Frege on Arithmetic

Frege is generally presented in philosophy of mathematics classes as the first logicist. He was. But the importance of his work in philosophy of mathematics goes far beyond that: he established what the problems of the philosophy of mathematics are, and many of the assumptions we take for granted, and his work led directly (via Russell) to analytic philosophy and (via Husserl) to Continental philosophy.

The key passage, for philosophy in general, is on p. 133:

The Context Principle

Only in the context of a sentence do words really have a meaning.

He says, disapprovingly, "we consider words individually and ask about their meaing [in isolation]."

The use of a word is its use in sentences. It may do something complicated to the use of a sentence that isn't specifiable as a separable use for the word in isolation.

Frege's motivations came from mathematics, and indeed the main example that got him started was the differential (dx). (The fraction line might be a more elementary example that makes the same point.) The point has been applied, however, throughout every area of philosophy.

Frege applies the context principle in trying to say what numbers are. Saying what numbers are is, for Frege, part of giving a philosophical account of "arithmetic." The word "arithmetic" has a much broader meaning for Frege and his contemporaries than it does for us: he means all of the mathematics of numbers of any kind, thus including the calculus, all the mathematics used in physics.

There is one more thing that Frege claims, that he argued in passages not in what we read, that I need to explain briefly: "numbers are (almost) properties of concepts. What does that mean? Suppose you point at an encyclopedia and ask "How many?" The answer might be

• 1 encyclopedia
• 23 volumes
• 12597 pages

Things, of themselves, do not have numbers. They need to be carved up or individuated somehow. Not only are numbers abstract, what has number is already abstract.

For Frege, a "concept" is not something psychological, but simply what a predicate expressed, just as an "object" is what a subject (grammatical) expresses. Some, but not all, concepts "pick out" some objects that fall under them. The number associated with that concept is the number of those objects. It is probably more common, following Russell, to take numbers to be numbers of a class or set, a collection, than of a concept, but that is also abstract.

Note that for Frege, in Quine's joke, "ontology recapitulates philology": the division into objects and concepts just follows the grammatical division into subjects and predicates. A "property" is denoted by a predicate, and hence it is not an object, but numbers are denoted by names, and so they are objects. Thus, when I said that a number is a property, I lied.

Consider "the number of planets=8", a rendition of the more normal "there are 8 planets. The numerical property a concept %$F$% has is "%$n$% is the number of things that fall under %$F$%." The number appears in the predicate, but it is only part of the predicate.

The reason I'm going through this in detail is that it is Frege's argument that numbers exist and that they are objects. That is all there is to the question for Frege. That should be familiar from almost every course in contemporary philosophy you have ever taken. (157)

Finally, what are numbers? It would be a mistake to think about number words and try to say what they stand for. (Context principle.) In particular, though we have mental pictures associated with many words, that is an accident of human psychology and never has much to do with how the words are used. What we need to know is how number words function in sentences, and the central sentences in which number words are used are equations (that is, sentences with a certain common use of "is.") He therefore says that if we can understand how number words function in equations, we will know what numbers are. So, how do we use the equation

The number of %$F$%s = the number of %$G$%s

? We can say without knowing what numbers are! The equation is true if the things that fall under %$F$% can be paired off with the things that fall under %$G$%, that is, in the standard but misleading term of art, if the %$F$%s and the %$G$%s are equinumerous. We know what an equation means before we know what numbers are. If we then recarve the meaning, we get numbers.

What do I mean by recarve? From "%$x$% murders %$y$%," we get "%$x$% murders %$x$%," the notion of suicide.

He uses the simpler example of directions to get his point across. The direction of (line) %$x$% is the direction of (line) %$y$% just if %$x$% is parallel to %$y$%.

Frege goes on to indicate a complete development of arithmetic (in our, restricted, sense), and later gave a development of arithmetic in his broad sense on the basis of these ideas and pure logic. In particular, he proves that every number has a successor.

Frege thought he had placed all of mathematics, including existence questions, on a basis of pure logic, hence the name "logicism" for his view.

Unfortunately, it doesn't work for two reasons:

• the Julius Caesar problem
• the mathematics is self contradictory (Russell's paradox)

The Julius Caesar problem ("is Caesar the number of planets?") is not a problem for doing mathematics.

Russell's paradox: Easy version. Is the class of all classes that are not members of themselves a member of itself?

-- ShaughanLavine - 16 Jan 2007