Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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From Topology to Logic.<br />
The Neural <strong>Reduction</strong> of Compositional Representation<br />
Markus Wern<strong>in</strong>g, Düsseldorf, Germany<br />
When we look at <strong>the</strong> structure of thought, what we f<strong>in</strong>d is<br />
logic. No matter what our start<strong>in</strong>g po<strong>in</strong>t is: <strong>the</strong> semantic<br />
analysis of l<strong>in</strong>guistic expressions, <strong>the</strong> psychology of<br />
cognition, or a philosophical <strong>the</strong>ory of reason<strong>in</strong>g, we<br />
usually arrive at some variant or extension of first order<br />
predicate logic that characterizes <strong>the</strong> underly<strong>in</strong>g structure<br />
of thought. However, when we look at <strong>the</strong> cortex, what we<br />
f<strong>in</strong>d is topology. The functional role of neurons is<br />
determ<strong>in</strong>ed by topological neighborhood relations. Given<br />
that <strong>the</strong> various k<strong>in</strong>ds of neurons are by <strong>and</strong> large<br />
homogenously distributed over <strong>the</strong> cortex, <strong>the</strong> major<br />
difference <strong>in</strong> <strong>the</strong> functional role of neurons is grounded <strong>in</strong><br />
which neurons are connected to each o<strong>the</strong>r, <strong>and</strong> which are<br />
not. In topological terms: Who’s <strong>in</strong> <strong>the</strong> neighborhood of<br />
whom. If we presuppose <strong>the</strong> materialist assumption that<br />
<strong>the</strong> cortex is what br<strong>in</strong>gs about thought, any reductive<br />
explanation has to show how <strong>the</strong> logical structure of<br />
thought is necessitated by <strong>the</strong> topological organization of<br />
<strong>in</strong>formation <strong>in</strong> <strong>the</strong> cortex.<br />
Unfortunately, classical textbook ma<strong>the</strong>matics is of<br />
little help here, even though <strong>the</strong>re are some <strong>the</strong>orems that<br />
l<strong>in</strong>k topology to logic. Stone’s representation <strong>the</strong>orem, e.g.,<br />
famously asserts <strong>the</strong> duality between <strong>the</strong> category of<br />
Boolean algebras <strong>and</strong> <strong>the</strong> category of totally disconnected<br />
compact Hausdorff spaces. We thus know how<br />
propositional logics is to be represented topologically.<br />
When our primary <strong>in</strong>terest is <strong>in</strong> thought, though, first order<br />
logic ra<strong>the</strong>r than propositional logic ought to be our ma<strong>in</strong><br />
concern. For, only first order logic (<strong>and</strong> its extensions)<br />
provides <strong>the</strong> means to represent <strong>and</strong> categorize objects.<br />
However, when it comes to first order logic, <strong>the</strong><br />
ma<strong>the</strong>matical l<strong>in</strong>ks to topology are sparse. Build<strong>in</strong>g on<br />
previous work, this paper provides an explanation of how<br />
<strong>the</strong> topological organization of <strong>the</strong> cortex yields a structure<br />
expressible by (some <strong>in</strong>tuitionist variant of) first order logic.<br />
The explanatory bridges are <strong>the</strong> Gestalt pr<strong>in</strong>ciples of<br />
perception <strong>and</strong> <strong>the</strong> physiological pr<strong>in</strong>ciples govern<strong>in</strong>g<br />
object-related neural synchronization.<br />
The Composition of Thought<br />
The view of thought I appeal to characterizes <strong>the</strong> triangle<br />
between language, m<strong>in</strong>d <strong>and</strong> world roughly as follows:<br />
L<strong>in</strong>guistic utterances are expressions of mean<strong>in</strong>g. Mean<strong>in</strong>gs<br />
are mental representations. More specifically, <strong>the</strong><br />
mean<strong>in</strong>gs of sentences are thoughts composed of concepts<br />
by logical connectives. Concepts aga<strong>in</strong> have an<br />
external content <strong>and</strong> this content is responsible for an utterance<br />
hav<strong>in</strong>g reference or denotation. The relation between<br />
concepts <strong>and</strong> <strong>the</strong>ir content is some relation of covariation<br />
– a causal-<strong>in</strong>formational relation of sorts (Fodor,<br />
1992). The denotation of an utterance is identical to (or<br />
o<strong>the</strong>rwise determ<strong>in</strong>ed by) <strong>the</strong> content of <strong>the</strong> concept <strong>the</strong><br />
utterance is an expression of. This view is captured by our<br />
first hypo<strong>the</strong>sis:<br />
Hypo<strong>the</strong>sis 1 (Covariation with Content). An expression<br />
has <strong>the</strong> denotation it has because <strong>the</strong> concept it expresses<br />
reliably co-varies with a content that is identical<br />
to <strong>the</strong> expression’s denotation.<br />
S<strong>in</strong>ce languages <strong>and</strong> fore <strong>and</strong> foremost first order<br />
languages have a rich constituent structure, it is ra<strong>the</strong>r<br />
plausible to assume that <strong>the</strong> structure of <strong>the</strong>ir mean<strong>in</strong>gs is<br />
complex, too, <strong>and</strong> that <strong>the</strong> structure of mean<strong>in</strong>gs <strong>in</strong> some<br />
way or ano<strong>the</strong>r resembles <strong>the</strong> structure of <strong>the</strong>ir<br />
expressions. Now, <strong>the</strong> most simple way to spell out this<br />
relation of resemblance is by means of a structural match,<br />
<strong>in</strong> technical terms: a homomorphism. This homomorphism<br />
is spelled out by <strong>the</strong> pr<strong>in</strong>ciple of <strong>the</strong> compositionality of<br />
mean<strong>in</strong>g:<br />
Hypo<strong>the</strong>sis 2 (Compositionality of Mean<strong>in</strong>g). The mean<strong>in</strong>g<br />
of a complex expression is a syntax-dependent function<br />
of <strong>the</strong> mean<strong>in</strong>gs of its syntactic constituents.<br />
It would be surpris<strong>in</strong>g, fur<strong>the</strong>rmore, if <strong>the</strong> covariation relations<br />
between primitive concepts <strong>and</strong> <strong>the</strong>ir contents should<br />
not <strong>in</strong> some way or ano<strong>the</strong>r contribute to <strong>the</strong> covariation<br />
relations between complex concepts <strong>and</strong> <strong>the</strong>ir contents.<br />
The quest for simplicity aga<strong>in</strong> leads us to <strong>the</strong> hypo<strong>the</strong>ses<br />
that <strong>the</strong> contents of <strong>the</strong> primitive concepts are <strong>the</strong> sole<br />
factors to determ<strong>in</strong>e <strong>the</strong> content of a <strong>the</strong>refrom comb<strong>in</strong>ed<br />
complex concept. Aga<strong>in</strong>, this is just what <strong>the</strong> pr<strong>in</strong>ciple of<br />
compositionality says for contents:<br />
Hypo<strong>the</strong>sis 3 (Compositionality of Content). The content<br />
of a complex concept is a structure-dependent function<br />
of <strong>the</strong> contents of its constituent concepts.<br />
The aim of this paper is to make out a neuronal structure<br />
that fulfills <strong>the</strong> three hypo<strong>the</strong>ses. The neuronal structure<br />
shall consist of a set of neuronal states <strong>and</strong> a set of<br />
<strong>the</strong>reon def<strong>in</strong>ed operations. S<strong>in</strong>ce <strong>the</strong> three hypo<strong>the</strong>ses<br />
may serve as (m<strong>in</strong>imal) identity criteria for concepts, <strong>the</strong>ir<br />
fulfillment by a neuronal structure will justify us <strong>in</strong> identify<strong>in</strong>g<br />
<strong>the</strong> neuronal structure with a structure of concepts. The<br />
three hypo<strong>the</strong>ses hence form <strong>the</strong> adequacy conditions for<br />
a neuronal reduction of concepts.<br />
The Topology of <strong>the</strong> Cortex<br />
For many attributes (color, orientation, size, speed, etc.)<br />
<strong>in</strong>volved <strong>in</strong> perceptual process<strong>in</strong>g one can anatomically<br />
identify cortical correlates. Those areas often exhibit a<br />
twofold topological structure <strong>and</strong> justify <strong>the</strong> notion of a<br />
feature map: (i) a receptor topology (e.g, ret<strong>in</strong>otopy <strong>in</strong> vision,<br />
somatotopy <strong>in</strong> touch): neighbor<strong>in</strong>g regions of neurons<br />
code for neighbor<strong>in</strong>g regions of <strong>the</strong> receptive field; <strong>and</strong> (ii)<br />
a feature topology: neighbor<strong>in</strong>g regions of neurons code<br />
for similar attribute values. Due to physiological facts, this<br />
twofold functional topology is reflected <strong>in</strong> <strong>the</strong> topography<br />
(<strong>the</strong> physical distance relations) of <strong>the</strong> cortex.<br />
With regard to <strong>the</strong> monkey, more than 30 cortical<br />
areas form<strong>in</strong>g feature maps are experimentally known for<br />
vision alone (Felleman & van Essen, 1991). In fact, <strong>the</strong><br />
majority view among neuroscientists now is that <strong>the</strong> cortical<br />
process<strong>in</strong>g of vision below hippocampus is entirely<br />
organized <strong>in</strong> <strong>the</strong> topological way described above. The<br />
attributes <strong>in</strong>volved can be very complex, though. Fig. 1<br />
shows a number of neural maps that relate to perceptual<br />
attributes.<br />
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