Gödel’s incompleteness theorems, free will and mathematical thought

Feferman, Solomon

doi:10.5871/bacad/9780197264898.003.0007

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…the Balslev-Combes theorem [32], which is vital to understanding and appreciating non-Hermitian quantum mechanics [33][34][35] and its consequences for the dynamics of resonance states embedded in the continuum and their properties for higher order dynamics. These ingredients impart a prompt reordering and exposition of acquiescent microscopic self-organization [9,20,36] with direct bearings on Artificial Intelligence, AI, [36][37][38][39][40] including the formulation of Gödel's theorems in an extended logic based on the fundamental teleonomic concept of communication, extending from the microscopic-to the mesoscopic-and macroscopic levels [9].…”

Section: Proposed Solutions To the Controversymentioning

confidence: 99%

The Origin and Evolution of Complex Enough Systems in Biology

Brändas

2017

Quantum Systems in Physics, Chemistry, and Biology

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Recent criticisms of Neo-Darwinism are considered and disputed within the setting of recent advances in chemical physics. A related query, viz., the ontological thesis, that everything is physical, confronts a crucial test on the validity of reductionism as a fundamental approach to science. While traditional 'physicalism' interprets evolution as a sequence of physical accidents governed by the second law of thermodynamics, the concepts of biology concern processes that owe their goal-directedness to the influence of an evolved program. This disagreement is met by unifying basic aspects of chemistry and physics, formulating the Correlated Dissipative Ensemble, CDE, as a characterization of a 'complex enough systems', CES, in biology. The latter entreats dissipative dynamics; non-Hermitian quantum mechanics together with modern quantum statistics thereby establish a precise spatio-temporal order of significance for living systems. The CDE grants a unitary transformation structure that comprises communication protocols of embedded Poisson statistics for molecular recognition and cellular differentiation, providing cell-hierarchies in the organism. The present conception of evolution, founded on communication with a built-in self-referential order, offers a valid argument in favour of Neo-Darwinism, providing an altogether solid response and answer to the criticisms voiced above. Introductory Remarks Reductionism in Natural ScienceA frequent understanding of natural sciences contends that biology reduces to chemistry and chemistry to physics. Even if the traditional analytic interpretation of mathematics as a language or tool to convert knowledge about nature, i.e. a formal science not being incorporated amongst the branches of physical or life sciences, some recent propositions impart that also physics can be reduced to mathematics [1]. Within such views, based on strict reductionnism, see more below, one argues that the present laws of physics are not only commensurate to biology, but also contain the necessary natural laws for expressing all the known biological facts.There are numerous differences of opinion regarding the scheme outlined above, e.g. the impossibility to derive the Coulomb Hamiltonian corresponding to a molecule from first principles and the general problem of quantum chemistry to treat nuclei and electrons on a more or less equivalent basis, see Löwdin [2]. Other arguments concern the concepts of biology as the laws of physics, at present, exclude a fundamental understanding of biological processes governed by an evolved program [3,4].Observation and deliberate experiments instigate rational, deductive theory, formulated in the language of mathematics, with the so accumulated knowledge discussed and contained by the methods of philosophy in general and the concepts of biology in particular. Recognizing the acquired demarcation between the philosophy of science, including biology, and the enactment of reductionism as embodied in the laws of physics, the endwise connections between them ...

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Section: Proposed Solutions To the Controversymentioning

confidence: 99%

The Origin and Evolution of Complex Enough Systems in Biology

Brändas

2017

Quantum Systems in Physics, Chemistry, and Biology

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“…And the most independent minds, despite general adoration before Gödel's proof of the incompleteness of formalized arithmetic, tried to restore the shaken authority of mathematical science by proving the completeness of formalized mathematical theories (Henkin, 1950;Hasenjaeger, 1953). The wave of publications on the significance of Gödel's theorems (Franzén, 2005), their philosophical interpretation (Rucker, 1995;Wang, 1997), the current demonstrative exposition (Smith, 2007), attempts to apply them far beyond the reasonable (Lucas, 1970;Feferman, 2011) continues to this day.…”

Section: Introductionmentioning

confidence: 99%

The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics

Djidjian¹

2017

wisdom

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The author of this article critically analyses the proof of Gödel's famous theorem on the incompleteness of formalized arithmetic. It is shown that Gödel's formalization of meta-mathematics provides a proof of the incompleteness not of mathematical science but of the system of formalized meta-mathematics developed by Gödel himself. The arguments against the idea of the formalization of meta-mathematics are presented. The article suggests also an interpretation of the essence of mathematical truth. It is noted that the refutation of Gödel's proof does not suggest returning to Hilbert's program of formalism since the formalization of an axiomatic theory can't exclude the appearance of paradoxes within its framework. It is shown that the use of self-referential Gödel's numbering in a formalized system leads to the emergence of a Liar type paradox -a self-contradictory formula that demonstrates the inconsistency of that same system.

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Brain and Mind

Leeuwen¹

2013

philosophiascientiae

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Le débat sur les relations esprit-cerveau a été centré sur des questions relatives au libre arbitre. J'examine ce débat et conclus que les neurosciences n'ont pas de raisons méthodologiques, ontologiques ou théoriques convaincantes, pas plus que de raisons empiriques, pour rejeter la notion de libre arbitre. Parallèlement, je reconnais que la question est très controversée, à la fois en science et dans la société. Le problème se situe dans l'incompatibilité entre notions scientifiques du cerveau et notions pré-scientifiques de l'esprit. Par conséquent, je propose d'envisager les relations esprit-cerveau sur une base empirique plus profonde et incontestée que l'on peut trouver dans la psychophysique. Je discute deux cas dans lesquels le contenu d'une expérience psychophysique, et sa dynamique, correspondent à la dynamique cérébrale. Dans un cas, la correspondance est une identité de type ; dans l'autre, il s'agit d'une instanciation multiple : plusieurs types radicalement différents d'activités donnent lieu à la même dynamique perceptive. Ces deux exemples illustrent que, bien que l'identité des types soit possible, elle peut être souvent hautement contextualisée. Par conséquent, on doit appréhender les relations esprit-cerveau une par une plutôt que les fonder d'un seul bloc.

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Gödel’s incompleteness theorems, free will and mathematical thought

Cited by 3 publications

References 17 publications

The Origin and Evolution of Complex Enough Systems in Biology

The Origin and Evolution of Complex Enough Systems in Biology

The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics

Brain and Mind

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