This piece appears in the first volume of Form & Content. It is some variation on a presentation that Duroux gave on 27 January 1965 during Lacan’s seminar at the Ecole Normale Superieure. It was meant as an introduction to Jacques-Alain Miller’s paper on Frege’s logic, and, more specifically, The Foundations of Arithmetic (1884). In this work, we are told, Frege developed a definition of the number zero and of the whole of numbers – finally, he described something called a ‘successor function’. The ‘successor function’ deals with the question: ‘how is it that we can count from 1 to 2, and from 2 to 3, and so on. We’ve seen the same question examined by Alain Badiou in his work. We’ve seen that Badiou wants to first describe the logic of succession (of the ‘count’) and second the logic of the axiom of second infinity (which allows us to break with the logic of succession itself).
In any case, we are dealing with whole numbers. And, to repeat Duroux’s questions, we are asking: (1) what is a number (eg., what is the number 1, what is the number 2?), (2) what is zero?, and (3) what is the successor?
These are the key questions. And, really, once we come to understand the answers to these questions we begin to realize that it is all very basic. But not immediately.
First: if we assume that zero is not separate from the rest of whole numbers then the real question is: what is a number?, and how do we pass from one number to another number? Another way of stating the first question is: how do we move from a collection of real things to a numeric representation of that collection of real things?
The first is a question of “collection” and the second is a question of “addition”.
The “unit” is an undifferentiated and undetermined element of any given “set”. I’m going to diagram this:
Undifferentiated/Undetermined <– Collection
Unit <– Set
A unit can also be the name of One, the number One.
One <– 1
One, One <– 2
When we say that there is one horse and one horse and one horse – each one horse is a unit. Each is an element of a set of the three horses.
Horse, Horse, Horse <– 3 Horses
There is nothing intrinsic to the units – one horse, one horse, one horse – that allows us to use the number 3 to identify the collection. There is nothing inside the units that gives rise to the number three for the set. The only way to use the number three is to impose it on the units by force.
One Horse + One Horse + One Horse = 3 Horses
Can only be said if we make two modifications: (1) we must presume that the “one” is a number, and (2) we must transform the “and” of “one horse and one horse and one horse” into an operation, given the symbol of the plus “+”.
What we therefore discover is that by repeating “one horse” we give rise to something new. It is through the repetition that a new signification comes into existence. We get something new which is not itself a repetition. For example, one plus one plus one, are a series of ones which give us a new number not already there: three.
We must suppose a subject here, one capable of operating on the succession of numbers and giving name to them.
Frege separates the domain of representation into the psychological or subjective side and the objective side. But this separation by Frege effaces the subjective side in favor of the objective side of laws and logic. And this, as we shall see, gives rise to a strange logic, one that, it seems, Frege does not himself acknowledge.
Frege provides the example “venus has no moon” What does “no” signify here? We do not, in this sentence, according to Frege, attribute the “no” to the object moon. This is because there is no moon. A moon does not exist, so we can not negate it. But this leaves open the question of “zero”. Is zero a negation? Can it be if there was nothing there to begin with? We have a concept of a “venus moon”. And this concept is related to the object “moon”. But the relation is such that there is “no moon”. Number, for Frege, belongs to “concept”.
Object <– Concept
No Moon <– 0 Moons
This does not give us an individual number though. There is no “the” one, “the” two, “the” three. For Frege, there is only “one one”, “one two,” “one three”, and so on. The numbers are not unique – they are all only “one”.
One, One, One <– One “3”
Frege moves forward. He gives another example of planets and moons: “Jupiter has four moons.” Here the number “four” is equal to the number of moons. It is identical to the number of Jupiter’s moons. To the concept of Jupiter’s moons there is the number four.
Frege posits a primordial relation of equivalence or identity. It describes a logical relation that enables one to order objects or concepts in a one-to-one correspondence.
“Socrates” = Object
“is a philosopher” = Concept
Socrates <– a philosopher
The two are equivalent – Socrates already is a philosopher, and a philosopher already is socrates. There is nothing remaining here, the latter is a designation of what is right there – it is not imposed from the outside, it is a natural consequence of a correspondence with Socrates.
One we posit the relation of equivalence we can arrive at a second definition of number: the number that belongs to the concept f is the extension of the concept equivalent to the concept F.
“one horse, one horse, one horse” – Object (F)
“three horses” – Concept f (extension of F)
This means we have posited a determined concept F; we have determined through the relation of equivalence, all the equivalences of this concept F; we then define the number as the extension of this concept equivalent to the concept F (all the equivalences of the concept F).
So the concept, three horses, is determined by its object, in this understanding.
determination (object) <– determined (concept)
How do we obtain the number zero?
Frege thinks the idea of that which is not-identical-to-itself. Initially, Frege imagines that any contradictory statement refers to zero. For example: “Socrates is a horse” has an Object which is “Socrates” and a concept which is “Horse” and this just is not true. So the concept is zero in this case.
Socrates <– Horse = 0
Now the question of addition.
One number follows another in a series if “this number is attributed to a concept under which falls an object” So, 1 follows 0 because 1 as an object is attributed to the concept “equal to zero”.
So there is a contradiction here in the succession from 0 to 1.
0 = contradiction (Socrates <– Horse)
1 = 0
0 (Socrates <– Horse) = 1 [ = 0]
First: zero is defined as a contradiction. It is the concept for contradiction.
Second: the movement from 0 to 1 is a contradiction of contradiction.
Thus, the motor of Frege’s addition is a ‘negation of negation’.