Irrational mechanics, draft Ch. 5
Incomplete interlude.
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I’ve been reading “Free Agents: How Evolution Gave Us Free Will” (2023), by Kevin Mitchell. The book covers important concepts in a way that is crystal clear and a pleasure to read.
Mitchell is a professor of genetics and neuroscience, and much of the book is about these things and evolutionary biology. But to me the most interesting parts of the book are those that cover the fundamental issue of free will in a universe determined by physical laws.
To Mitchell, a worldview that doesn’t have room for free will “is not just wrong - it’s wrong-headed.” I agree. Good science should help us better understand our experience, not deny the reality of our experience.
In the past, the prevailing view in physics was that the world is fully deterministic, without room for free will. But quantum physics comes to the rescue of free will because it “seems to have some fundamental indeterminacy right at the heart of it.” Today, Mitchell says (some disagree, but I agree), the prevailing view in physics is that “the indeterminacy observed in the evolution of quantum systems is real and fundamental.”
Fundamental chance, Mitchell argues, “undercuts necessity’s monopoly on causation.”
“The low-level physical details and forces are not causally comprehensive: they are not sufficient to determine how a system will evolve from state to state. This opens the door for higher-level features to have some causal influence in determining which way the physical system will evolve… physical indeterminacy creates causal slack in physical systems, which opens the door for what is known as ‘top-down causation.’”
This book is one of the first to cover and put in context the idea, recently proposed by Nicolas Gisin, to consider the initial conditions of mathematical physics as physical processes, sequences of bits that unfold in time. “It’s not just that we don’t know what they are - the universe also doesn’t know what they are, not yet.”
Mitchell argues that systems with genuine agency, natural or artificial, need “some indeterminacy in their low-level workings,” and quantum mechanics provides exactly that indeterminacy. This has implications for Artificial Intelligence (AI). AI agents will need “some causal slack at the bottom” to be free agents like us. “Otherwise, no matter how complex, the agent will be pushed around deterministically by its own components.” I think this insight suggests important links between quantum computing and AI research: future quantum computers could process human-like artificial agents better than today’s digital computers.
I’ve been reading “The Conservative Futurist: How to Create the Sci-Fi World We Were Promised” (2023), by James Pethokoukis.
Robin Hanson reviews “The Conservative Futurist” and the “Techno-Optimist Manifesto” by Marc Andreessen (Andreessen’s manifesto is republished in the Andreessen Horowitz (a16z) website).
Robin likes both, and so do I. But Robin also says:
“Yet, alas, I have to report that both seem to me to largely fail to inspire. I find it hard to imagine people moved to allegiance or action by these. These authors want to inspire, they try to inspire, and what they want to inspire you to is worthy of such elevation. And yet, they fail.”
I think Robin has a point here. To me and to people like me, both are very inspiring. But I guess both have that certain metallic taste of hard science fiction, and fail to inspire people with softer sensibilities.
My comment to Robin’s post: I agree that we (these two authors, you, I, and some - alas not too many - others) haven’t found yet a way to really inspire the masses. But let’s keep trying. Sooner or later some newcomers, probably much younger than us, will find a way.
So here’s a very early draft of Chapter 5 of my new book “Irrational mechanics: Narrative sketch of a futurist science & a new religion” (2024).
Note that this and the others draft chapters are very concise. At this point I only want to put down the things I want to say, one after another. Later on, when the full draft for early readers is complete, I’ll worry about style and all that.
5 - Incomplete interlude
Alfred Korzybski, a thinker whose work had a deep influence on some of the greatest science fiction writers [Nevala-Lee 2018], coined the often quoted quip “the map is not the territory” [Korzybski 1994].
This is a good illustration that no theoretical map can fully capture “the thickness of reality” [James 2018]. The territory of physical reality will always, I think, be richer than our theoretical maps, and full of unexpected surprises. There will always be little or big cracks in current theories, better theories hidden behind the cracks, then new cracks in the new theories, and so forth.
This also follows from Gödel’s incompleteness theorems. Perhaps not as a rigorous implication thereof, but certainly as a strong suggestion.
Kurt Gödel proved [Nagel 2001] that, if a mathematical system of axioms and rules is consistent (and powerful enough to deal with integer arithmetic), then it must be incomplete in the sense that it doesn’t permit proving some true statements.
For example, the Goldbach conjecture, according to which every even number is the sum of two prime numbers, has not been formally proved but holds for all the even numbers that have been examined. It is possible that the conjecture will be proved in the future, or that an even number will be found that is not the sum of two prime numbers. But it is also possible that the Goldbach conjecture is true but unprovable.
From within a mathematical system, true but unprovable statements seem “random” truths that just happen to be true for no reason. These random truths might be proved by a more powerful mathematical system, but then the new system would be also incomplete, with new random truths.
Incompleteness is the price that a mathematical system must pay for consistency. If a system is too weak it is incomplete, but if it is too strong it is inconsistent. Gödel proved that it’s one or the other.
Gödel also proved that the consistency of a mathematical system can’t be proved within the system.
Following Gödel, Alan Turing and Gregory Chaitin proved [Chaitin 2005, Rucker 2016, Rucker 2019], even more trenchantly, that consistent mathematical systems must be incomplete, with mathematical facts that don’t follow from elementary facts that are assumed to be true (axioms), but just happen to be true.
Besides the mathematical facts that can be reached by moving logically from the axioms upwards, there must be other true mathematical facts hanging in the air, so to speak. And if we add new axioms, perhaps we’ll reach those true mathematical facts that the old axioms couldn’t reach, but there will be new true but unreachable mathematical facts hanging in the air. And so forth.
What do incompleteness theorems say about physical reality? Do they imply that no finite mathematical model can capture physical reality? Could the indeterminism found in quantum and chaos physics have something to do with mathematical incompleteness?
Stanley Jaki was one of the first to propose a connection between incompleteness theorems and physics. Gödel’s theorem casts “serious doubts on any attempt aiming at an a priori mathematical theory that might claim to be the ultimate mathematical pattern of the physical world,” he said [Jaki 1966].
Erik Hoel suspects that there must be some kind of “scientific incompleteness” [Hoel 2023] analogous to mathematical incompleteness. A totally precise map of a territory, placed in the territory itself, must include the map itself. But then the map in the map must include another map and so forth in a regression without end. This sounds like the kind of reasoning used by Gödel and others in mathematical incompleteness proofs. Hoel doesn’t try to convert this insight into a rigorous scientific proof, but suspects that science is riddled with holes and “leaves facts about the universe on the table.”
Many other thinkers have said that there must be and there’ll always be Gödelian holes and cracks in our scientific understanding of the world. John Wheeler was persuaded that “the Gödel business has to come into physics, and physics into the understanding of it” [Gefter 2014]. Freeman Dyson, Stephen Hawking, and Roger Penrose argued that Gödel’s theorems have central implications for physics [Penrose 1994, Franzén 2005]. So does Rudy Rucker [Rucker 2016], whose “incompleteness theorem for the natural world” is published in [Rucker 2012]. Their arguments are suggestive rather than rigorous, but I find them persuasive enough.
Frank Tipler argued that “the indeterminism in quantum gravity is ontological and logically irremovable: it ultimately comes from Gödel's Incompleteness Theorem” [Tipler 1994]. Tipler’s argument is based on his Omega Point theory [Chapter 15 of Prisco 2020] and a Gödel-like mathematical theorem which states that “the topological equivalence problem for 4-manifolds” can’t be solved by any algorithm [Penrose 1994].
Gödel himself didn’t want to discuss “any possible connection between the undecidability he had discovered in the world of logic and the indeterminism that is a central feature of modern quantum mechanics” [Wheeler 2000], because “he did not believe in quantum mechanics.” Too bad Gödel died before the advent of chaos theory, which has its own indeterminism independent of quantum mechanics.
It seems (to me) intuitively obvious that mathematical incompleteness implies that no finite mathematical model can capture physical reality. But not everyone agrees. The custom-built unprovable statements used in Gödel’s and related proofs (precisely stated versions of “this sentence is false”) seem sort of artificial and far from applicable mathematics, let alone physics.
To me, mathematics is derived from physical experience [Lakoff 2000] and arithmetic is the physics of things that can be counted, so how could the Gödel business not come into physics eventually?
Here there’s an important clarification to make. If a mathematical system is entirely defined by axioms and logical rules, then it is entirely defined by those axioms and rules and that’s that. From this perspective, undecidable means false. If a mathematical conjecture is undecidable in the sense that it can’t be reached from the axioms, then it is false and unimportant.
But the choice of a specific mathematical system, a particular one of the very many abstract mathematical systems that can be conceived, is also based on the intuition that it - the mathematical system based on these (not those) specific axioms - could have something to do with physical reality, and the requirement that it should be a useful system that helps us deal with physical reality.
That two plus two equals four is experimental physics: if you put two little stones together with other two little stones, you get four little stones, and your mathematics must agree. Even numbers and prime numbers can be defined in terms of groups of little stones, so the Goldbach conjecture is a statement about physical reality and mathematics must have something to say.
From this perspective, undecidability is important. Geometry is a simple case in point. Civil engineers can use Euclidean geometry to design a bridge, but future spacetime engineers will have to use the non-Euclidean geometry upon which general relativity is based. One axiom (the parallels axiom, which is undecidable based on the other Euclidean axioms) makes the difference between Euclidean and non-Euclidean geometry, and the choice of the correct (useful) axioms depends on physical reality. Which geometry is true? The geometry that is more useful to model physical reality is true (yes, this sounds like William James). Of course, there’s the possibility that different geometries are needed for different aspects of physical reality.
I think other open issues in pure mathematics, such as those related to the axiom of choice and the continuum hypothesis [Cohen 2008, Rucker 2019], will be tackled on the same basis and will be very relevant to future science. Let Rudy Rucker’s preface to the 2019 edition of [Rucker 2019] blow up your mind with visions of “subquantum, subdimensional levels” of surreal space and time beyond the Planck scale.
Mathematics falls into Gödel’s trap when it talks about itself. We are parts of physical reality that talk about it from inside physical reality itself. It seems plausible that any theory of physical reality that we can formulate from the inside must be incomplete.