Frege’s Definitions of Number

Before settling on the definition he uses for the remainder of his logicist program, Frege formulates two other definitions of the concept of Number. The first, an attempt “to complete the Leibnizian definitions of the individual numbers”[1] which were given earlier, Frege brings up because it springs to mind so readily given earlier discussion. From the inability of that account to determine when the numbers a and b are identical, Frege comes up with a second strategy, in which the equality of numbers is related to the equinumerosity of the concepts to which they belong. This strategy alone, he objects, is still not sufficient for determining precisely what objects numbers are. He thus reformulates it slightly and defines numbers as the extensions of certain concepts about equinumerosity. It is this reformulation that he uses throughout the rest of his foundational program, but it may not be entirely free from all the problems Frege finds with the other formulations.

Frege accepts that “through such definitions [as Leibniz gives,] we reduce the whole infinite set of numbers to the number one and increase by one.”[2] The question, then, is how to define (zero,) one and increase by one. On his first attempt, Frege says that “the number 0 belongs to a concept, if the proposition that a does not fall under that concept is true universally, whatever a may be.” In logical notation, we might have

0FÛ"aØFa.

Then Frege goes on to describe the number one

1FÛ($aFaÙ"a"b(FaÙFbÞa=b))

and the process of increasing by one

(n+1)FÛ$a(FaÙnGÙGxÛ(FxÙx¹a).[3]

The first problem he finds with this account is that we don’t really understand nG (the number n belongs to the concept G) any more than we understand (n+1)F (the number n+1 belongs to the concept F), so defining the one in terms of the other is not illuminating. Frege believes that this difficulty prevents us from deciding “whether any concept has the number Julius Caesar belonging to it, or whether that same familiar conqueror of Gaul is a number or is not.”[4] There seems, however, to be an even greater difficulty—at least for Frege, who wants numbers to be objects. He goes on in the following section to assure us that the individual number, in an assertion about a number belonging to a concept, shows itself to be a self-subsistent object, but the definition does not make it entirely clear that numbers are anything other than properties of concepts. Just as we might say the number n belongs to a concept F, if such and such a proposition about F is true, we might similarly say that the property of being instantiated belongs to a concept, if the proposition that a does not fall under that concept is not true universally, whatever a may be. If we suppose that Frege has an unstated way of interpreting sentences like, “The concept of being a horse is instantiated,” as saying something meaningful about a concept rather than an object, then perhaps statements of number could say something meaningful about properties of concepts rather than about objects belonging to concepts. Much of his subsequent argument to the contrary turns not on the above account of statements of number, but rather on the uses to which we would like to put numbers for the purposes of science. As Frege thus has reason to believe, independently of his first attempt at defining them, that numbers are objects, we shall for the time being overlook this potential difficulty.

Granting that his first definition of numbers does identify them as self-subsistent objects, we still have the Julius Caesar problem, as well as the problem of being unable to prove that, “if the number a belongs to the concept F and the number b belongs to the same concept, then necessarily a=b.”[5] To solve the latter problem, Frege seeks to use the concept of identity in such a way that two numbers can be judged to be identical or not, an ability which in turn would fix the concept of Number. He starts out with a statement of Hume’s principle that numerical identity must involve one-to-one correlation, and then gives several related examples from geometry in order to illuminate the general concept of identity, which he takes to be already fixed. The directions of two parallel lines are identical, the orientations of two parallel planes are identical, and the shapes of two similar triangles are identical. These statements actually serve to introduce new expressions, namely those about direction, orientation, and shape, which Frege sees as less basic than the concepts of parallelism of lines and planes and similarity of triangles. Using Leibniz’s definition of identity,[6] he shows that these are nothing more than specific applications of the general law of universal substitutability. Indeed, because “it is possible, if line a is parallel to line b, to substitute ‘the direction of b’ everywhere for ‘the direction of a’,”[7] we are justified in saying these directions are identical.

Though Frege doesn’t explicitly spell it out at this point in the text, we are likely supposed to recall his earlier statement of Hume’s principle, from which we can conclude that, if there is some sort of one-to-one correspondence between two concepts, the numbers belonging to those concepts are identical. Unfortunately, this formulation still doesn’t solve the problem of picking out precisely what object a number is supposed to be. We can determine whether the number belonging to a concept F is identical with the number belonging to a concept G, but we are given no way to determine whether the number belonging to a concept F is identical with Julius Caesar, or in fact whether it is identical with anything not given in the form of “the number belonging to a concept G”. Being unable to determine whether a particular object is identical with a particular number, or whether it is a number at all, we are still without a general concept of Number, though we have at least made progress in that we are now able to determine whether the numbers belonging to two concepts are identical.

In order to more rigorously fix what objects, in particular, we are dealing with, Frege identifies them with the extensions of certain concepts. For instance, “the direction of line a is the extension of the concept ‘parallel to line a’,” and “the Number which belongs to the concept F is the extension of the concept ‘equal [equinumerous] to the concept F’.”[8] Two concepts are equinumerous when it is possible to correlate, one to one, the objects that fall under one with those that fall under the other. Frege later defines 0 as the extension of the concept of being equinumerous to the concept of not being self-identical, and the number immediately following n in the series of natural numbers as the extension of the concept of being equinumerous to the concept F, where there is some object a that falls under F and n is the extension of the concept of being equinumerous to the concept of ‘falling under F but not identical with a’.[9] We can now, it seems, determine whether some object x is a number by determining whether there is some concept G, such that x is the extension of the concept ‘equinumerous to the concept G’. If we are able to find such a concept, then for the number n that belongs to the concept F, we can determine whether n is identical with x by determining whether the concepts F and G are equinumerous. In particular, we should now be able to establish whether the number belonging to the concept F is identical with Julius Caesar, simply by checking for the existence of some concept G such that Julius Caesar is the extension of the concept ‘equinumerous to the concept G’. If there is such a concept, then we need to check for the equinumerosity of the concepts F and G, and if they are equinumerous, then the number n is identical with Julius Caesar. If they are not equinumerous, or no such concept G exists, then the number n cannot be identical with Julius Caesr.

By bringing it up as an objection to both of his previous attempts to define Number, Frege clearly indicated that he found the possibility of Julius Caesar being a number to be “a crude example”.[10] To think his final definition solved that problem and would strike readers as plausible, he must therefore have believed it to be obvious to those readers that Caesar is not the extension of any appropriate concept of being equinumerous to some particular other concept. It is unclear, however, why we should conclude that the conqueror of Gaul is not the extension of such a concept. Russell would conceivably have argued in favor of this conclusion by claiming that entities such as Caesar are of a different type than entities such as the number n. Even without a strictly formulated theory of types, one might point out that particular Roman Emperors are individuals whereas extensions of concepts are classes, and claim that the two are distinct sorts of things. Unfortunately, such arguments are not accessible to Frege, who shied away from talk about sets and classes and who seemed to want all objects to share the same level of his own ontological hierarchy (which includes objects, then first level concepts under which objects may fall, then second level concepts under which first level concepts may fall, and so on). Without resorting to sets, it is not clear that Frege could have made a distinction analogous to the one we would now make between Julius Caesar and {Julius Caesar}, the former being the late emperor himself and the latter perhaps being the extension of the concept ‘being a Roman emperor killed in 44 BC’. If he didn’t make this distinction, it is possible that Frege would have agreed that Julius Caesar himself is the extension of that concept, and thus could offer no argument against Caesar being an extension in the first place. This of course makes it more difficult to offer any argument against Caesar being the extension of a concept of the type required for a thing to be a number.

What would be the consequences if a particular number was somehow discovered to be identical with a man who has been dead for somewhat longer than two millennia? There would certainly be significant ramifications for the study of history, and classical historians the world over would have to seriously reconsider what they thought they knew about the identity of the first Caesar. Some might even be driven to abandon their careers altogether and become professors of philosophy or some such thing. The philosophy of mathematics would have to find a way to adapt to the concrete albeit former existence of one of a group of things that are widely regarded to be abstract at the very least, assuming they exist at all. But for the theory and practice of mathematics itself, it seems that consequences would be less severe. Supposing there were some way of dealing with the fact that one of the numbers was assassinated over 2000 years ago, other potential problems would sort of melt away. A Fregean construction of the natural numbers, along with the rigorous constructions of other sets of numbers built up from the natural numbers and from previously constructed sets, implies that if Caesar was in fact a number, he would be identical to some element of one of these other sets, and could thus be treated, for all mathematical purposes, exactly as one would treat any non-Caesar element of a set of numbers.

[1] Grundlagen, §55. Earlier Leibnizian definitions given in §6.

[3] §55. “Translation” into logical notation given primarily in the interest of saving space.

[6] “Things are the same as each other, of which one can be substituted for the other without loss of truth.” From §65.