In part I of this essay I have introduced and discussed the idea of reductionism from an epistemological point of view. In what follows we will go one step further, and discuss the idea of ontological reductionism . However, much of what follows will be actually devoted to rewriting the discussion of part I in a more formal framework, since this will provide us with a clearer picture of reductionism and allow us to discuss the idea of an ultimate fundamental theory, the so-called "theory of everything". The formal axiomatic framework will enable us to invoke Gödel's first incompleteness theorem to argue that such a theory cannot exist, thereby defeating any concept of ontological reductionism.
The axiomatic structure of any scientific theory is very complex, so complex that all axioms are virtually never spelled out explicitly. The reason for this is that there are too many of them, and that it is not always easy to figure out what is the minimal set of independent axioms underpinning any given theory. Nevertheless, any quantitative theory has the following gross axiomatic structure, with axioms classified into several groups:
Group (1) is typically the set of axioms of the first-order predicate calculus, establishing a formal language and rules which define what is meant by deductive reasoning within the theory. Group (2) typically consists of the axioms of Zermelo-Fraenkel set theory. Together with (1), these axioms establish the base for the rest of mathematics, necessary for the quantitative description of any theory. Group (3) represents the set of axioms concerning the kinematics of the theory, the properties of the basic physical quantities, i.e., the variables characterizing the theory, their postulated observability, and the experimental concepts necessary to observe them. Group (4) postulates what is the range of applicability of the theory, and introduces fundamental error bar estimates for the variables. Finally, group (5) are axioms about dynamics, the "meat" of the theory --- statements about the laws which Nature is supposed to uphold, in the context of all previous axioms.
Given the above structure for both an effective and its corresponding structure theory, one can reformulate reductionism as a simple formal statement: the effective theory is reducible to the structure theory if and only if all axioms of the effective theory are theorems in the structure theory, given appropriate approximation semantics. Axiom sets (1) and (2) are most often identical in both theories, so they are automatically theorems of the structure theory. The axioms in (3) tend to be different between the effective and the structure theory, and establishing the former as theorems of the latter amounts to specifying a consistent "vocabulary" between the two sets of variables, as I have discussed in part I. Proving that axioms (4) of the effective theory are theorems in the structure theory amounts to specifying a consistent approximation scheme and the choice of a set of parameters which can be considered suitable for asymptotic expansions. Finally, proving that axioms (5) of the effective theory are theorems in the structure theory (under the suitable constraints of the established approximation scheme) establishes that the dynamics of the effective theory is a rigorous consequence of the dynamics of the structure theory. This establishes the reductionism between the effective and the structure theory.
Such axiomatic description of reductionism can be useful to discuss some of its properties. For example, it is easy to see that reductionism can be regarded as a relation of partial order among theories. First, every theory is reducible to itself, since all its axioms are also theorems by definition. This establishes reflexivity. Second, if the effective theory is reducible to the structure theory, and in addition the structure theory is reducible to the effective theory as well, this means that both sets of axioms can be proved to be theorems of each other. This means that the two theories are in fact equivalent, which establishes antisymmetry. Finally, if some effective theory is reducible to some intermediate structure theory, which is in turn reducible to another structure theory, it follows that the first theory is also reducible to the third theory, given that its axioms can be proved by appealing to the "middle" theory as an intermediary step in the proof. This establishes transitivity.
Another aspect that can be usefully discussed in the axiomatic context is the issue of burden of proof for reductionism. Namely, given two sets of axioms, describing the effective and the structure theory, one cannot simply claim that the effective theory a priori must be reducible to the structure theory. It is not valid to just assume (or even worse, postulate as some metaphysical principle) that the axioms of the effective theory must always be theorems of the structure theory. This can hold only if one manages to prove that the axioms of the former are theorems of the latter. Moreover, this proof must be mathematically rigorous, or otherwise there might be substantial loopholes, as exemplified by the Solar neutrino problem discussed earlier. Therefore, the burden of proof is clearly on the one who claims that reduction holds, the criteria for such a proof are very high, and a priori one must always start from the assumption that reduction does not hold between the two theories. This is popularly phrased as the statement that a reductionist must "walk the walk", i.e., explicitly provide the proof for each pair of theories, before reductionism can be considered to hold.
One more useful aspect of the axiomatic definition of reductionism is the proof that a "theory of everything" cannot exist. Given the axiomatic structure (1)-(5) outlined above, this is a straight consequence of Gödel's first incompleteness theorem. In short, the theorem states that, given a set of axioms that defines some theory, if this theory meets some general requirements , there will always exist statements which are simultaneously both true and unprovable as theorems within that theory. These statements can be incorporated into the theory only as additional independent axioms, and there is infinitely many of them, which makes any set of axioms forever incomplete, loosely speaking. This is guaranteed already at the level of logic and set theory, and the existence of additional independent-but-true laws of physics (like the arrow of time) only provides additional source for this incompleteness. So-called Gödel-statements correspond to what we have described as strongly emergent phenomena --- if such a statement is added to the set of axioms of an effective theory, the latter becomes non-reducible to the structure theory, unless we add the same axiom to the structure theory as well.
One can rephrase these conclusions as follows: given that we have epistemological access to only a finite set of phenomena in Nature, there is no way we can construct a "theory of everything". At best, we can construct a "theory of everything so far", which is fundamentally incomplete in the sense that there will always exist strongly-emergent phenomena in Nature that have not been accounted for by the theory, and therefore are not reducible to our fundamental theory.
The prime example of such a strongly-emergent phenomenon is the arrow of time, as discussed in part I. It cannot be reduced to the behavior of individual elementary particles, and one must consider it as an additional axiom in a fundamental theory. One can postulate it as it stands, or through an initial condition at the Big Bang, or through the fine-tuning of the "inflaton potential" in some suitable inflationary model, but one way or another it has to be postulated. And this is just one of an infinitely many such strongly emergent phenomena, as guaranteed by Gödel's theorem.
So far we have discussed the topic of epistemological reductionism. I have described in what sense one theory can be said to be reducible to another and I have argued that both the effective and the structure theory must be well-defined as quantitative mathematical models which are not in contradiction with experiments within their respective domains of validity. I have given examples of cases where reductionism can and cases where it cannot be established, both with respect to quality, quantity and complexity. I have argued that the burden of proof lies with the claim of reduction --- any phenomenon must be considered strongly emergent until proved otherwise . I have demonstrated that Gödel's first incompleteness theorem excludes the existence of a "theory of everything", and allows only for an epistemologically incomplete "theory of everything so far". While we may try to keep redefining the fundamental theory by including each newly-discovered Gödel-statement again and again, this process does not converge, and therefore no well-defined theory of everything can exist. Finally, I have outlined a formal axiomatic treatment for all of the above.
As far as epistemological reductionism is concerned, the whole analysis has one final message: the program of establishing reductionism across all sciences is completely hopeless. Moreover, Gödel's theorem guarantees that it will remain hopeless in perpetuity, regardless of the level of mathematical proficiency we may ever reach in the future.
All that said, there is one more important issue to be addressed --- the possible ontological validity of reductionism. Namely, one could argue that the futility of epistemological reductionism does not imply the absence of ontological reductionism. In particular, one can claim that, despite Gödel's theorem, one could "in principle" imagine a theory containing the collection of all (infinitely many) Gödel-statements as axioms, thereby covering all phenomena that could ever exist in Nature, strongly emergent or otherwise. Just for the sake of the argument, and despite the fact that any expert in mathematical logic would immediately begin yelling at us, let's assume that such a theory can exist. As the last point of this essay I want to give an example to argue that this metaphysical assumption is intrinsically sterile and useless for any philosophical discussion.
The example goes as follows. Suppose that I get a flash of inspiration, and manage to mathematically formulate a theory of everything. Arguably, it will contain the specification of the most elementary "building blocks" of matter, the specification of all their interactions, the specification of all possible phenomena that can emerge from complexity, and the specification of, indulge me, eight uncomputable functions that provide the "input interface" for eight self-conscious Deities. The uncomputable functions are coupled to the rest of the theory in such a way that these eight Gods can influence any outcome of any physical process, as they see fit. For the sake of the argument, imagine that I can authoritatively claim that this is the ultimate theory of everything, describing our real world and all phenomena in it.
The most obvious feature of such a theory is its "anything goes" property. At best, it can be used to claim that ancient Greek religious mythology was wrong, since this mythology claims that there exist more than eight gods, which my theory demonstrates to be false. Outside of such silly arguments, it would be completely useless for any and all discussions whatsoever, including the most abstract metaphysical ones. But wait --- a naturalist reductionist might ask --- can we instead construct a theory which does not feature such a high level of arbitrariness, for example one which does not contain any deities? Well, that would mean that we should, say, remove the uncomputable functions from the theory. This in turn means that we are already restricting ourselves to a certain specific subclass of "all possible" theories (namely to the subclass of recursive theories) and any theory from that subclass runs into the danger of being incomplete in the sense of the Gödel's theorem. We then again run into the problem regarding the burden of proof --- we need to explicitly prove that these excluded properties of the theory (i.e., the presence of eight gods) are not necessary for its completeness. And any such proof is of course missing.
This example illustrates that one cannot consistently discuss a theory of everything while at the same time insisting on parsimony. Parsimony requires us to assume a smallest possible number of axioms for a theory, while any hypothetical theory of everything must contain infinitely many axioms, due to Gödel's theorem.
The moral of the story is that the concept of ontological reductionism is too elusive to be useful for anything --- we can either accept the anything-goes theory, which is useless, or try to be more specific about the properties of the fundamental theory, which is burdened by the absence of proof of reductionism, i.e., one cannot prove it to be the theory of everything. Thus, the only reasonable way out of this conundrum is to actually give up on any notion of ontological reductionism whatsoever. Together with the futility of epistemological reductionism, the overall argument of the article is that one should abandon the metaphysical idea that all sciences and Nature in general are reducible to any imaginable theory of fundamental physics. While it is important for our general knowledge to establish reductionism between various theories whenever possible, there are stringent criteria for doing so, and it is not possible in general.
At the end I would like to raise a friendly criticism regarding the proponents of reductionism in Nature. The conclusions of this essay stand in sharp contrast to the popular opinion among scientifically-oriented people (even some practicing scientists) that reductionism unquestionably holds in science. The reason for this popularity arguably lies mostly in the scientists' ignorance of the full axiomatic structure of the theories they study, and the lack of education in mathematical logic, especially its less trivial aspects. Despite being popular, the reductionist opinion is actually a heavy metaphysical assumption, virtually indefensible both on epistemological and ontological grounds. While reductionism can indeed be established in certain particular cases (which is always a useful thing to know), a sizable number of scientifically-oriented people generalize reductionism from these special cases to the full-blown level of scientific tautology (or something to that effect), completely disregarding a glaring lack of evidence and consistency. This was labeled by Dennett as "greedy reductionism". If anything, this approach can be labeled as "scientistic", since it demonstrates both an unwarranted overconfidence in scientific results, and a superficial level of knowledge about actual statements of science. Science tells us far less than what is being attributed to it by such people, and one must be careful not to get carried away when interpreting scientific results.
Giving up the idea of reductionism essentially amounts to accepting strong emergence as a fundamental property of Nature --- a physical system might display behavior that is more than the behavior of the sum of its parts. Proponents of reductionism might find this at odds with their favorite ideology (physicalism, naturalism, atheism, etc.), but there are actual examples of strong emergence in Nature, the arrow of time being the most prominent one. It would be interesting to see how many people would actually agree to change their minds when faced with this kind of approach, as giving up reductionism generally weakens the arguments that a physicalist may have against dualism, a naturalist against the supernatural, an atheist against religion, etc. Philosophy teaches one to keep an open mind, while science teaches one to appreciate the seriousness of experimental evidence. When these two combine to demonstrate that certain parts of a physicalist/naturalist/atheist belief system are just unfounded prejudices, even downright wrong, it would be interesting to see how many people will actually give them up. After all, these are precisely the people who boast about both open-mindedness and the scientific method, and invoke them to criticize dualists/supernaturalists/theists. Now they are challenged with giving up one of their cherished beliefs, and I would like to see how truly open-minded and scientific they can be in such a situation.
 I have not provided a precise definition of ontological reductionism, and there are many different attempts in the literature. But briefly, ontological reductionism is the assumption that there exists a fundamental "theory of everything" to which everything else could be epistemologically reduced, given enough effort and rigor. We might not be in possession of a full formulation of such a theory (so epistemologically it might be out of reach), but the claim is that it exists, in the sense that it can be approached as a limit by formulating epistemologically ever more precise fundamental theories of nature. The assumption of ontological reductionism is that such a limiting procedure is convergent. I argue that most of the definitions of ontological reductionism found in the literature boil down to this one, operationally.
 All theories discussed in physics and beyond are powerful enough to satisfy these requirements.
 One can draw a loose analogy with the principle of "innocent until proved guilty".
Marko Vojinovic holds a PhD in theoretical physics (with a dissertation on general relativity) from the University of Belgrade, Serbia. He is currently a postdoc with the Group of Mathematical Physics at the University of Lisbon, Portugal, though his home institution is the Institute of Physics at the University of Belgrade, where he is a member of the Group for Gravitation, Particles and Fields.