Frege on the Concept of Number

This has been a tremendously influential article, certainly in the philosophy of mathematics, but more importantly, and more surprisingly, in the philosophy of language.

The idea that has been so important is known as the "context principle":

p. 133 It is enough if the sentence as a whole has a sense; by means of this its parts also receive their content.

The context principle is a warning to not investigate the meaning of terms (words) independent of a sentence. Frege's larger project concerns sentence meaning in general. The worry is that if we consider the meaning of words in isolation from the sentence in which they appear it becomes natural to associate their meanings with ideas. As Frege is interested in the communicability and objectivity of sentence meaning this clearly will not work.

This replaces what is usually called compositionality (the meaning of a linguistic complex is composed of the meanings of its parts) with what might be called decompositionality.

How does he use the context principle? In order to tell us what numbers are, he doesn't have to give a definition of a number (at least not directly), instead, he has to explain how number sentences work.

Numbers are, he says, objects. The word object here is used in a technical sense; Frege doesn't think numbers are physical objects. Rather, numbers occupy the 'third realm', an order distinct from the physical and psychological world. That might seem controversial, but for Frege it's just a grammatical point. Number words function as nouns that take definite articles, and so they name objects.

How do you know what an object is? By being able to identify and reidentify it. It is not enough to know a name; you need to be able to recognize it under various circumstances.

Frege doesn't care how we know what an object is; he just cares about the metaphysical point: what an object is, whether or not we know anything about it. Thus, an object is given by a criterion for identifying and reidentifying it, independently of whether it is a humanly usable criterion.

Thus, to say what an object is, we just need to see how names for it function in identity sentences, that is, since we are doing mathematics, equations.

So, when is an equation like 2+2=4 true? If we can answer that without using numbers, we will have said what numbers are. Remember, we've been investigating sentence meaning because it is the entire sentence (from the context principle) that allows us to communicate the 'something' that can be expressed as truth or falsity.

Frege switches examples, to make life easier: Before telling us what numbers are, he tells us what directions (N,S,E,W) are: The direction of line A is the same as the direction of line B

is true if and only if

line A is parallel to line B.

Just as the context principle seems backwards, this method of introducing directions seems backwards: It seems much more natural to say that two lines are parallel if they go in the same direction, than the other way around. In fact though, we have independent characterizations of parallelism, but not of directions.

It may not seem like this is adequate as a definition, since it doesn't tell us what directions are, only how direction talk is used. Frege spends a significant portion of the article trying to dispel this impression.

There are lots of comparatively familiar cases (the Earth) in which different people have different pictures, but understand what they are talking about equally well. Thus, the pictures are a mere psychological convenience, and the content (or meaning) is of the kind Frege has supplied.

But directions have no location in physical space. Frege answers that what makes an object an object is that we can reidentify it in identity sentences. There is a particular kind of object in which we are especially interested that does always have an associated location: physical objects. But there are others, equally familiar: mental pictures do not have a location in physical space either. They aren't, despite the familiar metaphor, inside the head. What is inside the head is neurons, glial cells, and so on. Mental objects are not to the left or right of each other, have no distances between them, and so forth.

What is a number? First of all, whatever they are, they are second level in the sense that they apply to concepts.

Given an encyclopedia, the question "how many?" makes no sense: One encylopedia, twenty-five volumes, 12,500 pages, thirty pounds...

Number is not a property of objects; it is a property of concepts. To say that an encyclopedia 'has twenty-five volumes' is to say that twenty-five objects "fall under" the concept "a volume of this encyclopedia." Though this may sound abstract it is actually intuitively clear. Given three oranges in a bundle we say 'there are three' in virtue of a grouping, not by the objects themselves. We don't apply the term '3' to the objects per se, rather we apply it to the grouping term 'bundle', or the concept.

So, when is the number of objects that fall under the concept F equal to the number of objects that fall under the concept G?

The answer is Hume's: that is true when the objects that fall under F can be 'paired off' (placed into one-to-one correspondence) with the objects that fall under G.

This is 90% of the answer, but we still have a problem: We can't tell whether Julius Caesar equals 17.

To fix that he proposes that the number of objects that fall under a concept F is the extension of the concept "equinumerous with the objects that fall under the concept F"

A problem with this view, as Russell later pointed out, is both Frege's view of extensions of concepts as objects and his assumption that every concept must be defined for all objects. For example, consider the concept '() is the extension of a concept under which it does not fall'. Is the extension of this concept a concept or not? Try each and see!

-- ProfessorShaughanLavine? - 14 Jan 2005