One of the most striking discoveries in mathematics is the independence phenomenon within axiomatic set theory. It concerns set-theoretic sentences that are neither provable nor refutable from our best set theory, the axiom system called ZFC. The paradigmatic example of an independent sentence is the Continuum Hypothesis, which dates back to the 19th century in the work of Georg Cantor. Nevertheless, contemporary set theorists continue to discover new independent sentences all the time.

Some mathematicians, such as Shelah, Bell, and Scott, occasionally use probabilistic language in their writings on certain independent sentences. For example, they may say that an independent sentence φ has a very low probability, or a lower probability than another sentence ψ. A particular claim might be that the Axiom of Constructibility, viz. V = L, should be assigned the minimal probability of 0. In my talk, I model this probabilistic language with a non-standard theory of probability and indicate how this probabilistic modelling technique could be interpreted philosophically. My talk will therefore be both formal and philosophical.

The independence of a set-theoretic sentence φ is established by showing that there is a model of ZFC in which φ is true and another model of ZFC in which ¬φ is true. There are uncountably many such models to consider, which is why a non-standard probability theory called non-Archimedean probability theory (NAP) is applied. One of the distinctive features of NAP theory is that it involves infinitesimal quantities. NAP functions are defined according to a certain sample space, consisting of models of ZFC, a directed set, and an ultrafilter.

In our case, every axiom and theorem of ZFC has a probability of 1. Independent statements have non-extreme, possibly infinitesimal probabilities strictly between 0 and 1. One peculiarity of this approach is that the non-extreme probabilities depend heavily on the choice of the directed set and ultrafilter used to define the NAP function. For example, the probability of the Continuum Hypothesis can vary over every rational number strictly between 0 and 1. We call this the Problem of Indeterminacy.

However, some probabilistic claims from the literature can be verified via the structural properties of the underlying sample space, that is, the directed-set route, or via the choice of admissible ultrafilters, that is, the admissible-ultrafilter route. These two approaches to ‘mitigating’ the problem of indeterminacy are entirely general in that they provide means of investigating probabilistic claims about set-theoretic sentences that do not originate from the literature. Perhaps counterintuitively, I suggest that the Problem of Indeterminacy can be seen as a feature rather than a bug, as it enables the quantitative study of the phenomenon of independence to yield various interesting results.

The central question of my talk concerns the admissible-ultrafilter route, which, in my view, is more aligned with a belief-type interpretation of probability. (The directed-set approach is more suited to a frequency-type interpretation.) From a philosophical perspective, the following question arises: how do probability dynamics work in the belief-type interpretation of NAP functions, and what counts as evidence? I maintain that reasonable principles for probability dynamics can be formulated. Furthermore, I take a rather liberal stance on the question of what qualifies as mathematical evidence. On the formal side, I investigate the conditions that must be imposed on ultrafilters to ensure interesting or desirable properties of the NAP functions.

Finally, I ask whether the Problem of Indeterminacy should be understood as a problem specific to non-Archimedean probability theory, or whether it is better to treat it as a recurrence of a general problem in probability theory, namely the reference class problem. I advocate the latter view.

In evolutionary linguistics, a structure often encountered are so-called syntactic trees. Starting out with a graph-theoretic tree $T=(V,E)$, we can designate one specific vertex $v \in V$ as the root, which produces a rooted tree $(T,v)$. On rooted trees, one can define the notions of parent and child vertices. Next, an ordered tree is a rooted tree $(T,v)$ together with a total ordering on the children $\leq_u$ of any vertex $u \in V$. These local orderings then allow a linearization of the vertices, i.e.\ an algorithmic total ordering on all vertices. Using formal language theory, one can show that any context-free language produces derivation trees. Such trees are always ordered trees, and since many vertices may have the same label, one must work with labeled ordered trees.

This talk will showcase the structure of syntactic trees as neurolinguistically motivated mathematical problem. Starting with trees, the talk will show how rooting a tree gives it several implicit properties: Rooted trees have a preferred orientation given by orienting all edges away from the root, which induces a partial ordering $\leq$ on the vertices $V$. Moreover, to every vertex $u \in V$ one can assign its height $h(u)$, namely its distance to the root. Next, the talk will move to ordered trees and show off the lexicographical ordering, which corresponds exactly to performing depth-first search, as an example of a linearization algorithm that also preserves the local orderings. The talk will conclude with an answer to the mathematical problem by showing that performing depth-first search on the leaves of any linguistic syntactic tree, which is a special case of a labeled ordered tree, yields exactly the underlying sentence.