Abstract: Aristotle stated that the whole is more than the sum of its parts. Modern scientists came to the same conclusion. Phillip W. Anderson wrote an influential article titled “More is Different” in 1972 . Many other scientists joined him in the investigations of emergent phenomena. These studies showed that the emergent phenomena require distinct explanatory models that cannot be reduced to the models of the base level. Theories that are very successful at explaining the parts and their interactions have difficulty explaining the whole. This is the closure problem of emergence. There are weaker and stronger versions of the problem. The weaker version describes it as a theory construction problem. The stronger version claims that the root cause of the closure problem is the fact that the emergent level is ontologically distinct. This view is beautifully illustrated by the “3 worlds and 3 mysteries” of Roger Penrose. Howard H. Pattee, on the other hand, views the closure problem as a theory construction problem and proposes a solution known as the “semantic closure” (also known as “semiotic closure”). This article provides a short overview of these subjects and concludes with a philosophical commentary.
I will start by defining a term that may be less familiar: “ontologically distinct” means that the realm in question is categorically different therefore subject to different laws of nature.
In the strong version of emergence the emergent realm is ontologically distinct therefore a new theory is needed to explain the emergent behaviors. In the weaker versions of emergence the laws of the base level are assumed to be valid in the emergent realm as well, and the difficulty of explaining the emergent behaviors is attributed to the lack of computational power or lack of information about the initial conditions.
Organic life is emergent. If you are arguing for strong emergence you are claiming that organic life is ontologically distinct therefore subject to different laws of nature. If you are arguing for weak emergence you are expressing your hope that biology can be reduced to physics someday.
Now replace “organic life” with “individual mind” in the paragraph above. Strong emergence: “individual mind is subject to different laws of nature.” Weak emergence: “individual mind can be reduced to physics someday.” It gets more difficult to argue for weak emergence in the case of individual minds.
We don’t have to invoke life and mind to describe emergence. There are many examples of weak emergence in physics as described in Phillip W. Anderson’s classic article “More is Different” . For other examples of emergence in physics the website of the Dutch Institute for Emergent Phenomena can be consulted . More examples can be found in the publications of the Santa Fe Institute which has been very effective promoting emergence as a unifying theme  .
For a comprehensive but dense philosophical examination of emergence the SAP (Stanford Encyclopedia of Philosophy) article titled “Emergent Properties”  can be studied. A more readable review of emergence is in IEP (Internet Encyclopedia of Philosophy) . Interestingly, neither the SAP article nor the IEP article mentions the work of scientists Roger Penrose and Howard H. Pattee whose ideas have been very helpful for a clear understanding of the closure problem of emergence.
Roger Penrose invokes three ontologically distinct worlds to describe emergence.
“Platonic-mathematical, physical, and mental – has its own kind of reality, and where each is (deeply and mysteriously) founded in the one that precedes it (the worlds being taken cyclicly)” .
The biological world (organic life) is not shown in Penrose’s picture. The biological world is assumed to be part of the physical world. Penrose is discussing a cyclical relationship among the three worlds. That’s why he uses the term “preceding” instead of “lower” when he refers to three worlds.
As a side note, the reader might be interested in the debates  of Mark Alford, Max Tegmark, and Piet Hut. Their debate was inspired by the “3 worlds and 3 mysteries” of Penrose.
Closure problem of emergence
Penrose uses the word “mysteriously” to point to the closure problem. Individual mind emerges from the physical realm but it cannot be completely explained by physical processes. According to Penrose the root cause of the closure problem is the ontological distinction of the three worlds. This is an argument for strong emergence.
In weak emergence, as demonstrated in condensed matter physics, the models developed specifically for the emergent level have more explanatory and predictive power. Here, there is no claim about the ontological distinction of the emergent level but there is an admission of explanatory weakness of the models of the base level. The explanatory weakness is assumed to be due to lack of computational power or lack of information about the initial conditions. The explanatory weakness is sometimes attributed to an undiscovered law of nature. In summary, in weak emergence the explanatory weakness (closure problem) has an epistemological origin.
Howard H. Pattee’s approach to the closure problem
Howard H. Pattee      argued that the closure problem can be solved by proper theory construction in the domain of biology. His proposal is to treat rate-independent memory structures and rate-dependent dynamical laws as distinct categories and study their interactions.
Pattee originally referred to semiotic closure as “semantic closure.” The term “semantic” connotes with “meaning”. The concept of semantic/semiotic closure applies to systems that enclose their own meaning. If the system contains the definition of itself, in other words, if there is self-reference then the system can replicate itself. This opens the gates of biological evolution.
Pattee’s solution was inspired by the discovery of the DNA structure and its function which is an example of how a system can enclose its meaning within itself.
“…universal constructor architecture (UCA) first explored by von Neumann. In a UCA, machines interact with an abstract description of themselves to replicate by copying the abstract description and constructing the machines that the abstract description encodes. DNA-based replication follows this architecture, with DNA being the abstract description, the polymerase being the copier, and the ribosome being the principal machine in expressing what is encoded on the DNA. This architecture is semantically closed as the machine that defines what the abstract description means is itself encoded on that abstract description. ” 
Semiotic closure insists on the causal closure of the physical. Semiotic closure cannot be achieved if there is strong emergence. The underlying assumption of semiotic closure is the belief that Nature records the code (laws) of the emergent behavior on a physical substrate. By proper theory construction the “code” can be deciphered and its activation logic detailing how this code gets realized in space-time-matter can be explained. As long as the code (laws) of the emergent level is implemented on a physical substrate the causal closure of the physical is achieved. This seems to be a good argument in the biological world. But, can we generalize this? Does Nature always record the code of emergent behaviors on a physical substrate? My answer is no…not always.
Many mathematicians and physicists sense that there is an abstract realm and they have clues that the physical realm emerged from the abstract realm. Roger Penrose, Max Tegmark and many others call the abstract realm Platonic mathematical world. In the Eastern philosophies we find very sophisticated discussions of the abstract realm and its various subtlety levels. The abstract realm is currently a complete unknown to science. Regardless, if science discovers someday that the physical world emerged from the abstract realm then scientists can no longer claim that the individual mind is a physical expression.
Here’s why. Not all aspects of the emergent realm can be explained by the characteristics of the preceding realm. Some aspects of the emergent realm may be expressions of the prior realms in the progression. The individual mind emerges from the physical realm but it cannot be completely explained by the physical processes because the physical realm itself emerged from an abstract realm in the first place. The individual mind must have some elements exhibiting the characteristics of the abstract realm. Therefore, the individual mind is not entirely physical.
The concept of semiotic closure has limited applicability but I appreciate the lesson Howard H. Pattee teaches: the rate-independent memory structures and the rate-dependent dynamical laws should be treated as distinct categories.
 P. W. Anderson, “More is Different”, Science, New Series, Vol. 177, No. 4047. (Aug. 4, 1972) https://cse-robotics.engr.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf also https://www.jstor.org/stable/1734697
 The Dutch Institute for Emergent Phenomena, https://www.d-iep.org/emergence
 David Pines, “Emergence: A unifying theme for 21st century science”
 The Santa Fe Institute, https://www.santafe.edu/
 Timothy O’Connor, “Emergent Properties”, The Stanford Encyclopedia of Philosophy (Winter 2021 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/win2021/entries/properties-emergent
 Elly Vintiadis, “Emergence”, Internet Encyclopedia of Philosophy, https://iep.utm.edu/emergenc/
 Roger Penrose, “The Road to Reality: A Complete Guide to the Laws of the Universe”, Knopf (2005)
 Piet Hut, Mark Alford, Max Tegmark, “On Math. Matter and Mind”, https://philpapers.org/rec/HUTOMM also https://arxiv.org/abs/physics/0510188
 Howard H. Pattee (2001), “The Physics of Symbols: Bridging the Epistemic Cut”, Biosystems Vol. 60, pp. 5-21 https://www.informatics.indiana.edu/rocha/publications/pattee/pattee.html
 H.H. Pattee (1973), “Physical Basis and Origin of Hierarchical Control”, Hierarchy Theory, ed. Howard Pattee, George Braziller, New York
 H.H. Pattee (1982), “Cell Psychology: An Evolutionary View of the Symbol-Matter Problem”, Cognition and Brain Theory, v. 5, pp. 325-341
 H.H. Pattee (1995), “Evolving self-reference: matter symbols, and semantic closure”. Communication and Cognition Artificial Intelligence 12(1-2), 9-27.
 H.H. Pattee (2006), “The Physics of Autonomous Biological Information“, Biological Theory, Vol. 1, No. 3: 224–226. Also https://www.academia.edu/863897/The_physics_of_autonomous_biological_information
 Edward B. Clark, Simon J. Hickinbotham and Susan Stepney, “Semantic closure demonstrated by the evolution of a universal constructor architecture in an artificial chemistry”