Reductionism, emergence, and levels of abstractions

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Reductionism, emergence, and levels of abstractions

Russ Abbott

Computer Science, California State University, Los Angeles

Thirty five years ago Philip Anderson issued a challenge [1]. He didn’t disavow reductionism.

[The] workings of all the animate and inanimate matter of which we have any detailed knowledge are … controlled by the same set of fundamental laws [of physics]. … [W]e must all start with reductionism, which I fully accept.

But he challenged the claim that

the ability to reduce everything to simple fundamental laws … implies the ability to start from those laws and reconstruct the universe. … At each level of complexity entirely new properties appear. … Psychology is not applied biology, nor is biology applied chemistry. … [T]he whole becomes … very different from the sum of its parts.

Nearly 30 years earlier, Erwin Schrödinger made a similar point [2].

[L]iving matter, while not eluding the 'laws of physics' … is likely to involve 'other laws…,' hitherto unknown, which … will form just as integral a part of [the] science [of living matter] as the former.

Compare these statements with those of Steven Weinberg [3], a committed reductionist.

[A]ll of nature is the way it is … because of simple universal laws, to which all other scientific laws may in some sense be reduced. …

Every field of science operates by formulating … generalizations that are sometimes dignified by being called principles or laws. … But there are no principles of … chemistry that simply stand on their own, without needing to be explained reductively from the properties of electrons and atomic nuclei, and … there are no principles of psychology that … do not need ultimately to be understood through the study of the human brain, which in turn must ultimately be understood on the basis of physics and chemistry.

This too is an echo of a grand tradition. David Gross quotes Albert Einstein [4].

The supreme test of the physicist is to arrive at those universal laws of nature from which the cosmos can be built up by pure deduction.

I love this sentence. In one sentence Einstein asserts the strong reductionist view of nature: There exist universal, mathematical laws … from which all the workings of the cosmos can (in principle) be deduced, starting from the elementary laws and building up.

The remainder of this article, a summary and extension of [5], shows how computer science supports Anderson and Schrödinger against Weinberg, Gross, and Einstein.

The Game of Life Turing machine

Most CACM readers know that the Game of Life is a cellular automaton in which cells are either alive (on) or dead (off). At each time step:

  • any cell with exactly three live neighbors will stay alive or become alive;

  • any live cell with exactly two live neighbors will stay alive;

  • all other cells die.

Think of these Game of Life rules as a proxy for the fundamental laws of physics. They determine everything that happens on a Game of Life grid. Nevertheless there are “higher level” laws that are not derivable from them.

Certain Game of Life configurations create patterns. The most famous is the glider, a configuration of on and off cells that moves diagonally across the grid. Gliders and other patterns are epiphenomenal; they have no causal power. Only the Game of Life rules make cells go on and off. Nonetheless, Rendel [6] shows how to implement an arbitrary Turing machine by arranging such epiphenomenal Game of Life patterns.

A Game-of-Life Turing machine is a level of abstraction implemented using the Game of Life rules. Game-of-Life Turing machines are also epiphenomenal. But once it’s determined that a Turing Machine can be implemented as a Game-of-Life level of abstraction, the laws of computability theory apply. And computability theory is not derivable from the Game of Life rules. Thus while not eluding the Game of Life rules, new properties appear at the Turing machine level of abstraction—just as Anderson and Schrödinger said.

Furthermore, conclusions about Turing machines can be extended to conclusions about the Game of Life. Because the halting problem is unsolvable, it is unsolvable whether an arbitrary Game of Life configuration will ever reach a stable state. Call this “downward entailment”—not quite downward causation, but close.

Evolution as a property of a level of abstraction

Evolution by natural selection depends on (a) the possibility of heritable variation and (b) that survival and reproduction are affected by the relationship between an entity to its environment. Evolution is not a reductionist theory. It neither depends on nor is derived from lower level laws. Although evolution is implemented in part by DNA, Darwin didn’t know about DNA. He didn’t have to know about DNA; evolution occurs in any level of abstraction that satisfies the principles listed above.

The reductionist blind spot

Levels of abstraction are epiphenomenal.1 They have no causal power; the underlying physics turns the causal crank. Even so, the constraints2 embodied by levels of abstraction give rise to new laws. (It would be surprising if they didn’t.) Evolution and unsolvability are two such laws. Neither is derivable from nor dependent on the mechanisms used to implement the level of abstraction to which it applies. Each characterizes properties of a level of abstraction itself.

Epiphenomenal though they may be, reducing away a level of abstraction produces a reductionist blind spot. No equations over the domain of Game-of-Life grid cells can describe the computation performed by a Game-of-Life Turing machine unless the equations themselves model the Turing machine. The laws that characterize the regularities at a level of abstraction become meaningless when the abstractions are defined away.

Evolution and unsolvability illustrate how the level-of-abstraction—a concept familiar to most computer scientists but to few outside our field3—allows nature (and programmers) to build new regularities within a reductionist framework.

  • Emergence: the implementation—either statically (at equilibrium) or dynamically (far from equilibrium)—of a level of abstraction.

  • Generalized evolution: extant levels of abstraction are those whose implementations have arisen and whose environments support their persistence.


[1] Anderson, Philip W., “More Is Different,” Science, 4 Aug. 1972, (177, 4047), 393-396. 1972

[2] Schrödinger, Erwin, What is Life?, Cambridge University Press, 1944.

[3] Weinberg, Steven, “Reductionism Redux,” The New York Review of Books, October 5, 1995. Reprinted in Weinberg, S., Facing Up, Harvard University Press, 2001.

[4] Gross, David, “Einstein and the search for unification,” Current Science, 89/12, 25 December 2005, pp. 2035 – 2040.

[5] Abbott, Russ, “Emergence explained,” Complexity, Sep/Oct, 2006, (12, 1) 13-26. Preprint:

[6] Rendell, Paul, “Turing Universality in the Game of Life,” in Adamatzky, Andrew (ed.), Collision-Based Computing, Springer, 2002.,

[7] Laughlin, Robert B., A Different Universe, Basic Books, 2005.

1 Nonetheless, their reduced entropy and special mass properties render them objectively real.

2 Anderson called this broken symmetry.

3 Laughlin [7] used the term protectorate when he argued that Newtonian physics and the solid state of matter are both levels of abstraction.

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