Is there a notion of causality in physical laws? I was reading "A Few Useful Things to Know about Machine Learning" by Pedro Domingos and towards the end of the paper he makes this statement:
"Many researchers believe that causality is only a convenient fiction. For example, there is no notion of causality in physical laws. Whether or not causality really exists is a deep philosophical question with no definitive answer in sight..."
I was surprised by this statement because apart from Heisenberg's uncertainty principle, everything else (that I know of) in physics seems to operate under the assumption of causal relations. If you have an equation that describes a definitive outcome as the result of some input factors, then it is describing a causal relationship, is it not?
 A: Both in classical and in quantum physics you have causality meaning that in natural phenomena there is a relation of cause and effect. In the mathematical formulation you have initial conditions which determine the future development of a system as solutions of differential equations. Mathematically causality is often explicitly enforced in physical theories. Causality is essential for special relativity stating that a cause has to precede the effect in all inertial systems. In quantum mechanics you have causality in the time evolution of the wave function according to the Schrödinger's equation. Meaning that you have causality in the development of the probabilities of experimental outcomes of a system. However, you often don't have causality in individual events which seem to be random.  
A: All answers are good; I would like to add something missed though: Newtonian Mechanics is not exactly a deterministic theory. In other words, sometimes, even if there is not a cause, an effect can just occur. SeeNorton's dome.
In short, determinism (uniqueness theorem in Newton Mechanics) is a mathematical consequence of a few conditions/assumptions;  Nature does NOT have to always own these conditions.
A: In classical physics everything is deterministic, meaning if you know the initial conditions of a physical system, the physical laws with absolutely give you the state at any later time. In quantum theory things got a little confused because of the fact that it does not predict position and velocity exactly, as denoted in the uncertainty principle and the equations of quantum mechanics and quantum field theory. 
That has led to a lot of confusion in interpretations, where people have said, that the laws are non deterministic. But the laws do determine the quantum state of a system, expressed as its wave function or quantum field, exactly. Those then determine positions and velocities probabilistically. We deterministically predict the probabilities and within the laws know how to manipulate quantum systems. 
It is not unfair to call that deterministic. But it's all points of view. The laws of physics still denote cause and effect. Even in quantum theory an effect can only happen after a cause, with c the max speed at which any effect can propagate. I'd call that deterministic enough
Of course on systems that involve a large number of particles you can not compute it all and have to deal with statistical mechanics and macroscopic observations of average effects and fluctuations. 
A: Saying that there is "no notion of causality in the physical laws" is likely a gross misinterpretation of the lack of time-asymmetry in the majority of physics.  This is to say that many physical laws hold equally well given a flow of time in either the forward or reverse directions with respect to the flow of time we experience.
The concept of entropy, as described in the Second Law of Thermodynamics, serves somewhat to rectify this apparently non-physical balance in the laws of nature as described by science.  Really, I think we need take nothing more from this than that the majority of physical laws are not but convenient mathematical approximations whose scope of applicability dare not exceed their reach.  The science will go wrong long before the philosophy when you start trying to draw existential conclusions from mathematical relations forged of rather more modest constraints.
A: I find it very interesting that a scientific writer purporting to represent machine learning makes statements of this kind. Machine learning, insofar as it consists of adaptive algorithms executed on signals, stems at least in spirit, if not directly, from systems theory, specifically adaptive filter theory. (It's also worth pointing out that it has nothing in principle to do with physics, if you consider signal processing a mathematical discipline.)
In systems theory, a causal filter is one that only operates on values that are available before the present value. This includes values that are not generated by the filter itself, as well as values that can be determined to be in the past following a time evolution rule. (Normally this rule is the monotonic increase of some independent variable, which we just call 'time'.) Acausal and anticausal filters are still useful for things like image processing, because in that application you can expect to have the whole image available. On the other hand, a filter that is guaranteed to be causal, equipped with a simple update rule like the LMS filter is useful for operating in "realtime".
All this goes for machine learning algorithms (as they reduce to highly involved stochastic adaptive algorithms).
I say all this (unrelated to actual physics) to show two things:


*

*In a highly non-physical field such as signal processing, there still exists a well-defined notion of causality which has important implications for applied work.

*This well-defined notion is much simpler than the deeply rigorous and well-developed concepts of symmetry and invariance in physics (I know because I know from experience how much harder physics is than what I understand!)


So the author of that paper, if your restatement of his argument is accurate, spears himself on two horns:


*

*If causality is so fictitious in physics, then it is just as fictitious as defined in the field of adaptive algorithms. That calls into question the philosophical basis for the ability of such algorithms to be "smart" (i.e., to establish some sort of "truth").

*If causality in its "mere" formulation in adaptive algorithms has epistemologically substantive content (e.g., the theorems that can be developed about what is and what is not possible with causal and acausal filters), then the formulations of symmetry and invariance in physics, being at least as rigorous and explicative in the same sense (i.e., if you can employ theorems at that level of physics, you have enough mathematical "power" to work out adaptive algorithms), must also be at least as substantive. So there must be something to causality, and certainly a lot more of that "something" in physics at large than in "mere" adaptive algorithms.


Of course, when someone begins a sentence with "Many [people] believe..." and ends it with "...no definitive answer...", it's probably not worth criticizing, when it wouldn't even make it into Wikipedia.
A: Firstly, "an equation that describes a definitive outcome as the result of some input factors" is a description of a deterministic relationship between input and output, not needfully a causal one. Causality and determinism are not the same. What if the equation is invertible, so that one can swap the roles of input and output and describe the new equation in the same way?
Secondly, although it is hard to second guess what a writer is getting at, I don't think the denial of all notions of causality is quite as strong or deep as one might think from a first reading of this passage. I believe the writer is getting at something like the following. There is a large body of wholly experimental knowledge observing that the phases or sub-events within certain physical processes always have the same time ordering, that, for example, one always has to cook eggs before one can eat cooked and not raw eggs. I have never observed this process play out in anything other than this order. This observation of a fixed time ordering of this particular process and many others is all that we can rigorously say in describing a notion of causality. I discuss these bare observations and their role in a special relativistic notion of causality in this answer here.
What the author is most likely denying is a rigorous notion of a unifying, unambiguous notion of a causal relationship beyond these experimental observations. What defines notions like causal agent, effect and other supposed abstractions? When the author denies "causality", I think he/she is saying that a rigorous definition of these more abstract notions is likely impossible, and is probably the result of our projecting our humanly (but probably impossible to define rigorously) notions of will, wish, purpose and other teleological ideas onto physical processes. No philosopher, mathematician or physicist has so far been able to do this in an accepted,  universally accepted way.
So, in short, I don't believe the writer is denying the existence of causal filters or other fixed-time-ordered processes. He/she is simply saying that a rigorous definition of our human-centric ideas of causality is probably not possible and warning us not to expect that any rigorous notion of causality should resemble our human-centric expectations.
A: What we have in physics is experimental data and models attempting to describe the correlations found in the data. Then there is the more or less subjective and cultural meaning given to the models, often implicitly so. 
For example, suppose a law of physics showing that Earth orbits around the sun. Let's say such a law, based on the observation of the sky which shows consistent patterns (correlations), based also on earlier models having identified the sun with a specific object located in space (as opposed to, for example, a round patch of light glued to a celestial sphere), describes sun and Earth as being attracted in proportion to the product of their masses and in inverse proportion to the square of their distance. By manipulating this relation of attraction, one can show that Earth perfoms an elliptical orbit around the sun. 
Where is the causality here? 
Is it that the sun, because it is much more massive than Earth, is the "attractive cause" of Earth motion? That's how some people would make sense of the law, but note that this interpretation is no physics per se, but arises from the way people relate to the physical law by giving it a meaning. And in that case, a physicist could easily correct them by saying that Earth also attracts the sun.
Is causality part of the whole Hamiltonian picture: setting up the initial conditions of sun and Earth at specific points and with specific velocities, the law itself (translated in its potential form) is the cause of the evolving orbital system? Not quite, because, as other answers and comments pointed out, determinism is not causality: if the equations are time-reversible, the final state of the system is as much the cause of the initial state than the other way around, at least physics-wise. For an ordinary person such as myself, since time seems to flow from now to the future, it is the initial state that is the cause for future ones. But again, this is just me, not physics, speaking, trying to make sense of the equations.
Causality implies a narration: this made that happen. 
In physics, a narration always comes from an exterior conceptual framework: "I dropped the egg which broke" is a narration consistent with physical laws, but if we wanted to study what it is this sentence talks about, we would have to undertake the description of the egg, the ground, the gravitation field, using statistical quantities and finally invoking the second law of thermodynamics to explain that "because the egg hit the ground it was irreversibly smashed", and during this process some operations of simplification and coarse-graining have to be made, by which narration is injected into the (ideal) bare and raw physical description of the system, which is timeless and eventless, only relational.
A: There has to be. The only way we test laws is by doing experiments. If causality was not there then we would not have been able to prove any law experimentally. 
Doing actual calculations is a different story - Some processes are so random and complex that not only they appear to be non-deterministic, they practically are that way ("for us"). For looking at such process, we do not need to go quantum. We can not even calculate what day/time a particular molecule of water will evaporate from a swimming pool. Even though we understand the process of evaporation quite well.
"For us" means that the the complex and random processes are non deterministic for us, the observers, but not for the nature. The nature would be fully deterministic. Actually the randomness is caused by determinism of nature itself.
If causality was missing, we would not be able to experimentally prove any physical law. Simply because the cause and effect - "You do this experiment, and you will get this result" would not work for those laws. Any law itself may be defining causality in some way.
A: Most physical laws - no not by themselves. But perhaps if you combine several physical laws. Nothing particular woven into isolated physical descriptions that forbids the direction of time. But that shouldn't stop anyone from introducing additional equations, for example the laws of entropy to point time in the right direction. In physics, mathematics we often deal with isolated mathematical models that describe one particular aspect of reality but the truth is reality naturally involves many aspects of physics and so requires many models together to describe what's going on.
Try simulating the physics of a dynamic system and you'll often run into the issue of algebraic loops. The simulation software in a way is warning you that you've violated causality. Your differential equations may need to be recast as integral equations with initial conditions. Your attempt to simulate reality  is constrained by the order of computations, an arrow pointing you in the forward direction. Nature has natural feedback loops but there is always delay or filtering in feeding back information. Einstein discovered the ultimate limit in which information can be fed back, the speed of light. Simulation (and by analogy, nature) abhors algebraic loops and physical models that contain derivatives and devices that try to predict. Integration, delays and filtering seem to be the more natural way that movement evolves in the universe. 
Here is a question I posed that was unfortunately closed that asks a similar question. And another closed question here. 
A: I think the concept of causality is perfectly fine within a set of commonly understood parameters. However, in terms of our universe, causality/rationality is also built upon irrationality.
The basic most "building blocks" (axioms if you will) of our universe exist the way they do without a reason for being the way they are. It's not that we don't or can't know, it's that there is nothing to know. Such a conclusion seems inescapable.
What's important though is that irrationality and unpredictability are distinct concepts. Irrationality meaning without reason, unpredictability being something that cannot be predicted. They look the same on the surface but you can have something behave the way it does consistently enough to be predictable, but without any particular reason for behaving that way.
So while the parameters that we "build" causality upon may be irrational, they are predictable, and that consistency is what allows us build causal constructs on top of these basic assumptions. I know this is probably more philosophical than this site would like but that's how i look at things.
