and “Ontological Arithmetic. The Second Foot in the Door,” are the first early fruits, if “fruits” is the right word, of that thinking. Looking back further in time, the philosophy of mathematics was the area of my doctoral dissertation completed nearly four forty years ago; in the course of the intervening (long!) decades spent as a full-time academic administrator and an “off-and-on” adjunct professor of philosophy, I taught a wide variety of courses in philosophy (and others in Islamic Studies and a few in French), but none were immediately related to the philosophy of mathematics.

The reason that I am at this late date returning to the philosophy of mathematics is that even on the day I successfully, if that is what my being recognized as a doctor of philosophy in philosophy shows, defended my dissertation, I was deeply dissatisfied with it. And I am now even more dissatisfied with it. I will point to the reasons why below.

2. But first I’d like to note that the dissertation was not entirely without a few features that even now I think worth thinking about. One of them is evident in the dissertation’s title, *Quantity and Reality*, a title reflecting my admiration for the absolutely perfect title of Alfred North Whitehead’s *Process and Reality*, even though I was not, and am not, a “process philosopher.” More to point, however, it reflects the view that I had then and still have now, albeit perhaps with some precisions, that quantity is an attribute of real existents (or beings, things, or realities; in this context I use the four terms interchangeably, applying them to whatever is, in any way, whatsoever).

That is, first, I envisioned the theory I was seeking to articulate therein as a version of realism in the philosophy of mathematics, an initially apt enough definition of which has been set forth for us by Stewart Shapiro in his *Thinking about Mathematics. The Philosophy of Mathematics** (p. 25).

At least on the surface, this theorem [i.e., “the ancient theorem that for every natural number n, there is a prime number m > n,” from which “[i]t follows that there is no largest prime number, and so there are infinitely many primes”] seems to concern

numbers. What are these things? Are we to take the language of mathematics at face value and conclude that numbers, points, functions, and sets exist? If they do exist, are they independent of the mathematician, her mind, language, and so on? Definerealism in ontologyto be the view that at least some mathematical objects exist objectively, independent of the mathematician.

(I would have identified *realism in the ontology of mathematics*, rather than *realism in ontology*, as “the view that at least some mathematical objects exist objectively, independent of the mathematician,” for I distinguish between ontology and mathematics.)

One might say, for example, that the number two exists objectively, in independence from the mathematician.

3. But, second, realism in ontology *tout court* has traditionally, and with good reason, been divided into two sorts of realism, that of Platonism and that of Aristotelianism. Realism in the ontology of mathematics is, it has seemed to me, similarly to be divided. So, on the one hand, there is the realism of Platonism. Shapiro tells us (p. 27):

Realism in ontology does not, by itself, have any ramifications concerning the nature of the postulated mathematical objects (or properties or concepts), beyond the bare thesis that they exist objectively. What are numbers like? How do they relate to more mundane objects like stones and people? Among ontological realists, the more common view is that mathematical objects are acausal, eternal, indestructible, and not part of space-time. After a fashion, mathematical and scientific practice support this, once the existence of mathematical objects is conceded. The scientific literature contains no reference to the location of numbers or to their causal efficacy in natural phenomenon or to how one could go about creating or destroying a number. There is no mention of experiments to detect the presence of numbers or determine their mathematical properties. Such talk would be patently absurd. Realism in ontology is sometimes called ‘Platonism’, because Plato’s Forms are also acausal, eternal, indestructible, and not part of space-time.

To continue with the example, one might say that the number two exists, but nowhere in space and time.

Let us stipulate that Platonism is indeed and even by far the more common view of mathematical objects. I, however, was then and am now unable to accept Platonist realism either in ontology *tout court* or in the ontology of mathematics (in posts to come I will fulfill the obligation I have of explaining why). Moreover, I found it then and find it now necessary to accept an Aristotelian realism in ontology *tout court*, at least as I understood and understand what an Aristotelian realism should be (in a post to come I will fulfill the obligation I have here too of explaining why). I therefore thought, and think now, it necessary that an Aristotelian realism in the ontology of mathematics be worked out and presented; thus the subtitle of the dissertation, *The Bases of an Aristotelian Philosophy of Mathematics*.

4. The limitations of the dissertation are all too evident to me today. One thing that may be seen as a problem is the limitation of its scope. Though it did address at least in part a central problem, or confusion, inherent in the logicism of Gottlob Frege, Bertrand Russell, and Willard Van Orman Quine, in their identification of logic as the foundation of mathematics, the dissertation said very little about formalism and intuitionism, the other two of the doctrines that Shapiro identifies as “the big three” of the “major philosophical positions that dominated debates earlier” (p. vii) in the twentieth century (the “Contents” of *Thinking about Mathematics* shows the title of Part III as “THE BIG THREE.”) Nor did it say anything about the structuralism that is the perspective adopted by Shapiro himself; I was unaware of it.

But even within the scope of what I did take up there are real problems. One is that I was not able to work out the ontologicist, if I may, alternative to logicism. This is the theory that ontology, and not logic, provides the basic principles underlying arithmetic. Happily, I can report that I have been able recently to take a few initial steps in the working out of that ontologism; I offer in support of that claim that which I put forward in “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics” and “Ontological Arithmetic. The Second Foot in the Door.” I say, “a few initial steps’ advisedly; there are many more that need to be taken.

5. A second of the real problems falling within the scope of what I did take up in the dissertation has its basis in a difference that exists between, on the one hand, arithmetic and the theories, like algebra, that are the further developments of arithmetic and, on the other hand, geometry and its further developments. That is, on that one hand, some universal propositions of arithmetic, the true universal propositions of arithmetic, are exactly true, true without qualification, of any and all of the existents, beings, things, or realities denoted by their subjects, whether physical or not. Thus it is that, in his *An Introduction to Mathematics*** (p. 9), Alfred North Whitehead could say:

The first acquaintance which most people have of mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristics of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings. or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.

On that other hand, no universal propositions of geometry are exactly true, true without qualification, of any and all of the physical existents, beings, things, or realities purportedly denoted by their subjects. For but one example, not all physical lines are without thickness. Of course, one might reply that the lines that are properly the lines of geometry are ideal lines, and not physical. Then, however, one has the problem of determining what the relationship is between the ideal and the physical. Shapiro (pp. 69-70) sees here at least a potential problem.

There is a potential problem concerning the mismatch between real physical objects and geometric objects or properties. This, of course, is an instance of the mismatch between object and Form that motivates Platonism. Consider the brass sphere and the side of the ice cube. The sphere is bound to contain imperfections and the surface of the cube is certainly not completely flat. Recall theorem that a tangent to a circle intersects the circle in a single point…. This theorem is false concerning real circles and real straight lines. So what are we to make of Aristotle’s claim that ‘mathematical objects exist and are as they are said to be’, and the statement that ‘geometers speak correctly’?

This problem, as I will argue in a future post, is a problem that Aristotle and Aristotle never solved. It is also a problem that I have not yet solved.

6. To conclude this post: it is because I believe that I have, as the earlier posts mentioned above have indicated, gotten both of my feet in the door, albeit just barely, of the philosophy of arithmetic, while I have gotten neither foot at all in the door of the philosophy of geometry, that I have given the series of posts to which this one belongs the title, *Number and Reality*, as opposed to the *Quantity and Reality* of my dissertation. Correspondingly, I have been tempted to say that, if the series of post were to have a subtitle, that subtitle should be, “*The Bases of an Aristotelian Philosophy of Arithmetic*.” (It may be worth recognizing at the outset that, while I consider the philosophy of arithmetic I am in the process of working out to be an Aristotelian philosophy of arithmetic, the question of just how much it is in fact Aristotelian is one that perhaps many Aristotelians would consider to be at best, well, an open one.)

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. I will not take up all passages in *Thinking about Mathematics* or only passages of *Thinking about Mathematics*; for one thing, you may be sure that an adequate understanding of some of Shapiro’s passages will require, at least of me, some side journeys into a variety of blog posts, articles, and books.

I have, further, two hopes. The one, minimal, hope, is that I will have you as a reader and that you will follow the series as it unfolds. I invite you, then, if you have not already done so, to go to the bottom of this page’s right-hand panel to **Follow Blog via Email** and enter your email address.

But, of course, I hope for more than that. I hope that you will, not just follow, but also actively take part in the philosophical discussions among *After Aristotle*’s readers that the series of posts aims to have take place. I therefore invite you to ask for clarifications of, to challenge, to add to, or to otherwise comment on anything that I say in the posts to come.

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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**Alfred North Whitehead, *An Introduction to Mathematics* (Cambridge: Cambridge University Press, 1911). This book too is is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

You may wish to note that I do not anticipate returning to this book with any regularity.