If asked to define what an agent is, my usual answer -- one of them, anyway -- is “a worldline in Tegmark V”.

The Tegmark Level V Multiverse (the “V” here is a Roman numeral) is not defined by Max Tegmark (whose hierarchy goes only up to IV), but, as used in agent-foundations circles, it refers to the collection of not-necessarily-consistent mathematical universes, a.k.a. “impossible possible worlds”. It thus contains worlds in which 1+1 = 3, worlds in which triangles have four sides, and perhaps worlds with married bachelors -- in addition, of course, to all the more “ordinary” worlds in Tegmark Levels I-IV (and thus, in particular, us).

This definition of “agent” is intended to evoke the concept of an observer in physics (especially relativity), which is a worldline in physical spacetime. “Observer” is a more passive word than “agent”, corresponding to the fact that in physics, worldlines are determined by equations (the “equations of motion”) that represent the laws of physics; whereas the idea in agent theory is that the corresponding equations -- those that determine worldlines -- represent the preferences, or “caring structure”, of some entity other than (at any rate, not necessarily identical to) the physical universe.

(I am deliberately avoiding the terms “values” and “goals”, for obscure theoretical reasons that I won’t explain here.)

A worldline in Tegmark IV (to say nothing of V) would, almost by its very nature, suggest a higher degree of “agency” than the ordinary sort of worldline, because it would allow for the possibility that the “observer” or “agent” moves between universes with differing laws of physics. Were we ever to acquire the capacity of “hacking into” our universe and changing its physical laws, for example, this would be the sort of mathematical setting in which our activities would be appropriately modeled. The equations governing our trajectories in that case would be reasonably termed “laws of metaphysics” -- or, indeed (in the most general case at least), “laws of mathematics”.

Importantly, note that this setting enables us to reason counterfactually about physical laws (that is, about “metaphysical location”), in exactly the same way that ordinary physics allows us to reason counterfactually about physical location (e.g., “if I were here at this time, then I would be there at that time”).

The next step in this progression, Tegmark-V worldlines, might seem absurd at first: hacking into the laws of mathematics, and traveling into the world where 1+1=3? But the subject of logical uncertainty, and in particular logical counterfactuals, has indeed been receiving attention lately. Essentially, it arises out of situations in which agents need to construct models -- that is, maps that aren’t the territory -- not only of the physical world, but of mathematical truths: for example, to reason about how their own or other agents’ “source code” works. In other words, situations in which agents reason about agency.

Doing so is tantamount to reasoning counterfactually about the laws of mathematics. Consequently, reasoning about agency -- something that we would like agents to be able to do, as a result of our having done so -- is equivalent to reasoning about the “physics” of Tegmark V: meta-metaphysics, or metamathematics.