Meaning and Geometry

Concepts do not seem really to refer, at least not in the way definite descriptions seem to.  Not only do not concepts refer to their extensions, but extensions themselves are vague, incomplete, fluid, ambiguous and modally conditioned. Worse than that, we habitually make extensions into objects of reference, although in a qualified way. For instance 'all people alive today' may claim the title of extension of the concept 'people'.  Individuals themselves are problematic, knots and shifting meshes of unified meanings. But if we look at concepts qua meaning - something which is not new -  concepts in terms of comprehension, then there are some suggestive mathematical analogies.  The correspondence between radical ideals of a finitely generated $k$-algebra and algebraic sets in algebraic geometry. This is a kind of Galois correspondence between the lattice of such ideals and the poset of algebraic spaces (and there is a similar result for commutative $C^*$-algebras). A contravariant functor $M : \mathcal{A} \rightarrow \mathcal{B}$ so that what is 'greatest' in comprehension (i.e. a maximal ideal) corresponds to what is smallest in extension (a point); for $k$ an algebraically closed field we have that the whole space in turn  corresponds to the empty set.  In the case of a commutative ring $R$ this duality is somehow entirely contained within the ring itself; for we can look at $spec(R)$ as a space of ideals (meanings) or at the Zariski topology on $spec(R)$, making $spec(R)$ a topological space. Thus we can look at the elements of this topology (either open or closed set) as spaces or extensions.  A similar situation exists in Galois theory in which we are concerned with  subgroups of $\Gamma(N,K)$  for $N$ a separable normal extension of $K$.  The bigger the subgroup the smaller the corresponding fixed field $K< L < N$.  Group actions are perhaps the analogues of meaning while the spaces they act on (which can be the group itself) are like extensions (in particular group-action orbits are like extensions, they express the unfolding of the content, the possible interpretations of the group on a given point).  Also in forcing techniques it is usual, given a forcing poset $(\mathbb{P}, \leq, 1)$,  to write $p\leq q$ and interpret it as $p$ expressing more information than $q$ or as $p$ extending in a consistent way the information provided by $q$. The idea of $p$ being smaller stems from the fact that the range of possible 'ideal' objects corresponding to $p$ becomes smaller when $p$ conveys more information than $q$.

Modality involves a continuous variation, in particular an imaginary deformation of actual circumstance or patterns of experience. It is difficult not to compare this to the concept of moduli space and deformation in mathematics. If group actions express the unfolding of extensions of concepts then this is modally constrained. Hence it is more natural to consider moduli spaces of group actions (or deformations of rings) rather than merely the group action itself.

We also mention that some structures seems to have inexhaustible meaning. That is they generate in a canonical way potentially infinite (though often periodic) sequences. A famous example is the open problem of the higher homotopy groups of the sphere.

One of the biggest errors in philosophy is the idea that by formally analyzing the grammar of a language (and in particular modern English) somehow logical, semantic and ontological knowledge will be handed to us on a plate (and Frege was keenly aware that this is a mistake).