Sans objet

according to Frege "the number assigned to the concept F is the extension=
 of
the concept identical to the concept F". Frege's ternary system has as it=
s
effect that all that is left to the thing is the support of its identity
with itself, by which it is the object of the operative concept, and henc=
e
numerable.

The process that I have just set out authorizes me to conclude the follow=
ing
proposition, whose relevance will emerge later, - the unit which could be
called unifying of the concept in so far as it is assigned by the number =
is
subordinate to the unit as distinctive in so far as it supports the numbe=
r.

As for the position of the distinctive unit. its foundation is to be
situated in the function of identity which, conferring on each thing of t=
he
world the property of being one. effects its transformation into an objec=
t
of the (logical) concept.

At this point in the construction, you will sense all the importance of t=
he
definition of identity which I am going to present.

This definition which must give its true meaning to the concept of number=
,
must borrow nothing from it
[4]<http://www.lacan.com/symptom8_articles/miller8.html#_ftn4>-
precisely in order to ^ engender numeration.

_ This definition, which is pivotal to his system. Frege takes from Leibn=
iz.
It is contained in this statement: *eadem sunt quorum unum potest substit=
ui
alteri salva veritate*. Those things are identical of which one can be
substituted for the other *salva veritate* without loss of truth. Doubtle=
ss
you can estimate the crucial importance of what is effected by this
statement: the emergence of the function of truth. Yet what it assumes is
more important than what it expresses. That is, identity-with-itself. Tha=
t a
thing cannot be substituted for itself, then where does this leave truth?
Absolute is its subversion.

If we follow Leibniz's argument, the failing of truth whose possibility i=
s
opened up for an instant, its loss through the substitution for one thins=
 of
another, would be followed by its immediate reconstitution in a new
relation: truth is recovered because the substituted thing, in that it is
identical with itself, can be the object of a judgement and enter into th=
e
order of discourse: identical with itself, it can be articulated.

But that a thing should not be identical with itself subverts the field o=
f
truth, ruins it and abolishes it.

You will grasp to what extent the preservation of truth is implicated in
this identity with itself which connotes the passage from the thing to th=
e
object. Identity-with-itself is essential if truth is to be saved.

Truth is. Each thing is identical with Itself.

Let us now put into operation Frege's schema, that is, go through the
three-stage itinerary which he prescribes to us. Let there be a thing X o=
f
the world. Let there be the empirical concept of this X. The concept whic=
h
finds a place in the schema is not this empirical concept but that which
redoubles it, being "identical with the concept of X". The object which
falls under this concept is X itself" as a unit. In this the number, whic=
h
is the third term of the sequence, to be assigned to the concept of X wil=
l
be the number 1. Which means that this function of the number 1 is
repetitive for all things of the world. It is in this sense that this 1 i=
s
only the unit which constitutes the number as such. and not the 1 in its
personal identity as number with its own particular place and a proper na=
me
in the series of numbers.

Furthermore, its construction demands that. in order to transform it. we
call upon a thing of the world - which, according to Frege. cannot be: th=
e
logical must be sustained through nothing but itself.

In order for the number to pass from the repetition of the 1 of the
identical to that of its ordered succession, in order for the logical
dimension to gain its autonomy definitively, without any reference to the
real, the zero has to appear.

Which appearance is obtained because truth is. Zero is the assigned to th=
e
concept "not identical with itself". In effect, let there be the concept
"not identical with itself". This concept, by virtue of being a concept, =
has
an extension, subsumes an object. Which object? None. Since truth is, no
object falls into the place of the subsumed of this concept, and the numb=
er
which qualifies its extension is zero.

In this engendering of the zero. I have stressed that it is sup- ported b=
y
the proposition that truth is. If no object falls under the concept of
non-identical-with-itself. it is because truth must be saved. If there ar=
e
no things which are not identical with them- selves, it is because
non-identity with itself is contradictory to the very dimension of truth.=
 To
its concept, we assign the zero. It is this decisive proposition that the
concept of not-ldentical-with- itself is assigned by the number zero whic=
h
sutures logical discourse.

For, and here I am working across Frege's text. in the auto- nomous
construction of the logical through itself, it has been necessary, in ord=
er
to exclude any reference to the real, to evoke on the level of the concep=
t
an object not-identical-with-itself - to be subsequently rejected from th=
e
dimension of truth.

The zero which is inscribed in the place of the number con- summates the
exclusion of this object. As for this place, marked out by subsumption, i=
n
which the object is lacking, there nothing can be written, and if a o mus=
t
be traced, it is merely in order to figure a blank, to render visible the
lack.