## Ontological-epistemological formalizations

### April 4, 2016

It seems that the voluminous discussion going under names such as positivism vs. interpretivism, modernism vs. post-modernism, foundatonalism vs. non-foundationalism, and behaviorism, realism and pluralism could benefit from a formalization of the ontological and epistemological models implied by the various positions staked in the discussion. It may be predictably claimed by some of the discussants that the attempt at formalization stakes a position in the contested landscape rather than maps it. I tend to disagree, but do not address this issue. Below is a generic ontological-epistemological formalization framework which allows the description of a wide variety of ontological-epistemological models by setting different values for its parameters. A few specializations, corresponding to radically different situations on the positivist-interpretivist spectrum, are presented.

**Setup**

Let the “objective, impersonal” universe (that is, all that is observable other than other observers) be a time series of points in an observation space *R* – *r(t) ∈ R ^{T}*.

In addition the world contains a set of observers *I*, each with an internal state that develops over time *s _{i}(t) ∈ S, i ∈ I, t ∈ T*. The observations made by each observer at time

*t*depend upon the state of the objective world at this time and upon the state of the observer at the time:

*f*.

_{i}(t) = f(r(t), s_{i}(t))The observers can communicate between themselves. These communications are messages of finite length over a certain finite alphabet *A*, so the communications of observer *i* at time *t* is *b _{i}(t) ∈ A**. The communication of each observer at each time depend on their state at the time:

*b*.

_{i}(t) = b(s_{i}(t))Finally, each observer’s internal state, *s _{i}* can develop based on the observer’s observations and on the history of communications by other observers (and himself):

*∆s*, where

_{i}(t) = s'(s_{i}(t), f_{i}(t), B_{j}(t), j ∈ I)*B*is the concatenation of all the messages of observer

_{j}(t)*j*up to time

*t*:

*b*.

_{j}(t’), t’ ≤ t**Model I: objective observations**

Epistemologically, the simplest model – all observers see the same observations:

*f(r(t), s _{i}(t)) = f(r(t)).*

**Model II: reproducible subjective observations**

Here the observations of each observer depend on their state in some way, so that different observers can make different observations. However, observers can communicate to each other that part of their state upon which the observation depends, and the observers can match their state – or that part of it upon which the observations depend – to a communicated state so that two observers who matched their states (on the part of the state upon which the observation depends) will see the same observations.

**Model III: shareable states and shareable subjective observations**

It may be that the observations depend on the observer’s state, and that while both observations and the states upon which the depend can be described to other observers, those observers cannot reproduce those observations because they cannot match other observers’ states well enough to approximate their point of view. Thus while the observations cannot in general be reproduced, a mapping of reported states and reported observations can be constructed.

**Model IV: shareable observations, non-shareable states**

It may be that the observations depend on the observer’s state, and that while observations can be shared (at least partially), the states upon which they depend cannot be described accurately enough to provide useful information about the expected observations. In this case a state-to-observation mapping cannot be constructed, but the set of observations made (or at least those reported) at any point in time can be recorded.

**Model V: non-shareable observations, non-shareable states**

In this extreme case, each observer’s interaction with the non-personal universe is essentially isolated. Observers can neither describe what they observe nor their internal state. Still, however, by definition, communications are observable and shared, so discussions of the non-impersonal universe can be conducted. The subjects of such discussions are limited to “reinterpretation of communications”, but in fact this universe of communications is potentially as rich as the non-observable universe. The difference is that, by definition, and due to its discrete, digital nature, the basic observations are “objective” (i.e., are observed in the same way by all the observers).