搜索结果：Aristotles

We provide a writeup of a resolution of Erdős Problem #728; this is the first Erdős problem (a problem proposed by Paul Erdős which has been collected in the Erdős Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0<C_1<C_2$ and $0 < \varepsilon < 1/2$ there exist infinitely many triples $(a,b,n)\in\mathbb N^3$ with $\varepsilon n \le a,b \le (1-\varepsilon)n$ such that \[ a!\,b!\mid n!\,(a+b-n)!\qquad\text{and}\qquad C_1\log n < a+b-n < C_2\log n. \] The argument reduces this to a binomial divisibility $\binom{m+k}{k}\mid\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $ν_p\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ w

Aristotle is generally accepted as the father of logic. The ideas that he raised in his study of logical reasoning carried the development of science over the centuries. Today, in the era of AI, this title of the fatherhood of logic has a renewed significance. Behind it lies his original idea that human reasoning could be studied as a process and that perhaps there exist universal systems of reasoning that underly all human reasoning irrespective of the content of what we are reasoning about. In this article, we look into Aristotle's work on human thought, his work on reasoning itself but also on how it relates to science and human endeavor more generally, from a modern perspective of Artificial Intelligence and ask if this can help enlighten our understanding of AI and Science more generally.

Debate has been widely adopted as a strategy to enhance critical thinking skills in English Language Arts (ELA). One important skill in debate is forming effective argumentation, which requires debaters to select supportive evidence from literature and construct compelling claims. However, the training of this skill largely depends on human coaching, which is labor-intensive and difficult to scale. To better support students in preparing for debates, this study explores the potential of leveraging artificial intelligence to generate effective arguments. Specifically, we prompted GPT-4 to create an evidence card and compared it to those produced by human debaters. The evidence cards outline the arguments students will present and how those arguments will be delivered, including components such as literature-based evidence quotations, summaries of core ideas, verbatim reading scripts, and tags (i.e., titles of the arguments). We compared the quality of the arguments in the evidence cards created by GPT and student debaters using Aristotle's rhetorical principles: ethos (credibility), pathos (emotional appeal), and logos (logical reasoning). Through a systematic qualitative and quanti

Bayesian networks and causal models provide frameworks for handling queries about external interventions and counterfactuals, enabling tasks that go beyond what probability distributions alone can address. While these formalisms are often informally described as capturing causal knowledge, there is a lack of a formal theory characterizing the type of knowledge required to predict the effects of external interventions. This work introduces the theoretical framework of causal systems to clarify Aristotle's distinction between knowledge that and knowledge why within artificial intelligence. By interpreting existing artificial intelligence technologies as causal systems, it investigates the corresponding types of knowledge. Furthermore, it argues that predicting the effects of external interventions is feasible only with knowledge why, providing a more precise understanding of the knowledge necessary for such tasks.

This work discusses the concept of roulette, the generated curves that occur when one curve rolls without slipping along another, tracing the path of a fixed point. The coin paradox and Aristotle's wheel paradox are used as pedagogical motivations to discuss the parametric equations of epicycloids and hypocycloids, providing a geometrical intuition for the mathematical derivations and computational implementation of those curves. Python code is provided to motivate the application of the derived parametric equations, resulting in concrete visualizations and animations.

We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.

We applied computational methods to analyze references across 2,245 philosophical texts, spanning from approximately 550 BCE to 1940 AD, in order to measure patterns in how philosophical ideas have spread over time. Using natural language processing and network analysis, we mapped over 294,970 references between authors, classifying each reference into subdisciplines of philosophy based on its surrounding context. We then constructed a graph, with authors as nodes and textual references as edges, to empirically validate, visualize, and quantify intellectual lineages as they are understood within philosophical scholarship. For instance, we find that Plato and Aristotle alone account for nearly 10% of all references from authors in our dataset, suggesting that their influence may still be underestimated. As another example, we support the view that St. Thomas Aquinas served as a synthesizer between Aristotelian and Christian philosophy by analyzing the network structures of Aquinas, Aristotle, and Christian theologians. Our results are presented through an interactive visualization tool, allowing users to dynamically explore these networks, alongside a mathematical analysis of the ne

Extending the investigation of the presumed primordial comet as part of continuing work on a new model of the Kreutz sungrazer system, I confront a previously derived set of orbital elements with Aristotle's remarks in his Meteorologica to test their compatibility and determine the comet's perihelion time. The two translations of the treatise into English that I am familiar with differ at one point substantially from each other. Unambiguously, the year and season of the comet's appearance was early 372 BC (or -371). From Aristotle's constraint on the comet's setting relative to sunset, I infer that the probable date of perihelion passage was January 20, a date also consistent with the vague remark on frosty weather. On the day that Aristotle claims the comet was not seen, its head may have been hidden behind the Sun's disk or in contact with it. The observation that the `comet receded as far as Orion's belt, where it dissolved' is being satisfied by the tested orbit if the perihelion was reached between January 20 and February 10. Aristotle's third statement, which describes the tail as a streak 60 degrees in length, suggests a plasma feature stretching in space over 0.8 AU. The du

Aristotle vs. Ringelmann was a discussion between two distinct research teams from the ETH Zürich who argued whether the productivity of Open Source software projects scales sublinear or superlinear with regard to its team size. This discussion evolved around two publications, which apparently used similar techniques by sampling projects on GitHub and running regression analyses to answer the question about superlinearity. Despite the similarity in their research methods, one team around Ingo Scholtes reached the conclusion that projects scale sublinear, while the other team around Didier Sornette ascertained a superlinear relationship between team size and productivity. In subsequent publications, the two authors argue that the opposite conclusions may be attributed to differences in project populations, since 81.7% of Sornette's projects have less than 50 contributors. Scholtes, on the other hand, sampled specifically projects with more than 50 contributors. This publication compares the research from both authors by replicating their findings, thus allowing for an evaluation of how much project sampling actually accounted for the differences between Scholtes' and Sornette's resu

In the context of large language models (LLMs), current advanced reasoning methods have made impressive strides in various reasoning tasks. However, when it comes to logical reasoning tasks, major challenges remain in both efficacy and efficiency. This is rooted in the fact that these systems fail to fully leverage the inherent structure of logical tasks throughout the reasoning processes such as decomposition, search, and resolution. To address this, we propose a logic-complete reasoning framework, Aristotle, with three key components: Logical Decomposer, Logical Search Router, and Logical Resolver. In our framework, symbolic expressions and logical rules are comprehensively integrated into the entire reasoning process, significantly alleviating the bottlenecks of logical reasoning, i.e., reducing sub-task complexity, minimizing search errors, and resolving logical contradictions. The experimental results on several datasets demonstrate that Aristotle consistently outperforms state-of-the-art reasoning frameworks in both accuracy and efficiency, particularly excelling in complex logical reasoning scenarios. We will open-source all our code at https://llm-symbol.github.io/Aristotle

This paper has two goals. The first goal is to show how an extension of second-order logic is a natural framework to formalize portions of Aristotle's \emph{Topics} and to bring to the foreground the logical, linguistic and philosophical interest of this work, showing in particular that we are in the presence of a richly intensional and modal conception of logic. Aristotelian logic and its related traditions in antiquity are often held to have been equivalent to monadic predicate logic and as such inadequate to formalize mathematics as well as scientific and philosophical discourse in general. The second goal of this paper is to argue that on the contrary the logical theories of Aristotle (which we argue correspond to a variant of natural deduction) and ancient authors such as Galen and Boethius were in fact quite sufficient to account for the logically complex expressions and reasoning involving multiple generality fundamental to the aforementioned disciplines.

Regarding the famous Sea Battle Argument, which Aristotle presents in De Interpretatione 9, there has never been a general agreement not only about its correctness but also, and mainly, about what the argument really is. According to the most natural reading of the chapter, the argument appeals to a temporal concept of truth and concludes that not every statement is always either true or false. However, many of Aristotle's followers and commentators have not adopted this reading. I believe that it has faced so much resistance for reasons of hermeneutic charity denying the law of universal bivalence seems to be overly disruptive to logical orthodoxy the kind of logical orthodoxy represented by what we now call classical propositional logic, much of which Aristotle clearly supports in many texts. I intend to show that the logical-semantic theses that the traditional reading finds in De Interpretatione 9 are much more conservative than they may seem to be at first glance. First, I will show that they complement, and do not contradict in any way, the orthodox definitions of the concepts of truth and statement that Aristotle advances in other texts. Second, by resorting in an anachronis

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework -- named AI-Aristotle -- combines eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model, and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. While the current work focuses on the performance of AI-Aristotle based on synthetic data, it can equally handle noisy experimental data and can even be used for black-box identification in just a few minutes on a laptop. More broadly, our wor

When considering the opening part of 1800 short stories, we find that the first dozen paragraphs of the average narrative follow an action principle as defined in arXiv:2309.06600. When the order of the paragraphs is shuffled, the average no longer exhibits this property. The findings show that there is a preferential direction we take in semantic space when starting a story, possibly related to a common Western storytelling tradition as implied by Aristotle in Poetics.

According to Aristotle "time is the number of change with respect to the before and after". That's certainly a vague concept, but at the same time it's both simple and satisfying from a philosophical point of view: things do not change along time, but they do change and the measurement of such changes is what we call time. This deprives time of any attribute of substantiality, meanwhile depriving it of all problems in defining the properties of time as a substance. With the rise of Classical Mechanics, Aristotle's view is abandoned and Newton's concept of "true" and absolute time imposes itself; time flows independently on changes of any kind. Relativity will then radically modify our concept of time, but won't actually modify the fundamental idea: things keep changing along time -- changes do not make time. This work will argue Aristotle's thesis, showing how such an approach automatically leads to the principles of Special Relativity. An interesting consequence and, at least virtually, measurable will also be highlighted: the fact that synchronizing two clocks with a precision greater than a certain scale is impossible, estimating such scale around $10^{-22}$s.

I discuss Ren{é} Thom's approach to philosophy based on his mathematical background. At the same time, I will highlight his connection with Aristotle, his criticism of the modern view of science as a predictive process, his ideas on mathematical education, his position with respect to the French school of mathematics that was dominent in his time and his relationship with the philosophical community. I will also touch upon the connections between Thom's ideas and those of Leibniz, Riemann, Freud and others. The last version of this paper will appear as a chapter in the book Handbook of the History and Philosophy of Mathematical Practice (ed. Bharath Sriraman), Springer.

Aristotelian logic and its related traditions in antiquity are often held to have been equivalent to monadic predicate logic and as such inadequate to formalize mathematics as well as scientific and philosophical discourse in general. In this paper we argue that on the contrary the logical theories of Aristotle and ancient authors such as Galen and Boethius were in fact quite sufficient to account for the logically complex expressions and reasoning involving multiple generality fundamental to the aforementioned disciplines.

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