I need help understanding Frege's definition of number

Question

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much appreciated. I will write his definition down explicitly, then give write down a little bit of what he writes around this definition.

Number - The content of a statement of number is an assertion about a concept.

[source : The Foundations of Arithmetic, §55, Engl.transl. page 67]

Here is what he writes before and immediately after stating his definition:

Immediately Before: While looking at one and the same external phenomenon I can say with equal truth both "It is a copse" and "It is five trees," or both "Here are four companies" and "Here are 500 men." Now what changes here from one judgement to the other is neither any individual object, nor the whole, the agglomeration of them, but rather my terminology. But that is itself only a sign that one concept has been substituted for another.

[source : The Foundations of Arithmetic, §46, Engl.transl. page 59]

Immediately After: This is perhaps clearest with the number $0$. If I say "Venus has $0$ moons", there simply does not exist any moon oor agglomeration of moons for anything to be asseted of; but what happens is that a property is assigned to the concept "moon of Venus", namely that of including nothing under it. If I say "the King's carriage is drawn by four hourses", then I assign the number four to the concept "horse that draws the King's carriage"

[source : The Foundations of Arithmetic, §46, Engl.transl. page 59]

Edit: For those who are more inclined to reading logic. On part 2.5 of the following link it explains what Frege was doing. Unfortunately, I'm having difficulties understanding this as well. I have been exposed to some first order logic, but no second order logic whatsoever. I am not sure if that is necessary though.

http://plato.stanford.edu/entries/frege/#NatNum

Thank you in advance for anyone who can help!

Answer 1

Some hints

The context of Frege's assertion is his analysis of the "number assertions"; see :

• Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884), translated as The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number by J.L. Austin (1950).

Roughly speaking, Frege denies that when we use number-words in natural language assertions we express a property of objects; i.e. when I say "my hands are white" I'm asserting that each of my hands is white, while when I'm saying "I have two hands", I'm not asserting that each of my hands "is two".

Thus, Frege's conclusion is that the "number-words" are like adjectives for concepts, i.e. they express a property of a concept.

Example : there are two satellites of Mars : Phobos and Deimos. When we say "Mars has two satellites", we express the fact that the concept "satellite of Mars" is instantiated by two objects.

In modern expression, we have that :

$\exists x \exists y [Sat_M(x) \land Sat_M(y) \land x \ne y \land \forall z(Sat_M(z) \to (z=x \lor z=y))]$.

This is also clear from the analysis of the number $0$ :

"if I say "Venus has $0$ moons", there simply does not exist any moon or agglomeration of moons for anything to be asseted of; but what happens is that a property is assigned to the concept "moon of Venus", namely that of including nothing under it."

i.e.

$\lnot \exists x Moon_V(x)$.

This point has fundamental consequences in Frege's view : numbers are akin to quantifiers.

In Frege's terminology, both number and quantifiers are second-level concepts, i.e. they are concepts that express a property of first-level concepts (like the concept : "moon of Venus", i.e. concepts "applying" to objects).

This fact can be highlighted by the "formal" analysis; we have that Phobos satisfy the concept "Satellite of Mars", i.e. $Sat_M(Phobos)$ holds.

In Frege's terminology, the object Phobos "falls under" the concept "Satellite of Mars".

The concept "Satellite of Mars" is instantiated, i.e. there are objects that "fall under" it. Thus the second-level concept "existence" applies to the first-level concept "Satellite of Mars", i.e. $\exists x Sat_M(x)$ holds.

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We can "formalize" Frege's view (in a "philological" inaccurate way) as follows :

1) assume the universal quantifier $\forall$ as primitive;

2) define the existential quantifier $\exists$ as usual, as : $\lnot \forall \lnot$;

3) introduce the "numerical" quantifiers :

i) $\exists^0$ defined through : $\lnot \exists Fx$ : "there are no $F$'s"

ii) $\exists^1$ defined as the "usual" $\exists !$, i.e. as $\exists x Fx \land \forall y (Fy \to x=y)$ : "there is (exactly) one $F$"

iii) $\exists^2$ as $\exists x \exists y (Fx \land Fy \land x \ne y \land \forall z(Fz \to z=x \lor z=y))$ : "there are two $F$'s"

and so on.

These "numerical" quantifiers are the formal counterpart of "number-words".

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You can see :

• Kevin Klement, Frege’s Changing Conception of Number, Theoria (2012).

Answer 2

Frege develops his answer in several stages throughout the Grundlagen.

The first step (which you are asking about) is ascertaining what it is that number-statements are about.

For Frege the world divided into objects and concepts. An object is a specific entity that might be describable in one of several ways. A concept is like a way of thinking about or describing things. So ‘blue’ is a concept, as is ‘moons of the planet Jupiter’. Any object could be blue or could not; it could be a moon of Jupiter or not.

In the sections you refer to, Frege points out that you cannot point to single object and say it is (or has) the number x. You can talk about a concept and say it has x instantiations. So you could say ‘Jupiter has 4 moons’ which is similar to saying ‘the concept “moon of Jupiter” is instantiated by 4 entities’.

Frege’s point is that statements involving numbers are always being predicated of concepts not of objects.

Things can seem a little more confusing when you hit sections 55-57. There he makes it clear that he believes that numbers are objects and not concepts and that can seem confusing.

But the key is to distinguish number-statements from numbers. Statements that involve numbers, when used in real-world (ie not mathematical applications) such statements are always about concepts. But the number word itself is never a concept.

Frege points out that a) a number word often is used as an object (eg in mathematical statements like 1+1=2, or ‘the number 5 is a prime number’ b) number words never appear as complete predicates/concepts. You can never describe a concept with just the number 4, you will always have to incorporate the number-term into a large statement to turn it into a predicate (so you can’t say ‘the car is 4’ but you can say ‘the car has 4 doors’) c) statements involving number can always be re-parsed so that the number term is separated from the predicate sentence into a distinct object that is being referred to. Eg ‘Jupiter has 4 moons’ is logically equivalent to ‘the number of objects that satisfy the predicate “moon of Jupiter” is 4’.

So for Frege numbers are abstract objects. Therefore any translation if arithmetic into logic (his logicist project to provide a foundation for mathematics) must find a way to define numbers as objects, which means to be able to specify conditions under which something can be recognised as a number or not without appeal to human intuition. They are also the type of objects that, outside of mathematical contexts, will be used in second order statements, ie those that assert a concept of another concept (usually how many objects instantiate the concept).

Frege ends up by using ‘extensions of [certain] concepts׳ since extensions (think of sets of things) are (logically) objects. They are things you could talk about and potentially they could have more than one concept they instantiate.

To avoid circularity (defining numbers in terms of numbers) Frege uses the notion of ‘equinumerosity’ (an equal number) which sounds like it uses numbers but it doesn’t. The reason is that I don’t need numbers to know if something is equinumerous. Even if I don’t know how many knives or forks there are on the table, if I see that every plate has one of each then I see that they can be matched up therefore they are equinumerous.

So the number 0 is the abstract object of things that instantiate the concept ‘equinumerous with “not being identical with itself” ‘ So the steps are: (i) there is a concept “not being identical with itself” (ii) there are no objects that instantiate that concept. This is an example of ‘nothing’ that is generated out of logical notions and no intuitive or ‘in the world’ entities or concepts. It therefore belongs to the realm of logic and maths. In that realm there are concepts that have no objects that could ever instantiate them. (iii) now we have a concept ‘equinumerous with the concept “not identical with itself” ‘ (iv) so what is/are the object/s that instantiate that? You can either say it the object we call 0.

Now that we have a (what Frege thought was a) purely logical way to define zero, we can also define the number one in a similar way. (i) the concept “identical with 0” (ii) clearly there is only one object there, 0. Anything identical with an object is the same object! (iii) now we can talk about the concept ‘equinumerous with to the concept “identical with 0” ‘ that is a concept. (iv) now we talk about the instantiation of that concept, or the ‘extension of the concept…’ and the object that instantiates it is the number 1.

More generally Frege defines the successor relation such that if any number is defined its successor can also be defined, in a similar way. (i) assuming we defined an object that is the number n, we can then talk about the concept ‘identical with 0, or 1, or… , or n’ (ii) the set of objects that will instantiate that will be precisely the numbers from 0 to n ie the integers up to and including n. (iii) then we employ the concept ‘equinumerous to the concept “identical with…” ‘ that will be the concept of being in one-to-one correspondence with the aforementioned set. Note the set of numbers up to n includes zero so it has n+1 members. (iv) the ‘extension of that concept’ is the object that is the number n+1.

So by the time he has finished, each number can be put back into a sentence where it is part of a ‘concept phrase’ that predicates another concept.

Eg the sentence ‘The set of integers from 0 to n’ defines an object much as ‘the Planet Jupiter’ is an object. ‘Equinumerous with the set of integers from 0 to n’ is a concept that relates to the object, much as ‘moons of the planet Jupiter’ does. Then the following are equivalent: ‘The object that is the extension of the concept “equinumerous with…” is the number 4’ Equivalent to ‘The number of things “equinumerous with…” is 4’ Equivalent to ‘There are 4 members of the set ‘equinumerous…”’

Again in the Jupiter analogy ‘The object that is the extension of the concept “moons of Jupiter” is the number 4’ Equivalent to ‘the number of things that are moons of Jupiter is 4’ (or the number of moons of Jupiter is 4’) Equivalent to ‘There are 4 moons of Jupiter’

Which is why Frege set it up that way in the first place.