last revised October 12th, 2004.I think it is fair to say that it is only philosophers who get hung up ontologically about propositions, facts, or states of affairs. The rest of us don't take them too seriously, ontologically speaking, so in Loux's terms we are at heart nominalists of some sort about them. But we do tend to treat numbers and other bits of mathematics with more respect. For one thing, they are incredibly useful for us. We would have no science without what mathematics gives it; we didn't have much worth having until we mathematized our natural history, our elementary physics, etc. Some mathematics is very familiar - it is a staple of elementary schooling throughout the world - but even this familiar portion encompasses what can quickly seem like deep mysteries. If, as Kronecker said, "God made the natural numbers, all the rest is the work of man" he seems to have been pretty generous, since there is an infinite number of them, and we can easily find strange properties of them (there as just as many even numbers as integers, just as many cubes as integers,...).
It is also worth remembering that it has been actual mathematics that has generated many of philosophy's problems from the beginning. Plato was deeply impressed by the otherworldliness of mathematical entities. He might not have got so far just reflecting on propositions about ordinary life.
There are some brief remarks in Lowe ch. 20. Zalta has a complex article in SEP on Frege's Logic, theorem, and foundations for Arithmetic and another on Frege more generally.
Some definitions from Mathworld:
- Natural number/whole number = A positive integer 1, 2, 3, .... Unfortunately, 0 is sometimes also included in the list of "natural" numbers, and there seems to be no general agreement about whether to include it.
- Integer = One of the numbers ..., -2, -1, 0, 1, 2, ....
- Ordinal number = In common usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc.
In formal set theory, an ordinal number (sometimes simply called an "ordinal" for short) is one of the numbers in Georg Cantor's extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set. Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals as denoted using lower case Greek letters.
It is easy to see that every finite totally ordered set is well ordered. Any two totally ordered sets with k elements (for k a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, ..., i.e., the integers one less than the corresponding nonnegative integers.
The first transfinite ordinal, denoted ω, is the order type of the set of nonnegative integers. This is the "smallest" of Cantor's transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers.- Cardinal number = In common usage, a cardinal number is a number used in counting (a counting number), such as 1, 2, 3, ....
In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has ℵ0 (aleph-0) members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set.
It is usual to think that no one really had any clear idea what numbers are until Frege's work. Zalta tells us
In what has come to be regarded as a seminal treatise, Die Grundlagen der Arithmetik (1884), Frege began work on the idea of deriving some of the basic principles of arithmetic from what he thought were more fundamental logical principles and logical concepts. Philosophers today still find that work insightful. The leading idea is that a statement of number, such as ‘There are nine planets’ and ‘There are two authors of Principia Mathematica’, is really a statement about a concept. Frege realized that one and the same physical phenomena could be conceptualized in different ways, and that answers to the question ‘How many?’ only make sense once a concept F is supplied. Thus, one and the same physical entity might be conceptualized as consisting of 1 army, 5 divisions, 20 regiments, 100 companies, etc., and so the question ‘How many?’ only becomes legitimate once one supplies the concept being counted, such as army, division, regiment, or company (1884, §46).
We have already met the idea that existence, or at least an answer to the question "how many?", should be seen as a second-order property of a property. This is how Frege took numerical adjectives to be functioning. He showed that one could define the finite numerical adjectives using only the tools of what we now think of as the predicate calculus. What we may call condition (0): Nothing falls under F, is easily written as ¬∃xFx. One can go on to state other conditions, for instance, condition (2) that exactly two things fall under F, thus:
∃x∃y(x ≠ y & Fx & Fy & ∀z(Fz → z = x ∨ z = y))
Relying further on Zalta, we may note how Frege moves from the adjectival to the substantive use of number words. "If concepts P and Q are both concepts which satisfy one of these conditions, then there is a one-to-one correspondence between the objects which fall under P and the objects which fall under Q. That is, ... every object falling under P can be paired with a unique and distinct object falling under Q and, under this pairing, every object falling under Q gets paired with some unique and distinct object falling under P. (By the logician's understanding of the phrase ‘every’, this last claim even applies to those concepts P and Q which satisfy Condition (0).) Frege would call such P and Q equinumerous concepts (1884, §72)."
"With this notion of equinumerosity, Frege defined ‘the number of the concept F’ (or, more informally, ‘the number of Fs’) to be the extension or set of all concepts that are equinumerous with F (1884, §68). For example, the number of the concept author of Principia Mathematica is the extension of all concepts that are equinumerous to that concept. This number is therefore identified with the class of all concepts under which two objects fall, as this is defined by Condition (2) above. Frege specifically identified the number 0 as the number of the concept not being self-identical (1884, §74). It is a theorem of logic that nothing falls under this concept. Thus, it is a concept that satisfies Condition (0) above. Frege thereby identified the number 0 as the class of all concepts under which nothing falls, since that is the class of concepts equinumerous with the concept not being self-identical. Essentially, Frege identified the number 1 as the class of all concepts which satisfy Condition (1). And so forth. But though this defines a sequence of entities which are numbers, this procedure doesn't actually define the concept natural number (finite number)."
Classes are particular abstract objects. Frege's account of particular numbers is, then, that they are a kind of class, and so a particular.
You may consult Zalta or others to see how Frege moved on to the elaboration of a general account of the natural numbers.
We have seen that Frege identified particular natural numbers with particular classes or sets. Set theory has become the lingua franca for most if not all of mathematics, and so it is tempting to think that one has reduced numbers to sets. But there are many alternative ways of "identifying" numbers with sets, all of which work perfectly well, but which seem to make different ontological claims. Thus one can align 0 with the empty set ∅, 1 with the singleton of 0 {∅}, 2 with the singleton of 1, {{∅}}, and so on. But one can equally well do it von Neumann style: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, .... And there are other ways as well. If numbers simply are sets, which ones are they?
Lowe (in his The Possibility of Metaphysics, pp. 220-227, rather than the Survey) argues for a different approach to the natural numbers. Rather than see them as a type of set, he sees them as kinds of sets, i.e. kinds whose instances are sets. Numbers are not sets, any more than the kind dog is a dog. The number 2, say, is the kind of two-membered sets; each two-membered set is 'a' 2, just as each particular dog is a dog. Lowe in fact reverses the usual approach and takes set to be defined in terms of number: a set is a number of things.
The general term 'natural number' denotes a kind, but on Lowe's view each particular number is also a kind. 2 is a kind, whose instances are sets, not a set of a certain kind. Lowe mentions the fact that when teaching elementary arithmetic, we have to present a child with two twos to show him or her that 2 + 2 = 4. If two were an abstract particular what would it mean to add it to itself? He later makes the point that whereas Frege needs to talk about the class of all two-membered classes (and one may wonder whether that is a determinate notion), Lowe's view merely requires us to recognise some two-membered set to guarantee that the kind exists.
I want to conclude with some earlier ideas of W.C. Kneale ('Numbers and Numerals', British Journal for the Philosophy of Science 23 (3): 191-206 (1972)), who sticks pretty close to Frege's manner of exposition. Kneale emphasises the fundamental nature of the adjectival or operator use of numerical words, 'there are 2 Fs'. Noun forms, 'the number 2', are, he thinks, a form of quotation of such operators (quotation like Sellars' dot quotation). 'The number of things which are F' is to be analysed as 'the number n such that just n things are F', which avoids Frege's use of classes. It is this definition that provides the link between the adjectival and the substantival use of numerals, a link that Kneale considers makes it impossible to identify numberhood simply with membership in a progression that obeys Peano's axioms.
Kneale's view is that a number, 2 say, is a kind to which a token of the numeral "2" belongs when it presents a "quotity" or "extent of exemplification. I think Lowe is saying that the number 2 is the kind to which these quotities belong.
© Ed Brandon, 2004.
last revised October 12th, 2004.
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