PRN: Philosophy of Mathematics & Logic (Topic)最新文献

{"title":"On an Extension of Stein's Lemma: A Refutation","authors":"Moawia Alghalith","doi":"10.2139/ssrn.3653224","DOIUrl":"https://doi.org/10.2139/ssrn.3653224","url":null,"abstract":"We prove that the assertion of Genest (2020) is incorrect and irrelevant. First, there is no claim (in the paper he is referring to) regarding the dependence of the non-arbitrary constant. That paper did not make any claim that would justify the emergence of Genest (2020). Furthermore, these are ”linearized functions” based on convergence as n → ∞. Moreover, though it becomes irrelevant and needless given the above discussion, the author still made incorrect claims and we provide refuting examples.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"433 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127797944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory","authors":"Vasil Penchev","doi":"10.2139/ssrn.3638937","DOIUrl":"https://doi.org/10.2139/ssrn.3638937","url":null,"abstract":"A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I.Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them.Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines.The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV.Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129477402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Propositional Logic Applied to Three Contradictory Definitions of the Zeta Function, and to Conditionally Convergent Series","authors":"Ayal Sharon","doi":"10.2139/ssrn.3450279","DOIUrl":"https://doi.org/10.2139/ssrn.3450279","url":null,"abstract":"The paper discusses the following contradictions: <br><br>(1) Contradictory definitions of the Zeta function. These three definitions of the Zeta function contradict each other: the Dirichlet series, Riemann’s Zeta function, and a third definition that is conditionally convergent throughout the critical strip and divergent throughout half-plane Re(s)≤0. If the Dirichlet series definition and one of the other two definitions are both true, then there is a contradiction regarding convergence/divergence of Zeta in the critical strip. If Riemann’s Zeta function and the third definition are both true, then there is a contradiction regarding convergence/divergence of Zeta in half-plane Re(s)≤0. And if only the Dirichlet series definition is true, than all theories that falsely assume that the other definitions are true are rendered logically unsound.<br><br>(2) The Hankel contour’s contradiction of the preconditions of Cauchy’s integral theorem. The derivation of the Riemann Zeta function uses both of these,but the contradiction between the two renders the Riemann Zeta function invalid.<br><br>(3) Conditionally convergent series. According to the Riemann series theorem, any conditionally convergent series can be rearranged to be divergent. This contradicts the associative and commutative properties of addition. It also means that all conditionally convergent series are paradoxes, and that any argument that uses a conditionally convergent series (e.g. the third definition of the Zeta function, and the definition of the Euler-Mascheroni constant) violates the Law of Non-Contradiction (LNC) in logic, and triggers Ex Contradictione Quodlibet (ECQ), also called the \"Principle of Explosion\".","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126355768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Keynes Had No ‘Hidden Method’ in the A Treatise on Probability (1921): Keynes's Method Is an Explicit Inductive Logic Built on Inexact Measurement and Approximation, Which Was Openly Based on Boole’s Non Linear, Non Additive Approach Using Interval Values Probability","authors":"M. E. Brady","doi":"10.2139/ssrn.3474080","DOIUrl":"https://doi.org/10.2139/ssrn.3474080","url":null,"abstract":"J M Keynes’s method was explicitly introduced and used in the A Treatise on Probability in Parts II, III and V. Keynes’s method is an inductive logic built on the mathematical logic and algebra of George Boole. Boole introduced non linearity and non additivity into his approach using interval valued probability that used lower and upper bounds. Boole’s approach, like Keynes’s, deals explicitly with problems like non comparability, non measurability and incommensurability that can’t be dealt with by additive and linear probability representations. Keynes initially introduced a brief discussion of these problems in chapter III of the A Treatise on Probability and Chapter 4 of the General Theory.<br><br>Keynes called this method inexact measurement and approximation in chapter 15 of the A Treatise on Probability. It is impossible for Keynes to be anti-mathematical,anti-formalist, anti logicist, or a rationalist, given that, building on Boole, he created an inductive logic. Rationalists, by definition, do not accept the concept of induction, which is why they are called deductivists and not inductivists. Keynes, of course, took certain elements of the rationalist perspective and combined these elements (a priori thought, intuition, etc.) with certain elements of empiricism to create an early version of logical empiricism. Similarly, it is impossible for Keynes to have been anti logicist, anti formalist and anti mathematical because his work in Parts II and V of the A Treatise on Probability is logical,formal,and mathematical analysis built directly on the mathematical logic and algebra of George Boole. <br><br>The belief that Keynes was a rationalist and/or anti logicist, anti formalist and anti mathematical is maintained by economists, such as A. Carabelli and R. O’Donnell, who never read or understood Parts II, III and V of the A Treatise on Probability or chapters 16-21 of G Boole’s 1854 The Laws of Thought. It is impossible for Keynes to be a rationalist and proclaim at the end of chapter 26 of the A Treatise on Probability that probability is the guide to life since all rationalists reject inductive logic.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126122472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Markets of Thought","authors":"A. Flis","doi":"10.2139/ssrn.2916524","DOIUrl":"https://doi.org/10.2139/ssrn.2916524","url":null,"abstract":"We need a proper market model free of rigid mathematics. The best we can do is a truthful approximation to reality that coarsely identifies key elements, reflecting freedom of human thought and behavior. We then consider modern problems.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"109 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126014750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical Economics Comes to America: Charles S. Peirce's Engagement with Cournot's Recherches Sur Les Principes Mathematiques De La Théorie Des Richesses","authors":"J. Wible, K. Hoover","doi":"10.2139/SSRN.2302686","DOIUrl":"https://doi.org/10.2139/SSRN.2302686","url":null,"abstract":"Although Cournot’s mathematical economics was generally neglected until the mid-1870s, he was taken up and carefully studied by the Scientific Club of Cambridge, Massachusetts even before his “discovery” by Walras and Jevons. The episode is reconstructed from fragmentary manuscripts of the pragmatist philosopher Charles S. Peirce, a sophisticated mathematician. Peirce provides a subtle interpretation and anticipates Bertrand’s criticisms.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130453423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exclusive Disjunction and the Biconditional: An Even-Odd Relationship","authors":"J. S. Fulda","doi":"10.1080/0025570X.1993.11996097","DOIUrl":"https://doi.org/10.1080/0025570X.1993.11996097","url":null,"abstract":"Two identities are proved by mathematical induction. The identities relate exclusive disjunction and the biconditional (material equivalence).The publisher has made this available /gratis/.","PeriodicalId":257462,"journal":{"name":"PRN: Philosophy of Mathematics & Logic (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125723677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}