Reduction and Elimination in Philosophy and the Sciences

From Topology to Logic.
The Neural Reduction of Compositional Representation
Markus Werning, Düsseldorf, Germany
When we look at the structure of thought, what we find is
logic. No matter what our starting point is: the semantic
analysis of linguistic expressions, the psychology of
cognition, or a philosophical theory of reasoning, we
usually arrive at some variant or extension of first order
predicate logic that characterizes the underlying structure
of thought. However, when we look at the cortex, what we
find is topology. The functional role of neurons is
determined by topological neighborhood relations. Given
that the various kinds of neurons are by and large
homogenously distributed over the cortex, the major
difference in the functional role of neurons is grounded in
which neurons are connected to each other, and which are
not. In topological terms: Who’s in the neighborhood of
whom. If we presuppose the materialist assumption that
the cortex is what brings about thought, any reductive
explanation has to show how the logical structure of
thought is necessitated by the topological organization of
information in the cortex.
Unfortunately, classical textbook mathematics is of
little help here, even though there are some theorems that
link topology to logic. Stone’s representation theorem, e.g.,
famously asserts the duality between the category of
Boolean algebras and the category of totally disconnected
compact Hausdorff spaces. We thus know how
propositional logics is to be represented topologically.
When our primary interest is in thought, though, first order
logic rather than propositional logic ought to be our main
concern. For, only first order logic (and its extensions)
provides the means to represent and categorize objects.
However, when it comes to first order logic, the
mathematical links to topology are sparse. Building on
previous work, this paper provides an explanation of how
the topological organization of the cortex yields a structure
expressible by (some intuitionist variant of) first order logic.
The explanatory bridges are the Gestalt principles of
perception and the physiological principles governing
object-related neural synchronization.
The Composition of Thought
The view of thought I appeal to characterizes the triangle
between language, mind and world roughly as follows:
Linguistic utterances are expressions of meaning. Meanings
are mental representations. More specifically, the
meanings of sentences are thoughts composed of concepts
by logical connectives. Concepts again have an
external content and this content is responsible for an utterance
having reference or denotation. The relation between
concepts and their content is some relation of covariation
– a causal-informational relation of sorts (Fodor,
1992). The denotation of an utterance is identical to (or
otherwise determined by) the content of the concept the
utterance is an expression of. This view is captured by our
first hypothesis:
Hypothesis 1 (Covariation with Content). An expression
has the denotation it has because the concept it expresses
reliably co-varies with a content that is identical
to the expression’s denotation.
Since languages and fore and foremost first order
languages have a rich constituent structure, it is rather
plausible to assume that the structure of their meanings is
complex, too, and that the structure of meanings in some
way or another resembles the structure of their
expressions. Now, the most simple way to spell out this
relation of resemblance is by means of a structural match,
in technical terms: a homomorphism. This homomorphism
is spelled out by the principle of the compositionality of
meaning:
Hypothesis 2 (Compositionality of Meaning). The meaning
of a complex expression is a syntax-dependent function
of the meanings of its syntactic constituents.
It would be surprising, furthermore, if the covariation relations
between primitive concepts and their contents should
not in some way or another contribute to the covariation
relations between complex concepts and their contents.
The quest for simplicity again leads us to the hypotheses
that the contents of the primitive concepts are the sole
factors to determine the content of a therefrom combined
complex concept. Again, this is just what the principle of
compositionality says for contents:
Hypothesis 3 (Compositionality of Content). The content
of a complex concept is a structure-dependent function
of the contents of its constituent concepts.
The aim of this paper is to make out a neuronal structure
that fulfills the three hypotheses. The neuronal structure
shall consist of a set of neuronal states and a set of
thereon defined operations. Since the three hypotheses
may serve as (minimal) identity criteria for concepts, their
fulfillment by a neuronal structure will justify us in identifying
the neuronal structure with a structure of concepts. The
three hypotheses hence form the adequacy conditions for
a neuronal reduction of concepts.
The Topology of the Cortex
For many attributes (color, orientation, size, speed, etc.)
involved in perceptual processing one can anatomically
identify cortical correlates. Those areas often exhibit a
twofold topological structure and justify the notion of a
feature map: (i) a receptor topology (e.g, retinotopy in vision,
somatotopy in touch): neighboring regions of neurons
code for neighboring regions of the receptive field; and (ii)
a feature topology: neighboring regions of neurons code
for similar attribute values. Due to physiological facts, this
twofold functional topology is reflected in the topography
(the physical distance relations) of the cortex.
With regard to the monkey, more than 30 cortical
areas forming feature maps are experimentally known for
vision alone (Felleman & van Essen, 1991). In fact, the
majority view among neuroscientists now is that the cortical
processing of vision below hippocampus is entirely
organized in the topological way described above. The
attributes involved can be very complex, though. Fig. 1
shows a number of neural maps that relate to perceptual
attributes.
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