May 2004
Causal modeling (with eternals)
Jaime Glaría
Department of Electronic Engineering, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaiso, Chile
___________________________________________________________________________________________________________
1. FREEDOM
Assume a variable x (you, yourself, perhaps?) with two
numbered alternatives: either 1 or 2 (choosing left or right?)
Assume another variable y (a hummingbird, perhaps?) with
two alternatives: either 3 or 4 (choosing up or down?)
Put both variables in a set: x and y (you and the hummingbird,
fooling around?)
Roughly, the set would have four alternatives: either 1 and 3;
or 1 and 4; or 2 and 3; or 2 and 4. That is, if the variables were
free from each other; i.e., if x could choose 1 or 2 when y chose
3, and also when y chose 4, and if y could choose 3 or 4 when x
chose 1, and also when x chose 2. In fact, this conceptually
defines freedom (within the alternatives that are available for
the available variables).
Now assume a third variable z with two alternatives: either 5
or 6.
Put all three variables in a set: x, y and z.
If the three variables were free from each other, this set would
have eight alternatives: either 1, 3 and 5; or 1, 3 and 6; or 1, 4
and 5; or 1, 4 and 6; or 2, 3 and 5; or 2, 3 and 6; or 2, 4 and 5; or
2, 4 and 6. And this defines freedom again.
“Freedom”, “Freiheit”, “liberté”, “libertad”: the concept is
widespread.
2. CORRELATION
When some alternatives are denied to a set of variables,
freedom is turned off (and rebellion might be turned on).
Correlation appears between the variables.
It can be nasty; e.g., when only these two alternatives are left
for the set of variables x, y and z: either 1, 4 and 5; or 2, 3 and 6.
A fundamental question is “how many correlations do you
see?”; e.g,:

one such that the alternatives for x, y and z are either 1, 4
and 5, or 2, 3 and 6?;

or one such that the alternatives for x and y are either 1 and
4, or 2 and 3; and one such that those for y and z are either
4 and 5, or 3 and 6?;

or one such that the alternatives for y and z are either 4 and
5, or 3 and 6, and one such that those for z and x are either
5 and 1, or 6 and 2?;

or one such that the alternatives for z and x are either 5 and
1, or 6 and 2; and one such that those for x and y are either
1 and 4, or 2 and 3?;

or one such that the alternatives for x and y are either 1 and
4, or 2 and 3; and one such that those for x, y and z are
either 1, 3 and 5, or 1, 3 and 6, or 1, 4 and 5, or 2, 3 and 6?;

…?
?
?
?
?
…?
Mathematical tables, formulas and graphs deal with
correlations.
As seen from these five constructs, mathematical tables,
formulas and graphs involve arbitrariness; not necessarily in
small amounts.
To reduce arbitrariness, use Occam’s razor and “shave off”
unnecessary complexity 1; e.g., discard the fifth construct unless
you have a good excuse to keep it.
Then think on the first four constructs and answer this: which
one is less complex? It is not obvious. Mathematically, the first
construct is the best candidate… to be shaved off: it is simpler
because it involves a single relation instead of two, but it is
more awkward because it includes no subset with two variables
free from each other.
Arbitrariness (freedom?) is hard to die and you still have three
constructs left.
In Nature and various human circumstances,
the denial of alternatives is not explicitly
declared and correlation is a conjecture
imagined because of some repetitive
observations.
Assume that you are in a silent cage, a bell rings outside and
food is offered to you closely after; the next day the bell rings
and food is offered to you again; and so on day after day,
without exception, until statistics appeal to something like your
scientific imagination and you begin predicting food offering as
soon as the bell rings. Later on, probably at the edge of insanity,
you could name the condition of the bell as x, the ringing of the
bell as 1, the lack of ringing as 2, the condition of the food as y,
the offering of the food as 4 and the lack of offering as 8, and
you could mathematically state y=4x. This lack of freedom
between x and y would probably be nice to you (and there
would be no rebellion from the bell and the food).
1
William of Occam (1300?-1349)
3. DEPENDENCE
The conjecture may go a little further, until it incorporates the
perception of your own will and the dream of participating in
some stages of the sequences, on purpose; e.g., the dream of
ringing the bell for bringing food.
Assume that one day the cage happens to be unlocked and you
grab the bell abandoning your scientific observations to move
into the engineering of food bringing. Then the imagined
relations can:

languish with the contradiction of your predictions, which
blesses you with the loss of innocence (i.e., if you ring the
bell and food, alas, is not offered; you loose faith and, by
the way, you realize that there is no point in trying to get
food in order to hear the bell ringing before);

or blossom with the fulfillment of your predictions, which
blesses you with rosy technical promises: since you can
ring the bell at will and food offering depends on the
ringing of the bell, food depends on you now and maybe
the whole world will some day!
Such is the essence of causality: the ambition to vary reality
predictably by knowing what parts obey one’s will locally, what
others depend on them, which else are driven by these and so on
up to the parts one wants to vary, and then by acting on those
mentioned first to rule those mentioned last.
The word “dependence” involves the conjecture, beyond pure
correlation, that some variables locally depend on others that
dominate them; i.e., it involves a conjecture on orientation: in
the story, the condition of the food, y, seems to depend on that
of the bell, x; not the other way around:
in spite that somewhere else, in revenge at some other untold
part of the story, x might depend on y
Since the essence of the conjecture is technical rather than
scientific, the eventual orientation of a dependence is definitely
elucidated, not by only observing, but by enforcing alternatives
on the hypothetically dominant variables and then observing the
supposedly dependent ones; if these vary as thought, one can
decide that the hypothetically dominant variables are truly so;
otherwise, one can decide that they are not.
However, this is a callous job rising a number of questions.
To enforce alternatives on any hypothetically
dominant variable one must act dominantly on
it; specifically, if one wants to enforce an
alternative on x in the last construct, expecting
to vary y indirectly, one must act in the role of z
in the following construct:
Can one really act that way?
Technically, the question looks rather philosophical (i.e.,
irrelevant) because one has to trust that one can vary reality:
you can ring a bell, can’t you?. But the case is far from trivial,
and one must accept that many hypothetically dominant
variables are very hard to reach.
Besides, isn’t one compelled to believe that one can also be
varied by reality; e.g., when observing?
The gracious answer is that the dependences through which
one dominates reality appear separate from those by which one
depends on reality: (some of) the dependences mentioned first
are supplied by the muscles, and (some of) those mentioned last
are provided by the senses.
Terrifyingly, though, illness and death menace both (from
reality) either separately or jointly.
On the other side, the hand menaces reality: if one enforces
alternatives, doesn’t one vary dependences and degrade
knowledge? (Observe the prospect of dependences being
variable).
The main hope along with the conjecture on orientation is that
variables depend and dominate through separate dependences
and enforcing alternatives on variables varies the dependences
through which those variables depend, but not necessarily those
by which they dominate.
Thus, the variables depending on one self allow a predictable
access to wider reality if they dominate others through other
local dependences, since one can assume that these dependences
persist when one enforces alternatives on the reachable
variables, and the corresponding knowledge remains valid for
predicting.
Now, if one can act as said, the contradiction of predictions in
the case of the bell and the food means that someone in the
untold part of the story might have put up a farce for you by
acting both on the bell and on the food in the next role of z: as a
puppeteer:
Another question emerges immediately with the loss of
innocence: does reality ever behave as you thought before, or is
there a Great Puppeteer juggling everything, including the food,
the bell, the puppeteer and yourself?
This question is devastating.
It shakes the technician’s faith on his or her capability to vary
reality.
It troubles the sense of will and ethical liability.
And it disrupts sentences such as this present one: if the
answer is positive, except for the Great Puppeteer nothing
disrupts, but is juggled into a disruption; nobody rings a bell,
but is juggled into a bell ringing; nobody speaks, but is juggled
into a speech.
In the question posed last, orientation is still present.
What if dependences are not oriented at all; i.e., if they are
mere correlations?
This new question is very difficult to answer, too.
It also bothers sentences (is correlated with the bothering),
unsettles will and liability (is correlated with the unsettlement),
and challenges the technician’s strive (is correlated with the
challenge).
Should one ignore the issue on sentences claiming that it is
grammatical and, hence, only of human nonsense? Should one
disregard the issue on will and liability alleging a similar
reason? Should one put the technician’s strive aside too? Should
one also cast away the idea that variables can depend and
dominate through separate dependences as anthropomorphic?
Should one stick to dehumanized observation and description of
variations for the sake of Science?
How?
One cannot do the above ignoring, disregarding, putting aside,
casting away and sticking if dependences are disoriented: one
can only be correlated with those acts. And “should” is a
disregarded word if one is involved in the disregarding of
ethical liability.
Let us bet on standard sentences (such as this one) and on
oriented dependences (back and forth separately, if necessary).
There is an anthropomorphic side here, no doubt. And one is
allowed to get nervous because anthropomorphism is dubious
for understanding the world (which on the whole does not seem
to rebel as easily as one self against lack of freedom, for
instance). But anthropocentrism is much worse, and one is
permitted to get more nervous with the arrogant idea that the
world puts up a domination-and-dependence farce precisely for
us, and behaves differently for the rest.
At the end, we must perceive our selves and act upon our
selves as we perceive the rest of the world and act upon it, and
vice versa.
4. BECAUSE
The callous job of reducing arbitrariness in correlations and
orienting them, moreover, can be helped with some further
conjectures and bets.
It seems that the dependent variables are only those that do
not vary before the others, in time. This can save some
strenuous enforcement tryouts.
However, the cases in the last two figures seem the same if in
the next-to-the-last one y obeys x after some time delay and x
obeys z after a similar postponement, while in the last one x
obeys z after a resembling delay but y obeys it after a double
wait. Moreover, both cases seem the same as the one in the prior
figure if z is not perceived.
Confusion is complete when variations are either cyclic or
suspended, in equilibrium, since “after” and “before” cannot be
discerned.
Therefore, a clue that helps the job of orienting dependences
is the conjecture that a variable convincingly varying after some
others does not dominate them. This helps shaving off but is,
obviously, often insufficient.
Other clues come from the physicists, who observe that the
speed of light constraints all physical speeds: one can also say
that a variable varying before some others does not dominate
them physically if it is too far (considering how long before it
varied). Moreover, if one observes that circumstance but
receives a message from a fast-moving observer saying that the
variable mentioned first varies after some of the others, one
should not be astonished and can also say that such variable
does not dominate those others physically even if it is not too
far (under the prior consideration).
Mathematicians mess around with domination, precedence in
time and sequencing in reading and in writing, because they
have to pull all the strings, as puppeteers: one can vary a letter
such as x by erasing it from a mathematical paper and writing
another symbol like 1 instead; but it would be mind-blowing if
another letter on the paper, such as y, varied without one doing
anything else. In fact, Mathematics includes some narrow and
very slippery slots to mimic causal orientation, just in case: by
convention, in any single-entry table involving a truly
dependent variable, the right column (the last one according to
Western writing) should be devoted to that dependent variable;
in any multiple-entry table involving a truly dependent variable,
the inner space should be devoted to that dependent variable; in
any equation involving a truly dependent variable, the left
member (the last one according to Middle Eastern writing)
should be devoted to that dependent variable; and in any graph
involving a truly dependent variable, the vertical axis should be
devoted to that dependent variable.
This brings us to reading from tables, formulas and graphs;
i.e., to verbalizing them, out loud.
And also to writing them for others that should read.
There is a big barrier there; and not just for unintelligent
people.
Algebra started in the 9th-century Middle East with alKhwarizmi’s equivalence between “z=x+y” and “x=z-y” 2. That
equivalence bluntly ignored causality, maybe because of the
callousness of orienting dependences. A funny thing is that,
almost at the same time and very near, the Banu Musa brothers
devised a feedback-loop mechanism for automatically
regulating the water level in horse ponds, and feedback consists
on back-and-forth dependences rather than a mere correlation 3.
Another funny thing is that, later on but very near again, alHazen decided that the hand is sent out but the eye is not, which
was a novelty, means that the dependences through which one
dominates reality are different from those by which one depends
on reality, and allows one to be part of doubly-oriented
feedback loops within reality, beyond disoriented correlations 4.
Be it. Take Newton’s law II for a body with given mass (say,
2), “a=f/2”, recall the prior conventions and read: “the
acceleration (a) of the body varies because of the force
impressed upon it (f); the acceleration grows if the force grows”
5. Then take the archetypical equation “f=2a”, which is
mathematically equivalent, and read “the force impressed upon
the body varies because of the acceleration; the force grows if
the acceleration grows”. Obviously, this sounds awkward. One
says “the force varies” because the acceleration varies and
because the acceleration varies because of the force. Entangled?
Consider that the force varies because of other variables
elsewhere (e.g., because of proximity to another body) and you
get the picture: “f=2a” deserves to be shaved off with Occam’s
razor unless you have a good excuse.
Now, take the microeconomic approximation for a hypothetic
merchandise in a given case “qd=3/P”, read “the demanded
quantity (qd) varies because of the price (P); the quantity shrinks
if the price grows”, consider the equivalent equation “P=3/qd”,
and read “the price varies because of the demanded quantity; the
price shrinks if the quantity grows”. Of course, something is
awkward in the last sentence. While you prepare the razor, take
Boyle’s law for a hypothetic gas in a given case “p=4/V”, read
“the pressure (p) varies because of the volume (V); the pressure
shrinks if the volume grows”, consider the equivalent equation
“V=4/p”, read “the volume varies because of the pressure; the
volume shrinks if the pressure grows” 6. Here, awkwardness is
less evident until the question “which pressure?” arises. There is
a pressure from the outside and a pressure from the inside. They
should coincide at equilibrium. Okay. But, how? Equilibrium is
a bad revealer of dependences. How does the gas depicted by
Boyle’s law adjust the pressure from the inside? Use Occam’s
razor and shave off “P=3/qd”, “Pqd=3”, “V=4/p”, “pV=4” and
the like. The pressure from the inside (p) varies because of the
volume (V). The volume varies because of the difference
between the pressure from the inside and that of the outside; but
that is an untold part of the story.
Next, take the thermodynamic equation for a hypothetic gas
“u=90+8T-5p” and read “the internal specific energy of the gas
(u) varies because of its temperature (T) and its pressure (p); the
specific energy grows if the temperature grows and the pressure
stands, and shrinks if the temperature stands and the pressure
grows”. The internal energy shrinks if the pressure (from the
inside) grows? Add the ideal gas law “pv=0.01T”. Happy?
Would you like Occam’s razor?
2
Abu Ja-far Muhammad ibn Musa al-Khwarizmi (780?-850?)
Ja-far Muhammad ibn Musa ibn Shakir (800?-873), Ahmad ibn Musa
ibn Shakir (805-873?) and al-Hasan ibn Musa ibn Shakir (810?-after
873)
3
4
Abu Ali al-Hasan ibn al-Haytham (965?-1038)
Isaac Newton (1643-1727)
6
Robert Boyle (1627-1691)
5
?
?
Looking at thermodynamic tables, which devote their borders
to the temperature (T) and the pressure (from the inside, p) and
their inner space (not that of the gas!) to both the specific
volume (v) and the specific energy (u), one could think that the
original version of the ideal gas law deserves the razor: it should
be “v=0.01T/p”.
But the original flaw involving u is pending. And the last
version of the ideal gas law is not confirmed by the way van der
Waals and later investigators chose 7: the ideal gas law should
rather be “p=0.01T/v”. And this is confirmed by some
thermodynamic graphs, which are three dimensional and show p
in their vertical axis.
What is it then: can one choose any two variables to dominate
the others?
What about the razor? There is a pending flaw regarding the
internal energy.
Read “the pressure (from the inside) varies because of the
temperature and the specific volume; the pressure grows if the
temperature grows and the specific volume stands, and shrinks
if the temperature stands and the specific volume grows”. Is
anything wrong with that?
Write “T=100vp” and read “the temperature varies because
of the specific volume and the pressure (from the inside); the
temperature grows if the specific volume grows and the
pressure stands, and grows if the specific volume stands and the
pressure grows”. The temperature grows if the specific volume
grows? Okay: bring the razor. And bring Mathematics along
with it.
A gas is trapped in a cylinder; you squeeze the piston pushing
it; the piston moves in the direction of the push, confining the
gas more tightly; the gas sees its vital space menaced in spite of
its own pushing from the inside, receives mechanical power that
way, becomes courageous because of both circumstances and
fights back, pushing harder; however, you squeeze harder too
and the gas surrenders: cowardly, it turns into a liquid; it has
less volume and energy now, but it is more slim and desperate
and really able to put up a fight. Are you going to beat it once
more? Wait. If the gas pushed while withdrawing, it always
received power. Where did accumulated energy go? Did it go as
mechanical power (i.e., “work” flow)? In this case, the gas
should have pulled the piston at some stage, in the direction of
the shrinkage; i.e., it should have manifested some negative
pressure. Is there such a pressure? How can the gas grab the
piston to pull it? Probably it cannot and got rid of energy during
its flee by means of the temperature: it was the temperature that
7
Johannes van der Waals (1837-1923)
shoved power out, abandoning tranquility because of the
energetic and volumetric circumstances; energy went away as
thermal power (i.e., “heat” flow). What will the liquid do now
that you are about to beat it again? Can it pull the piston if you
force it to retreat to its last defense: turning into a solid? Solids
do pull. They have to grab, though, with sticky fingers.
Spooky, huh?
What we have at the core of this tale is that energy and
volume dominate pressure and temperature.
Give Mathematics an opportunity to redeem itself. Take
“u=90+8T-5p” and “pv=0.01T”, shake them algebraically and
turn them into “p=((u-90)/5)(0.00625/(v-0.00625))” and
“T=((u-90)/8)(v/(v-0.00625))”. Then read “the pressure from
the inside varies because of the internal energy and the specific
volume; the pressure grows if the energy grows and the volume
stands, and shrinks if the energy stands and the volume grows”
and “the temperature varies because of the internal energy and
the specific volume; the temperature grows if the energy grows
and the volume stands, and shrinks if the energy stands and the
volume grows”. That’s it. Use Occam’s razor to shave off
“u=90+8T-5p” and “pv=0.01T”, “v=0.01T/p”, “p=0.01T/v”,
“T=100vp” and the like.
Energy and volume are free from each other here; pressure
and temperature depend on them. Energy and volume depend on
powers and forces elsewhere; forces and powers depend on
temperature and pressure somewhere else still. The whole
includes feedback and reaches equilibrium, but with the aid of
untold parts.
5. ETERNALS
One can think on other constructs, instead of the last one in
the previous section; e.g., to rescue van der Waal’s approach,
which did not seem to have anything wrong, some additional
algebraic shaking can yield “T=((u-90)/8)(v/(v-0.00625))” and
“p=0.01T/v”.
?
The question, though, is “how comes that the pressure
depends on the temperature?”
The answer is difficult if one thinks that temperature is an
intra-molecular phenomenon, and pressure an extra-molecular
one.
There is a reason explaining how the pressure depends on the
energy and the volume; it was suggested by Boltzmann and his
group 8: the pressure from the inside of the gas is the
manifestation of the bouncing of gas molecules on the surface
surrounding the gas and, therefore, of the change of their
momentums, which lets forces (momentum flows) emerge; the
bigger the internal extra-molecular energy of the gas, the bigger
8
Ludwig Boltzmann (1844-1906)
the momentums, the changes of momentums, the forces and the
pressure; the smaller the volume, the more frequent the bounces
and the bigger the pressure.
Maybe there is a subtler phenomenon explaining that the
pressure depends on the temperature, or vice versa, and
Mathematics (the villain and the hero) will help tuning a proper
construct.
For the moment, since arbitrariness is hard to die and the last
construct in the previous section seems to be the less complex,
let us stick to it.
Let us also claim that arbitrariness is much less than at the
starting “u=90+8T-5p”.
The specific internal energy (u) depends on the internal
energy and the mass: u=U/m. Energy is an eternal. “Next”
energy inside any region depends on the present energy, on the
incoming and outgoing powers (energy flows) and on the
difference from the present to the “next” time: Un=U+witwot. Mass is an eternal (at low speed compared with that of
light). “Next” mass inside any region depends on the present
mass, on the incoming and outgoing mass flows and on the
difference from the present to the “next” time: mn=m+qitqot. All this is an untold part of the story and prevents u from
being dependent in the told part.
Eternals (energy, mass, momentum, electric charge,…) have a
privilege that is valid no matter if it is told or untold: inside any
region, they depend on themselves and on their incoming and
outgoing flows, and that leaves all other constructs that treat
them as dependent aside.
They set a causal bias that helps reducing the callous job of
orienting correlations to see dependences.
6. CONCLUSIONS
When some alternatives are denied to a set of variables,
freedom is turned off and correlation appears locally between
the variables.
Correlation is often a conjecture that involves arbitrariness:
several constructs are possible.
The conjecture can go a little further, until it includes an
orientation meaning that some variables locally depend on
others that dominate them.
The orientation is definitely elucidated, not by only observing,
but by enforcing alternatives on the hypothetically dominant
variables and then observing the supposedly dependent ones.
This is a callous job that can be helped with
some clues: without strenuous enforcement
tryouts, variables that vary before others can be
shaved off as candidates to depend on those
others; on the other hand, variables that vary too
soon before others (considering the distances
and the speed of light) can be shaved off as
candidates to dominate those others.
Eternals set a causal bias that helps reducing the job of
orienting correlations to see dependences: inside any region,
they depend on themselves and on their incoming and outgoing
flows, and that leaves all other constructs that treat them as
dependent aside.