A Formal Expression of Causality

 

Introduction:

Causality.

It is perhaps the single, most intuitive phenomenon of all human experience.

Something (anything!) happens because something else 'caused' it to happen.

Should you happen upon any mundane occurrence, say a lit candle, your immediate instinct is to presuppose how it came to be (in this case, someone must have placed the candle there and lit it).

So fundamental and intrinsic causality is to our understanding of the world, that every entity we regard as sentient...people, children, infants, animals, even machines...adhere to the principle of causality. Nothing, it seems, can happen without a cause.

Precisely for this reason, it's extremely odd to me that such a core phenomenon of our physical world does not have a formal definition.

Anyone can define 'causality' in simple words. Anything 'happens' because something else 'caused' it to happen. Yet, there isn't any law in physics that governs causality or explains it formally.

Formal Systems: 

"A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules."

That's a technical way of saying that a formal system is any logical construct which operates according to set of inviolable rules.

Math, for example, is the most commonly known formal system. It has rules (such as arithmetic operations...addition, subtraction, division, multiplication etc.) according to which it works. The syntax of a computer program is another example. Chemistry, Electromagnetism, Biology are all abstract formal systems derived from Physics. And Physics is the core formal system of the physical universe itself.

Another fundamentally intuitive phenomenon within the human experience is gravity. We are bound to the floor we stand on because of it. Gravity is explained precisely via Isaac Newton's law of gravitational attraction:

F= Km1m2/r2

That is, every object in the universe attracts every other object with a force that is directly proportional to the product of their mass and inversely proportional to the distance between them. All of Astronomy, Celestial Mechanics, Astronautics, Rocket Science, even Architecture is based on this law.

This 'law' or formula, is a formal expression of gravity. It formalises the phenomenon such that it can be understood in accordance to precise rules.

Yet, no such formal expression exists for causality.

Motive:

Even though our ancestors knew gravity as a fact of life, they were unable to use it functionally to,  say....calculate the motions of heavenly bodies or calculate ballistic trajectories (just a couple of trivial examples) because they lacked a formal expression. 

What follows is an a priori (deductive) formalisation of causality as we understand it.

Premise:

Consider a simplistic setup of a causal event...a moving ball on a pool table striking another, stationary ball, causing the latter to also move.

We understand that:

• Nothing happens without a cause (a stationary ball cannot move unless something causes it to move)
• Cause and Event always occur at the same time and place (the moving ball strikes the stationary ball both at the same place at the same time when the former causes the latter to move)
• And, the resulting event always occurs AFTER the cause (the struck ball will only move after it has been struck)

This simple understanding can be formalised thusly:

Formalisation:

Setting: A pool table

Causal Event: Cue ball strikes the 8 ball, causing it to move

Legend:

o   SL(tn) = Local State at time tn. It’s the set of all material entities within the local area (including radiation and latent energy, although we'll disregard it in this scenario for simplicity), with information about the mass ‘m’/energy ‘e’/spatial coordinates ‘x’, ‘y’, ‘z’ (position) of those entities at time tn

o   En(a) = Entity a (cue ball)

o   En(b) = Entity b (8 ball)

o   En(a, tx)[m(a), e(a), x(a), y(a), z(a)] = mass/energy/position of cue ball at time 'tx'

o   En(b, tx)[m(b), e(b), x(b), y(b), z(b)] = mass/energy/position of 8 ball at time 'tx'

Local State at time t1: SL(t1) = En(a, t1)[x(a), y(a), z(a), m(a), e(a)], En(b, t1)[x(b), y(b), z(b), m(b), e(b)]... En(n, t1)[x(n), y(n), z(n), m(n), e(n)]

 

Events, in sequence:

·        Ev(a,t1) = Cue ball’s movement immediately before strike, at time t1

·        Ev(b,t2) = 8 ball’s movement (nil) immediately before strike, at time t2

·        Ev(b,t3)= 8 ball’s movement immediately after strike, at time t3

·        Ev(a, t4) = Cue ball’s movement immediately after strike, at time t4

Note: the alphabet within parenthesis indicates the entity, whereas the numerical within parenthesis indicates the time

 

Since the above events are in sequence, hence t4 > t3 > t2 > t1.

In other words: the time at event Ev(a, t4) is later than the time at event Ev(b, t3) is later than the time at event Ev(b, t2) is later than the time at event Ev(a, t1).

 

With relation to the Local States at t1 and t4:

• Event Ev(a, t1) = d/dt*SL[En(a, t1)] = Change in Local State SL(t1) with relation to time, and specifically with relation to the mass/energy/position of cue ball immediately before strike

• Event Ev(b, t3) = d/dt*SL[En(b, t3)] = Change in Local State SL(t3) with relation to time, and specifically with relation to the mass/energy/position of 8 ball immediately after strike

 

CAUSAL EVENT 'CE' (cue ball striking 8 ball causes it to move) occurs at t3

• Local State at t3 = SL(t3)
• Local State at t2 = SL(t2)

 

Hence, CE = d/dt*[SL(t3)-SL(t2)]

(i.e. Causal Event CE with regards to the local states immediately before and after the event is the difference, with relation to time, between local state at t3 and local state at t2)

 

The operative states within CE are Ev(a1), that caused Ev(b3).

 

Coincidence: The coincidence of entities En(a, tx) and En(b, ty) during any instantaneous time of transition from tx to ty is the coincidence or equality of any parameter from the set [m, e, x, y, x]. I.e. either their mass, energy, or any spatial coordinate is the same at that time (as would be the case at the time of contact between the balls. The table effectively being a 2-dimensional plane, the planar coordinates x and y of the balls would have the same value at the point of contact)

Thus, CO (coincidence) = d/dt[En(b, ty) - En(a, tx)]

As established within the premise, Ev(a, t2) causes Ev(b, t3) ONLY when all three conditions below are true:

è Ev(b, t3) exists ONLY when Ev(a, t2) occurs

AND

è CO[En(a, t2), Ev(b, t3)] = d/dt[En(b, ty - En(a, tx)] = TRUE

AND

è Ev(a, t3) precedes Ev(b, t3)

 

Corollary:

GIVEN

è SL(tx) = [Ev(a, tx), Ev(b, tx), Ev(c, tx)… Ev(n, tx)] is the local state at time tx, which is the set containing events Ev(a) at time tx, Ev(b) at time tx, Ev(c) at time tx and so on

 

è SL(ty) = [Ev(a, ty), Ev(b, ty), Ev(c, ty)… Ev(n, tx)] is the local state at time ty, which is the set containing events Ev(a) at time ty, Ev(b) at time ty, Ev(c) at time ty and so on

WHEN

è CO[En(a, tx), En(b, ty)] = d/dt[En(b, ty - En(a, tx)] = TRUE

AND

è SL(tx) - Ev(a, tx) != SL(ty) - Ev(b, ty) (!= means not equal to, implying a change has occurred)

AND

è ty > tx

THEN

Ev(a, tx) caused Ev(b, ty)

Prologue:

Such an expression yields a formal tool to make sense of any causal scenario, no matter how simplistic or complex. 

So say one had a coop of hens where all hens were initially pecking peacefully on the ground. But then one of them jumps up and started flapping about, startling the adjacent hens to get do the same. Pretty soon all the hens would be aloft or in a state of agitation. Intuitively, any observer would know this was caused by something. But the formal expression would let you trace the sequence of events, in chronological order, to the initial trigger (or offending hen in this case).

Or perhaps more usefully, a set of magnets levitating on a superconducting surface would all move or vibrate seemingly chaotically. Yet the movement of each one can be traced to a disturbance caused by the movement of adjacent magnets. If you knew the local state of the set at the start of observation, you could predict the end state of every part therein PLUS the outcome of the entire set at any time using this formal expression.

An example of theory (formalisation) predicting reality. As is the entire goal of science.

After all, reality is but a manifestation (formalisation) of the rules of our world.