An example: the existence of numbers

We've already discussed one large group of propositions that are true in all the possible worlds: they are the logical truths and all the other necessary truths.

In 3.11, I pointed out both that logical truths were necessary and that some necessary truths—“the Morning Star is the Evening Star,” for example—are not logical truths. So there's more to what is true in all the possible worlds than just logic. Most philosophers think, for example, that the truths of mathematics are necessary; but, despite serious attempts in the late nineteenth and early twentieth centuries to prove that all mathematics was really logic, it is now widely agreed among mathematicians and philosophers of mathematics that that is not so. Logicism, which is the name for the position that tries to derive all mathematics from logic (plus definitions), has not been successful.

If mathematical truths are necessary, then since it's true that there's a prime number between 17 and 23 (it's 19), it's also true that there's a prime number between 17 and 23 in every possible world. I can prove that there is a prime number between 17 and 23 by proving that 19 is a prime number—which I can do by showing that it's not divisible without remainder by any whole number between 2 and 10—and then by proving that 19 is greater than 17 and less than 23. Investigating the nature of numbers, then, isn't a matter for physics—because the numbers exist in nomically impossible worlds—and so maybe that could be a possible metaphysical subject. And, in fact, it is: the nature of numbers—what it means to say that numbers exist—is one central metaphysical question.

It's important to insist here that when I say “numbers” I don't mean the numerals—that is, the signs, like the symbol “9” or the Roman “IX” that we use to talk about numbers.

As I said in the introduction, we shouldn't confuse using a word with mentioning it. If I were to say that “9” existed, that would be plainly true. Obviously numerals exist. The interesting question is whether the numerals refer to actual objects and, if so, what kinds of objects they are. Whether 9 exists and whether “9” exists are very different questions. (Hobbes, you will recall, got this right, in the passage I quoted in 3.2, when he distinguished “number” and “the names of numbers.”)

Our normal ways of speaking are not very helpful here. We use the word “number” to refer both to the numeral and to the mathematical object, both “9” and 9. And we also use it both to refer to individual inscriptions of the numeral, like the “9” on the next line,

and to make claims about all such inscriptions, such as the observation

The upright stroke of “9” is often written at right angles to the top of the page.

Each inscription of the numeral “9” is a token of “9” the general type of inscription, just as each individual man is a token of the type man. This distinction is helpful in sorting out some possible confusions about numerals. For example, it makes sense to ask where an individual token of the type “9” is—there's one to the left of the last quotation mark—but it doesn't make sense to ask where the type is, at least until you've said a bit more about how to characterize a type. You might, for example, want to identify the type with the class of all the tokens—all the objects that share the property of being the same numeral; and then you might want to say that a class was an abstract object distinct from its members and thus that it didn't have a spatial location at all, even though there is nothing mysterious about the spatial location of each inscription.

Numbers and other mathematical objects are not the only things that one might suppose to exist in worlds other than the nomically possible worlds.

One other obvious candidate for metaphysical examination is the possible worlds themselves. What is it for a possible world to exist? Are there any impossible worlds, worlds, say, in which it rains and doesn't rain at the same time? These are also important and challenging metaphysical questions.

But there are also many other things that exist in the nomically impossible worlds about which philosophical exploration looks enticing. There are, for example, people, objects, events, times, and places in nomically impossible worlds. So the nature of people, events, objects, times, and places is not a matter just for natural science. Furthermore, as I said a little while ago, there are nomically impossible worlds where g is different, but there's no reason to think that any of the objects mentioned just now couldn't exist in some of those worlds. The mere fact that gravity was slightly different surely wouldn't have guaranteed that there would be no people or no material objects. You and I could still have existed, for example, and so could that tree. So there are possible worlds in which we exist but g has a different value. But what is it for a person or a material object to exist? Or for it to endure through time, to occupy a place, and participate in events?

I argued just now, in effect, that since there is a prime number between 17 and 23, it follows that there exists at least one number. There are lots of interesting arguments of this form, and some of them imply the existence of persons. It's true, for example, that Romeo loved Juliet, isn't it? So presumably it's true that there was someone that Romeo loved (namely, Juliet). But that means that the person Juliet existed! Now, most of us think that Juliet didn't exist, because she's a fictional character. But if she didn't exist, how can there be any truths about her? Do fictional characters exist in some other possible worlds? And if so, is that what determines what's true about them?

Questions such as these—about what persons or objects are or whether numbers or fictional entities exist somewhere—are ontological questions. They are questions about what exists—what there is—and about the nature of that existence.

We have already discussed a number of ontological questions in the course of this book: in 4.7 and 4.8, for example, we asked whether we have reason to believe that postulated theoretical entities exist. And just now I assumed that a mathematical proof that there was a prime number between 17 and 23 showed that that number existed. Many mathematicians and philosophers do think that mathematical entities, such as numbers, exist, Plato, famously, among them—which is why this ontological view about mathematical objects is called “Platonism.” Plato thought that numbers and many other abstract things—such as goodness, Truth, and Beauty, for example—existed in a sort of perfect realm of their own as Ideas or Forms. Good things and true things and beautiful things in the world that we experience were pale reflections of these Ideas of the Good and the True and the Beautiful. (Plato's critics had some fun with this, because the theory seemed also to require that actual mud, say, was a pale reflection of the Idea of Mud; and that made the realm of Ideas seem somehow less pure!) What made it true that I had five physical fingers was that the fingers of my hand participated somehow in the Idea of Five. Modern Platonists do not tend to think of numbers or any other abstract entities as existing in a special sort of place; they don't suppose that goodness or 9 are anywhere, any more than Descartes supposed that our thoughts had spatial locations. But they follow Plato in insisting that we can only make sense of the world if we suppose that numbers (and other mathematical objects) are in some sense real.

Many philosophers, however, have doubted that numbers really exist, at least in the way that tables and chairs do. And so they have sought to show that when we say, “There is a number... “ what we mean can be translated into some other sentence that doesn't imply that numbers exist. The American philosopher W.V.O. Quine argued that you were committed to the existence of anything about which you said (or believed) that it satisfied an open sentence.

Or, as he put it, “to be is to be the value of a variable.” (I introduced this terminology in 3.5. An object that satisfies an open sentence is a value of the variable that replaces the blank.) So if, to use an example of Quine's, I asked you what the number of the planets was and you said “9,” then you would be committed to the existence of the number 9 because you are saying that 9 satisfies the open sentence

------- is the number of the planets.

As a result, if you don't think numbers really exist you have to find a way of translating

P: 9 is the number of the planets

that doesn't have this ontological commitment. One simple way to do this would be to say that P just means

P': There are nine planets.

But then, of course, you would have to explain what P' means in a way that didn't bring numbers in again by the back door! Obviously, for example, it wouldn't do to say that P' means:

P": There are as many planets as there are numbers between 1 and 9.

For then someone could say that that the open sentence

P": There are as many planets as there are numbers between 1 and

was satisfied by the number 9, which could be so only if 9 existed.

Along with Frege, Bertrand Russell, the great twentieth-century British philosopher and mathematician, developed an account of what “there are n X's” means (where “n” is replaced by a numeral) that was meant to avoid commitment to numbers. If you had asked Russell how to say there were exactly two planets, without using the numeral “2”, he would have offered the following translation:

FR: There exists an X and there exists a Y such that X is a planet and Y is a planet and Y is not the same thing as X and every planet is identical to X or to Y.

If you develop a general method of getting rid of any natural numeral, like “2” or “17,” in this sort of way, then you have avoided ontological commitment to numbers.

The basic idea of Frege and Russell's treatment of numbers was to identify one with the class of all one-membered classes, two with the set of two-membered classes, and so on. They proposed this as an analysis of what numbers really were. But you might start from this idea and develop instead the view that the fact that we could eliminate reference to numbers by formulas such as FR entitled you to conclude that numbers didn't exist. This would be a form of nominalism about numbers: it would hold that while the numerals made sense, they didn't refer to anything. So the numerals were real (“nominalism” comes from the Latin word “nomen,” which means “name”), but the numbers were not.

Once you start thinking about it, in fact, there seem to be very many questions like these about the natures of things—including many ontological questions—that are not about the nomically possible worlds alone. And I'm going to be able to introduce you to discussions of only a few of the many interesting and important topics in metaphysics. I have chosen, in fact, to consider some of the questions that arise in the context of thinking about an issue that has been central to philosophical discussions for more than two thousand years: namely, the nature and existence of God.

8.3

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Source: Appiah Kwame Anthony. Thinking It Through: An Introduction to Contemporary Philosophy. Oxford University Press,2003. — 425 p.. 2003

An example: the existence of numbers

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