The reductionist blind spot: Russ Abbott Department of Computer Science California State University, Los Angeles three examples

Living matter, while not eluding the ‘laws of physics’ … is likely to involve ‘other laws,’ [which] will form just as integral a part of [its] science. Why is there anything except physics? — Fodor [Starting with the basic laws of physics] it ought to be possible to arrive at … the theory of every natural process, including life, by means of pure deduction. All of nature is the way it is … because of simple universal laws, to which all other scientific laws may in some sense be reduced. 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 are free-standing. [Starting with the basic laws of physics] it ought to be possible to arrive at … the theory of every natural process, including life, by means of pure deduction. — Einstein Living matter, while not eluding the ‘laws of physics’ … is likely to involve ‘other laws,’ [which] will form just as integral a part of [its] science. — Schrödinger. All of nature is the way it is … because of simple universal laws, to which all other scientific laws may in some sense be reduced. 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 are free-standing. — Weinberg The ability to reduce everything to simple fundamental laws [does not imply] the ability to start from those laws and reconstruct the universe. — The ability to reduce everything to simple fundamental laws [does not imply] the ability to start from those laws and reconstruct the universe. — Anderson Why is there anything except physics? — Fodor

Why won’t a square peg fit into a round hole? If a square peg can be “reduced” to the elementary particles that make it up, why can’t those particles fit through a hole of any shape? Because its shape isn’t compatible with the shape of the hole. Common sense. Right? First example

Is it quantum mechanics or solid geometry? Describe a particular square peg and a particular round hole by characterizing the positions of the elementary particles that make them up.  Will be very different depending on materials: metal, glass, wood, ….  Argue that the forces among particles when in a "peg" and "hole" configuration require them to satisfy various invariants: the geometric relationships among the peg particles are fixed (it doesn’t change shape); the hole has rotational symmetry. Conclude that the forces, invariants, and symmetries prevent the particles that represent the peg from moving to a position that would be described as being “in” the hole created by the round hole particles. A similar argument must be made for each peg-hole combination. Quantum mechanics

Is it quantum mechanics or solid geometry? Describe the abstract geometrical characteristics of square pegs and incompatible round holes, namely that the diagonal of the face of the square peg is greater than the diameter of the round hole.  Solid geometry is a level of abstraction. Discussed further later. If we require that non-interpenetrable solids do not intersect, we can conclude that any (abstract) square-peg/round-hole pair with incompatible dimensions will not fit one within the other. Whenever nature constructs entities that instantiate such solid geometry abstractions, they too will not fit one within the other. Solid geometry

Is it quantum mechanics or solid geometry? Is solid geometry reducible to physics?  Is solid geometry just a convenient generalization—something that captures multiple physics cases in a convenient package?  Or is it an independent domain of knowledge? My answer is that it’s an independent domain of knowledge. But this example may seem somewhat borderline. The more basic question is whether the square peg and round hole are ontologically real entities and if so whether reducing them away loses anything. My answers:  They are ontologically real—for a number of reasons.  Reducing them away loses something. One loses both the entities themselves and the theory—solid geometry—that describes how they behave. Reducible or not?

Similar arguments: The theory that explains how an atom at the end of my nose got from LAX to Indianapolis.  I’m an entity; the airplane is an entity.  We are governed by laws from Newtonian physics and aerodynamics. The trajectory of Roger Sperry’s nail on the rim of a wheel.  The wheel is an entity governed by laws from Newtonian physics and plane geometry.

By suitably arranging GoL patterns, one can simulate a Turing machine. Can conclude that the Game of Life halting problem is undecidable. By suitably arranging GoL patterns, one can simulate a Turing machine. Can conclude that the Game of Life halting problem is undecidable. Turing machines and the Game of Life A 2-dimensional cellular automaton. The Game of Life rules determine everything that happens on the grid. A dead cell with exactly three live neighbors becomes alive. A live cell with either two or three live neighbors stays alive. In all other cases, a cell dies or remains dead. The “glider” pattern Second example

Is this attributable to the Game of Life rules or to computability theory? I can’t think of an argument that attributes GoL undecidability to the GoL rules. There may be a direct argument, but the only obvious argument is to construct a TM and argue from computability theory? Yet reductionism holds! The GoL rules control everything that occurs on a GoL grid—just as fundamental physics controls everything that happens in our world. But computability theory is independent of—and was even developed prior to—the GoL.

Is this attributable to the Game of Life rules or to computability theory? The undecidability of the TM halting problem—and hence of the GoL halting problem—is a consequence of computability theory, not of the GoL rules. When a GoL configuration implements a Turing machine, computability theory applies to that implementation—and hence to the universe in which the implementation occurs. Again, isn’t this just common sense?

Is this strange?  The unsolvability of the TM halting problem entails the unsolvability of the GoL halting problem. We import a new and independent theory into the GoL world and use it to draw conclusions about the GoL. Downward causation entailmentDownward causation GoL Turing machines are causally reducible but ontologically real.  You can reduce them away without changing how a GoL run will proceed.  Yet they obey higher level laws, not derivable from the GoL rules. Reducing everything to the level of physics, i.e., naïve reductionism, results in a blind spot regarding higher level entities and the laws that govern them. GoL Turing machines are causally reducible but ontologically real.  You can reduce them away without changing how a GoL run will proceed.  Yet they obey higher level laws, not derivable from the GoL rules. Reducing everything to the level of physics, i.e., naïve reductionism, results in a blind spot regarding higher level entities and the laws that govern them. This is called “reduction” in Computer Science. We reduce the question of GoL unsolvability to the question of TM unsolvability by constructing a TM within a GoL universe.

The same argument was used for both the TM and the pegs and the holes True for pegs and holes made from electrons, protons, and neutrons. True for Turing machines made from GoL patterns. If I build something that’s governed by a special theory i.e., something that realizes an abstraction that is governed by that theory my construction i.e., my implementation of the abstraction is governed by that theory. If I build something that’s governed by a special theory i.e., something that realizes an abstraction that is governed by that theory my construction i.e., my implementation of the abstraction is governed by that theory. In both cases the constructed entity follows the rules that govern the abstraction it implements. Raises the question: what is an entity? The lower-level rules determine whether one can build an instantiation of the abstraction at all—and if so how. In both cases the constructed entity follows the rules that govern the abstraction it implements. Raises the question: what is an entity? The lower-level rules determine whether one can build an instantiation of the abstraction at all—and if so how.

The same argument was used for both the TM and the pegs and the holes True for pegs and holes made from electrons, protons, and neutrons. True for Turing machines made from GoL patterns. If I build something that’s governed by a special theory i.e., something that realizes an abstraction that is governed by that theory my construction i.e., my implementation of the abstraction is governed by that theory. If I build something that’s governed by a special theory i.e., something that realizes an abstraction that is governed by that theory my construction i.e., my implementation of the abstraction is governed by that theory. In both cases the constructed entity follows the rules that govern the abstraction it implements. Raises the question: what is an entity? The lower-level rules determine whether one can build an instantiation of the abstraction at all—and if so how. In both cases the constructed entity follows the rules that govern the abstraction it implements. Raises the question: what is an entity? The lower-level rules determine whether one can build an instantiation of the abstraction at all—and if so how.

Evolution It doesn’t seem possible to talk about either biological entities or their evolution in terms of elementary particles.  Evolution through natural selection is about biological entities. Nature can build biological entities from elementary particles, but how does one talk about them in the language of elementary particle physics?  The evolutionary process involves environmentally influenced differential survival and reproduction along with combination and mutation of inherited properties. Again, the language of elementary particle physics has no means to express these concepts. Even if we assume that it’s (theoretically) possible to trace how any state of the world—including the biological organisms in it— came about by tracking elementary particles (plus quantum randomness), it doesn’t seem possible to express the theory of evolution in terms of those particles. Third example

The language problem? There is no mechanism to extend the language of elementary particle physics beyond the entities of that universe. Computer science has mechanisms for building and talking about new entities. Mathematics has definitions and axioms. Physics can use definitions but not axioms? Science builds new sciences. But the language only reduces downward; it doesn’t build upward. Is that the fundamental problem? Nature builds entities but science can’t build its language upwards to talk about them. Besides science tends not to acknowledge that higher level entities exist.

Abstract data type & level of abstraction void push(stack: s, : e) pop(stack: s) top(stack: s) top(push(stack: s, : e)) = e pop(push(stack: s, : e) = s Stack 1.Zero is a number. 2.If A is a number, the successor of A is a number. 3.Zero is not the successor of a number. 4.Two numbers of which the successors are equal are themselves equal. 5.(Induction axiom) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. Peano’s axioms. A collection of “types” (categories), operations that may be applied to entities of those types, and often constraints that are required to hold. Typical examples: stack, naturals. Every computer program, e.g., PowerPoint, implements a level of abstraction—typically including a number of abstract data types. The things you can manipulate What you can do (and can’t) do with them

Level of abstraction: the reductionist blind spot A collection of concepts and relationships that can be described independently of its implementation. Every computer application creates one. A collection of concepts and relationships that can be described independently of its implementation. Every computer application creates one. A level of abstraction is causally reducible to its implementation.  You can look at the implementation to see how it works. A level of abstraction is causally reducible to its implementation.  You can look at the implementation to see how it works. Its independent specification—its properties and way of being in the world—makes it ontologically real.  How it interacts with the world is based on its specification and is independent of its implementation.  It can’t be reduced away without losing something Its independent specification—its properties and way of being in the world—makes it ontologically real.  How it interacts with the world is based on its specification and is independent of its implementation.  It can’t be reduced away without losing something

Three kinds of entities Static entities: created by energy wells.  Atoms, molecules, solar systems, most engineered artifacts. Dynamic entities: require energy to persist.  Biological and social entities; hurricanes. Symbolic and abstract entities.  Abstract: instances of abstract types.  Symbolic: implemented by computers.

Backups

How are levels of abstraction built? By adding persistent constraints to what exists.  Constraints break symmetry by limiting the possible transformations. Symmetry is equality under a transformation.  Easy in software.  Software constrains a computer to operate in a certain way.  Software (or a pattern set on a Game of Life grid) “breaks the symmetry” of possible sequences of future states. A constrained system operates differently (has additional laws—the constraints) from one that isn’t constrained. I’m showing this slide to invite anyone who is interested to work on this with me. Isn’t this just common sense? Ice cubes act differently from water and water molecules.

How are levels of abstraction built? How does nature build levels of abstraction? Two ways.  Energy wells produce static entities. Atoms, molecules, solar systems, …  Activity patterns use imported energy to produce dynamic entities. The constraint is imposed by the processes that the dynamic entity employs to maintain its structure. Biological entities, social entities, hurricanes. A constrained system operates differently (has additional laws—the constraints) from one that isn’t constrained. I’m showing this slide to invite anyone who is interested to work on this with me. Isn’t this just common sense? Ice cubes act differently from water and water molecules.

How macroscopic behavior arises from microscopic behavior. Emergent entities (properties or substances) ‘arise’ out of more fundamental entities and yet are ‘novel’ or ‘irreducible’ with respect to them. Stanford Encyclopedia of Philosophy Emergence: the holy grail of complex systems The ‘scare’ quotes identify problematic areas. Plato Emergence: Contemporary Readings in Philosophy and Science Mark A. Bedau and Paul Humphreys (Eds.), MIT Press, April 2008.

Are there autonomous higher level laws of nature? The fundamental dilemma of science How can that be if everything can be reduced to the fundamental laws of physics? The functionalist claim The reductionist position It can all be explained in terms of levels of abstraction. My answer Emergence

Gliders are causally powerless.  A glider does not change how the rules operate or which cells will be switched on and off. A glider doesn’t “go to an cell and turn it on.”  A Game of Life run will proceed in exactly the same way whether one notices the gliders or not. A very reductionist stance. But …  One can write down equations that characterize glider motion and predict whether—and if so when—a glider will “turn on” a particular cell.  What is the status of those equations? Are they higher level laws? Gliders Like shadows, they don’t “do” anything. The rules are the only “forces!”

Amazing as they are, gliders are also trivial.  Once we know how to produce a glider, it’s simple to make them. Can build a library of Game of Life patterns and their interaction APIs. By suitably arranging these patterns, one can simulate a Turing Machine. Paul Rendell. The Turing machine and the Game of Life A second level of emergence. Emergence is not particularly mysterious.

Causally reducible but ontologically real GoL Turing machines are causally reducible but ontologically real.  You can reduce them away without changing how a GoL run will proceed.  Yet they obey higher level laws, not derivable from the GoL rules. GoL Turing machines are causally reducible but ontologically real.  You can reduce them away without changing how a GoL run will proceed.  Yet they obey higher level laws, not derivable from the GoL rules.

Evolution Darwin and Wallace’s theory of evolution by natural selection is expressed in terms of  entities  their properties  how suitable the properties of the entities are for the environment  populations  reproduction  etc. These concepts are a level of abstraction. The theory of evolution is about biological entities. Let’s assume that it’s (theoretically) possible to trace how any state of the world—including the biological organisms in it— came about by tracking elementary particles Even so, it is not possible to express the theory of evolution in terms of elementary particles. Reducing everything to the level of physics, i.e., naïve reductionism, results in a blind spot regarding higher level entities and the laws that govern them. Darwin and Wallace’s theory of evolution by natural selection is expressed in terms of  entities  their properties  how suitable the properties of the entities are for the environment  populations  reproduction  etc. These concepts are a level of abstraction. The theory of evolution is about biological entities. Let’s assume that it’s (theoretically) possible to trace how any state of the world—including the biological organisms in it— came about by tracking elementary particles Even so, it is not possible to express the theory of evolution in terms of elementary particles. Reducing everything to the level of physics, i.e., naïve reductionism, results in a blind spot regarding higher level entities and the laws that govern them. Third example