While wondering if I should talk about how we solve differential equations this week or plunging right in to the solutions of the system of equations I wrote down last week to describe pandemics, I realized I was missing a huge opportunity to swerve into a bit of philosophy. And in the course of writing this--which I thought was going to be mostly dashing off an argument I've been making in one way or another for thirty years--I've come to see my old understanding of the philosophy involved was insufficient, so this represents new ground, freshly ploughed, and possibly not quite right.
As I said last week, differential equations relate causes to effects. This is the conventional way of putting it, and it's a bit of a weasel, because it leaves the nature of the relationship vague.
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"Differential equations equate causes with effects" is a more accurate way to put it. The meaning of that equation, that equating, that equality, is interesting.
Philosophers sometimes claim dynamical laws are tautologies: restatements of the same fact by way of replacing words with definitions. This is a problem because tautologies are supposedly vacuous: they have no empirical content. In Kantian terms, they are "analytic": their "truth value" can be determined by simply analysing the definitions involved.
This would mean that what we call "scientific laws" are all, apparently, tautologies.
Fortunately for us, tautologies--or something very like them--are in fact among the most powerful empirical statements we can make, and even the most apparently empty "tautology" has enormous empirical implications when we stop thinking about language as something isolated from reality.
Language is not a game. It is a tool that knowing subjects use to describe reality. Treating it as a game of formal manipulation isn't exactly an error--logicians and mathematicians need love too--but insisting that because you enjoy treating it as a game that it is only a game is wrong.
It's when we remove differential equations from the context of language games and treat them as meaningful statements about cause and effect that they become powerful.
Consider the most famous, and most famously tautological, differential equation of them all, Newton's second law:
F = ma
or in explicitly differential terms:
d2x/dt2 = F/m
As I said last week, the second derivative with respect to time is called acceleration, the "a" in F = ma. Newton's second law is a differential equation that says the acceleration an object experiences is equal to a cause (the force, F) times a constant of proportionality (1/m, where m is the object's mass).
Another way of writing this differential equation is:
dp/dt = F
where p is the object's momentum, equal to its velocity times its mass. Want to know the force acting on an object? See how fast its momentum is changing with time.
But look, we're just equating force with the rate of change of momentum! Whatever the rate of change of momentum, we just infer that there is a force sufficient to cause it. How boring is that? Empty, vacuous, meaningless! Or is it?
The significance of this kind of equation is that we are relating a cause to its effect with an equals sign. Causes are not effects, so are these really tautologies? Or something else?
"Tautology" means literally something like "to say the same". "Tauto" is a contraction of "to auto", where "to" means "the". "Auto" without an article means "self" but with an article it means "the same", so "tautologos" comes out as "the same word". "A is A": tautology.
But when the first A is a cause and the second is an effect, the meaning of the statement changes even though its form does not. Again: this is how language as a tool for thought, which is not a formal game, works.
Maybe these things aren't tautologies at all, but "otiologies", which means something like "because words" (the first letter would normally be transliterated from the Greek to Latin alphabet as "ho" but "hotiologies" sounds like we're on a different subject entirely.)
Cause equals effect.
This is the fundamental dynamical law. It is not a statement of pure logic. It’s much more powerful than that, and we know that when dynamical laws and pure logic disagree, as in the case of Bell’s Inequalities, dynamical laws win. The "equals" is the equality of causality, not identity. The equality is ontological, not logical.
That is, while "cause equals effect" is true, "cause is effect" is false. Obviously: causes are not their effects. The force of gravity is not my rate of change in momentum as I fall to Earth.
If we identified causes and effects with each other it would collapse the whole edifice of causal explanation, which would be great for those philosophers who have always resented scientist's success in this regard, but not so good for anyone else.
Dynamical laws assert that given a particular cause, a particular effect will occur.
But doesn't this just mean that whatever effect we see we can find--or invent--a cause that will satisfy the appropriate dynamical law?
The reason for this is that causes have an ontology, a way of being. What that is varies depending on a the dynamical law, but it's always the case.
One extreme case of this is what I call "Darwin's Theorem", which is the statement that: "Imperfectly reproduced traits that endow offspring with differential reproductive fitness result in a population that explores a volume of individual-configuration space limited only by physical law, and settles sub-populations with multi-generational stability in the vicinity of local dynamical optima." That's a hell of a handful, but I'm pretty sure in the hands of a competent mathematician (which I am not) it could be turned into a formal statement that demonstrated the necessity of Darwinian evolution from a combination of probability theory and measurable constants of DNA replication fidelity.
This does not mean that whatever species we find we are free to simply make up a just-so story about reproductive fitness. Quite the opposite, in fact: it commits us to finding the specific, concrete, selective pressures that make a given population stable while adjacent possibilities remain unoccupied.
For example, crows and ravens are quite similar birds, but there is a very significant gap in size between them. Ravens are over 60 cm beak-to-tail and have a mass of more than 1 kg, while crows are rarely over 50 cm long and typically around 500 g.
Why the gap? Darwin's Theorem, far from letting us simply identify two distinct populations with "some kinda selective pressure" places on us the commitment to find the ontological differences that produce the difference between them. If we can't--if we could find distinct populations that remain distinct despite neither one having any reproductive advantage over intermediate forms--then Darwinian evolution would have a problem. The fact that I personally can't imagine such a situation means nothing: what we can imagine has no power over what is real.
We ran into just that kind of problem with the laws of Newtonian dynamics in the early 1900s.
Newtonian dynamics is usually stated in terms of three laws:
1) Absent external forces, objects in motion remain in uniform linear motion, and objects at rest remain at rest
2) F = ma
3) For every action there is an equal and opposite reaction.
The first law sets out the system as being in opposition to the once-prevailing belief that circular motion was "natural" and that objects left to themselves tended to slow to a stop. Contrary to this, the first law declares that not only are forces a cause of motion, they are the only cause of motion.
The third law expresses the ontology of Newtonian physics, because it commits us to the existence of something that reacts. When the momentum of one mass changes, there must be an equal and opposite change in the momentum of another mass. More generally, it commits us to the idea that motions have material causes.
For a system of laws that sometimes gets mocked as tautological by philosophers, that's an extremely strong ontological commitment: when we seen an effect (the acceleration of a mass, or equivalently a mass' momentum changing with time) we are committed by the otiologies of Newtonian dynamics to go searching for entities whose reaction compensates for it.
This precise compensation is an instance of a conservation law: a quantity whose total value is fixed within the system under study. Last week I mentioned the "Law of Conservation of People" as an important constraint in the creation of our dynamical model. The law of conservation of momentum--which is what the third law describes--is equally vital to our dynamical understanding of classical reality.
In the early 20th century we found that there were cases where motions occurred, but the required material cause of Newtonian ontology did not exist.
The "luminiferous ether" that explained motions near the velocity of light proved to be undetectable, and electrons in atoms seemed to be involved in motions that were constrained by something entirely outside of the scope of Newtonian ontology. Looking at the forces alone gave predictions that disagreed with reality.
In both cases new dynamical theories--relativity and quantum mechanics--were created that deviated from Newtonian ontology within their domains of application. Important aspects of motion in these domains are not caused by Newtonian forces, but space-time transformations (relativity) and quantization conditions (quantum mechanics).
So there is a brief summary of my thinking on the philosophy of dynamics, which underlies my thinking on dynamical models.
Dynamical models are systems of differential equations that relate cause and effect by what I am calling "causal equality." These equations are not tautologies, as sometimes claimed by philosophers, but otiologies, which carry with them strong ontological commitments.
When those commitments cannot be satisfied--when we can't find the entities our causal relations have committed us to--the dynamical model is broken, and we have to go looking for a better one. That may be simply a better, more complete or accurate, model within an existing theory, or it may involve creating an entirely new dynamics, which happens rarely, but often enough that we should never let it entirely slip our minds. But both historical reality and the theory described here show us that causal equality can be violated by reality in a way that tautologcial identity cannot. We can’t just make up any old thing to keep our description consistent with reality, particularly when operating under the tight constraints of conservation laws.
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I think this is pretty obvious. And I don't mean that in a bad way. I am distinctly NOT saying "So what"? This is a very important point you are making here. It's one of those posts I want to read again, and I suggest you elaborate more on. The idea that some kinds of equivalent explanations have more legitimacy is a very strange but true aspect of human life. It's a point on which my understanding differs a lot from other people's. But it's very important for me to develop a stranger understanding of the idea of explanatory equivalency, and this post is a step forward in that direction.