Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of (...) rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinsteinís group identification problem. (shrink)

This paper critically engages Philip Mirowki's essay, "The scientific dimensions of social knowledge and their distant echoes in 20th-century American philosophy of science." It argues that although the cold war context of anti-democratic elitism best suited for making decisions about engaging in nuclear war may seem to be politically and ideologically motivated, in fact we need to carefully consider the arguments underlying the new rational choice based political philosophies of the post-WWII era typified by Arrow's impossibility theorem. A distrust (...) of democratic decision-making principles may be developed by social scientists whose leanings may be toward the left or right side of the spectrum of political practices. (shrink)

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite (...) descent. The infinite descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for (...) “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it. (shrink)

The previous two parts of the paper demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure (...) of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum (...) contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

Suppose that a majority of jurors decide that a defendant is guilty (or not), and we want to know the likelihood that they reached the correct verdict. The French philosopher Marquis de Condorcet (1743-1794) showed that we can get a mathematically precise answer, a result known as the “Condorcet Jury Theorem.” Condorcet’s theorem isn’t just about juries, though; it’s about collective decision-making in general. As a result, some philosophers have used his theorem to argue for democratic forms (...) of government. This essay explains Condorcet’s theorem and how philosophers have used it as an argument for democracy. (shrink)

In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, non-dictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or implication preservation. We fully characterize the agendas for which (...) there are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (2002), Nehring (2006), Dietrich and List (2007a) and Dokow and Holzman (2010a). (shrink)

My aim in this paper is to explain what Condorcet’s jury theorem is, and to examine its central assumptions, its significance to the epistemic theory of democracy and its connection with Rousseau’s theory of general will. In the first part of the paper I will analyze an epistemic theory of democracy and explain how its connection with Condorcet’s jury theorem is twofold: the theorem is at the same time a contributing historical source, and the model used by (...) the authors to this day. In the second part I will specify the purposes of the theorem itself, and examine its underlying assumptions. Third part will be about an interpretation of Rousseau’s theory, which is given by Grofman and Feld relying on Condorcet’s jury theorem, and about criticisms of such interpretation. In the fourth, and last, part I will focus on one particular assumption of Condorcet’s theorem, which proves to be especially problematic if we would like to apply the theorem under real-life conditions; namely, the assumption that voters choose between two options only. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Although written in Japanese, this article handles historical and technical survey of Gödel's incompleteness theorem thoroughly.

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good (...) measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental. (shrink)

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in RCA0. This approach yields a proof of Arrow's (...) class='Hi'>theorem in RCA0, and thus in PRA, since Arrow's theorem can be formalised as a Π01 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0 over RCA0. (shrink)

A survey of more philosophical applications of Gödel's incompleteness results.

Abstract: In this paper, the definition of triple Aboodh transform of fractional order α, where α ϵ [0, 1], is introduced for functions which are fractional differentiable. We also present several properties of this transform. Furthermore, some main theorems and their proofs are discussed.

In this short survey article, I discuss Bell’s theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest.

In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we (...) thereby provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71.. (shrink)

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically (...) impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

In recent work Allan Gibbard claims to be both a local quasi-realist, in Blackburn’s sense, and a global expressivist. His local quasi-realism rests on an argument that for naturalistic discourse but not ethical discourse, the semantic relation of denotation and the causal relation of tracking can and should be identified; that denoting simply is tracking, for naturalistic vocabulary. I argue that Gibbard’s case for this conclusion is unconvincing, and poorly motivated by his own expressivist standards. I also argue that even (...) if it were successful, it is doubtful whether the resulting position would count as global expressivism, as Gibbard and I both use that term. (shrink)

Two of the most important ideas in the philosophy of law are the “Coase Theorem” and the “Prisoner’s Dilemma.” In this paper, the authors explore the relation between these two influential models through a creative thought-experiment. Specifically, the paper presents a pure Coasean version of the Prisoner’s Dilemma, one in which property rights are well-defined and transactions costs are zero (i.e. the prisoners are allowed to openly communicate and bargain with each other), in order to test the truth value (...) of the Coase Theorem. In addition, the paper explores what effect (a) uncertainty, (b) exponential discounting, (c) and elasticity have on the behavior of the prisoners in the Coasean version of the dilemma. Lastly, the paper considers the role of the prosecutor (and third-parties generally) in the Prisoner’s Dilemma and closes with some parting thoughts about the complexity of the dilemma. The authors then conclude by identifying the conditions under which the Prisoner’s Dilemma refutes the Coase Theorem. (shrink)

Riker (1982) famously argued that Arrow’s impossibility theorem undermined the logical foundations of “populism”, the view that in a democracy, laws and policies ought to express “the will of the people”. In response, his critics have questioned the use of Arrow’s theorem on the grounds that not all configurations of preferences are likely to occur in practice; the critics allege, in particular, that majority preference cycles, whose possibility the theorem exploits, rarely happen. In this essay, I argue (...) that the critics’ rejoinder to Riker misses the mark even if its factual claim about preferences is correct: Arrow’s theorem and related results threaten the populist’s principle of democratic legitimacy even if majority preference cycles never occur. In this particular context, the assumption of an unrestricted domain is justified irrespective of the preferences citizens are likely to have. (shrink)

I point out a sign mistake in the GHZ variant of Bell's theorem, invalidating the GHZ's claim that the premisses of the EPR argument are inconsistent for systems of more than two particles in entangled quantum states.

Bell’s theorem has fascinated physicists and philosophers since his 1964 paper, which was written in response to the 1935 paper of Einstein, Podolsky, and Rosen. Bell’s theorem and its many extensions have led to the claim that quantum mechanics and by inference nature herself are nonlocal in the sense that a measurement on a system by an observer at one location has an immediate effect on a distant entangled system. Einstein was repulsed by such “spooky action at a (...) distance” and was led to question whether quantum mechanics could provide a complete description of physical reality. In this paper I argue that quantum mechanics does not require spooky action at a distance of any kind and yet it is entirely reasonable to question the assumption that quantum mechanics can provide a complete description of physical reality. The magic of entangled quantum states has little to do with entanglement and everything to do with superposition, a property of all quantum systems and a foundational tenet of quantum mechanics. (shrink)

Fermat’s Least Time Principle has a long history. World’s foremost academies of the day championed by their most prestigious philosophers competed for the glory and prestige that went with the solution of the refraction problem of light. The controversy, known as Descartes - Fermat controversy was due to the contradictory views held by Descartes and Fermat regarding the relative speeds of light in different media. Descartes with his mechanical philosophy insisted that every natural phenomenon must be explained by mechanical principles. (...) Fermat on the other hand insisted an end purpose for every motion. For example, least time of travel and not the least distance of travel is the end purpose for motion of light. This implied a thinking nature, which was rejected by Descartes. Surprisingly, with contradictory assumptions regarding the relative speeds of light in different media, both Descartes and Fermat came to the same result that the ratio of sines of angles of incidence and refraction is a constant. Fermat’s result came to be known as the ‘Fermat’s least time principle’. We show in this article that Fermat’s least time principle violates a fundamental theorem in geometry – the Ptolemy’s theorem. That leads to the invalidity of Fermat’s principle. -/- . (shrink)

A probability aggregation rule assigns to each profile of probability functions across a group of individuals (representing their individual probability assignments to some propositions) a collective probability function (representing the group's probability assignment). The rule is “non-manipulable” if no group member can manipulate the collective probability for any proposition in the direction of his or her own probability by misrepresenting his or her probability function (“strategic voting”). We show that, except in trivial cases, no probability aggregation rule satisfying two mild (...) conditions (non-dictatorship and consensus preservation) is non-manipulable. (shrink)

This essay deals with Gödel's Theorems in relationship to Philosophy of Science; firstly, in outlining Ludwig Wittgenstein's position on the limits of philosophical truth that we can derive from Gödel (and how this in turn impacts modern-philosophical conceptions of science), and secondly, the deeper uncertainty about consciousness that Gödel's theorems point to, most notably elucidated by Sir Roger Penrose.

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These (...) ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (shrink)

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these (...) versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic. (shrink)

According to Aristotle, democracy and oligarchy are empirically the most widespread and analytically fundamental ‘constitutions’. I analyse how in different places in his Politics Aristotle ‘positively’ defines and differentiates between democracy and oligarchy. At the same time, I substantiate in detail a new interpretation of Aristotle’s view that significantly differs from the current interpretation. ‘Combining’ the elements, procedures, and principles of democracy and oligarchy gives rise to mixed ‘constitutions’, a special place among which is occupied by the (...) polity or republic, which is the best regime ‘for most states and for most people’. I show the ways in which, according to Aristotle, it is possible to form such a regime. Carl Schmitt and, later somewhat differently, Bernard Manin draw a link between Aristotle’s mixed regime and the representative democracies of today. (shrink)

A criticism of Dorothy Edgington's attempt to make Gibbard's problem for indicative conditionals apply to counterfactuals.

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice (...) function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

Leibniz proposed the ‘Most Determined Path Principle’ in seventeenth century. According to it, ‘ease’ of travel is the end purpose of motion. Using this principle and his calculus method he demonstrated Snell’s Laws of reflection and refraction. This method shows that light follows extremal (local minimum or maximum) time path in going from one point to another, either directly along a straight line path or along a broken line path when it undergoes reflection or refraction at plane or spherical (concave (...) or convex) surfaces. The extremal time path avoided the criticism that Fermat’s least time path was subjected to, by Cartesians who cited examples of reflections at spherical surfaces where light took the path of longest time. Thereby it became the standard method of demonstration of Snell’s Laws. Ptolemy’s theorem is a fundamental theorem in geometry. A special case of it offers a method of finding the minimum sum of the two distances of a point from two given fixed points. We show in this paper that Leibniz’s calculus proof of Snell’s Laws violates Ptolemy’s theorem, whereby Leibniz’s proof becomes invalid. (shrink)

Abstract— In this paper, we introduce the definition of triple Aboodh transform, some properties for the transform are presented. Furthermore, several theorems dealing with the properties of the triple Aboodh transform are proved. In addition, we use this transform to solve partial differential equations with integer and non-integer orders.

This paper makes a novel linkage between the multiple-computations theorem in philosophy of mind and Landauer’s principle in physics. The multiple-computations theorem implies that certain physical systems implement simultaneously more than one computation. Landauer’s principle implies that the physical implementation of “logically irreversible” functions is accompanied by minimal entropy increase. We show that the multiple-computations theorem is incompatible with, or at least challenges, the universal validity of Landauer’s principle. To this end we provide accounts of both ideas (...) in terms of low-level fundamental concepts in statistical mechanics, thus providing a deeper understanding of these ideas than their standard formulations given in the high-level terms of thermodynamics and cognitive science. Since Landauer’s principle is pivotal in the attempts to derive the universal validity of the second law of thermodynamics in statistical mechanics, our result entails that the multiple-computations theorem has crucial implications with respect to the second law. Finally, our analysis contributes to the understanding of notions, such as “logical irreversibility,” “entropy increase,” “implementing a computation,” in terms of fundamental physics, and to resolving open questions in the literature of both fields, such as: what could it possibly mean that a certain physical process implements a certain computation. (shrink)

In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .

This paper has three main goals. First, to motivate a puzzle about how ignorance-expressing terms like maybe and if interact: they iterate, and when they do they exhibit scopelessness. Second, to argue that there is an ambiguity in our theoretical toolbox, and that exposing that opens the door to a solution to the puzzle. And third, to explore the reach of that solution. Along the way, the paper highlights a number of pleasing properties of two elegant semantic theories, explores some (...) meta-theoretic properties of dynamic notions of meaning, dips its toe into some hazardous waters, and offers characterization theorems for the space of meanings an indicative conditional can have. (shrink)

Whether it was John Searle’s Chinese Room argument (Searle, 1980) or Roger Penrose’s argument of the non-computable nature of a mathematician’s insight – an argument that was based on Gödel’s Incompleteness theorem (Penrose, 1989), we have always had skeptics that questioned the possibility of realizing strong Artificial Intelligence (AI), or what has become known by Artificial General Intelligence (AGI). But this new book by Landgrebe and Smith (henceforth, L&S) is perhaps the strongest argument ever made against strong AI. It (...) is a very extensive review of what building a mind essentially amounts to drawing on insights and results from biology, physics, linguistics, computability, philosophy, and mathematics. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, (...) say, Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s school. I (...) shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (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)

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical (...) instrumentalism are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by J.P. Burgess and G. (...) Rosen). Examples are discussed. The historical development of written mathematics from algorithms on clay tablets to theorems with proofs is said to show that application of algorithms to specific problems and theorems to scientific objects is like allegorical interpretation. Mathematics fits into the modern scientific context because the sciences, beginning with Galileo, have been constructed in imitation of mathematics. Viewing mathematics this way does not solve any ontological problems, but it does show how mathematics avoids them. Epistemological problems, insuperable for object realism, are simplified. For example, we have access to relations among any objects that we can consider objectively, first physical objects and then mathematical objects of greater and greater abstraction. (shrink)