# Multigraphs, hypergraphs, and the epistemic regress

Some definitions (from what I can tell):

1. A multigraph is a graph where a node can connect via multiple edges.
2. A hypergraph is a graph where a single edge can connect more than two nodes. Alternatively, there are nodeless edges available in this domain (not sure if nodeless edges occur in the base of graph theory).

And so there are presumably fusions, etc. In the base theory of graphs, it seems as if implementing the sethood relation by whatever is formally possible "at hand" gives us images of foundationalistic, coherentistic, and infinitistic epistemic graphs, and then there are hybrid positions (e.g. infinite-step loops). But what, if any, are the forms of solutions to the epistemic regress, that can be defined from multigraph and hypergraph theory in turn?

For example, use a graph with a finite number of nodes (let's just use two for ease of representation) but say that there are 200 multigraphical edges connecting them. Contrast this with a pair of nodes with only 10 such edges. The epistemic flavor of the pair is foundationalistic, but would judging the first pair to be a better solution to a local regress problem be more coherentistically seasoned? Or, rather, is the relationship between multigraphical concepts and the forms of epistemic regresses much of an aside from questions like foundationalism-vs.-coherentism?

Nodeless edges, and then multiplicities of these edges, also seem to represent a much more precise, if not totally alternative, form of possible solutions to regress problems, at least assuming that all such graph-theoretic concepts can be used to characterize epistemic concepts. But again, I'm not (yet) seeing much of a direct link to the base epistemic types.

Another way to frame the question: Alessio Moretti says that his "geometry of logic" is not just a newfangled graph theory. However, I don't know that he does or even can rule out multigraphical/hypergraphical characterizations of his theory. In fact, a universal Moretti object in which all the Moretti crystal-fractals for each type of logic occur "as one" would seem to be profitably describable in such terms, with a crystal (on whichever level) with n nodes (for n logical operators on that level) for a given logic-type LT1 having the crystal for some other LT2, with < n nodes, embedded in it as a subcrystal, and the embedding being via some of the overcrystal's nodes being multigraphically connected (the different edges carrying the operational differences between the logics of the overcrystal and its subcrystals).

NOTE: I have used the foundations-of-mathematics tag for this post on account of the role that graph theory plays in foundational studies of mathematics, e.g. in the formation of set theory, and because if graph theory is relevant to the regress problem in the given manner, then graph theory is relevant to the form of the very question of mathematical foundations anyway.

EDIT: so far, I have found this paper about something they call "uncertainty theory," which seems to be an epistemological matter on some level. But so they tie in hypergraph theory with this matter of uncertainty.

2: one of the first linked essays does bring up hypergraphs in this connection, although I didn't recall that it does (it's been a while since I read the full texts). See also Conifold's comment below for a possible example from C. S. Peirce's work (and keep Peirce's work in graph theory in mind overall).

• Your coherentism link uses directed hypergraphs (section 5), and argues that they provide a new version of coherentism resistant to circularity objections. It reminded me Peirce's idea from Some Consequences of Four Incapacities:"trust rather to the multitude and variety of its arguments than to the conclusiveness of any one... reasoning should not form a chain which is no stronger than its weakest link, but a cable whose fibers may be ever so slender, provided they are sufficiently numerous and intimately connected." Mar 1 at 10:37
• Added modeling because MG/HGs would likely be used to describe epistemic examples.
– J D
Mar 1 at 17:18

As a preface and to affirm your justification, category theory is absolutely foundations of mathematics, and graphs and sets are models both generalized over by it; when you talk about the semantic interpretation of categories in terms of natural language semantics such as epistemic regress, then it should be obvious that category theory is a model of epistemic concerns, and an alternative way to formulate epistemological matters symbolically. Onward!

[W]hat, if any, are the forms of solutions to the epistemic regress, that can be defined from multigraph and hypergraph theory in turn?

Sounds to me you're trying to create an isomorphism between epistemic regress, a semantic concept of natural language, and abstractions of graph theory. If such an isomorphism exists (and I don't see why you couldn't argue it exists since concept maps are instances of graph-theoretic abstractions that are in wide currency), it would have to be understood in terms of creating an isomorphism as extending the model to MG/HGs. Let's see if we can't kick around some sort of example that relates MG/HGs to real world epistemic challenges. I'll do it with computer science which is concerned with manipulating Turing machines through sequences, iterations, and conditionals of operations in procedural languages.

I was just having a back and forth with Frank the other day about the circularity of the implementation of factorial functions as an example of impredicativity. His chosen response was to argue (in foundationalist manner), that n! being defined by n*(n-1)! wasn't circular because n! is not identical to (n-1)! and that if one follows the chain of reason down to the base case (the foundation), then clearly there is no circularity in meaning nor is there infinite regress. And yet, his analysis smells fishy to me because there is something both circular and potentially infinite in recursion. For instance, one could define n! in terms of n*(n!/n) which is tautological and which would certainly run on a machine infinitely, since each call to n! as a procedure would never return a value for n!. Now, can we use MG/HGs in our computational infinite loop as a model? Intuitively, I say yes.

In a standard application of graph theory, we would simply draw a directed loop edge, an edge directed away from and back to the same node for a recursive procedure. The directed loop would be the call to the functional definition and the node would represent an instance of the function executing. Of course, no real change in state is implicated by the node returning to itself. But, this graph-theoretic model can be improved to include more detail. The infinite loop doesn't happen once, temporally, but an infinite times, so tracing motion along a path, there could be an infinite number of directed loop edges, one for each call to the procedure at some time. That is, we could have an infinite number of directed loop edges paired with (R,t) where R stands to the reference of the call on the stack and t stands for the time at which the call is made. And if we have an infinite number of these directed loops (drawn slightly larger and concentric with the first call), clearly we're dealing with a MG! If the node never changes state, it's a good idea to use the same node (as opposed to a call to (n-1)! which would require more nodes given that the argument passed from one node to the next as a parameterized function call would change the value the procedure is working on).

Obviously, we have multiple sorts of formalisms involved in this example: the pictoral graph itself is graph-theoretic, a set of relations (R,t) which is set-theoretic, a computer science semantics of your choice such as algebraic or operational semantics could be used to model the execution looking for logical correctness of state transition. But, that's only half. We are now modeling with MGs. Let's see if we can't extend the natural language semantics to allow for the incorporation of a HG.

If epistemic regress can be modeled by the lambda calculus, and the lambda calculus provides practical solutions to problems associated with epistemic regress, then yes one could say that using a MG/HGs, for instance, since PLACING CONSTRAINTS on it is semantically equivalent to solving epistemic problems associated with regress. In the case of the infinite loop above, if having a foundation is a necessary condition in the reasoning process (for instance, having a base case TO AVOID infinite regress in computation in this factorial example), then constraints on the number of directed loops (max==10) in a real-world problem like the execution of code provides an exit condition, which is a semantic event. HGs simply correspond to a conditional assessment of state such as having an accumulator reach a state that causes the regress to terminate with an error message (ie "ErrorInfiniteLoop") that has a natural language semantics useful to the programmer (ie "Pay attention because it seems like you don't have a terminating condition).

But we don't have to stop there. In a HG, syntactic conditions which appear mathematically as piecewise functions or case-statments/nested-ifs in computer-science-speak would have not just forks (either-this-or-that executes), but could have any number of transitions in state. So max==5, max==10, max==100 could be used to conditionally branch to other procedures. On 5 iterations, execute a warning to the user ("5 iterations and it hasn't terminated"). On 10 iterations, check to see if user wants to continue, and then on 100 iterations terminate with the error message. Viola! We now have a MG/HG because the state in the procedure (represented by a node) maps to outcomes of a generalized arity. The following data structures would represent the MG/HGs edges.

(R1,t1) CALLING PROCEDURE (initial point of edge)
{(state1,...),...} STATE USED IN CONDITIONAL (unique IDs of edges)
{(R2,t2),...} TARGET PROCEDURES BASED ON STATE LOGIC (set of terminal points of edges)

So, we have a node that has many directed loops to itself, but we also have a series of directed edges out to a potentially infinite number of nodes based on state. The directed loop represents recursive calls that potentially never terminate (n!->n!), and the set of n directed edges represents multiple exit conditions. Anyway, this is just an off-the-cuff sketch of how MG/HGs can be used to model natural language semantic spaces. I feel like your question is probative in that it wants to understand what, if any, relationship is there between MG/HGs and epistemic natural language centered around traditional issues of foundationalism. It seems to me MG/HGs can be used to model methods of avoiding the problems traditionally found in epistemic discourse.

• Thought I'd take a stab at it. If something doesn't sit right, point it out. I haven't thought it through.
– J D
Mar 1 at 17:17
• You're on point. From reading about how multisets showed up as "bags" in older-school programming/computer science, and with multigraphs involving the multiset relation, I can see that your references to computational structures are apropos. I do wish I'd learned more actual programming back when I had the free time (the most I did was the in-game object-oriented programming language for the really old computer game ZZT, and some in-line stuff for a spinoff/remake/successor called Megazeux). Mar 1 at 18:10
• There seems to be a lot of discussion about whether brains are Turing machines, but Turing very clearly and explicitly modeled machines on the brains of human mathematicians. It think in retrospect (that of us contemporaneous thinkers) that Howard-Curry-Lambek isomorphism is essentially a philosophic endorsement of Turing's choice. Computer science clearly has been an effort to address epistemic problems. Even the nomenclature is similiar. They're simply called decision problems in computability; Turing went right to the Entscheidungproblem.
– J D
Mar 1 at 18:18
• @KristianBerry According to WP: "On January 28, 2023, the original source code for ZZT 3.0 (sans third party content) was uploaded to GitHub under the MIT License with permission of Tim Sweeney" Looks like you could relive your youth there. Sounds like a fascinating game I never heard of. And, man, you never have to justify your vote or management question to me. Your questions always are fun to read and when I can, I like to stretch the braincells and try an answer. Thanks for your participation in this forum. It adds a real philosophical spirit.
– J D
Mar 1 at 19:55
• Yeah my goal eventually is to cite specific Q&As on this site, in an actual attempt-at publication. I've seen published/refereed/w/e essays about mathematics that will directly cite the MathOverflow so I think it's a fair reference to this site (when the answers have the technical character required for these kinds of citations). Mar 1 at 19:57