So what is Fundamental?
A theory is fundamental if it cannot be derived from another, more complete, theory. More complete means the theory is applicable to a larger range. Note that a fundamental theory can be derivable from another theory if both are equivalent to each other (though one could plausibly argue then one should consider both the same theory).
Throughout history, the search and discovery of more fundamental theories in the natural sciences has lead to a tremendous amount of progress. That however is not a guarantee it will continue to be the path to progress. The issue is in the expression “cannot be derived” which could mean three different things:
Cannot be derived, version I: not possible in principle.
It might not be possible because it is not possible. Believers in reductionism think this is not the case for the laws of Nature we presently know: they should all follow from one most fundamental "Theory of Everything." While it is true that reductionism proved to be very useful and we thus have good reasons trying to continue it, there is no knowing the laws of Nature always allow a reduction. We would then be left with layers of theories that describe Nature on various scales that cannot ever be derived from each other, and thus have to be considered equally fundamental. While we presently don’t have evidence for this, it is a self-consistent point of view.
In the previous post on Emergence and Reductionism, I explained this is known as “strong emergence:” Emergent features on a higher level require a theory that cannot be derived from the underlying one. We previously discussed the paper “More really is different,” in which Gu et al offer an example for a system that does have emergent features, but it can be proved these are not derivable from the underlying theory. Granted, the system they consider isn’t particularly natural (see discussion on earlier post), but it gives you an impression of what this case means.
Cannot be derived, version II: not possible in practice
It might not be possible to derive emergent features from a more fundamental theory because of practical constraints. For example, it might take more computing power than we will ever have available, or more time than the lifetime of the universe to do it. It might take infinitely precise knowledge of initial conditions; it would make it necessary to measure parameters more precisely than we can plausibly expect ever; it would take a detector the size of the galaxy; etc etc.
Cannot be derived, version III: not yet possible
We might simply not have a derivation because
Okay, now what is fundamental?
The problem is that at any one time we might not know which of these 3 cases we are dealing with. The exception is if we had an actual proof for the impossibility of a derivation. (But then a proof is only as good as its assumption.) We are thus left with our assessment of the situation, which might change with better understanding of the theories we have. In some cases there is a pretty clear consensus on whether a law is fundamental, in other cases it might not be so clear.
- Take for example the Tully-Fisher relation. It relates the luminosity of a spiral galaxy with the 4th power of its rotational velocity. It is a useful heuristic relation, extracted from data, and has predictive power. There is no derivation of that relation; yet I doubt any physicist would argue it is a fundamental law. Instead, with increasing understanding of astrophysical processes, we will finally be able to derive it.
- Stefan came up with an interesting historical example, the Titius-Bode law according to which the distance of planets to the sun grows exponentially with their order in the sequence. The law works pretty well up to Uranus and fails with Neptune, but the far out planets were not known when the law was suggested. People once thought the planets' orbits are fixed by fundamental principles, but with better understanding about the gravitational interaction, the "law" was downgraded to a "rule," or possibly just a coincidence. Though with further knowledge about the dynamics relevant for the formation of solar systems the approximate validity of the relation might be an "emergent" feature one can expect to approximately be valid.
- Then there is of course the often discussed question whether it is in principle possible to derive all of biology, psychology, sociology and economics from physics and thus physics is the most fundamental of all sciences. Many physicists believe this to be the case. For that reason, one of my profs used to refer to physics as “the queen of sciences” (physics is a female noun in German). But we are far away from practically achieving such a derivation, and we thus do not actually know which of the three cases of “cannot be derived” we are dealing with. Already at the level of proteins things get murky, and we should be considering the option that indeed biology might be as fundamental as physics in the sense that it cannot be derived - cannot be derived in principle, not ever.
One of the reasons why the first case might apply even though reductionism has worked so well over a large range of scales is that in some areas of science the separation of scales might no longer work, and/or there might be no scale that can be used for separation. In physics typically the scale is energy, and we are used to neglect things that happen at energies much higher (wavelengths much smaller) than what we are probing. We know this is a safe procedure backed up by the framework of effective field theories. In contrast, a system like our societies does not simply have higher level organizations constituted out of smaller elements, such that these smaller elements define the "emergent" properties. Instead, these organizations also act back on the elements that they are built of and change their behaviour.
Coming back to physics, there are of course the questions that are hotly discussed at the front of research today, those asking what is fundamental in our present theories. Can the masses of particles in the Standard Model be derived from a more fundamental theory? Are space and time themselves emergent from an underlying theory (generally expected to marry quantum mechanics with general relativity). Is quantum mechanics fundamental, or can the quantization procedure and the measurement prescription be derived from a more complete theory?
I don’t know. But I really, really want to know.
Aside: Some weeks ago Clifford also wrote about the question what is fundamental, anyway? Since he sent me the link to make sure I don’t miss it, I can’t get away without mentioning it. Clifford is mostly concerned with people who use the label “more fundamental” to mean their work is more relevant. While that might happen, people using superlatives to claim their own work (life, opinion) is “more this” or “more that” than others’ is hardly remarkable, and certainly not specific to theoretical physics. The other point Clifford makes is that “Nature recycles good ideas,” meaning that the framework of fundamental theories can often also be found to be useful in non-fundamental areas - and the other way 'round. It is an interesting point, but it addresses more the question where one can find inspiration, not what is actually fundamental.
A theory is fundamental if it cannot be derived from a more complete theory, yet there are different reasons for why we may not be able to derive it: It might not be possible in principle, it might not be possible in practice, or we might not yet have the sufficient knowledge to do it. In general, we do not know which case we are dealing with. Misjudgement of the situation can waste a lot of time and hinder progress. If we wrongly believe a property is not fundamental, we risk searching forever for a more fundamental explanation that doesn't exist. On the other hand, if we believe something is fundamental even though it isn't, our understanding of Nature will remain limited. What is sure though is that understanding always starts with a question.