The following is an informal exposition of some mathematical concepts from Topology via Logic, with special attention to philosophical implications. Those seeking more technical detail should simply read the book.
There are, roughly, two ways of doing topology:
- Point-set topology: Start with a set of points. Consider a topology as a set of subsets of these points which are "open", where open sets must satisfy some laws.
- Locale theory: Start with a set of opens (similar to propositions), which are closed under some logical operators (especially and and or), and satisfy logical relations.
What laws are satisfied?
- For point-set topology: The empty set and the full set must both be open; finite intersections and infinite unions of opens must be open.
- For local theory: "True" and "false" must be opens; the opens must be closed under finite "and" and infinite "or"; and some logical equivalences must be satisfied, such that "and" and "or" work as expected.
Roughly, open sets and opens both correspond to verifiable propositions. If X and Y are both verifiable, then both "X or Y" and "X and Y" are verifiable; and, indeed, even countably infinite disjunctions of verifiable statements are verifiable, by exhibiting the particular statement in the disjunction that is verified as true.
What's the philosophical interpretation of the difference between point-set topology and locale theory, then?
- Point-set topology corresponds to the theory of possible worlds. There is a "real state of affairs", which can be partially known about. Open sets are "events" that are potentially observable (verifiable). Ontology comes before epistemology. Possible worlds are associated with classical logic and classical probability/utility theory.
- Local theory corresponds to the theory of situation semantics. There are facts that are true in a particular situation, which have logical relations with each other. The first three lines of Wittgenstein's Tracatus Logico-Philosophicus are: "The world is everything that is the case. / The world is the totality of facts, not of things. / The world is determined by the facts, and by these being all the facts." Epistemology comes before ontology. Situation semantics is associated with intuitionist logic and Jeffrey-Bolker utility theory (recently discussed by Abram Demski).
Thus, they correspond to fairly different metaphysics. Can these different metaphysics be converted to each other?
- Converting from point-set topology to locale theory is easy. The opens are, simply, the open sets; their logical relations (and/or) are determined by set operations (intersection/union). They automatically satisfy the required laws.
- To convert from locale theory to point-set topology, construct possible worlds as sets of opens (which must be logically coherent, e.g. the set of opens can't include "A and B" without including "A"), which are interpreted as the set of opens that are true of that possible world. The open sets of the topology correspond with the opens, as sets of possible words which contain the open.
From assumptions about possible worlds and possible observations of it, it is possible to derive a logic of observations; from assumptions about the logical relations of different propositions, it is possible to consider a set of possible worlds and interpretations of the propositions as world-properties.
Metaphysically, we can consider point-set topology as ontology-first, and locale theory as epistemology-first. Point-set topology starts with possible worlds, corresponding to Kantian noumena; locale theory starts with verifiable propositions, corresponding to Kantian phenomena.
While the interpretation of a given point-set topology as a locale is trivial, the interpretation of a locale theory as a point-set topology is less so. What this construction yields is a way of getting from observations to possible worlds. From the set of things that can be known (and knowable logical relations between these knowables), it is possible to conjecture a consistent set of possible worlds and ways those knowables relate to the possible worlds.
Of course, the true possible worlds may be finer-grained than these consistent set; however, it cannot be coarser-grained, or else the same possible world would result in different observations. No finer potentially-observable (verifiable or falsifiable) distinctions may be made between possible worlds than the ones yielded by this transformation; making finer distinctions risks positing unreferenceable entities in a self-defeating manner.
How much extra ontological reach does this transformation yield? If the locale has a countable basis, then the point-set topology may have an uncountable point-set (specifically, of the same cardinality as the reals). The continuous can, then, be constructed from the discrete, as the underlying continuous state of affairs that could generate any given possibly-infinite set of discrete observations.
In particular, the reals may be constructed from a locale based on open intervals whose beginning/end are rational numbers. That is: a real r may be represented as a set of (a, b) pairs where a and b are rational, and a < r < b. The locale whose basis is rational-delimited open intervals (whose elements are countable unions of such open intervals, and which specifies logical relationships between them, e.g. conjunction) yields the point-set topology of the reals. (Note that, although including all countable unions of basis elements would make the locale uncountable, it is possible to weaken the notion of locale to only require unions of recursively enumerable sets, which preserves countability)
If metaphysics may be defined as the general framework bridging between ontology and epistemology, then the conversions discussed provide a metaphysics: a way of relating that-which-could-be to that-which-can-be-known.
I think this relationship is quite interesting and clarifying. I find it useful in my own present philosophical project, in terms of relating subject-centered epistemology to possible centered worlds. Ontology can reach further than epistemology, and topology provides mathematical frameworks for modeling this.
That this construction yields continuous from discrete is an added bonus, which should be quite helpful in clarifying the relation between the mental and physical. Mental phenomena must be at least partially discrete for logical epistemology to be applicable; meanwhile, physical theories including Newtonian mechanics and standard quantum theory posit that physical reality is continuous, consisting of particle positions or a wave function. Thus, relating discrete epistemology to continuous ontology is directly relevant to philosophy of science and theory of mind.