Related papers: Deep Disagreement in Mathematics
We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…
Understanding how humans revise their beliefs in light of new information is crucial for developing AI systems which can effectively model, and thus align with, human reasoning. While theoretical belief revision frameworks rely on a set of…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…
Aligning large language models with expert judgment is especially difficult in subjective evaluation tasks, where experts may disagree, rely on tacit criteria, and change their judgments over time. In this paper, we study expert alignment…
Deep learning is very effective at jointly learning feature representations and classification models, especially when dealing with high dimensional input patterns. Probabilistic logic reasoning, on the other hand, is capable to take…
Mathematical reasoning is essential for problem-solving in education, science, and industry, serving as a crucial benchmark for evaluating artificial intelligence systems. As Large Language Models (LLMs) improve their reasoning…
Discrete mathematics is the foundation of computer science. It focuses on concepts and reasoning methods that are studied using math notations. It has long been argued that discrete math is better taught with programming, which takes…
Large language models (LLMs) have demonstrated impressive performance on reasoning tasks, including mathematical reasoning. However, the current evaluation mostly focuses on carefully constructed benchmarks and neglects the consideration of…
In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal')…
A series of monte carlo studies were performed to compare the behavior of some alternative procedures for reasoning under uncertainty. The behavior of several Bayesian, linear model and default reasoning procedures were examined in the…
Within computational neuroscience, informal interactions with modelers often reveal wildly divergent goals. In this opinion piece, we explicitly address the diversity of goals that motivate and ultimately influence modeling efforts. We…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
Doubts are raised concerning the usual interpretation of the alleged failure, by quantum mechanics, of the distributive law of classical logic. The difficulty raised by incompatible sets of observables is overcome within an epistemic…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to…
This chapter presents a brief review of complexity research in mathematics education. We argue how research on complexity, as it pertains to mathematics education, can be viewed as an epistemological discourse, an historical discourse, a…
The Weihrauch degrees are a tool to gauge the computational difficulty of mathematical problems. Often, what makes these problems hard is their discontinuity. We look at discontinuity in its purest form, that is, at otherwise constant…
Current conceptions of expert problem solving depict physical/conceptual reasoning and formal mathematical reasoning as separate steps: a good problem solver first translates a physical Current conceptions of quantitative problem-solving…
Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in…