# Chomsky and Levi-Strauss

Thu, 14 Dec 1995 00:35:00 PST

Fischer has commented that Levi-Strauss and Chomsky are neither
derivative from the other. Let me add a couple more comments.
Chomsky and Levi-Strauss share the idea of structure, but are using
two quite different notions of structure. For Chomsky, structure
in the form of a grammar becomes a set of production rules that
begins with a completely abstract object; e.g. the symbols S (which
we can think of standing for Sentence) and ending up with a
concrete object; i.e., a particular sentence, via rules such as: S
--> Noun Phrase + Verb Phrase; Noun Phrase --> Noun; Verb Phrase --
> Very + Object; Noun --> {John, Mary}, Verb --> {hits, throws},
Object --> {the ball, the balloon}. So via these rules we can
generate sentence such as: John hits the balloon, Mary throws the
ball. Structure here is captured via the transformational rules.

In contrast, structure for Levi-Strauss is more like the example I
posted a while back about Friends and Enemies:

@ F -----> E @
<-----

where @ is the best I can do for an arrow that starts at a symbol
and then points to the same symbol. We can characterize this
structure several ways. One is as a permutation group on two
symbols. There are two possible permutations of two symbols: p1
leaves each symbol alone, and p2 interchanges the two symbols.
These are conventionally denoted as follows:

p1 = ( F E ) p2 = (F E)
F E E F

where the second row indicates how the symbols are interchanged by
the permutation. Graphically, p1 would be the "@" symbol, p2 the
"----->" and "<------" symbols. To say this is a group means that
under products of permutations, the set {p1, p2} is closed (e.g.,
p1 X p1 = P1, p1 X p2 = p2, p2 X p1 = p2, p2 x p2 = p1--now think
of p1 as Friend, P2 as Enemy and these equations become: Friend of
a Friend is a Friend, Friend of an Enemy is an Enemy, Enemy of a
Friend is an Enemy, Enemy of an Enemy is a Friend). Further, the
permutation p1 acts like an identity element, and p2 has an
inverse, namely itself since p2 x p2 = p1.

What all of these means is that the two permutations, p1 and p2,
along with a binary product of the permutations, forms an abstract
structure.

Another way to view this structure (and this will look a lot like
the idea of a binary opposition) is to assume each symbol has
associated with it a set of attributes and further, each attribute
associated with the first symbol can be paired with an attribute
for the second symbol which is the "opposite" attribute. Thus, if
we associate with the symbol Friend the attribute "helpful" we
might very well agree that the attribute "harmful" is an attribute
we would associate with Enemy. Similarly, we might agree that "to
be trusted" is an attribute we associate with the concept of Friend
whereas the attribute we associate with the concept of Enemy is
"not to be trusted."

So we have: Friend Enemy
A1 ~A1
A2 ~A2
A3 ~A3
etc etc

where "~A1" is the "opposite" of A1, etc. (This procedure was used
by El Guindi to determine what categories in Zapotec culture are in
opposition and she determined that House and Field are categories
in opposition as native ascribed attributes for House are the
"opposite" of attributes ascribed to Field.)

If we think of "O" as standing for opposition, we can formulate a
characterization of what is meant by "binary opposition" in the
following manner.

We can define two mappings: I maps an attribute to itself, and O
maps an attribute to its "opposite". Thus, I: A1 --> A1, I: ~A1 --
> ~A1, etc. and O: A1 --> ~A1, O: ~A2 --> A2, etc. The structure
generated by I and O is isomorphic to the structure generated by p1
and p2, by Friend and Enemy, by +1 and -1 using multiplication, by
0, 1 using binary addition, etc.

The point to be noted here is that this is a very different notion
of structure than that used by Chomsky. There is no "production"
in the sense used by Chomsky.

Rather, Levi-Strauss's notion of structure (and this is even
clearer for the notion of structure used by Piaget) comes, I suggest,
from the work of the Bourbaki school of French mathematicians. The Bourbaki
school has had as its goal the reduction of all of mathematics to,
roughly, axiomatically defined structures where the notion of
structure is captured in the ways in which symbols may be combined,
or "multiplied," not in the content of the symbols per se.

While it is true that Levi-Strauss's clearest associations are with
the structural linguists such as Troubetzkoy, it is evident that
the algebraic notion of structure developed by the Bourbaki school
has had substantial influence as well. It is no coincidence that
Levi-Strauss asked a mathematician, Andre Weil, to provide an
appendix to his Les Structures Elementaire de la Parente, to show
how a marriage system such as practiced by the Murngin constitutes
an abstract structure that can be precisely modeled using the
notion of axiomatically defined algebraic structures.

Levi-Strauss comments, with regard to Troubetzkoy the following:
"N. Troubetzkoy, the illustrious founder of structural linguistics,
himself furnished the answer to this question [What will be the
revolutionary role played by structural linguistics analogous to
the role that nuclear physics played for the physical sciences].
In one programmatic statement, he reduced the structural method to
four basic operations. First, structural linguistics shifts from
the study of conscious linguistic phenomena to study of their
unconscious infrastructure; second, it does not treat terms as
independent entities, taking instead as its basis of analysis the
relations between terms; third, it introduces the concept of
system" and Levi-Strauss also comments: "structures are models,
the formal properties of which can be compared independently of
their elements" [e.g., the Friend-Enemy structure, the permutation
structure, the -1, +1 structure, etc. are isomorphic as structures
even though what the arrows "mean" and what is the content of the
symbols is different from one structure to the other]. The problem
with Levi-Strauss is that his definitions are (in my opinion)
"bastardized" versions of formal definitions of structures as used,
for example, by the Bourbaki school of mathematics (and throughout
modern algebra). Thus, Levi-Strauss writes: "what kind of model
deserves the name 'structure.' This is not an anthropological
question, but one which belongs to the methodology of science in
general. Keeping this in mind, we can say that a structure consists
of a model meeting with several requirements. First, the structure
exhibits the characteristics of a system. It is made up of several
elements, none of which can undergo a change without effecting
changes in all the other elements. [This is not exactly crystal
clear!] Second, for any given model there should be a possibility
of ordering a series of transformations resulting in a group of
models of the same type. [This appears to be taken directly from
the algebraic notion of structures.] Third, the above properties
make it possible to predict how the model will react if one or more
of its elements are submitted to certain modifications. [Again,
this is problematic depending upon what kind of structure one
imagines Levi-Strauss has in mind.] Finally, the model should be
constituted so as to make immediately intelligible all the observed
facts. [Problematic here is WHAT set of 'observed facts' are
relevant."

In short, it is not a very clear notion of structure; e.g., in part
it sounds like structure in the sense of a system; in other parts
it clearly is coming from the notion of structure developed by the
Bourbaki school. But what ever is meant, it definitely is NOT the
same as the notion of structure used by Chomsky.