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 [15]. 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.

*Axiomatic structure*

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:

(1) axioms of logic

(2) axioms of set theory

(3) axioms about correspondence of basic theoretical quantities to experiment

(4) axioms about theory’s range of validity

(5) axioms about laws that Nature upholds

Group (1) is typically the set of axioms of the first-order predicate calculus [16], 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 [17]. 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 [18]. In short, the theorem states that, given a set of axioms that defines some theory, if this theory meets some general requirements [19], 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.

*Ontological reductionism*

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 [20]. 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 [21]) 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.

*Conclusion*

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” [22]. 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.

_____

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.

[15] 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.

[16] First-order logic.

[17] Zermelo–Fraenkel set theory.

[18] Gödel’s incompleteness theorems.

[19] All theories discussed in physics and beyond are powerful enough to satisfy these requirements.

[20] One can draw a loose analogy with the principle of “innocent until proved guilty.”

[21] Recursive language.

[22] Greedy reductionism.

Categories: essay

Hi

Marko,It seems to me that your entire thesis, despite claiming to be about

ontologicalreductionism, is actually entirely aboutepistemology. In Note 1 you define: “ontological reductionism is the assumption that there exists a fundamental “theory of everything” to which everything else could beepistemologicallyreduced”, so your thesis is actually about epistemology.Second, Godel’s incompleteness theorem is all about the incompleteness of what we can know and prove, and thus is again epistemological.

Overall, in neither of your two parts are there arguments against what I would call “ontological reductionism” or “supervenience physicalism”. All of your arguments are against fairly strong forms of

epistemologicalreductionism (and in refuting those I entirely agree with you).I don’t think it is a popular opinion that “reductionism”

as you have defined it“unquestionably holds in science”. Can you point to many actual claims that reductionismas you have defined itdoes hold in science?It is indeed a popular opinion that supervenience physicalism holds, but that is a much weaker thesis.

Re-iterating from Part 1, I still disagree. All you need is for the low-level particle behaviour to be in some degree probabilistic rather than fully deterministic. You yourself have argued that that is so! As I’ve pointed to, here is a demonstration of weakly emergent second-law behaviour using only Newtonian rules and a “rand” function.

I don’t see that you’ve given any arguments at all in favour of strong ontological emergence, or against the sufficiency of supervenience physicalism and weak emergence. Epistemology is, of course, a whole ‘nother matter.

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Marko,

I enjoyed it a lot. Very clear.

“Just for the sake of the argument, and despite the fact that any expert in mathematical logic would immediately begin yelling at us”

! 🙂

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Godel was incomplete. There is One mathematical equation that defines everything absolutely and completely, reductionism is a Way to see it, Nature is too. =

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First of all, this series has to be my favorite on SS so far. I expected the clarity of the writing to induce people to engage with the issue as you have structured it, but that was probably unwarranted optimism.

There’s one thing that’s not clear to me, though. In the last part, where you move from epistemological to ontological reductionism, you are still talking about theories. I thought the distinction between epistemological and ontological reductionism was that the former is about theories while the latter is about “whatever is really existent that we are describing with theories”. From that view, ontological reductionism would be about whether, for lack of a better word, anything is really “caused” by non-fundamental entities. Have I got this wrong?

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Hi Marko,

For me, the practical meaning of a framework like this lies in its effects on our behavior. And so I wonder, what would a scientist who understands your framework do differently?

As a neuroscientist I am particularly interested in a specific type of reduction: the possibility of reducing mind and consciousness to the operations of brain cells. Does your framework tell me anything about that? Does it say that I should just give up on the idea, or point in the direction of a particular way of attacking it? Or does it just warn me that the problem is likely to be hard, and perhaps unsolvable?

Even my strongest hopes don’t reach the level of axiomatizing the mind-brain relationship, so perhaps the type of reduction I am interested in is cruder than the type you are talking about here.

Best regards, Bill

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Coel,

” All you need is for the low-level particle behaviour to be in some degree probabilistic rather than fully deterministic.”

I can’t speak at your or Marko’s level of math and physics, but I think Marko is arguing that all particular cases may be reducible (of which the ultimate requirement is that we have to rewrite all laws), but at the same time any particular non-reduced case may be a case of the non zero number of cases that are not reducible.

(on epistemology / ontology, I can say nothing because I’m still not comfortable with the distinction, though I’ve progressed to the point of not having to look up the terms every time someone uses them)

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Re: Burden of proof“Therefore, the burden of proof is clearly on the one who claims that reduction holds… a priori one must always start from the assumption that reduction does not hold between the two theories.”–the “therefore” is unfounded. your argument looks like this: proving reduction is hard, therefore burden of proof is on reductionists and we should for now assume nonreductionism. Analogy: proving the mind is physical is hard, therefore the burden of proof is physicalists and we should assume dualism. Your footnote “innocent until proven guilty” just begs the question at issue. Do you think it is more reasonable to assume that there are only a finite number of prime quadruplets? (reducto)

Re: theory of everything, Godel“So-called Gödel-statements correspond to what we have described as strongly emergent phenomena… 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”— Why can’t we be nominalists about them? They might emerge in our theories (i.e.

only ifwe assume our fundamental theory must consistent), but not in anything observable. When we actually observe and measure seemingly emergent phenomena, then we have reason to suppose they exist. Just as we don’t have to suppose numbers are fundamental, I don’t have to assume Godel-statements are emergent “in Nature”. Indeed, adding ‘in nature’ seems completely unjustified.time“The prime example of such a strongly-emergent phenomenon is the arrow of time”–Or, we could say time, or spacetime, is part of our fundamental level. What is the argument against making time (or some 2nd law equivalent) part of our fundamental level? Reductionism requires effective theories to be reduced to structural theories. I don’t know any serious thinker who goes on to claim that the structural theory has to be formally complete. I see a strawperson.

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Hi Marko,

Again, I must praise the clarity of your writing, but while I may agree with you that reductionism

as you have defined itis untenable, I don’t at all agree with your other conclusions, particularly regarding the naïveté of scientists. I’m going to concentrate on what Gödel says about a theory of everything.I think it is useful to have a paradigmatic case of a Gödelian true statement which cannot be proven. Unfortunately, by definition, we can have no such case, because if we could really be sure it were true then that would amount to a proof. Nevertheless it helps to have a hypothetical example. You have proposed the arrow of time, which I dispute because I think this can indeed be derived from the underlying structural theory.

So my hypothetical example is Goldbach’s conjecture. Let’s imagine it is true but that it can never be proven in ZF set theory.

To you, this means that Goldbach’s conjecture cannot be reduced to ZF set theory, and so it is strongly emergent. Indeed it is by your definitions.

But I think this bears little relationship to what the community of scientists are talking about when they postulate a theory of everything. A theory of everything, in my view, is supposed to be a structural theory which entails all the macroscopic behaviour we see in the universe given certain initial conditions. It is not required to allow us to prove unprovable mathematical statements or even to show that the macroscopic laws we perceive are universally true at all times (as a proof would). Indeed, it may be the case that some of the macroscopic laws we perceive (such as the 2nd law of thermodynamics) are only statistically or usually true. If that is so then a proof of their truth would appear to be ruled out — Goldbach’s conjecture is disproved if there exists even a single prime which disobeys it, even if it holds for all others.

Rather, it is enough to show that what we observe at a macroscopic level is

typicalof what we should see given the structural theory. If the expected macroscopic behaviour we should see given the structural theory corresponds to what we observe, then our observations have been explained (if not reduced,as you define it).If we suppose our effective theory is that even numbers greater than two are always expressible as the sum of two primes, then a structural theory (ZF) which shows that all even numbers

ever testedhave this property explains our effective theory even if we can’t prove the effective theory is universally true (which we didn’t know anyway).So, ontologically, a theory of everything is a mathematical description of the universe that entails all the effective theories we have based on observation. Epistemologically, even if we had such a description we could never know it to be correct, and in particular we can’t be sure of the correctness of our effective theories.

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OK, the argument presented by Marko is basically the one I used in a comment before to demolish reductionism. Its weak point as DM more or less alluded to, is that the Godel argument(s) do not fabricate (= compute) explicitly propositions that are beyond the reach of axiomatics (as those Godel arguments consist essentially in redeploying Cantor’s diagonal proof, which is not constructivist).

Marko speaks of axiomatics, and then naturally uses the Incompleteness Theorem(s). However, there is a little point he does not mention, that is even more basic in any mathematical logic framework. I already mentioned it. Any logical theory is not just made of axioms. It is also made of a “UNIVERSE”.

Clearly the different scientific theories we have now (say Proteomics, Plate tectonics, Entanglement Physics, Plasma Physics, Spider Ethology, etc.) belong to different universes. So they clearly do not reduce, to each other. Axiomatics is not everything.

All these arguments about reductionism would be clearer in Category Theory. Theory A reduces to B when A is a full subcategory of B.

http://en.wikipedia.org/wiki/Subcategory

Why is reductionism clearer in category theory? First, because I gave a one sentence definition.

But there is a more subtle point, a point that often baffle students of the Incompleteness Theorems. In all honesty, as I already said, Godel arguments are not constructivist (so I was a bit perverse when I alluded to them). One can, and I would even say, one ought to consider only constructivist arguments in science (the debate is a century old in logic Russell, and then Brouwer contributed).

Category theory provides with a more flexible way of thinking. Without getting tangled in mock precision and mock computation (sorry, Godel!).

By the way, I did not answer the interesting juxtaposition of “energy is not conserved” except if “one considers gravitational potential energy” (Coel). I did not find the time yet. However, if General Relativity is about comoving frames, what happens to potential gravitational energy? Well, it disappears (according to some). This is the logical breach in which the dark forces of the Multiverse rush.

Notice the paradox: logic is all about multiple universes. But physics ought to be just about the universe. The physical universe is the union of all possible logical multiverses (among other objects).

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You run into a number of problems – unnecessarily – in your final paragraph. First, physicalism, naturalism, and atheism come in a variety of positions having multiple arguments. Your thesis damages a number, but leaves quite a number unscathed. E.g., it is entirely possible to see mind as strongly emergent from body without lapsing into dualism: The body generates the mind as performing effectively biological processes, e.g., finding food, etc., but the mind becomes an ‘overgrowth’ developing into a system of its own greater than the sum of its parts. This does not free up the mind from the body in a dualistic manner, since the biological function would remain primordial and enjoy the highest priority. For such a position, your argument presents no threat, and indeed could be used to support it.

For your argument to be a strong criticism of naturalism as a rejection of any supernaturalism, I think supernatural phenomena would need to be proposed as somehow strongly emergent in some way from some prior system, but the only systems your argument concerns are physical and mathematical, and it’s hard to see how supernaturalism can emerge from a physical system without obviously being natural on the basis of its arising from the natural.

In general then, one problem here is that simply because a system strongly emerges from another, does not mean that it somehow enters a different category of being thereby. The macrocosm may be strongly emergent from some quantum level microcosm, but both are physical systems, clearly. (If the supernatural strongly emerges from the natural, then it is not super-natural, it is simply natural at a higher and more complex level.)

Finally, while there are atheists that have argued systematically, and so possibly present a kind of ‘belief system,’ the base of any rejection of theism is simply that – disbelief in theism. This needs no systematic foundation. My own atheism is founded on a question of relevance – do I need belief in a deity to explain anything? to guide my ethical behavior? to comfort me in any way? Not that I can see. Thus I can demand evidence of a divinity should the question arise, because it’s relevance is lacking. Thus the possibility of its existence could only be supported by evidence (beyond personal testimony, which raises questions of trustworthiness, even when contained in an ancient text).

So the positions you raise criticism of, at the end of your article, do not necessarily need reductive arguments to be maintained in some form or other. So I think it was a tactical error to bring them up. You argue that reductionism carries the burden of proof, and in many matters it does. But then you remark issues where that simply isn’t so, and this unnecessarily undercuts the body of your argument.

I am quite sympathetic to that argument. But one doesn’t need a closed mind to find reasons to reject dualism, theism, or the supernatural.

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Outstanding two essays!! Thank you!

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Here is yet another philosophical essay that claims that physicists do not know what they are doing. The core of your argument is 2 claims.

I. If you define the Second Law in a way that makes it false, then it is not reducible to things that are true.

II. If you define a theory of everything to include ZFC, then it includes well-known paradoxes of ZFC.

You blame belief in scientific reductionism on a “lack of education in mathematical logic”. The consequences of Goedel’s theorems have been well-understood for 80 years. Goedel would not agree with you, and neither would those who actually have an education in math logic, as far as I can see.

This is like complaining that the equations of general relativity cannot explain cosmology because they contain pi, and pi is irrational. Yes, the irrationality of pi may seem paradoxical to people who first learn it, but it has almost nothing to do with scientific reductionism. It will only cause a problem if you give some completely artificial requirement that pi must be reduced to rational numbers.

This is the first essay with SciSal’s new policy of restricting to academics. I expected to see philosophers explaining what they have published. Instead we have someone with training in relativity who makes an anti-science argument that has nothing to do with his training.

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schlafly, two things: first, once again, your comment – dripping of contempt as it does – very barely made it through my filter. Please do not make me ban you from this site. Second, you may have not paid attention to Marko’s credentials: he is not a philosopher, he is a theoretical physicist. And his understanding of relativity is certainly pertinent to his argument. Thirdly, why on earth would you think that the essay is “anti-science”? So now reductionism, clearly and obviously and indisputably either an epistemological or an ontological notion, is part and parcel of science so that even questioning it is amount to talking creationism??

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I say anti-science because Marko’s arguments, in his own words, “stand in sharp contrast to the popular opinion among scientifically-oriented people”. I do not question his expertise in relativity, but his essay is about philosophy, emergence, and math logic, not relativity.

I do not know why you keep threatening to ban me. I am expressing mainstream science views. If an essay makes claims that are contrary to standard science textbook knowledge, then someone should call the author out on those points. What else are the comments for?

Questioning reductionism like this is a little like talking creationism. If Marko wanted to argue that dark matter might be new form of matter that only emerges at a galactic level and cannot be reduced to some microscopic theory, then I would let it pass as a legitimate intellectual possibility. But it would be a little like saying that dark matter is supernatural stuff that God put there in order for the galaxies to form. My objection is that he claims to have disproved reductionism with his examples and his Goedel argument.

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schlafly, plenty of people have criticized Marko’s positions (e.g., Coel, DM), but have done so with none of the harsh and contemptuous tone that characterizes your comments. And if you think that questioning reductionism is like advocating for creationism you have a serious misunderstanding of one or the other — or both.

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Schlafly:

-I say anti-science because Marko’s arguments, in his own words, “stand in sharp contrast to the popular opinion among scientifically-oriented people”.-

It seems to me that saying things that stand in sharp contrast to popular opinion among scientifically-oriented people is, often times, part of the way science progresses (especially in the case of paradigm shifts).

Noam chomsky’s claim that we have innate information about syntactic properties that facilitate language acquisition stood in sharp contrast to popular opinion at the time (behaviorism), however nobody would claim what he did was anti-science.

Similar kinds of moves that stood in sharp contrast to popular opinion in scientifically oriented people but were scientifically helpful were made by Brian Scholl and Chaz Firestone (There are no interesting top-down effects on visual perception), Bert Vogelstein (cancer research: refined version of the multi-hit hypothesis that claims that specific types of mutations are necessary for cancer growth and that these mutations don’t have to occur in any particular order. Not to mention his 2015 paper on how 85% of your risk of getting cancer has nothing to do with environmental or genetic factors but instead, simply bad luck), and some might say Einstein and Newton made claims that were contrary to popular scientific opinion..

So, I suppose I am not clear on why making a supported claim that “stands in sharp contrast to the popular opinion among scientifically-oriented people” makes somebody anti-science. In many cases, it actually is how science progresses (science friendly/helpful).

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I think that

DM‘s analogy is useful to illustrate the difference between ontological and epistemological reductionism, and between weak and strong forms of reductionism.DM supposes that Goldbach’s Conjecture is both true (under ZF axioms) and Godel-unproveable (under ZF axioms).

This means that a mathematical “simulation” based on ZF will always exhibit GC-compliant behaviour.

Ontologically, that model will produce “output” primes that comply with GC. There will never “exist” a prime, in that model, which violates GC. Thus, ontologically, GC is, de facto, weakly emergent.But

epistemologicallywe can never know that there will never be such a prime, though we can note that there has been no such prime so far. This lack ofepistemologicalcompleteness is indeed already accepted as an unavoidable aspect of science itself (not just of any reductionist scheme). Thus, the strong form of epistemological reductionism (as Marko has defined it) does indeed fail.That is not, though, a refutation of ontological reductionism (which, ex hypothesi, holds).

Nor is it a refutation of a “theory of everything”, if by that we mean a unified and self-consistent theory that is

entirely empirically adequate. It does rule out a known-to-be-fully-true-and-complete ToE, but then, again, we’ve long accepted that empirical adequacy is the best we can hope for and know about.What would amount to strong ontological emergence would be — in the above scheme — if the “ensemble” built upon ZF started throwing up primes that violated GC, even though GC were true under ZF. (If that sounds contradictory it’s because strong ontological emergence is a rather weird concept.)

[Note that I’m treating the above as an

analogyto supervenience physicalism, with ZF axioms playing the role of basic particles. DM, as a mathematical Platonist, might go further than that (!), but even as an analogy it is useful and illustrative.]LikeLike

Coel, why do I get the sneaky suspicion that applying the concept of reduction to mathematical conjectures, rather than empirical theories, is a category mistake? In DM’s example math is all there is. In science, well, the issue is the external world, and math is just a convenient way to think about it. (And no, let’s not bring in the mathematical universe, okay?…)

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Schafly,

“

My objection is that he claims to have disproved reductionism with his examples and his Goedel argument.”Some of us think he has done a good job of showing the weakness of reductionist arguments. You are welcome to give carefully reasoned answers in rebuttal. But the following statements decidedly do not qualify!

“

claims that physicists do not know what they are doing”“

makes an anti-science argument”“

Questioning reductionism like this is a little like talking creationism.”To make these extraordinary, unfounded claims is not to give an answer but to admit your answers are lacking in substance. Physicists are free to question each other’s work and propose alternative hypotheses. That does not make them anti-science but it does make them good scientists. Questioning reductionism is healthy science and has nothing whatsoever to do with creationism.

Marko mentions burden of proof and this is a good place to take you up on that. When you make the extravagant claims that I have quoted above you also assume the burden of proof. Can you meet this burden of proof, by, for example, showing how Marko’s words are anti-science or that he is talking creationism? Can you quote actual words to this effect? I certainly can’t recall any.

Finally, accusations of anti-science and creationism have become a little shop worn lately. They have started to sound like the slogans used to punish the politically incorrect. Needless repetition and careless use debases the currency.

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Hi

Marko, while this essay makes an interesting case, I think I agree withCoelthat it is mainly a continuation of the former (epistemological) argument. Admittedly I have not put a lot of effort into Goedel, but my conception is that he was explicitly dealing with limits of proof/knowledge and not what might exist regardless of our knowledge. Reading your piece and the link did not shift this impression. Maybe I am missing something?I am also not sure that because a theory of everything might be useless for explanation (an interesting point), that means the underlying reality (if true) would be powerless to produce the results one is looking to explain!

Finally, I am not sure it is kosher to use epistemology to define ontology, and so by refuting epistemology take down ontology. I get that if ontological reductionism were true, that would theoretically grant the ability to model that reductionist reality. However, isn’t it possible for there to be rules regarding knowledge (or building models) which mean such can’t be made, but does not impinge on how reality can work. I think that is especially true when we are talking about building in safeguards such that a part of that reality (us) can be ‘sure’ of the correctness of their model. That may call for greater stringency.

This is why I generally don’t try to travel beyond epistemology. It seems to bring in elements which can play by other rules, and I’m not sure how to frame our knowledge (worth speaking about) outside of epistemology.

Still I think your end point stands for people who adhere to ontological reductionism in the face of never being able to achieve some complete epistemic reductionism. I mean you really seem to have put a pretty heavy kibosh on the latter.

To claim something exists for which one can never make a demonstrative case would seem an article of faith.

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I thought these two posts were excellent!

I hadn’t seen the concept of emergence framed so clearly in this way before, and I feel It was communicated in a way that a layman such as myself could comprehend the argument. I do agree however with ejwinner regarding the logical implications for beliefs in the supernatural vs atheism.

Thanks (Marko & Massimo) for making these arguments so accessible.

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Hi Massimo,

> why do I get the sneaky suspicion that applying the concept of reduction to mathematical conjectures, rather than empirical theories, is a category mistake?

You may be right, Massimo, but in that case the mistake was made by Marko when he decided to apply Gödel to the problem of reduction of physical theories. Gödel’s theorems only apply to mathematical statements which are necessarily true or false and it is not at all clear to me that they can be applied to the laws of a high-level (effective) theory which are just descriptions of how the natural world usually appears to behave. My illustration used mathematics because it is the only way I can make sense of an argument from Gödel. I honestly cannot think of a single plausible example of an analogous physical law which would have to be necessarily true given fundamental physics and yet in principle underivable from fundamental physics (I reject Marko’s example of the arrow of time because it is not hard to derive, as argued previously).

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In “Turing’s Ideas and Models of Computation” [1], Eugene Eberbach, Dina Goldin, and Peter Wegner identify three principles underlying the emergence of

super-Turingcomputing [2] in an open system: interaction with the world, infinity of resources, and evolution of system.Perhaps a TOE

could bedeveloped in such an environment.[1] http://www.cse.uconn.edu/~dqg/papers/turing04.pdf

[2] http://www.ewp.rpi.edu/hartford/~eberbe/projects/superTuring.html

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Daniel, anyone who talks about science progressing by paradigm shifts is anti-science also. The paradigm shift is defined to be a change in views that has no rational or measurable advantages. It is an idea that is popular among philosophers, but not scientists. You only hear a scientist talking about paradigm shifts when he is promoting an idea that has no evidentiary support.

Maybe Marko is the new Chomsky, but that Chomsky theory was not only against popular opinion at the time, it is still against popular opinion today. And people do attack Chomsky for being anti-science when he says things like: “Newton came along and he did not exorcise the ghost in the machine: he exorcised the machine and left the ghost intact. So now the ghost is left and the machine isn’t there. And the mind has mystical properties.”

Labnut, you are quoting my response to SciSal. My responses to Marko are here, here, and here. I claim to show Marko where he is wrong. If you or he can give some error in my reasoning, go ahead.

Coel, there is no such thing as a statement being “both true (under ZF axioms) and Godel-unproveable (under ZF axioms).” There is no such thing as primes that comply with or violate the GC. You seem to be going after some sort of mathematical reductionism that has little to do with scientific reductionism.

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Schlafly wrote:

Maybe Marko is the new Chomsky, but that Chomsky theory was not only against popular opinion at the time, it is still against popular opinion today.

———————————-

This is incorrect. Find me a linguistics department that is teaching behavorist — i.e. Skinnerian — linguistics and not generative grammar. Chomsky is absolutely the mainstream in linguistics.

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Your view is the majority view in the mainstream physics community.

I do agree with you on one thing; whatever the emergent-process which produces this universe to its current stage cannot be ‘trace-backed’ with the reductionism in terms of either philosophy or physics.

Marko Vojinovic: “I have argued that the burden of proof lies with the claim …”

Amen! But, this is not the case for the majority of physics theories, as tens billions dollars are spent for some theories which are not making any contact to any known physics.

It is nice to use the term (disagree with you) in our discussion. But, as the burden of proof is on me, I must use the word (wrong) about some of your key points.

“… to invoke Gödel’s first incompleteness theorem to argue that such a theory cannot exist, thereby defeating any concept of ontological reductionism.”

This is the total anti-realism. But, it is wrong. In the book {Linguistics Manifesto, (ISBN 978-3-8383-9722-1), see https://scientiasalon.wordpress.com/2015/01/05/apa-2014-4-emergence-and-complex-systems/comment-page-1/#comment-10709 }, this universe emergent out with three steps.

One, an initial condition (a formal system)

Two, the Gödel process (goes to ad infinitum)

Three, the Life (not restricted as bio-life) process (terminates the ad infinitum).

Thus, the Gödel process is by no means of any hindrance for the ‘manifestation’ of an ontological ‘ultimate’ reality. As this is an issue of the whole book, I will not go into it any further, as the book is available in many great university-libraries.

Byron Jennings (a very prominent physicist, an administrator at TRIUMF, Canada) blogs at Quantum Diaries (an official CERN blog) often. He is the loud voice in the mainstream physics community on this total anti-realism.

He wrote: “… The underlying problem is the fine tuning of the fundamental constants needed in order for life as we know it to exist. …There are two popular ideas for why the Universe is fined tuned. One is that the constants were fine-tuned by an intelligent designer to allow for life as we know it. … An alternate is that there are many possible universes, all existing, and we are simply in the one where we can exist.”

Indeed, his two choices cannot be reduced to, and it leads to anti-realism. But, I commented there, saying “There is another alternative; all nature constants {the Cabibbo and Weinberg angles, Alpha, electric charge, mass-charges, ħ (Planck constant), etc.} and many measured parameters {such as, Planck data (dark energy = 69.2; dark matter = 25.8; and visible matter = 4.82)} can be ‘derived’ (calculated); details, see http://www.quantumdiaries.org/2015/01/09/string-theory/#comment-1788717126 .”

As all nature constants can be calculated and all the calculations are totally self-proofing, I must say that your total anti-realism is totally wrong. More of your wrongs, next.

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MarkoThank you for another great essay! As an agnostic on the issue of reductionism, these essays and corresponding commentary has provided a lot of interesting points to consider. I had one question, you wrote:

I’m confused by this statement because the way I read it, it sounds like your saying physicalism, naturalism, and atheism are strongly dependent on a a reductionist point of view. I can see reductionism being related to those topics but I don’t see how it is essential, can you elaborate on this point? For example, I have read theories of naturalism that are non-reductionistic and seem to be perfectly reasonable theories.

Also, how does giving up reductionism weaken the arguments against dualism, supernaturalist and religion? Are you arguing that we should accept these views by just assuming they emerge? Seems to me we would have to argue in the positive for these concepts, even if we hold to an emergence view. Actually, I’m not even sure how reductionism was involved in the criticism of these topics as the little I have read on this topic rarely bring up reductionism as proof. Dualism is not rejected because it “mind” is emergent but rather because saying “You don’t know, therefore non-physical mind” is just fallacious reasoning. Saying the mind is a different substance doesn’t offer an explanation at all, it just adds a label. None of this seems to require a reductionist view point.

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Daniel TippensI wouldn’t go as far as saying Chomsky is anti-science but he’s probably not the best example to bring up, considering that he never tried to establish his LAD theory by doing good science and he didn’t seem particularly interested in what science had to say about the topic (and ironically as Schlafy pointed out, he was ultimately found to be wrong about this issue).

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@DM

I can’t say I agree much with your general take on mathematics and the universe (I’m pretty firmly in the camp of embodied mind anti-realism), but you are entirely correct here. If we operate under the assumption that mathematics and empirical science are two fundamentally different human enterprises that require two fundamentally different modes of inquiry and approaches, Godel’s contributions find themselves only in the proper context in the former camp — and not the latter one.

It’s NOMA 2.0, really.

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Actually. what both Coel and Dm are relying on is the presumption that ‘simulation of X = X’ which is simply ridiculous

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Shafly.

The issue here is one of rhetoric

.

It really doesn’t matter whether you’re wrong or right; – can you *persuade* us to take you seriously?

We all want ‘truth’ to ‘shine forth’ and convince others – but that’s not how reality works.

Attacking those you don’t agree with does not persuade us to your cause, but suggests you don’t want to be a participant in this community.

Please reconsider how you present your opinions here.

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You make it sound like those are the only two choices.

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Schlafly,

Um… I am pretty sure there is.

Obviously there are unprovable statements under ZF, otherwise we have to wipe out nearly a hundred years of mathematics. An unprovable statement can be true or false (as opposed to an undecidable statement).

It has often been conjectured that the Riemann Hypothesis might be true unprovable.

Coel,

I am not sure that we can meaningfully talk of the ontology of numbers and mathematical theorems. (I am sure Aravis will tell me what Carnap said about this, but even so…)

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Possibly you are quibbling about the terminology “Goedel-unprovable”, well OK. I am assuming Coel is referring to true and unprovable statements under ZF. There is not, as far as I am aware, any special category of “Goedel-provability”.

But something being both true and unprovable under ZF, I can see no problem with that.

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Hi DM,

We can know that such a statement exists and specify it nearly completely.

The example I usually use is this. For some natural number

n, for some Chaitin constantomega, there are two statements “the nth binary digit of omega is 0” and “the nth binary digit of omega is 1” and one of these statements will be true and unprovable.If your example, the Goldbach Conjecture, is true then it has an in principle proof by cases (with infinitely many cases) and so has a sort of quasi empirical demonstration.

But if your example could be considered an analogy for ontological reducibility then mine is an analogy for ontological irreducibility

One of that pair of statements (we could never know which) is a true statement which is immune from proof and immune.even in principle to the kind of quasi empirical demonstration you.have for the Goldbach Conjecture.

So I think that even if your argument is not a category mistake, as Massimo suggests, there are still problems with it.

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Schafly,

“

Labnut, you are quoting my response to SciSal.”Quite so, but that is no reason for making such unfounded accusations. I think you owe Marko a retraction. People with good arguments do not have to resort to personal attacks.

Imzasirf,

“

it sounds like your saying physicalism, naturalism, and atheism are strongly dependent on a a reductionist point of view.”That is not the main subject of Marko’s excellent and lucid essay but even so it is worth discussing.

It is no accident that activist atheists are ardent advocates of reductionism. Essentially they argue, that, as a consequence of reductionism:

R1. all that exists is, ultimately, observable and explainable by science;

R2. we observe no boundaries that demarcate the natural from the supernatural(or unexplainable);

R3. therefore there is no supernatural;

R4. therefore there is no deity.

And this, they think, is a knockout blow for deism.

That would seem to be a nice satisfying conclusion (to atheists), except that it is wrong. The supernatural is a folk concept, deeply rooted in narrative mythologies. It has an important cultural role but we must not confuse narrative mythologies, or aetiological narratives, for that matter, with a careful definition of God.

There are two errors in the reasoning. The first is to argue that (R3) follows from (R2). The second is to argue that (R4) follows from (R3). To see why this is so, consider two possibilities for a postulated deity, granting (R1) and (R2):

P1. God is the author of the Laws of Nature, creating them in order to create the Universe. The Laws of Nature are the way that God intended the Universe to evolve and operate and thus everything is subject to them. In this case God is supernatural (in the simple sense of being above or outside of nature) but unobservable because science is confined to the domain of the Laws of Nature. We live in a closed system, enclosed by the Laws of Nature and where everything operates according to the Laws of Nature, whereas God, as their creator, is ‘outside’ the Laws of Nature.

alternatively

P2. The Laws of Nature are properties of God. The universe evolves and operates in accordance with the Laws of Nature because that is the nature of God. In this case there is no such thing as the supernatural. Everything we see in nature is an aspect of God.

Whichever postulate one chooses, P1 or P2, the effect is the same. Science can only ever observe the natural. In the first case, God, the supernatural, is beyond observation, since we inhabit a closed system. So we cannot conclude (R3) from (R2). In the second case, there is no supernatural, by definition, and so one cannot conclude (R4) from (R3).

These conclusion apply, regardless of whether we believe in reductionism or strong emergentism. Activist atheists prefer reductionism because it would seem to make (R1) and (R2) stronger. But that is immaterial because their arguments fail at (R3) and (R4).

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This is the right (current) approach to defining theories and their relations, and then making it concrete (constructive) with type theory [1][2].

[1] http://ncatlab.org/nlab/show/relation+between+type+theory+and+category+theory

[2] http://arxiv.org/abs/1408.0054 (Quantum Gauge Field Theory in Cohesive Homotopy Type Theory)

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Again, sorry for the delay in answering, I’ve spent the whole day yesterday at various airports… 🙂

Coel,Your comment is making me regret that I didn’t define what I mean by “ontology” anywhere in the article. I have made a small comment in footnote [15] (note that footnote counting continues from part I of the article), but I can see it isn’t enough.

Ok, I agree with your criticism that you may consider ontology to entail more than what I have addressed in the article. Fair enough. But I should at least explain the reasons behind my “weak” definition of ontology — being a physicist, the only distinction between epistemology and ontology that I can effectively discuss is the following:

“Epistemology” entails the knowledge about Nature (i.e. existing theories) that we have accumulated so far. “Ontology” entails the knowledge about Nature (i.e. possible theories) that we may ever hope to accumulate, not just in practice but in principle.

I can acknowledge that the above idea of “ontology” is something you could claim to be still only epistemology, just a slightly stronger version of.

Note that any stronger notion of ontology is out of the scope of physics, and is firmly in the domain of philosophy, metaphysics in particular. So I cannot really debate such strong notions as a physicist. There is no way I can discuss what “really happens out there in Nature”, since all that physics can possibly tell me is how to describe it (i.e. map experiments to some theory).

That said, the idea of reductionism (the flavor I am discussing in the article) is conceptually connected to science, and is being used in science, rather than metaphysics. You are welcome to debate any notion of “metaphysically ontological reductionism” (I am guessing that is what you are calling “supervenience physicalism”), but any such notion is completely unrelated to science, and I am unable to even define it scientifically, let alone debate it. Therefore, any type of metaphysical reductionism is out of the scope of the article. Looking back, I should have said that in the article itself.

So I agree with you on this point. 🙂 But I also think it does not devalue the article — see below.

Oh, yes I can! And this is where things get completely surreal! 🙂

Example one — Sean Carroll, a practicing physicist and a well-acknowledged popularizer of science, claimed [23] that a “soul” cannot exist, because the behavior of the human brain is reducible to the Dirac equation, no less! In doing so, he implicitly appealed to reductionism (in line with my definition) as an established truth — not only that he didn’t provide a rigorous mathematical proof that all aspects of the human brain can be completely described by the Dirac equation (or to “some” established theory of elementary particles, if we read him charitably enough), but he even dismissed any such potential proof as unnecessary! Talk about the burden of proof…

[23] Physics and the Immortality of the Soul

Example two — again Sean Carroll, arguing that the laws of everyday physics are completely understood [24]. He wrote no less than four articles on this topic. Again he assumes that all “everyday world” phenomena are reducible to that single equation, basing this on reductionism as an established truth. Granted, in third paragraph he is careful enough to say that this doesn’t include “all the complex collective phenomena of macroscopic reality” — which is actually most of the everyday world phenomena! But the worse problem is that people don’t read much into those small remarks in parentheses in the middle of the text. The gist of his articles come accross as a completely legitimate statement that everything in everyday life is known to be reducible to that single equation, a statement which is actually scientifically completely unfounded, and an expression of belief, not fact.

[24] The World of Everyday Experience, In One Equation

Example three — Steven Weinberg has also made claims [25] that reductionism is an established truth. In more than several situations I have argued against reductionism with various people, only to be replied on the lines of “But Steven Weinberg says that reductionism is true, he surely knows what he is saying, right?”… And then I need to argue against the argument-from-authority fallacy, where the authority figure is no less than a Nobel prize winner, whose name is associated to the creation of the Standard Model of elementary particles! Unbelievable!

[25] I don’t have a reference handy, but Massimo has talked to him in person during the “Moving naturalism forward” meeting a few years back, and I guess he can tell us.

Massimo, I remember you have written about this on your old blog, can you provide us a link?Finally, example four — there are a slew of statements (just use Google!) that claim something on the lines of the following: a God (even if exists) cannot interact with the real world, since this would violate the known laws of elementary particle physics (if you look at equations, they do not contain any account for such an interaction, so…). Again this is an appeal to reductionism as an established truth, and ignorance of any burden of proof.

All of these examples appeal to reductionism precisely as I have defined it, either epistemological or ontological flavor, whatever. And these are authoritative statements by established fellow physicists. I won’t even mention the ways in which everybody else (non-physicists) are abusing reductionism in their arguments.

These examples were one part of the motivation for me to write this article to begin with.

I posted another comment in the part I section, discussing this. I agree with you in general, but I do not agree that QM as we know it today (epistemology!) is a strong enough structure theory that we could prove your statement within its realm. My article on lack of determinism in nature did not appeal to QM in particular, but to certain experimental facts and some math results. There is no contradiction between the claims in that article and the claim that QM (again, as we understand it today) cannot account for the second law of thermodynamics. 🙂

I hope we can agree on this and put the matter to rest. 🙂

(More replies follow!)

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I shouldn’t suggest that the position Coel and DM seem to hold is ‘simply ridiculous,’ please excuse that. Off-hand comments are easily misunderstood; they come too easily, don’t they? and never really say what must be said.

Coel and DM have a strong command of their fields of interest; hence the strength of their arguments. But at base they are trusting the power of mathematical and computer models to give us direct access to the ontological structures of reality, and I see that as deeply flawed. I would argue that computer simulations provide no ontological claims on the entities they reproduce schematically. At best they give us shadow-plays of the real drama, so to speak. Reality is richer than any model can suggest; theories are simply not the stuff they model. I understand that scientists need to trust that they are; but the fact remains that theories are human constructs for human understanding. If they work they are useful – at least until we get theories that work better. But exactly because of that, it is probably a mistake to commit to any theory so strongly as to close off the possibility of looking at the world in a different way than hitherto.

What Marko is suggesting is that if epistemological reductionism, applied to physics, fails, then ontological reductionism is unjustified. I think to be fully convincing, the argument would need to be stronger. However, Marko’s argument does not rely on any one of its component parts; all he needs to do here is present enough evidence to weaken the strong reductionist claim that all higher order systems are merely epiphenomenal to base systems, so then can be completely explained by those base systems. I think he has been fairly successful in this.

Scientists will continue to search for a complete Theory of Everything, because that’s what scientists do. But it may be well to bear in mind that such Theory is really not possible, for a host of mathematical, logical, epistemological, and probably ontological reasons. We should always remain humble when confronting the universe, it is rather larger than we are.

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ej, it’s not ridiculous, just, well, very fanciful…

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I’d like to comment on some of the remarks made about Chomsky in the comments here.

On one hand: The most important contribution that Chomsky made was to invent a methodology known as transformational grammar — a tool for describing the deep structure of a language in a systematic way. His invention revolutionized linguistics, and virtually all modern linguists know it and use it. So in that sense, every modern linguist is a Chomskian.

On the other hand: Chomsky has advocated some theoretical ideas whose purport is that the structure of human language is almost entirely preconstructed in our brains, and that we are born with a neural “Language Acquisition Device” whose function is to acquire the pieces of information we are not born with. That theory is popular with some linguists, but the majority think it is oversimplified or wrong. So in this theoretical sense, few modern linguists are Chomskians.

Bottom line: Nearly every modern linguist is a Chomskian at the level of methodology; few modern linguists are full-fledged Chomskians at the level of theory. To a working scientist, methodology is probably more important than theory, because it determines what you spend your day doing.

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Correction to my interpretation of Marko’s thoughts, I just noticed his response to one of my questions in Part I:

“It’s not like there is one particular phenomenon that cannot ever be reduced to any structure theory. I am not saying that … Instead, given any particular strongly emergent phenomenon, we can always reformulate the structure theory to make it reducible and weakly emergent. However, there can always be other phenomena that are still strongly emergent even for that new structure theory. So any particular example of strong emergence can be made reducible, but there are infinitely many such examples.”

That feels better with my overall understanding, I’ll have to let it sink in (now I see why I felt it was better for me to say non-zero instead of infinite).

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Hi

yaryaryarThanks. While (unlike you, apparently) I would be willing to entertain arguments from mathematics in the context of philosophy of science, at least we agree that it doesn’t seem to work in this particular case. I also agree with you that science and mathematics are (mostly) very different enterprises employing different modes of enquiry. The distinction becomes more blurry in theoretical physics.

Hi

ejwinner> Actually. what both Coel and Dm are relying on is the presumption that ‘simulation of X = X’ which is simply ridiculous

Actually, I don’t think we are. I have never (knowingly) made this presumption, certainly not in this conversation, and this is even farther from Coel’s physicalist views.

I may use the example of simulation as a means of illustrating what it means for, e.g. Newtonian mechanics to entail the arrow of time without any of Marko’s strong reductionism, but nowhere do I assume that a simulation of a room full of gas particles IS a room full of gas particles. It is enough for my point to note that it shows similar (in the sense of analogous) behaviour.

Hi

Robin,While I like and accept your example with the Chaitin constant, I think it may be bordering on too technical. I think Goldbach’s conjecture is much easier to get a handle on because it is so simple. Either there is an even number (greater than 2) which cannot be expressed as the sum of two primes or there isn’t.

I also don’t agree with your interpretation that it makes a fundamentally different point. Chaitin’s constant is an uncomputable number which is related to the Halting Problem, which involves determining if a computer program will ever halt. Your problem with the Goldbach conjecture is that we can just test all the infinite even numbers and see if we find a counter example. One can make the same point about the halting problem — to determine if a program halts we can just run the program for infinite time and see if it halts. The two problems are therefore very much of the same character in my view and I think both relate more to epistemic irreducibility than ontological irreducibility. The problems demonstrate something like epistemic irreducibility because we can never prove the truths of these statements, but are not analogous to ontological irreducibility because the statements really are necessarily true given the axioms. Unlike the axiom of choice, for instance, we are not really free to decide arbitrarily whether we take them to be true or not.

But I think that’s too much of a sidetrack to get into any further.

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Marclevesque,Thanks, I am really flattered that so many people like the article! 🙂

Asher Kay,Yes, I wasn’t fully clear on that. See my response to Coel above, hopefully it will clear things up.

Bill Skaggs,My framework tells you to be careful not to claim success of such a reduction, unless you have a rigorous proof. And given the complexity of the problem at hand, a rigorous proof is hopelessly intractable, for our lifetime (and I suspect even longer). One should certainly keep studying the possibility (and potential limits, which may exist!) of such a reducion problem, as much as possible, since we can certainly learn a lot about functioning of the brain by looking at the pieces. But always be mindful that the pieces may not explain all aspects of the whole brain, and that criteria for claiming success for such reduction are extremely high.

I don’t think that any cruder type of reductionism can be justifiable — the Solar neutrino problem essentially dusted one whole sector of the Standard Model of elementary particle physics, merely because of an unaccountable factor of three in one of the observables. Unless you have a firm quantitative control over both the effective and the structure theory, reductionism can only be a mirage. 🙂

Dyami Hayes,No, my argument is that a statement (that something is reducible to something else) can be said to be valid (i.e. to be a theorem of a formal system) only after you have provided a rigorous proof. Ask any mathematician. Being hard or easy has nothing to do with it.

Because a priori, when faced with some observable phenomenon in Nature, we do not know whether or not it corresponds to a Goedel-statement in our “theory of everything so far”. Virtually everything is a potential Goedel-statement, until (and if!) we manage to prove it is not. And the theorem tells us that the set of such statements is infinite, so we always have to suppose they exist. See also my response to DM below.

There is no argument against that, except that our fundamental level “so far” doesn’t contain it. It can certainly be added as an additional axiom. That is what I call strong emergence. The issue is that the arrow of time is but one of such phenomena, and we can never include all of them into our fundamental theory.

DM,The Goldbach conjecture is the statement in ZF, which entails the axiom groups (1) and (2) from the article. If it were a Goedel-statement (which we don’t know, but for the sake of the argument let’s suppose it is) then it would indeed be strongly emergent (according to the definition) in ZF (but note, not necessarily in ZF plus something more!).

That said, the Goldbach conjecture is not related to any statement involving axiom groups (3), (4) or (5), in the sense that a purely mathematical statement is not an “observable” in Nature. But there may be other Goedel-statements in our formal system, those which do involve observable quantities specified in (3), (4), (5) — those are the ones which are unprovable, but can be true (i.e. experimentally no counterexamples can be found). Those kind of statements are important for the article — given any structure theory, there exist observable phenomena in Nature that (a) we can observe, (b) we never observe the opposite, and (c) they are not provable within our structure theory. Those are the really relevant strongly emergent phenomena, while the Goldbach conjecture is irrelevant, in a sense, since it doesn’t involve any quantities postulated in axioms (3), (4), (5).

Note that these additional axioms (3), (4), (5) open the avenue for “experimentally true” statements, a notion of truth that is independent of the one that exists in pure math (axioms (1) and (2)). For example, the dark matter can be experimentally verified to exist, which is a distinct notion from mathematical truth. But on a formal level of axiomatic structure, any such distinction is only semantical — Goedel’s theorem applies equally well both to statements that are experimentally true and to those that are mathematically true, as long as they are not provable within our system of axioms.

If you show me a physical system that reproducibly violates the second law of thermodynamics, it could be used to construct a perpetuum mobile of the second kind, and solve almost all energy problems of human civilization. Until you do that, I stand by the claim that the second law of thermodynamics holds universally, with no exceptions. So it is not just usually true, but rather always true, as far as we know.

Besides, reductionism is a relationship between theories, say thermodynamics and Newtonian mechanics. And thermodynamics postulates that the second law holds universally. So if you say that the second law may be violated, you are not saying that thermodynamics can be reduced to Newtonian mechanics, but rather you are saying that thermodynamics is experimentally wrong, in a very profound way. Such a statement has nothing to do with reductionism anymore, it’s a completely different topic.

Patrice,Agreed. But not many people are familiar with the notions and inner workings of category theory, so it is not very useful for communicating the ideas of the article to general audience. Maybe one day, when category theory becomes a part of the mainstream math course in high-schools and common knowledge among educated public. 🙂

Ejwinner,I have a feeling you are reading into the article more than I have actually stated. I never said that lack of reductionism completely destroys the arguments of physicalists, naturalists and atheists. I merely said it

weakenstheir position, just like you said — some of their arguments fail (due to their unfounded appeal to reductionism), others do not.Ray,Thank you too! 🙂

(more to follow!)

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Coel,Massimo,DM,Yaryaryar,Massimo said:

I wouldn’t say it is a category mistake. Goedel’s theorem applies to any formal system of axioms (provided that the assumptions of the theorem are fulfilled), including scientific theories.

I think that the confusion arises only due to the different notions of “truth” being used around. The moral of the Goedel’s theorem is that there is a difference between truth and provability, in a given axiomatic system. Here, “provability” is a well-defined concept within the first-order predicate logic (axioms (1)), but “truth” is something that has to be established independently of the formal system. As far as math goes (axioms (1) and (2) only) the truth can be established by the Goedel-statement itself (self-referencing statement), or in some more intricate way (non-self-referential Goedel-statements do exist). But if we add axioms of physics, (3), (4) and (5), we gain another “arbiter” of truth (experimental one), which is also independent of the axiomatic system.

In other words, one could say that the Goedel’s theorem establishes the existence of statements that are (a) unprovable within a given axiomatic system, and (b) also “true” in that axiomatic system, given any notion of truth the axiomatic system may be compatible with. Any axiomatic system (1)-(2) allows at least for a “self-referencing truth”, or more. And axioms (3)-(4)-(5) specifically open the door for additional “experimental truth”. Given all axioms together, Goedel’s theorem then applies to all these types of truth.

Note that all this is a fine-print point that can be inferred from the exact statement of the theorem, which is highly technical. So I am not going to attempt to defend this further at a blog for general audience (in other words — either trust me, or don’t trust me, or learn the relevant math and check for yourself). 🙂

I have also touched on this in my previous response to DM.

Dbholmes,I agree as well, for a definition of “ontology” that involves metaphysical stuff (i.e. what “really” exists). See my response to Coel — I concede that both of you have a point regarding this, since my operative definition of “ontology” is much weaker than the (usual?) metaphysical one. I should have noted that in more detail in the text of the article.

That said, and if you look at the examples of people charitably using reductionism left, right and center accross science, the “weak ontology” that I discuss in the article is more than enough to state the argument against reductionism. One doesn’t need metaphysics for that. 🙂

Seth,Thanks! 🙂

Imzasirf,Thanks! 🙂

I would not say that they are completely dependent on reductionism — it’s just that people sometimes invoke reductionism as one of the arguments that all sorts of things don’t exist. Things like mind, consciousness, soul, God, angels, qualia, unicorns, zombies (specifically unconscious ones), etc.

Arguably there is a whole arsenal of arguments either in favor or against these notions, reductionism in science being one usually used against them. In that sense, my statement is that reductionism cannot really be used in this context without the burden of proof — and the proof requires level of rigor that is usually impossible to reach. In other words, all of the above notions may be considered strongly emergent phenomena in Nature, compatible but independent of the laws of physics (known either potentially or actually), until you manage to prove otherwise.

Other arguments of the “arsenal” in favor or against those notions are independent of reductionism, and I do not address any of them in any way. Not in this article. 😉

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Marko,

“the existence of additional independent-but-true laws of physics (like the arrow of time) only provides additional source for this incompleteness.”

In your part 1, you made the following comment; “Dynamics is all about “restrictions” — laws that a physical system must obey.”

Given the inherent inertia of a dynamical system, it would seem the first restriction would be the asymmetry of time, given the energy of the process propels it in one direction/dynamic and not any other. If we were to ignore this basic property, then reality would be entirely random.

What first attracted me to study physics was that foundational processes, such as thermodynamic convection cycles, were evident in every aspect of existence, from plate tectonics, to the business cycle. Which suggests some validity to reductionism, so maybe it is taken out of context.

For one thing, there seems to be an assumption that scale is a determinant of what is fundamental. The smaller it is, the more fundamental it must be. So chemistry must be reducible to quantum mechanics, etc. Given the earth is locally flat and globally spherical, would that mean it must be structurally flat and only effectively spherical? Could it be the laws most universal are not ones of scale and it is therefore something of a dichotomy, where the smaller are as much an isolated part of the larger, as it is the larger are the sum of the smaller? Top down and bottom up as two sides of a larger whole.

To me what seems a basic relation is between energy and form. Energy will always express some degree of form, as Marko points out that dynamics is about restrictions, as form must be manifest by some amount of energy. There are no platonic forms in the void. The most evident proof of this relationship is that as evolved beings, we have a central nervous system to process information and the digestive, respiratory and circulatory systems to process energy.

Also it goes to the issue of time, since energy is dynamic and conserved, at least locally, while information is static and transient, so it is the process of energy creating and dissolving these forms which creates the effect of time.

As for reductionism, what is it, but distillation? We know it is not the whole picture and that is its use, to extract that which is most useful out of the larger whole. Not only do we extract energy from our environments, but this environment is a process of energy extraction. Plants photosynthesize it from sunlight and turn it into fiber, oils and other nutrients, animals consume it and further concentrate it into proteins, other animals consume them and eventually everything not radiated back out turns into carbon and eventually geologic processes concentrate some of the carbon into diamonds, or squeeze it out as oils. Yet most gets radiated back out in the process.

Now compare this to the process of human intellectual reductionism; We see light reflecting off the environment around us and our sight distills out particular spectrums of the light, in useful concentrations, focused at particular distances and directions. These then collect as sequences of observations. From these and the other senses, we extract patterns useful to our survival, including acquisition of sufficient energy. We then form these patterns in our minds and learn to associate them with particular noises other such beings make and learn to communicate. Then we associate these noises and relationships with our perception, to particular marks organized in sequences and recorded in storable fashion. Then different uses develop for these marks. Some tell narratives and some organize patterns. We keep recording ever more such narratives and patterns synthesis and learn to make associations between them and how they relate. Many are discarded or lost and a few become intellectually and culturally foundational.

Keep in mind that astronomy and astrology developed simultaneously, as both description of and explanation for cosmic order. Naturally we can directly observe that order, but its source is not readily apparent, so we revert to our stores of prior knowledge and make associations and connections. With the ancients, it make good sense to relate these cosmic patterns to the diverse biological environment and so the skies seem populated with any number of beasts and people. So then these two sides of this process when their separate directions, with astrology becoming one of the pillars of religion and astronomy as one of the primary sources for geometry. Yet even with epicycles, explaining the observed order became a function of associating it with the mechanical processes that were cutting edge of the time and so it became the clockwork universe of giant cosmic gearwheels.

Since this could go much beyond the word limit, I want to conclude by pointing out that reductionism is a foundational device for how we function, but it is a limiting and defining process. What is discarded is important, as it is the energy used to create the forms we hold onto. The energy expands and the mass contracts. Reductionism/distillation is part of the function of creating form from energy and much energy goes to creating little form.

As to how the convection cycles produced by this expansion of energy and concentration of mass might give us a useful theory to at least hold onto……

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Hi

Marko, it would have been useful to highlight you were not planning to deal with metaphysical issues when arguing against ontological reductionism. But I agree that this did not undercut the utility of your essay(s).Wrapping up the previous essay… you are correct that the arrow of time issue mirrored the brain modeling issue I brought up. As I said in my first reply in that thread, I didn’t quite get that argument. Now I get it.

So overall two well structured arguments against common forms of reductionism. This second part really seemed to add a bigger spike.

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Reblogged this on emerging mind.

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Aravis: Others have explained that Chomsky’s claim about a language acquisition device is still contrary to popular opinion. See also this article saying that his belief in a universal grammar is also contrary to a majority of linguists.

Labnut: You twice accuse me of making an unfounded accusation by saying that Marko makes an anti-science argument, and you say that I “owe Marko a retraction”. In fact I supported that statement here, noting that Marko himself admitted that his arguments “stand in sharp contrast to the popular opinion among scientifically-oriented people”.

I do say that Marko is wrong, and I have very specific reasons. Marko has not even responded.

I am trying to be polite enough to not get banned, but Marko is saying stuff here that is contrary to every logic textbook of the last 80 years when he says:

No, there is only one notion of truth in an axiomatic system. That is, provability from the axioms.

Yes, there are unprovable statements in ZF. Such a statement will be either undecidable or provably false. An undecidable statement is true in some models of ZF, and false in others.

Goedel proved the existence of undecidable statements in ZF, assuming consistency of ZF. Those statements have a metaphysical interpretation in which they are true, but they are certainly not true in ZF, as Goedel proved that there are models of ZF in which they are false.

Almost everything Marko says about Goedel is wrong.

Beyond that, Marko wants to prove that Weinberg’s reductionism is wrong. There are several problems. If Goldbach’s conjecture turns out to be undecidable, then why would that have any physical implications? And if it did, then some physical experiment ought to decide it, thereby eliminating the issue as any impediment to reductionism. So there is no actual connection between Goedel and reductionism, except to confuse readers with red herrings.

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Marko Vojinovic: “… strong emergence in Nature, the arrow of time being the most prominent one.”

In your language, the above statement is saying that the emergent process of arrow of time (AOT) is unknown, or impossible to know. If this statement is true, then all the critiques on your article are just bickering hot airs.

If the emergent process of AOT is not known, there is no way to calculate any of the Nature constants, such as Alpha.

For showing this AOT emergent process, I will start with your ‘Effective/Structure’ scheme (ESS) which is “seemingly” rationally constructed, with many successful examples. Thanks for also pointing out a few failures. You are “seemingly” recognizing some limitation of that ESS, as you said, “The axiomatic structure of any scientific theory is very complex, so complex that all axioms are virtually never spelled out explicitly.” Yet, your final conclusion is not about the badness of your ESS but is the total anti-realism.

Your ESS is useful only if both effective and structure theories are clearly defined and in ‘completion’. How can any two pieces have some good match if there are some chunks missing on both? How can anyone conclude that the failure is not caused by the badness of the ESS but is the result of the destination being a non-reality?

Well, all the above is just the talking talks. Let’s show it with physics.

The first important thing is that many laws play very minimum role in the ‘fundamental’ level. You have pointed out that the ‘energy conservation law’ governs only a ‘local’ structure (symmetry) but plays no role globally.

For the past 100 years, almost all physicists who try to unified quantum with gravity use General Reality (GR) as the ‘right’ theory for gravity. The result is a total failure, and there could be two causes.

C 1: all those physicists are too dumb.

C 2: the GR is a bad stuff.

In this short comment, I cannot show you that how bad this GR is. But, I can assure you that both GR and SR play zero role in the calculation for any nature constants. GR is one ‘expression (the useless one)’ of the gravity which misses out the essence (the instantaneous action). GR “seemingly” has two great importance.

One, it defines a ‘causal’ universe. But, ‘photon’ does a better job on this.

Two, the ‘cosmology constant (CC)’ defines the cosmos. But, the CC is not a ‘theoretical’ consequence of the GR. Furthermore, as a non-zero zero; it will do the same by adding to any equation: Einstein, Weinstein or else.

GR is not wrong. But, anyone who wants to use GR as gravity in the quantum unification better has an eternal life, and even that he will not get a successful result.

The quantum/gravity unification can only be done when the ‘instantaneous action’ mechanism is understood. Then, there comes the AOT. Next.

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