Taking the BCS Hamiltonian written in second-quantized form, a modified form of Umezawa's self-consistent field theory method is applied, and a unitarily nonequivalent representation is selected in which the Hamiltonian obviously describes a superconducting system. This result is not at all obvious, since the original Hamiltonian is completely symmetric, and there is no reason a priori for expecting it to describe an asymmetric superconducting configuration. All higher order terms are accounted for, and in doing so, one finds the existence of (...) the energy-gap condition for Cooper pairs. The representation is picked out without using the adiabatic theorem or pair approximation, as is typically done in the self-consistent method for superconductivity. (shrink)

Bolzano reduced inferential validity of the inference (from premise judgements to conclusion judgment) to the holding of logical consequence between the propositions (in themselves) that serve as contents of the respective judgements. This explicit reduction of inferential validity among judgements to logical consequence among propositions (or, alternatively, to logical truth of certain implicational propositions) has been largely taken over by current logical theory, say, by Wittgenstein’s Tractatus, by Hilbert and Ackermann, by Quine, and by Tarski also. Frege, though, stands out (...) among those who did not adopt such an account. Under the Bolzano reduction also some blind inferences, with no epistemic support, are deemed valid, which is unacceptable. The Completeness Theorem reduces inferential validity to the logical truth of a certain well-formed formula, whence the unacceptable blindness phenomena are retained. Accordingly, the interest of the Completeness Theorem for logic, construed as the theory of inference, is nugatory. (shrink)

In a previous work (M. Campisi. Stud. Hist. Phil. M. P. 36 (2005) 275-290) we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the Generalized Helmholtz Theorem. It was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat theorem and it is an adiabatic invariant. This property explains the ``equal'' sign in Clausius principle ($S_f \geq S_i$) in a purely mechanical way and suggests that (...) the volume entropy might explain the ``larger than'' sign (i.e. the Law of Entropy Increase) if non adiabatic transformations were considered. Based on the principles of microscopic (quantum or classical) mechanics here we prove that, provided the initial equilibrium satisfy the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external sources of work on a insulated system. This can be regarded as a rigorous quantum mechanical proof of the Second Law. We discuss how this result relates to the Minimal Work Principle and improves over previous attempts. The natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature. Actually the Law of Entropy Decrease holds in artificially prepared negative temperature systems. (shrink)

Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

Protective measurements illustrate how Yakir Aharonov's fundamental insights into quantum theory yield new experimental paradigms that allow us to test quantum mechanics in ways that were not possible before. As for quantum theory itself, protective measurements demonstrate that a quantum state describes a single system, not only an ensemble of systems, and reveal a rich ontology in the quantum state of a single system. We discuss in what sense protective measurements anticipate the theorem of Pusey, Barrett, and Rudolph (PBR), (...) stating that, if quantum predictions are correct, then two distinct quantum states cannot represent the same physical reality. (shrink)

We prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.

In [This Zeitschrift 25 , 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order Pavelka's logic (...) is preserved under some direct product of lattices of truth values. (shrink)

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many (...) questions related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that—if one is willing to assume a thesis due to Donald Knuth—the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent—including a not-necessarily infallible human agent—satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Self-comprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers. (shrink)

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system SCD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of SCD, we extend SCD to the system SPCD for the first-order classical logic with disjunctive syllogism. It (...) can be shown that SPCD is conservative extension to SCD. Then, the normalization theorem and the consistency of SPCD are given. (shrink)

A running commentary is offered on the first half of Frege’s Grundlagen der Arithmetik, §17, and suggests that Frege anticipated the method of demonstration used by Paul Bernays for the Deduction Theorem.

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a (...) rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work. (shrink)

Previously we saw that racist prejudice is supported by false history. The false history of the Greek origins of mathematics is reinforced by a bad philosophy of mathematics. There is no evidence for the existence of Euclid. The “Euclid” book does not contain a single axiomatic proof, as was exposed over a century ago. Such was never the intention of the actual author. The book was brazenly reinterpreted, since axiomatic proof was a church political requirement, and used in church rational (...) theology adopted during the Crusades, as a counter to Islamic rational theology. Deductive proofs are MORE fallible than inductive or empirical proofs. Even a validly proved mathematical theorem, such as the “Pythagorean” theorem, is invalid knowledge in the real world. There is no concept of approximate truth in formal mathematics. Nevertheless, the myth of “superior” axiomatic proofs in the “Euclid” book continues to be reiterated by Western historians, and colonial education teaches axiomatic mathematics. Actually, superior practical value comes from the two “Pythagorean” calculations well known toIndian/Egyptian tradition, but unknown to Greeks. The advantage of related decolonized courses in mathematics has been pedagogically demonstrated. But understanding and political will are needed to change colonial/church education. (shrink)

The fundamental principle of the theory of possible worlds is that a proposition p is possible if and only if there is a possible world at which p is true. In this paper we present a valid derivation of this principle from a more general theory in which possible worlds are defined rather than taken as primitive. The general theory uses a primitive modality and axiomatizes abstract objects, properties, and propositions. We then show that this general theory has very small (...) models and hence that its ontological commitments—and, therefore, those of the fundamental principle of world theory—are minimal. (shrink)

The theorem of Bell states that certain results of quantum mechanics violate inequalities that are valid for objective local random variables. We show that the inequalities of Bell are special cases of theorems found 10 years earlier by Bass and stated in full generality by Vorob’ev. This fact implies precise necessary and sufficient mathematical conditions for the validity of the Bell inequalities. We show that these precise conditions differ significantly from the definition of objective local variable spaces and as (...) an application that the Bell inequalities may be violated even for objective local random variables. (shrink)

Justification logics are logics of knowledge in which explicit reasons are formally represented. Standard logics of knowledge have justification logic analogs. Connecting justification logics and logics of knowledge are Realization Theorems. In this paper we give a new, constructive proof of the Realization Theorem connecting S5 and its justification analog, JS5. This proof is, I believe, the simplest in the literature.

The ultraproduct construction is generalized to p‐ultramean constructions () by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments of continuous logic and are very close to the constructions in real analysis. A powermean variant of the Keisler‐Shelah isomorphism theorem is proved for. It is then proved that ‐sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.

We state and prove the Translation Theorem. Then we apply the Translation Theorem to Soare's Extension Theorem, weakening slightly the hypothesis to yield a theorem we call the Modified Extension Theorem. We use this theorem to reprove several of the known results about orbits in the lattice of recursively enumerable sets. It is hoped that these proofs are easier to understand than the old proofs.

The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.

This chapter sets out the theorems, and presents some examples that show in a rough way how the theorems work. It explains separability precisely, and states the theorems. The chapter starts the work of interpreting the theorems, and also explains the significance of their conclusions from a formal, mathematical point of view. It then discusses a significant assumption that is used in the proofs of the theorems. The published proofs of both the separability theorems depend on an assumption that may (...) be objectionable in some contexts. These theorems are about an ordering defined on a field of alternatives. If an ordering can be represented by an additively separable utility function, the theorem shows it can be represented by a whole family of additively separable functions. Each member of the family is an increasing linear transform of the others, and the family includes every increasing linear transform of each of its members. (shrink)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...) a counting argument. We extend his method to arbitrary similarity types. (shrink)

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] $\prec$ [b] $\prec$ [a] there exists [c] ∈ R/M such that [c] $\prec$ [a] and [a] = [b] ∨ [c].

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a (...) way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

This paper examines Boltzmann’s responses to the Loschmidt reversibility objection to the H-theorem, as presented in his Lectures on Gas Theory. I describe and evaluate two distinct conceptions of the assumption of molecular disorder found in this work, and contrast these notions with the Stosszahlansatz, as well as with the predominant contemporary conception of molecular disorder. Both these conceptions are assessed with respect to the reversibility objection. Finally, I interpret Boltzmann as claiming that a state of molecular disorder serves (...) as a necessary condition for the application of probabilistic arguments. This in turn offers a way to bridge the conceptual gap between the H-theorem and his combinatorial argument. (shrink)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend (...) his method to arbitrary similarity types. (shrink)

We give a short proof of the fundamental theorem of central element theory. The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.

In this paper, we present two proofs of the reciprocal butterfly theorem. The statement of the butterfly theorem is: Let us consider a chord PQ of midpoint M in the circle Ω(O). Through M, two other chords AB and CD are drawn, such that A and C are on the same side of PQ. We denote by X and U the intersection of AD respectively CB with PQ. Consequently, XM = YM. For the proof of this theorem, (...) see [1]. The reciprocal of the butterfly theorem has the following statement: In the circle Ω(O), let us consider the chords PQ, AB and CD which are concurrent in the point M≠O, such as the points A and C are on the same side of the line PQ. Let X and Y respectively be the intersections of the chord PQ with AD and BC respectively. If XM = YM, then M is the middle of the chord PQ. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

We revisit the analogy suggested by Madelung between a non-relativistic time-dependent quantum particle, to a fluid system which is pseudo-barotropic, irrotational and inviscid. We first discuss the hydrodynamical properties of the Madelung description in general, and extract a pressure like term from the Bohm potential. We show that the existence of a pressure gradient force in the fluid description, does not violate Ehrenfest’s theorem since its expectation value is zero. We also point out that incompressibility of the fluid implies (...) conservation of density along a fluid parcel trajectory and in 1D this immediately results in the non-spreading property of wave packets, as the sum of Bohm potential and an exterior potential must be either constant or linear in space. Next we relate to the hydrodynamic description a thermodynamic counterpart, taking the classical behavior of an adiabatic barotopric flow as a reference. We show that while the Bohm potential is not a positive definite quantity, as is expected from internal energy, its expectation value is proportional to the Fisher information whose integrand is positive definite. Moreover, this integrand is exactly equal to half of the square of the imaginary part of the momentum, as the integrand of the kinetic energy is equal to half of the square of the real part of the momentum. This suggests a relation between the Fisher information and the thermodynamic like internal energy of the Madelung fluid. Furthermore, it provides a physical linkage between the inverse of the Fisher information and the measure of disorder in quantum systems—in spontaneous adiabatic gas expansion the amount of disorder increases while the internal energy decreases. (shrink)

By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...) also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)

The possibility of spectrum inversion has been debated since it was raised by Locke and is still discussed because of its implications for functionalist theories of conscious experience . This paper provides a mathematical formulation of the question of spectrum inversion and proves that such inversions, and indeed bijective scramblings of color in general, are logically possible. Symmetries in the structure of color space are, for purposes of the proof, irrelevant. The proof entails that conscious experiences are not identical with (...) functional relations. It leaves open the empirical possibility that functional relations might, at least in part, be causally responsible for generating conscious experiences. Functionalists can propose causal accounts that meet the normal standards for scientific theories, including numerical precision and novel prediction; they cannot, however, claim that, because functional relationships and conscious experiences are identical, any attempt to construct such causal theories entails a category error. (shrink)

Although skeptical of the prohibitive power of no-hidden-variables theorems, John Bell was himself responsible for the two most important ones. I describe some recent versions of the lesser known of the two (familiar to experts as the "Kochen-Specker theorem") which have transparently simple proofs. One of the new versions can be converted without additional analysis into a powerful form of the very much better known "Bell's Theorem," thereby clarifying the conceptual link between these two results of Bell.

Here is Hilbert is his famous address of 1900: The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.

The PBR theorem gives insight into how quantum mechanics describes a physical system. This paper explores PBRs’ general result and shows that it does not disallow the ensemble interpretation of quantum mechanics and maintains, as it must, the fundamentally statistical character of quantum mechanics. This is illustrated by drawing an analogy with an ideal gas. An ensemble interpretation of the Schrödinger cat experiment that does not violate the PBR conclusion is also given. The ramifications, limits, and weaknesses of the (...) PBR assumptions, especially in light of lessons learned from Bell’s theorem, are elucidated. It is shown that, if valid, PBRs’ conclusion specifies what type of ensemble interpretations are possible. The PBR conclusion would require a more direct correspondence between the quantum state ) and the reality it describes than might otherwise be expected. A simple terminology is introduced to clarify this greater correspondence. (shrink)