Purported Proof Research Articles

Towards the exact complexity of realizability for Safety LTL

We study the realizability and strong satisfiability problems for SafetyLTL, a syntactic fragment of Linear Temporal Logic (▪) capturing safe formulas. While it is well-known that realizability for this fragment lies in ▪, the best-known lower bound is ▪-hardness. Surprisingly, closing this gap has proven an elusive task. Previous works have claimed first ▪-completeness [1] and later ▪-completeness [2] for this problem, but both of these proofs turned out to be incorrect.We revisit the problem of the exact classification of the complexity of realizability for ▪ through the lens of seemingly weaker fragments. While we cannot settle the question for ▪, we study a subfragment of it consisting of formulas of the form ▪, where α is a present formula over system variables and ψ contains Next as the only temporal operator. We prove that the realizability problem for this new fragment, which we call ▪, is ▪-complete, and observe that this fragment is equirealizable to existing more expressive fragments, such as the class ▪[3].Furthermore, we revisit the techniques used in the purported proof of ▪-completeness of Arteche and Hermo [1], and observe that, while incorrect in their original claims, their proofs can be modified to classify the complexity of strong satisfiability, a necessary condition for realizability introduced by Kupferman, Sadigh, and Seshia [4]. We prove that, with regards to strong satisfiability, the fragments ▪ and ▪ are in fact equivalent under polynomial-time many-one reductions.

Formalization of Mathematical Proof Practice Through an Argumentation-Based Model

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project.

Riemann’s Hypothesis New Proof

After some papers proving the RC (short for “Riemann’s Conjecture”, also known as the “Riemann’s Hypothesis”, RH), now the author provides a new proof, using the “Spira Criterion” that states “The RH is equivalent to the statement that if s>0.5 and t> 6.5 then |z(1-s)|> |z(s)|”. We use the concept of “transfer function” for control systems. This new proof is so simple that the author wonders why a great mathematician like Riemann did not see it; therefore F. Galetto thinks that somewhere in the purported proof there should be an error.

Shielding Probabilistically Checkable Proofs: Zero-Knowledge PCPs from Leakage Resilience.

Probabilistically Checkable Proofs (PCPs) allows a randomized verifier, with oracle access to a purported proof, to probabilistically verify an input statement of the form “” by querying only a few proof bits. Zero-Knowledge PCPs (ZK-PCPs) enhance standard PCPs to additionally guarantee that the view of any (possibly malicious) verifier querying a bounded number of proof bits can be efficiently simulated up to a small statistical distance. The first ZK-PCP construction of Kilian, Petrank and Tardos (STOC 1997), and following constructions employing similar techniques, necessitate that the honest verifier makes several rounds of queries to the proof. This undesirable property, which is inherent to their technique, translates into increased round complexity in cryptographic applications of ZK-PCPs. We survey two recent ZK-PCP constructions—due to Ishai, Yang and Weiss (TCC 2016-A), and Hazay, Venkitasubramaniam and Weiss (ITC 2021)—in which the honest verifier makes a single round of queries to the proof. Both constructions use entirely different techniques compared to previous ZK-PCP constructions, by showing connections to the seemingly-unrelated notion of leakage resilience. These constructions are incomparable to previous ZK-PCP constructions: while on the one hand the honest verifier only makes a single round of queries to the proof, these ZK-PCPs either obtain a smaller (polynomial) ratio between the query complexity of the honest and malicious verifiers or obtain a weaker ZK guarantee in which the ZK simulator is not necessarily efficient.

Gambling by the Numbers: Frain du Tremblay and Barbeyrac on the Moral Uses of Probability

The invention of probability mathematics in the seventeenth century added a novel element to the long European cultural discussion surrounding gambling. While the mathematicians regularly used gambling games in setting up and solving problems, they were often reluctant to make broader moral claims regarding the practice beyond establishing if a game was fair. Two French-language moralists, however, drew on probability in their explorations of games and the morality of play, although in contradictory ways. Jean Frain du Tremblay’s Conversations morales sur les jeux et les divertissements attempted to use probability mathematics in order to argue that chance did not exist. Overstating probability’s claims, Frain du Tremblay argued that the results of games of chance were predictable and that they thereby demonstrated God’s order. By contrast, Jean Barbeyrac’s Traité du jeu criticized the inability of probability to predict unerringly what would happen next. Based on this perceived flaw, Barbeyrac similarly rejected probability’s claim to establish the morality of games of chance, although his purported proofs show his own inability to master the implications and mechanisms of the new ‘geometry of chance.’ Together the two authors show the struggle for non-mathematicians to understand the implications of probability for early modern concepts of chance.

An Asynchronous Online Forum that Supports Undergraduate Students in Proof Validation

We discuss how an asynchronous online forum can be used to support the learning of proof validation. In this study, one face-to-face class session was replaced with an online session in a first-year university introduction to proof course with 19 volunteer research participants. The online session was centered around an online forum that took the form of gallery that contained snapshots of five different purported proofs. The purported proofs could be viewed collectively by swiping back and forth and students could leave comments and heart/like the individual snapshots. The online forum was augmented by providing students with criteria of a “good” purported proof and having students complete and view the results of an online class poll. Here, we discuss how the online session enhanced learning by sharing students’ decision about proof validation before and after they participated in the online forum and students’ experiences of the online session.

Acceptance criteria for validating mathematical proofs used by school students, university students, and mathematicians in the context of teaching

Although there is no generally accepted list of criteria for the acceptance of proofs in mathematical practice, judging the acceptability of purported proofs is an essential aspect of handling proofs in daily mathematical work. For this reason, school students, university students, and mathematicians need to hold certain acceptance criteria for proofs, which are closely tied to their epistemology, notion of proof, and concept of evidence. However, acceptance criteria have so far received little academic attention and results have often been related to mathematicians’ acceptance criteria in a research context. Still, as the role of proof changes during mathematical education and differs from research, it can be assumed that acceptance criteria differ substantially between educational levels and different communities. Thus, we analyzed and compared acceptance criteria by school students, university students, and mathematicians in the context of teaching. The results obtained reveal substantial cross-sectional differences in the frequency and choice of acceptance criteria, as well as major differences from those acceptance criteria reported by prior studies in research contexts. Structure-oriented criteria, regarding the logical structure of a proof or individual inferences, are most often referred to by all groups. In comparison, meaning-oriented criteria, such as understanding, play a minor role. Finally, social criteria were not mentioned in the context of teaching. From an educational perspective, the results obtained underline university students’ need for support in implementing acceptance criteria and suggest that a more explicit discussion of the functions of proof, their evidential value, and the criteria for their acceptance may be beneficial.

Some highs and lows of hylomorphism: on a paradox about property abstraction

We defend hylomorphism against Maegan Fairchild’s purported proof of its inconsistency. We provide a deduction of a contradiction from SH+, which is the combination of “simple hylomorphism” and an innocuous premise. We show that the deduction, reminiscent of Russell’s Paradox, is proof-theoretically valid in classical higher-order logic and invokes an impredicatively defined property. We provide a proof that SH+ is nevertheless consistent in a free higher-order logic. It is shown that the unrestricted comprehension principle of property abstraction on which the purported proof of inconsistency relies is analogous to naive unrestricted set-theoretic comprehension. We conclude that logic imposes a restriction on property comprehension, a restriction that is satisfied by the ramified theory of types. By extension, our observations constitute defenses of theories that are structurally similar to SH+, such as the theory of singular propositions, against similar purported disproofs.

A Crisis of Identification

Almost a decade after first publishing a purported proof of the abc conjecture, the prospects for Shinichi Mochizuki’s work remain unclear. Indeed, Mochizuki’s ideas are still met with puzzlement by many mathematicians. David Michael Roberts examines recent efforts by Peter Scholze and Jacob Stix to resolve the impasse and ponders the ultimate fate of Mochizuki’s proof.

Clocks and Chronogeometry: Rotating Spacetimes and the Relativistic Null Hypothesis

Abstract Recent work in the physics literature demonstrates that, in particular classes of rotating spacetimes, physical light rays in general do not traverse null geodesics. Having presented this result, we discuss its philosophical significance, both for the clock hypothesis (and, in particular, a recent purported proof thereof for light clocks), and for the operational meaning of the metric field. 1Introduction2Fletcher's Theorem2.1Maudlin on the clock hypothesis in special relativity2.2Fletcher’s result in special relativity2.3Fletcher’s theorem in general relativity3Electromagnetism and the Geometrical-Optical Limit 3.1Maxwell’s equations in curved spacetime3.2The geometrical-optical limit3.3Rotating spacetimes3.4Aren’t Gödel spacetimes unphysical?4The Clock Hypothesis and Chronogeometry 4.1Natural and mathematical observations4.2Clock registry discord4.3Chronogeometry5Conclusion

Comment on “Comments on ‘The Euclidean gravitational action as black hole entropy, singularities and space-time voids’” [J. Math. Phys. 50, 042502 (2009)]–Schwarzschild black hole lives to fight another day

In a comment published several years ago in this journal, Mitra [J. Math. Phys. 50, 042502 (2009)] has claimed to prove that a neutral point particle in general relativity as described by the Schwarzschild metric must have zero gravitational mass, i.e., the mass parameter M0 of a Schwarzschild black hole necessarily vanishes. It is shown that the purported proof is incorrect. The error stems from a basic misunderstanding of the mathematical description of coordinate volume element in a differentiable manifold.

Truthmaking, Evidence Of, and Impossibility Proofs

Beginning with Zagzebski (The Philosophical Quarterly 44:65–73, 1994), some philosophers have argued that there can be no solution to the Gettier counterexamples within the framework of a fallibilist theory of knowledge. If true, this would be devastating, since it is believed on good grounds that infallibilism leads to scepticism. But I argue here that these purported proofs are mistaken and that the truthmaker solution to the Gettier problems is both cogent and fallibilist in nature. To show this I develop the notion of evidence of a state of affairs, a crucial concept in the truthmaker theory. I also argue that a common principle of the transmission of evidence through entailment is false, and the cause of much of the trouble.

On Mathematicians' Different Standards When Evaluating Elementary Proofs

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

On Mathematicians' Proof Skimming: A Reply to Inglis and Alcock

We recently reported a study in which undergraduate students and research mathematicians were asked to read and validate purported proofs (Inglis & Alcock, 2012). In our eye-movement data, we found no evidence of the initial skimming strategy hypothesized by Weber (2008). Weber and Mejía-Ramos (2013) argued that this was due to a flawed analysis of eye-movement data and that a more fine-grained analysis led to the opposite conclusion. Here we demonstrate that this is not the case, and show that their analysis is based on an invalid assumption.

Expert and Novice Approaches to Reading Mathematical Proofs

This article presents a comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians. Participants' eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short purported proofs; (b) compared with mathematicians, undergraduate students spend proportionately more time focusing on “surface features” of arguments, suggesting that they attend less to logical structure; and (c) compared with undergraduates, mathematicians are more inclined to shift their attention back and forth between consecutive lines of purported proofs, suggesting that they devote more effort to inferring implicit warrants. Pedagogical implications of these results are discussed, taking into account students' apparent difficulties with proof validation and the importance of this activity in both schooland university-level mathematics education.

Why College or University Students Hate Proofs in Mathematics?

Problem Statement: A proof is a notoriously difficult mathematical concept for students. Empirical studies have shown that students emerge from proof-oriented courses such as high-school geometry, introduction to proof, complex and abstract algebra unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof and cannot determine whether a purported proof is valid. A proof is a convincing method that demonstrates with generally accepted theorem that some mathematical statement is true and each proofs step must follow from previous proof steps and definition that have already been proved. To motivate students hating proofs and to help mathematics teachers, how a proof can be taught, we investigated in this study the idea of mathematical proofs. Approach: To tackle this issue, the modified Moore method and the researcher method called Z.Mbaïtiga method are introduced follow by two cases studies on proof of triple integral. Next a survey is conducted on fourth year college students on which of the proposed two cases study they understand easily or they like. Results: The result of the survey showed that more than 95% of the responded students pointed out the proof that is done using details explanation of every theorem used in the proof construction, the case study2. Conclusion: From the result of this survey, we had learned that mathematics teachers have to be very careful about the selection of proofs to include when introducing topics and filtering out some details which can obscure important ideas and discourage students.

Why College or University Students Hate Proofs in Mathematics?

Problem Statement: A proof is a notoriously difficult mathematical co ncept for students. Empirical studies have shown that students emerge f rom proof-oriented courses such as high-school geometry, introduction to proof, complex and abstra ct algebra unable to construct anything beyond very trivial proofs. Furthermore, most university s tudents do not know what constitutes a proof and cannot determine whether a purported proof is valid . A proof is a convincing method that demonstrates with generally accepted theorem that some mathematical statement is true and each proofs step must follow from previous proof steps and definition tha t have already been proved. To motivate students hating proofs and to help mathematics teachers, how a proof can be taught, we investigated in this study the idea of mathematical proofs. Approach: To tackle this issue, the modified Moore method and the researcher method called Z. Mbaitiga method are introduced follow by two cases studies on proof of triple integral. Next a survey is conducte d on fourth year college students on which of the proposed two cases study they understand easily or they like. Results: The result of the survey showed that more than 95% of the responded students pointe d out the proof that is done using details explanation of every theorem used in the proof cons truction, the case study2. Conclusion: From the result of this survey, we had learned that mathemat ics teachers have to be very careful about the selection of proofs to include when introducing top ics and filtering out some details which can obscur e important ideas and discourage students.

Modeling Modal Talk in Quantum Mechanics

In this paper, modal and counterfactual logical connectives are defined in an extended framework of branching space-time (Belnap, N. D. (1992). Branching space-time. Synthese92, 385–434). It is shown that a variety of definitions of the counterfactual can be given. The validity of certain modal statements occurring in quantum mechanics depends on the choice of definition. These considerations can be applied to an analysis of Stapp’s premises LOC1 and LOC2 from his purported proof of non-locality (Stapp, H. P. (1997). Nonlocal character of quantum theory. American Journal of Physics65, 300–304). It is shown that while the validity of LOC1 depends on the choice of the definition of the counterfactual, LOC2 is absolutely invalid.

Validations of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument Proves a Theorem?

This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited-perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.

Hardness of Approximate Hypergraph Coloring

We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCPs). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to "satisfy" the verifier on every random string; i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and, in particular, (hyper)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple not-all-equal check on the four bits it reads. This enables us to prove that, for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption.