This paper critically examines color relationalism and color relativism, two theories of color that are allegedly supported by variation in normal human color vision. We mostly discuss color relationalism, defended at length in Jonathan Cohen's The Red and the Real, and argue that the theory has insuperable problems.

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) (...) with different accuracies. The traditional and the new approaches are compared and discussed. (shrink)

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of (...) Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s (...) axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide (...) whether or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence (...) of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

In the present paper, these authors argue on actual reasons why Hilbert’s axiomatic program to unify gravitation theory and electromagnetism failed completely. An outline of plausible resolution of this problem is given here, based on: a) Gödel’s incompleteness theorem, b) Newton’s aether stream model. And in another paper we will present our calculation of receding Moon from Earth based on such a matter creation hypothesis. More experiments and observations are called to verify this new hypothesis, albeit it is (...) inspired from Newton’s theory himself. (shrink)

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the (...) developments in 19th-century geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer. (shrink)

David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which (...) have so far received little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)

We present a new approach to the invariant subspace problem for complex Hilbert spaces.This approach based on nonconservative Extension of the Model Theoretical NSA. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace.

McQueen and Vaidman argue that the Many Worlds Interpretation (MWI) of quantum mechanics provides local causal explanations of the outcomes of experiments in our experience that is due to the total effect of all the worlds together. We show that although the explanation is local in one world, it requires a causal influence that travels across different worlds. We further argue that in the MWI the local nature of our experience is not derivable from the Hilbert space structure, but (...) has to be added to it as an independent postulate. This is due to what we call the factorisation-symmetry and basis-symmetry of Hilbert space. (shrink)

The philosophy of information (PI) is a new area of research with its own field of investigation and methodology. This article, based on the Herbert A. Simon Lecture of Computing and Philosophy I gave at Carnegie Mellon University in 2001, analyses the eighteen principal open problems in PI. Section 1 introduces the analysis by outlining Herbert Simon's approach to PI. Section 2 discusses some methodological considerations about what counts as a good philosophical problem. The discussion centers on Hilbert's (...) famous analysis of the central problems in mathematics. The rest of the article is devoted to the eighteen problems. These are organized into five sections: problems in the analysis of the concept of information, in semantics, in the study of intelligence, in the relation between information and nature, and in the investigation of values. (shrink)

The paper addresses the problem, which quantum mechanics resolves in fact. Its viewpoint suggests that the crucial link of time and its course is omitted in understanding the problem. The common interpretation underlain by the history of quantum mechanics sees discreteness only on the Plank scale, which is transformed into continuity and even smoothness on the macroscopic scale. That approach is fraught with a series of seeming paradoxes. It suggests that the present mathematical formalism of quantum mechanics is (...) only partly relevant to its problem, which is ostensibly known. The paper accepts just the opposite: The mathematical solution is absolute relevant and serves as an axiomatic base, from which the real and yet hidden problem is deduced. Wave-particle duality, Hilbert space, both probabilistic and many-worlds interpretations of quantum mechanics, quantum information, and the Schrödinger equation are included in that base. The Schrödinger equation is understood as a generalization of the law of energy conservation to past, present, and future moments of time. The deduced real problem of quantum mechanics is: “What is the universal law describing the course of time in any physical change therefore including any mechanical motion?”. (shrink)

Arthur Clark and Michael Kube–McDowell (“The Triger”, 2000) suggested the sci-fi idea about the direct transformation from a chemical substance to another by the action of a newly physical, “Trigger” field. Karl Brohier, a Nobel Prize winner, who is a dramatic persona in the novel, elaborates a new theory, re-reading and re-writing Pauling’s “The Nature of the Chemical Bond”; according to Brohier: “Information organizes and differentiates energy. It regularizes and stabilizes matter. Information propagates through matter-energy and mediates the interactions of (...) matter-energy.” Dr Horton, his collaborator in the novel replies: “If the universe consists of energy and information, then the Trigger somehow alters the information envelope of certain substances –“. “Alters it, scrambles it, overwhelms it, destabilizes it” Brohier adds. There is a scientific debate whether or how far chemistry is fundamentally reducible to quantum mechanics. Nevertheless, the fact that many essential chemical properties and reactions are at least partly representable in terms of quantum mechanics is doubtless. For the quantum mechanics itself has been reformulated as a theory of a special kind of information, quantum information, chemistry might be in turn interpreted in the same terms. Wave function, the fundamental concept of quantum mechanics, can be equivalently defined as a series of qubits, eventually infinite. A qubit, being defined as the normed superposition of the two orthogonal subspaces of the complex Hilbert space, can be interpreted as a generalization of the standard bit of information as to infinite sets or series. All “forces” in the Standard model, which are furthermore essential for chemical transformations, are groups [U(1),SU(2),SU(3)] of the transformations of the complex Hilbert space and thus, of series of qubits. One can suggest that any chemical substances and changes are fundamentally representable as quantum information and its transformations. If entanglement is interpreted as a physical field, though any group above seems to be unattachable to it, it might be identified as the “Triger field”. It might cause a direct transformation of any chemical substance by from a remote distance. Is this possible in principle? (shrink)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important (...) programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

Recently Dimitria Electra Gatzia has argued that individual colour variations present difficulties for both colour objectivism and colour subjectivism (Gatzia 2010). On the objectivist side, she has singled out Byrne and Hilbert’s colour physicalism for criticism (Byrne and Hilbert 2003; 2007). The aim of this paper is to defend Byrne and Hilbert’s colour physicalism against an argument of Gatzia’s.

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then (...) elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (shrink)

The brain is composed of electrically excitable neuronal networks regulated by the activity of voltage-gated ion channels. Further portraying the molecular composition of the brain, however, will not reveal anything remotely reminiscent of a feeling, a sensation or a conscious experience. In classical physics, addressing the mind–brain problem is a formidable task because no physical mechanism is able to explain how the brain generates the unobservable, inner psychological world of conscious experiences and how in turn those conscious experiences steer (...) the underlying brain processes toward desired behavior. Yet, this setback does not establish that consciousness is non-physical. Modern quantum physics affirms the interplay between two types of physical entities in Hilbert space: unobservable quantum states, which are vectors describing what exists in the physical world, and quantum observables, which are operators describing what can be observed in quantum measurements. Quantum no-go theorems further provide a framework for studying quantum brain dynamics, which has to be governed by a physically admissible Hamiltonian. Comprising consciousness of unobservable quantum information integrated in quantum brain states explains the origin of the inner privacy of conscious experiences and revisits the dynamic timescale of conscious processes to picosecond conformational transitions of neural biomolecules. The observable brain is then an objective construction created from classical bits of information, which are bound by Holevo’s theorem, and obtained through the measurement of quantum brain observables. Thus, quantum information theory clarifies the distinction between the unobservable mind and the observable brain, and supports a solid physical foundation for consciousness research. (shrink)

In this paper I discuss the Newman problem in the context of contemporary epistemic structural realism (ESR). I formulate Newman’s objection in terms that apply to today’s ESR and then evaluate a defence of ESR based on Carnap’s use of Ramsey sentences and Hilbert’s ε-operator. I show that this defence improves the situation by allowing a formal stipulation of non-structural constraints. However, it fails short of achieving object individuation in the context of satisfying the Ramsified form of a (...) theory. Thus, while limiting the scope of Newman’s argument, Carnap sentences do not fully solve the problem. (shrink)

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also be proved (...) by means of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present (...) such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

This collection of articles explores a wide range of subject, from Godel’s incompleteness theorem, to possible technocalypse and neutrofuturology. Articles on historical debates on irrational number to electroculture, on vortex particle, or on different Neutrosophic applications are included.

A principle, according to which any scientific theory can be mathematized, is investigated. Social science, liberal arts, history, and philosophy are meant first of all. That kind of theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather (...) a metamathematical axiom about the relation of mathematics and reality. The main statement is formulated as follows: Any scientific theory admits isomorphism to some mathematical structure in a way constructive. Its investigation needs philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. The sketch of the proof is organized in five steps: a generalization of epoché; involving transfinite induction in the transition between Peano arithmetic and set theory; discussing the finiteness of Peano arithmetic; applying transfinite induction to Peano arithmetic; discussing an arithmetical model of reality. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. The present paper follows a pathway grounded on Husserl’s phenomenology and “bracketing reality” to achieve the generalized arithmetic necessary for the principle to be founded in alternative ontology, in which there is no reality external to mathematics: reality is included within mathematics. That latter mathematics is able to self-found itself and can be called Hilbert mathematics in honour of Hilbert’s program for self-founding mathematics on the base of arithmetic. The principle of universal mathematizability is consistent to Hilbert mathematics, but not to Gödel mathematics. Consequently, its validity or rejection would resolve the problem which mathematics refers to our being; and vice versa: the choice between them for different reasons would confirm or refuse the principle as to the being. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being is discussed again in a new way A few directions for future work can be: a rigorous formal proof of the principle as an independent axiom; the further development of information ontology consistent to both kinds of mathematics, but much more natural for Hilbert mathematics; the development of the information interpretation of quantum mechanics as a mathematical one for information ontology and thus Hilbert mathematics; the description of consciousness in terms of information ontology. (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...) means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)

Panpsychism, the view that consciousness is present everywhere at the fundamental level of reality, has established itself as an increasingly popular option in the philosophy of mind. Situated between substance dualism and reductive physicalism, panpsychism aims to capture the intuitions behind both, integrating consciousness into the physical world without explaining it in terms of purely physical facts. In this thesis, I offer a defence of panpsychism. -/- First, I examine influential arguments against physicalism, such as Thomas Nagel’s (1974, 1979) perspective-based (...) proposal, Frank Jackson’s (1982) knowledge argument, David Chalmers’ (1996, 2010) conceivability argument, as well as vagueness arguments (Antony 2006a; Tye 1996). I argue that none of them offers a knockdown case against physicalism, particularly because of the threat presented by the phenomenal concept strategy (Chalmers 2007; Papineau 2007a), though they suggest panpsychism as a promising alternative that avoids many common objections. -/- Second, I discuss independent arguments for panpsychism, including the Russellian monist view (Goff 2017; Russell 1927/1992), the intrinsic nature argument (Seager 2006; Strawson 2006), recent developments in the literature as presented by Hedda Hassel Mørch (2018, 2020) and Luca Dondoni (2022), Tye’s (2021b) representational panpsychism based on the vagueness argument, as well as cosmopsychism (Nagasawa & Wager 2016; Shani 2015) – an alternative to panpsychism that shares its aims whilst claiming to avoid its shortcomings. -/- Third, I address the most pressing objection to panpsychism: the combination problem or the question of how fundamental consciousness can combine to lead to complex consciousness (James 1890). Focusing especially on the subject-summing version of the combination problem or the issue of how small subjects can combine to form complex subjects, I go over various proposals for either solving or avoiding it, such as Philip Goff’s (2016) phenomenal bonding relation and William Seager’s (2010) combinatorial infusion proposal, arguing against both and agreeing at the end with Sam Coleman (2014), who advocates that subject-summing is demonstrably incoherent. -/- However, as I argue in the fourth and final chapter, the panpsychist still has an option that avoids the subject-summing problem, while also circumventing the objections faced by the alternative view of cosmopsychism. Inspired though not reliant on the philosophy of Gottfried Wilhelm Leibniz (1902/2005), I introduce what I call monadic panpsychism, the view that the phenomenal character of any given consciousness-bearing physical ultimate is determined by its relations to other such ultimates featuring in the relevant causal structure. -/- The theory consists of two key claims. The first is that phenomenal qualities are relational rather than intrinsic. This view has precedent in literature on the philosophy of mind, such as in Rudolf Carnap (1928), Gottlob Frege (1956), Moritz Schlick (1959), as well as more recently in David Hilbert and Mark Kalderon (2000) and Austen Clark (2000), though not in the context of panpsychism. -/- The second claim is that physical ultimates only possess a very basic form of consciousness necessary for experiencing as such. There are many ways to conceptualise this basic consciousness: non-relational or intrinsic, rudimentary and non-specific, a point of view without experiences, the minimal subject, a mere conduit for experience, phenomenal space, etc. Russellian panpsychists offer a more detailed proposal in which every physical ultimate has an appropriate associated type of experience. For example, all quarks have the same quark-type of experience, assuming for the sake of discussion that quarks are ultimates. Still, arguing for a more minimal notion, as I do, requires making less claims and should not be controversial to any panpsychist. -/- So, what follows from these two claims? I propose that the relational constitution of phenomenal qualities has its endpoint at the fundamental level rather than at the level of a combined complex or emergent subject. Put simply, physical ultimates guarantee the basic consciousness or the point of view that I occupy, while the relational structure constitutes my rich and full human experience. So to speak, relationally constituted phenomenal qualities ‘anchor’ themselves in physical ultimates. As long as these relations result in a phenomenally unified experience, which is an uncontroversial thesis, then no matter where that process reaches the end of its causal chain, it will necessarily be experienced from a subjective point of view. -/- Since only qualities but not subjects need to combine, my proposal avoids the subject-summing version of the combination problem, which is arguably an insurmountable challenge to other forms of panpsychism. Because of this, I conclude that my theory contributes to the field, serving as a good starting point for further developments of neither combinatory nor emergent panpsychism with regard to subjects. (shrink)

The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I (...) believe, that we consider how space itself relates to reality, and how other so-called "spaces" used in physics relate to reality. In chapter 2, I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the "property view" of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter 3, prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman. In chapter 4, some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the "measurement problem" of quantum mechanics: this problem and its proposed solution are the primary subjects of chapter 4. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber. Finally, in chapters 5 and 6, I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time. (shrink)

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function but by a density matrix. I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories with (...) a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research. (Bibliographic note: the thesis was submitted for the Master of Science in mathematics at Rutgers University.). (shrink)

One of the criteria to a strong principle in natural sciences is simplicity. This paper claims that the Free Energy Principle (FEP), by virtue of unifying particles with mind, is the simplest. Motivated by Hilbert’s 24th problem of simplicity, the argument is made that the FEP takes a seemingly mathematical complex domain and reduces it to something simple. More specifically, it is attempted to show that every ‘thing’, from particles to mind, can be partitioned into systemic states by (...) virtue of self-organising symmetry break, i.e. self-entropy in terms of the balance between risk and ambiguity to achieve epistemic gain. By virtue of its explanatory reach, the FEP becomes the simplest principle under quantum, statistical and classical mechanics conditions. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are (...) category-theoretically dual to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

This paper concerns “human symbolic output,” or strings of characters produced by humans in our various symbolic systems; e.g., sentences in a natural language, mathematical propositions, and so on. One can form a set that consists of all of the strings of characters that have been produced by at least one human up to any given moment in human history. We argue that at any particular moment in human history, even at moments in the distant future, this set is finite. (...) But then, given fundamental results in recursion theory, the set will also be recursive, recursively enumerable, axiomatizable, and could be the output of a Turing machine. We then argue that it is impossible to produce a string of symbols that humans could possibly produce but no Turing machine could. Moreover, we show that any given string of symbols that we could produce could also be the output of a Turing machine. Our arguments have implications for Hilbert’s sixth problem and the possibility of axiomatizing particular sciences, they undermine at least two distinct arguments against the possibility of Artificial Intelligence, and they entail that expert systems that are the equals of human experts are possible, and so at least one of the goals of Artificial Intelligence can be realized, at least in principle. (shrink)

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we (...) likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes but leaves the theory’s basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory permits assuming that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive (...) picture of states and Hilbert-space trajectories to the more realistic case of open quantum systems despite the generic development of entanglement. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure—which we call subsystem space—that we believe merits further study. We conclude with a summary, a list of criteria for future work on quantum foundations, and further research directions. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text. (shrink)

This paper outlines the intellectual biography of Walter Dubislav. Besides being a leading member of the Berlin Group headed by Hans Reichenbach, Dubislav played a defining role as well in the Society for Empirical/Scientific Philosophy in Berlin. A student of David Hilbert, Dubislav applied the method of axiomatic to produce original work in logic and formalist philosophy of mathematics. He also introduced the elements of a formalist philosophy of science and addressed more general problems concerning the substantiation of human (...) knowledge. What set Dubislav apart from the other logical empiricists was his expertise in the history of logic and exact philosophy which enabled him to elucidate and advance the thinking in both disciplines. In the realm of logic proper, Dubislav is best known for his pioneering work in theory of definitions. What is more, he did original work on the so called ‘quasi truth-tables’ which aided Reichenbach in developing his logic of probability. Dubislav also elaborated an influe.. (shrink)

Walter Dubislav (1895–1937) was a leading member of the Berlin Group for scientific philosophy. This “sister group” of the more famous Vienna Circle emerged around Hans Reichenbach’s seminars at the University of Berlin in 1927 and 1928. Dubislav was to collaborate with Reichenbach, an association that eventuated in their conjointly conducting university colloquia. Dubislav produced original work in philosophy of mathematics, logic, and science, consequently following David Hilbert’s axiomatic method. This brought him to defend formalism in these disciplines as (...) well as to explore the problems of substantiating (Begründung) human knowledge. Dubislav also developed elements of general philosophy of science. Sadly, the political changes in Germany in 1933 proved ruinous to Dubislav. He published scarcely anything after Hitler came to power and in 1937 committed suicide under tragic circumstances. The intent here is to pass in review Dubislav’s philosophy of logic, mathematics, and science and so to shed light on some seminal yet hitherto largely neglected currents in the history of philosophy of science. (shrink)

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (besides the above (...) only mathematical equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrödinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation. (shrink)

Quantum complementarity is interpreted in terms of duality and opposition. Any two conjugates are considered both as dual and opposite. Thus quantum mechanics introduces a mathematical model of them in an exact and experimental science. It is based on the complex Hilbert space, which coincides with the dual one. The two dual Hilbert spaces model both duality and opposition to resolve unifying the quantum and smooth motions. The model involves necessarily infinity even in any finitely dimensional subspace of (...) the complex Hilbert space being due to the complex basis. Furthermore, infinity is what unifies duality and opposition, universality and openness, completeness and incompleteness in it. The deduced core of quantum complementarity in terms of infinity, duality and opposition allows of resolving a series of various problems in different branches of philosophy: the common structure of incompleteness in Gödel’s (1931) theorems and Enstein, Podolsky, and Rosen’s argument (1935); infinity as both complete and incomplete; grounding and self-grounding, metaphor and representation between language and reality, choice and information, the totality and an observer, the basic idea of philosophical phenomenology. The main conclusion is: Quantum complementarity unifies duality and opposition in a consistent way underlying the physical world. (shrink)

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...) of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)

The main algebraic foundations of quantum mechanics are quickly reviewed. They have been suggested since the birth of this theory till up to last years. They are the following ones: Heisenberg-Born- Jordan’s (1925), Weyl’s (1928), Dirac’s (1930), von Neumann’s (1936), Segal’s (1947), T.F. Jordan’s (1986), Morchio and Strocchi’s (2009) and Buchholz and Fregenhagen’s (2019). Four cases are stressed: 1) the misinterpretation of Dirac’s algebraic foundation; 2) von Neumann’s ‘conversion’ from the analytic approach of Hilbert space to the algebraic approach (...) of the rings of operators; 3) Morchio and Strocchi’s improving Dirac’s analogy between commutators and Poisson Brackets into an exact equivalence; 4) the recent foundation of quantum mechanics upon the algebra of perturbations. Some considerations on alternating theoretical importance of the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices to group theory and C*-algebras but has not led to a definition of the foundations of physics; in particular, an algebraic formulation of QM organized as a problem-based theory and an exclusive use of constructive mathematics is still to be discovered. (shrink)

Putnam’s most famous contribution to mathematical logic was his role in investigating Hilbert’s Tenth Problem; Putnam is the ‘P’ in the MRDP Theorem. This volume, though, focusses mostly on Putnam’s work on the philosophy of logic and mathematics. It is a somewhat bumpy ride. Of the twelve papers, two scarcely mention Putnam. Three others focus primarily on Putnam’s ‘Mathematics without foundations’ (1967), but with no interplay between them. The remaining seven papers apparently tackle unrelated themes. Some of this (...) disjointedness would doubtless have been addressed, if Putnam had been able to compose his replies to these papers; sadly, he died before this was possible. In this review, I do my best to tease out some connections between the paper; and there are some really interesting connections to be made. (shrink)

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one can (...) get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)

Given the hard problem of consciousness, there are no brain electrophysiological correlates of the subjective experience (the felt quality of redness or the redness of red, the experience of dark and light, the quality of depth in a visual field, the sound of a clarinet, the smell of mothball, bodily sensations from pains to orgasms, mental images that are conjured up internally, the felt quality of emotion, the experience of a stream of conscious thought, or the phenomenology of thought). (...) However, there are occipital and left temporal brain electrophysiological correlates of the subjective experience. Notwithstanding, as an evoked signal, the change in event-related brain potential phase (frequency is the change in phase over time) is instantaneous, that is, the frequency will transiently be infinite: a transient peak in frequency (positive or negative), if any, is instantaneous in electroencephalogram averaging or filtering that the event-related brain potentials required and the underlying structure of the event-related brain potentials in the frequency domain cannot be accounted for, for example, by the Wavelet Transform or the Fast Fourier Transform analysis, because they require that frequency be derived by convolution rather than by differentiation. However, as I show, one suitable method for analysing the instantaneous change in event-related brain potential phase and accounting for a transient peak in frequency (positive or negative), if any, in the underlying structure of the event-related brain potentials is the Empirical Mode Decomposition with post-processing Ensemble Empirical Mode Decomposition. (shrink)

In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of—primarily state-supported—mathematics: (a) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that first-order Set Theory can be treated as the lingua franca of mathematics, since its theorems—even if unfalsifiable—can be treated as ‘knowledge’ because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (b) the slowly emerging, but (...) powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as ‘knowledge’ which are, further, categorically communicable as Factually Grounded Beliefs—hence as comprehensible pre-formal ‘mathematical truths’—by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing under-estimation of the value to society of ‘semantic arithmetical truth’, and an increasing over-estimation of the value to society of ‘syntactic set-theoretical provability’; a chasm which must be narrowed and bridged explicitly. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical ‘provability’ and mathematical ‘truth’ need to be interdependent and complementary, ‘evidence-based’, assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical ‘proofs’ may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical ‘truths’ must be viewed as seeking to ensure that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the post-graduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent—and hitherto unsuspected—unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a ‘mathematical truth’; (ii) what we can evidence as a ‘mathematical truth’; and (iii) what we ought not to believe is a ‘mathematical truth’; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be ‘arithmetical truth’, not ‘set-theoretical proof’. (shrink)

We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth (...) to the compound formulas of PA over N---I_PA(N; SV ) and I_PA(N; SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both I_PA(N; SV ) and I_PA(N; SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under I_PA(N; SV ), then this assignment corresponds to the classical non-finitary putative standard interpretation I_PA(N; S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment I_PA(N; SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Goedel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which---under interpretation---are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas' Goedelian argument is validated if the assignment I_PA(N; SV ) can be treated as circumscribing the ambit of human reasoning about `true' arithmetical propositions, and the assignment I_PA(N; SC) as circumscribing the ambit of mechanistic reasoning about `true' arithmetical propositions. (shrink)

Color-vision defects constitute a spectrum of disorders with varying degrees and types of departure from normal human color vision. One form of color-vision defect is dichromacy; by mixing together only two lights, the dichromat can match any light, unlike normal trichromatic humans, who need to mix three. In a philosophical context, our titular question may be taken in two ways. First, it can be taken at face value as a question about visible properties of external objects, and second, it may (...) be interpreted as the more intangible question of “what it’s like” to be color-blind. (shrink)

A survey of arguments for and against the view that colors are physical properties.

A survey of color science and color vision.

In a recent Philosophy of Science article Gerhard Schurz proposes meta-inductivistic prediction strategies as a new approach to Hume's. This comment examines the limitations of Schurz's approach. It can be proven that the meta-inductivist approach does not work any more if the meta-inductivists have to face an infinite number of alternative predictors. With his limitation it remains doubtful whether the meta-inductivist can provide a full solution to the problem of induction.

The New Evil Demon Problem presents a serious challenge to externalist theories of epistemic justification. In recent years, externalists have developed a number of strategies for responding to the problem. A popular line of response involves distinguishing between a belief’s being epistemically justified and a subject’s being epistemically blameless for holding it. The apparently problematic intuitions the New Evil Demon Problem elicits, proponents of this response claim, track the fact that the deceived subject is epistemically blameless for (...) believing as she does, not that she is justified for so believing. This general strategy—which I call the “unjustified-but-blameless maneuver”—is motivated, in part, by the assumption that the distinction between epistemic justification and blamelessness is merely an extension of the familiar distinction between moral justification and blamelessness. In this paper, I consider three ways of drawing the distinction between justification and blamelessness familiar from the moral domain: the first in terms of a connection with reactive attitudes, the second in terms of the distinction between wrongness and wronging, and the third in terms of reasons-responsiveness. All three ways of drawing the distinction, I argue, make it difficult to see how an analogous distinction in the epistemic domain could help externalists explain away the intuitions which underwrite the New Evil Demon Problem. Motivating the unjustified-but-blameless maneuver, I conclude, is a much less straightforward task than its proponents tend to assume. (shrink)