The Incompleteness of Logic, the Incompleteness of Physics, and the Primitive Sourcehood of Rational Human Animals.
By Robert Hanna
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The Incompleteness of Logic, the Incompleteness of Physics, and the Primitive Sourcehood of Rational Human Animals*
What follows is my attempt to provide a compact, clear, distinct, and philosophically defensible reformulation and extension of an important line of argument that has been developed since the early 1960s, in various ways, by J.R. Lucas, Roger Penrose, and others.[i]
To be sure, these various developments of this important line of argument are all somewhat controversial; so my aspiration is that by standing on the shoulders of these giants, I can see further than they did.
It’s an argument by formal analogy.
The formal analogy is between (i) the incompleteness of Principia Mathematica-style systems of mathematical logic[ii] that are rich enough to contain the Peano axioms for arithmetic and the primitive recursive functions over the natural numbers, and (ii) the incompleteness of the Standard Models in contemporary physics.
By the Peano axioms for arithmetic I mean: (i) 0 is a number, (ii) the successor of any number is a number, (iii) no two numbers have the same successor, (iv) 0 is not the successor of any number, and (v) any property which belongs to 0, and also to the successor of every number which has the property, belongs to all the numbers.
By the primitive recursive functions over the natural numbers I mean: the successor function, addition, multiplication, exponentiation, and so-on.
And by the Standard Models in contemporary physics I mean the current Standard Model of Cosmology (SMC), together with a proper sub-part of SMC, the current Standard Model of Particle Physics (SMPP), as per the following non-technical gloss by B.A. Robson:[iii]
The current Standard Model of Cosmology (SMC), also called the “Concordance Cosmological Model” or the “ΛCDM Model,” assumes that the universe was created in the “Big Bang” from pure energy, and is now composed of about 5% ordinary matter, 27% dark matter, and 68% dark energy.[iv]
[T]he SMC is based primarily upon two theoretical models: (1) the Standard Model of Particle Physics (SMPP),[v] which describes the physics of the very small in terms of quantum mechanics and (2) the General Theory of Relativity (GTR),[vi] which describes the physics of the very large in terms of classical mechanics; it also depends upon several additional assumptions.
The main additional assumptions of the SMC are: (1) the universe was created in the Big Bang from pure energy; (2) the mass energy content of the universe is given by 5% ordinary matter, 27% dark matter, and 68% dark energy; (3) the gravitational interactions between the above three components of the mass energy content of the universe are described by the GTR; and (4) the universe is homogeneous and isotropic on sufficiently large (cosmic) scales.
Unfortunately, both the SMPP and the GTR are considered to be incomplete in the sense that they do not provide any understanding of several empirical observations. The SMPP does not provide any understanding of the existence of three families or generations of leptons and quarks, the mass hierarchy of these elementary particles, the nature of gravity, the nature of dark matter, etc.[vii] The GTR does not provide any understanding of the Big Bang cosmology, inflation, the matter-antimatter asymmetry in the universe, the nature of dark energy, etc.
Furthermore, the latest version of the SMC, the ΛCDM Model is essentially a parameterization of the Big Bang cosmological model in which the GTR contains a cosmological constant, Λ, which is associated with dark energy, and the universe contains sufficiently massive dark matter particles, i.e., “cold dark matter.” However, both dark energy and dark matter are simply names describing unknown entities.
By the completeness of a logical system I mean the systemic property such that all the logical truths of the system (i.e., tautologies or valid sentences) are provable (i.e., theorems), hence the incompleteness of a logical system is the systemic property such that not all of its logical truths are provable, i.e., that some of its logically true sentences are unprovable sentences.
By the completeness of a physical theory I mean the systemic property such that all its true empirical sentences are predictable, hence the incompleteness of a physical theory is the systemic property such that not all of its true empirical sentences are predictable, i.e., that some of its true empirical sentences are unpredictable sentences.
And by predictable sentence, I mean that a sentence S in a physical theory PT is predictable if and only if (i) S is explicable or understandable in terms of PT, (ii) S is or would be entailed by PT if certain contingent empirical conditions were met, and (iii) S is self-consistent and non-paradoxical — hence unpredictable sentences violate one or more of those necessary conditions.
Every system of mathematical logic describes a corresponding real mathematical system and/or a class of such real systems: for example, Principia Mathematica-style systems that are rich enough to include the Peano axioms for arithmetic and the primitive recursive functions over the natural numbers describe Peano arithmetic.
So too, every physical theory describes a corresponding real physical system and/or a class of such real systems: for example, the Standard Models of cosmology and particle physics describe closed deterministic or indeterministic physical systems that obey the First and Second Laws of Thermodynamics, i.e., equilibrium entropic thermodynamic systems with reversible time and computable basic quantities.
Let’s call such systems mechanical physical systems.
In 1931, Kurt Gödel proved (i) that all Principia Mathematica-style systems rich enough to contain the Peano axioms for arithmetic are incomplete because they contain some unprovable sentences that are equivalents of the Liar Paradox (i.e., a self-referring sentence such that necessarily, it’s true/provable if and only if it’s false/uprovable; such sentences are also undecidable, hence uncomputable[viii]), and also (ii) that all such systems cannot prove their own consistency or contain their own truth-definition, on pain of containing unprovable sentences that are equivalents of the Liar Paradox, and thereby being inconsistent systems, hence that the truth-definitions for and consistency of such systems must be established outside those systems.[ix]
Let’s call all this logico-mathematical incompleteness.
Now, just as no Principia Mathematica-style system rich enough to contain the Peano axioms for arithmetic can contain its own truth-definition, or demonstrate its own consistency, without generating equivalents of the Liar Paradox and its own inconsistency, so too no mechanical physical system can contain its own initial conditions, spacetime-architecture, or the nomological framework (i.e., the framework of causal natural laws) that determines the evolution of states within the system, without generating experimental situations that produce physical paradox in the form of empirically true but unpredictable sentences.
For example, as Robson puts it:
Unfortunately, both [quantum mechanics] and the [General Theory of Relativity] are considered to be incomplete in the sense that they do not provide any understanding of several empirical observations. [Quantum mechanics] does not provide any understanding of the existence of three families or generations of leptons and quarks, the mass hierarchy of these elementary particles, the nature of gravity, the nature of dark matter, etc.[x] The [General Theory of Relativity] does not provide any understanding of the Big Bang cosmology, inflation, the matter-antimatter asymmetry in the universe, the nature of dark energy, etc.
Moreover, the Standard Models in contemporary physics, via General Relativity and quantum mechanics, also generate these further examples of physical paradox: the slowing-down of time for objects/observers traveling at speeds close to that of light, quantum indeterminacy, quantum superposition, quantum decoherence, Schrödinger’s cat,[xi] quantum particle/wave duality, quantum complementarity, quantum entanglement, quantum non-locality, and quantum non-separability.[xii]
Let’s call all this physico-mechanical incompleteness.
Now, just as, if a system of mathematical logic is incomplete, then necessarily its corresponding real mathematical system and/or class of such systems inherits that logico-mathematical incompleteness, so too, if a mechanical physical theory is incomplete, then necessarily its corresponding real physical system and/or all the members of the class of such real physical systems inherit that physico-mechanical incompletness.
Just as Alfred Tarski’s semantic conception of truth (aka model-theoretic semantics) theoretically bears witness to logico-mathematical incompleteness by systematically locating truth-definitions and consistency in logical meta-languages that are outside of logico-mathematical systems construed as object-languages,[xiii] so too General Relativity and quantum mechanics theoretically bear witness to physico-mechanical incompleteness by systematically operationalizing the fundamental concepts of physical entities, physical properties, physical relations, physical forces, physical quantities, etc., in physico-mechanical systems, strictly in terms of what can be represented by experimental measuring devices.[xiv]
But just as the semantic conception of truth doesn’t tell us how the truth-definitions and consistency are known — that’s what Benacerraf’s Dilemma[xv] is all about — so too General Relativity and quantum mechanics don’t tell us how the initial conditions, spacetime architectures, and nomological frameworks are created.
Correspondingly, just as truth-definitions and consistency for every Principia Mathematica-style system rich enough to contain the Peano axioms for arithmetic must be semantically and epistemically established outside that system in an inherently non-logical way (let’s call that mathematical creativity), so too the initial conditions, spacetime architecture, and nomological framework for every mechanical physical system must be causally established outside that system in an inherently non-deterministic, non-indeterministic, non-equilibrium, negentropic, time-irreversible, uncomputable thermodynamic way (let’s call that natural creativity).
Therefore, just as the fact of logico-mathematical incompleteness entails the existence of mathematical creativity, so too the fact of physico-mechanical incompleteness entails the existence of natural creativity.
Given the multiplicity of mathematical axioms and mathematical truths, there must be a multiplicity of primitive sources of mathematical creativity, including brilliant mathematicians but not restricted to them — indeed, including all mathematical a priori knowers of mathematical axioms and mathematical truths.
For example, you, I, and the people living next door can all know the Peano axioms for arithmetic and that 2+2=4.
So too, given the multiplicity of mechanical physical systems, there must be a multiplicity of primitive sources of natural creativity, including the Big Bang Singularity, but not restricted to it — indeed, including all non-deterministic, non-indeterministic, non-equilibrium, negentropic, time-irreversible, uncomputable thermodynamic systems (aka “Little Bangs”) that causally establish the initial conditions, spacetime architecture, and nomological framework for each local mechanical physical system, i.e., every mechanical system that is smaller than the natural cosmos or universe itself.
For example, you, I, and the people living next door, by being rational human essentially embodied minded animals,[xvi] whose living organismic bodies are embedded in egocentrically-centered three-dimensional orientable local spaces, in irreversible time, with unique non-deterministic, non-indeterministic, non-equilibrium, negentropic thermodynamic causal powers, can all causally establish the local initial conditions, local spacetime architecture, and local nomological framework for their own intentional body-movements, which are then described — with physico-mechanical incompleteness — by the Standard Models in contemporary physics.
Therefore, all rational human animals are not only primitive sources of mathematical creativity, as mathematical a priori knowers of mathematical axioms and mathematical truths, arguably by means of mathematical intuition,[xvii] but also primitive sources of natural creativity, as rational human free agents, arguably by means of deep freedom of the will and non-instrumental practical agency.[xviii]
NOTES
* The references for the two images at the head of the essay are: (image 1) K. Gödel, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” in J. Van Heijenoort (ed.), From Frege to Gödel (Cambridge, MA: Harvard Univ. Press, 1967), pp. 596–617; and (image 2) National Aeronautics and Space Administration, aka NASA, “Wilkinson Microwave Anisotropy Probe,” available online at URL = <https://map.gsfc.nasa.gov/>.
[i] See J.R. Lucas, “Minds, Machines, and Gödel,” Philosophy 36 (1961): 112–127; R. Penrose, The Emperor’s New Mind (Oxford: Oxford Univ. Press, 1990); T. Deacon, Incomplete Nature: How Mind Emerged from Matter (New York: Norton, 2013); and L. Smolin, Time Reborn: From the Crisis in Physics to the Future of the Universe (New York: Mariner/Houghton Mifflin Harcourt, 2014).
[ii] See A.N. Whitehead and B. Russell, Principia Mathematica to *56 (2nd edn., Cambridge: Cambridge Univ. Press, 1962).
[iii] B.A. Robson, “Introductory Chapter: Standard Model of Cosmology,” available online at URL = <https://www.intechopen.com/books/redefining-standard-model-cosmology/introductory-chapter-standard-model-of-cosmology>.
[iv] Ade PAR et al. (Planck Collaboration). Planck 2013 results. I Overview of products and scientific results. Astronomy and Astrophysics. 2014;571:A1, 48pp. (Footnote 1 in Robson, “Introductory Chapter: Standard Model of Cosmology.”)
[v] Gottfried K, Weisskopf VF. Concepts of Particle Physics. Vol. 1. New York: Oxford University Press; 1984. 189pp. (Footnote 2 in Robson, “Introductory Chapter: Standard Model of Cosmology.”)
[vi] Einstein A. The basics of general relativity theory. Annals of Physics. 1916;49:769–822. (Footnote 3 in Robson, “Introductory Chapter: Standard Model of Cosmology.”)
[vii] Robson BA. The generation model of particle physics. In: Kennedy E, editor. Particle Physics. Rijeka: InTech; 2012. pp. 1–28. (Footnote 4 in Robson, “Introductory Chapter: Standard Model of Cosmology.”)
[viii] See A. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society, series 2, 42 (1936): 230–265, with corrections in 43 (1937): 644–546; and G. Boolos and R. Jeffrey, Computability and Logic (3rd edn., Cambridge, Cambridge Univ. Press, 1989).
[ix] Gödel, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.”
[x] Robson BA. The generation model of particle physics. In: Kennedy E, editor. Particle Physics. Rijeka: InTech; 2012. pp. 1–28. (Footnote 4 in Robson, “Introductory Chapter: Standard Model of Cosmology.”)
[xi] See, e.g., J.D. Trimmer, “The Present Situation in Quantum Mechanics: A Translation of Schrödinger’s ‘Cat Paradox’ Paper,” Proceedings of the American Philosophical Society, 124 (1980): 323–338.
[xii] See, e.g., W. Myrvold, “Philosophical Issues in Quantum Theory,” in E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), available online at URL = <https://plato.stanford.edu/archives/fall2018/entries/qt-issues/>.
[xiii] See A. Tarski, “The Concept of Truth in Formalized Languages,” in A. Tarski, Logic, Semantics, and Metamathematic (Oxford: Oxford University Press, 1956), pp. 152–278; and A. Tarski, “The Semantic Conception of Truth and the Foundations of Semantics,” Philosophy and Phenomenological Research 4 (1943): 342–360.
[xiv] See, e.g., R. Hanna, “Frame-by-Frame: How Early 20th Century Physics Was Shaped by Brownie Cameras and Early Cinema” (Unpublished MS, May 2021), available online HERE.
[xv] P. Benacerraf, “Mathematical Truth,” Journal of Philosophy 70 (1973): 661–680. Benacerraf’s Dilemma say that if we plausibly assume both (i) a “standard” Tarskian model-theoretic semantics for mathematics, which yields platonic (i.e., non-spatiotemporal, acausal) mathematical objects as truth-makers, and also (ii) a “reasonable” epistemology based on direct reference to objects causally given in sense perception, then either mathematical truths are unknowable, or we have to give up our standard semantics or our reasonable epistemology.
[xvi] See R. Hanna and M. Maiese, Embodied Minds in Action (Oxford: Oxford Univ. Press, 2009).
[xvii] See R. Hanna, Cognition, Content, and the A Priori: A Study in the Philosophy of Mind and Knowledge (THE RATIONAL HUMAN CONDITION, Vol. 5) (Oxford: Oxford Univ. Press, 2015), chs. 6–8.
[xviii] See R. Hanna, Deep Freedom and Real Persons: A Study in Metaphysics (THE RATIONAL HUMAN CONDITION, Vol. 2) (New York: Nova Science, 2018).
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