Related papers: Accepted proofs: Objective truth, or culturally ro…

For obtaining causal inferences that are objective, and therefore have the best chance of revealing scientific truths, carefully designed and executed randomized experiments are generally considered to be the gold standard. Observational…

The accuracies for many pattern recognition tasks have increased rapidly year by year, achieving or even outperforming human performance. From the perspective of accuracy, pattern recognition seems to be a nearly-solved problem. However,…

Formal methods have been largely thought of in the context of safety-critical systems, where they have achieved major acceptance. Tens of millions of people trust their lives every day to such systems, based on formal proofs rather than…

With the relentless rise of computer power, there is a widespread expectation that computers can solve the most pressing problems of science, and even more besides. We explore the limits of computational modelling and conclude that, in the…

For a newcomer, paraconsistent logics can be difficult to grasp. Even experts in logic can find the concept of paraconsistency to be suspicious or misguided, if not actually wrong. The problem is that although they usually have much in…

Improvements in large language models have led to increasing optimism that they can serve as reliable evaluators of natural language generation outputs. In this paper, we challenge this optimism by thoroughly re-evaluating five…

In logic there is a clear concept of what constitutes a proof and what not. A proof is essentially defined as a finite sequence of formulae which are either axioms or derived by proof rules from formulae earlier in the sequence.…

Human knowledge is made up of the conceptual structures of many communities of interest. In order to establish coherence in human knowledge representation, it is important to enable communication between the conceptual structures of…

In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce…

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie "non-computational" proofs from probability theory,…

Formally verified compilers and formally verified static analyzers are a solution to the problem that certain industries face when they have to demonstrate to authorities that the object code they run truly corresponds to its source code…

This is a brief review of the experimental and theoretical quantum computing. The hopes for eventually building a useful quantum computer rely entirely on the so-called "threshold theorem". In turn, this theorem is based on a number of…

Algorithmic decision-making is replacing idiosyncratic human judgment in domains such as hiring, lending, and criminal justice. This shift promises increased consistency, but many scholars worry that it can go too far. They warn of the…

Good problems grab us. They invite us to find patterns, make conjectures, and prove-or perhaps disprove-a conjecture. When I first taught, I saw my work as tantalizing students with structures just beyond their reach, so that I could elicit…

This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…

The ability of humans to create and disseminate culture is often credited as the single most important factor of our success as a species. In this Perspective, we explore the notion of machine culture, culture mediated or generated by…

In recent months, large language models (LLMs) have made significant progress in mathematical proof generation, but further advancement is hindered by the lack of a large-scale, high-quality dataset of human-evaluated proofs. While…

Real data are rarely pure. Hence the past half-century has seen great interest in robust estimation algorithms that perform well even when part of the data is corrupt. However, their vast majority approach optimal accuracy only when given a…

Combining a standard proof search method, such as resolution or tableaux, and rewriting is a powerful way to cut off search space in automated theorem proving, but proving the completeness of such combined methods may be challenging. It may…

The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an…