# In The Armchair

## Determinism I: Definitions of Determinism

Posted in Armchair Ruminations by Armchair Guy on November 21, 2008

In a previous post, I said that determinism and causality are incompatible.  The argument was that in a universe where everything is determined at every time point, it makes no sense to speak of alternatives or counterfactuals of the sort: If A had happened instead of B, then C would have happened instead of D…A could never happen, so we are predicating on something impossible: logically, “if A happens instead of B” is like “if 1 = 2”; if that happens, every statement is vacuously true.

I’ve since rethought my ideas about this.  The issue is the exact definition of determinism.  I used an unusual definition, so I got the unusual result that causality and determinism are incompatible.  Here’s my attempt to structure my current understanding of determinism.

It appears that a definition of determinism is contingent on how we describe the evolution of the universe over time.

Let’s assume that at each time point, a system can be described by a state which belongs to some prespecified set of states.  A system here could mean something like the universe, and a state could be a position and momentum for every particle in the universe (assuming the universe only contains particles).  The set of states is the collection of all possible configurations of particles in the universe.  In a simple universe containing only two particles that move only in one dimension, a state would look like ((p1, m1), (p2, m2)), where p1, m1 are the position and momentum of the first particle, respectively, and p2, m2 are the position and momentum of the second particle, respectively.  p1, m1, p2, m2 are all single real numbers here.  (If we were in 3-d space, p1, m1, p2, m2 would be vectors like (x, y, z).)

Definition of Evolution of the System in Time and Definition of Determinism. When we speak of a description of the evolution of the system over time, we are not talking about what actually happens in the system (unless, of course, the system is deterministic in which case what actually happens is the same as what might happen). Instead, we are talking about potential occurrences i.e. predictions about the future.  We might say that if the system is in state X at time 1, it could be in either state Y or state Z at time 2.  This does not mean the system will be in both states Y and Z simultaneously at time 2.  It means that the system will be in one of those two states at time 2; we don’t know which one.  The description is from the viewpoint of an extra-system observer, unaffected by the system’s timeline, who knows what might happen, but not what actually will happen (unless the universe is deterministic).

To describe the system’s evolution, we have to provide an evolutionary tree of some sort.  It might be of the following form:

That is, a tree depicting potential states at each time point.  Providing a particular evolutionary tree with X1 being the actual (observed) state at the beginning of the universe is one option.  Providing “transition functions” that specify what the potential states at the following time point are given any state at a particular time point is another way.

1. [Invalid Definition] (E1) Suppose a description of the evolution of the system over time consists of a single evolutionary tree.  (D1) A system is deterministic if the state of the tree has no forks; otherwise it is non-deterministic.
2. [Valid Definition] (E2) An alternative description of the evolution of the system over time consists of a collection of transition functions (“the laws of physics”).  Given any state at a time, the transition functions can be applied to calculate the possible states of the system at any point in the future (i.e., calculate an evolutionary tree starting at that time point).  (D2) Here we would say the system is deterministic if there is exactly one state possible at every point in the future (i.e., the calculated tree has no forks).

These two definitions look almost identical.  However, the first definition only specifies one possible tree rooted at X1. The second definition lets us calculate the potential states at time 2 once we know the actual state at time 1, using the transition functions.  In other words, we can substitute another state, say X8, at time 1 and still compute what possibilities that universe would have.

Even in the deterministic case, they are different for the above reason.  The second definition is constructive and so tells us what will happen in the case of interventions.  That is, if an external (from outside the system) agent sets the state of the system to some value at some time point, the second definition allows us to calculate the new states in the future of that time point.  The first definition doesn’t.  For example, in the figure, if the state at time 2 is X2, we know that the possibilities for time 3 are X4 and X5.  But what if the state at time 2 is X6?  Under (E1) we have no way of knowing; under (E2) we can calculate the possibilities.

### 2 Responses

1. Stephen Anastasi said, on June 4, 2009 at 3:28 am

To deal with determinism one must first consider Kant’s Prologemena. The world with which we are acquainted is a human centred view. Imagine instead that (for the sake of simplicity) that the world can be described in terms of mathematics. But we know that every function can be described in Fourier transforms, and every part of the Fourier transform can be further reduced. Which of these descriptions is the proper description of the world? They all might be. Just which is, and whether any are, is not accessible from an empiricist standpoint, for this reduces to no more than guesses and description (a Feynmanian view perhaps). If the mathematics is founded on infinite concepts, meaning that all numbers exist, up into transfinite numbers, then certainly determinism is a bother. But what if there is a smallest number? Then the gap between identifiable points, within which further division is meaningless, implies a level of looseness in the universe.

2. Armchair Guy said, on June 4, 2009 at 6:23 pm

@Stephen

Thank you for your comment. I confess I wasn’t familiar with Kant’s Prolegomena until I saw your comment. On looking at the wikipedia article, Prolegomena seems epistemological rather than descriptive of the world. Could you clarify what the connection with determinism is?

On your point about description, as long as different descriptions are functionally equivalent (i.e. events in one description when translated to another have the same consequences), it seems the description doesn’t matter. I agree that empirical inference of the truth isn’t really possible, but do you mean something beyond this when you say “proper description”?

If a smallest number does not exist, then we’d need a continuous time version of the collection of transition functions and it’s not obvious how to construct these. Is this what you refer to? I haven’t thought this through, wouldn’t any continuous time Markovian process be an example of such a description?