Related papers: Why Do We Believe Theorems?
In this paper we discuss contrastive explanations for formal argumentation - the question why a certain argument (the fact) can be accepted, whilst another argument (the foil) cannot be accepted under various extension-based semantics. The…
Belief functions are a powerful and popular framework for the mathematical characterisation of uncertainty, in particular in situations in which lack of data renders learning a probability distribution for the problem impractical. The first…
A key assumption fuelling optimism about the progress of large language models (LLMs) in accurately and comprehensively modelling the world is that the truth is systematic: true statements about the world form a whole that is not just…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
Argumentation is the process of constructing arguments about propositions, and the assignment of statements of confidence to those propositions based on the nature and relative strength of their supporting arguments. The process is modelled…
Mathematical proof aims to deliver confident conclusions, but a very similar process of deduction can be used to make uncertain estimates that are open to revision. A key ingredient in such reasoning is the use of a "default" estimate of…
In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions,…
Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be…
Large Language Models (LLMs) tend to be unreliable in the factuality of their answers. To address this problem, NLP researchers have proposed a range of techniques to estimate LLM's confidence over facts. However, due to the lack of a…
The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which…
Large Language Models (LLMs) have demonstrated impressive capabilities in structured reasoning and symbolic tasks, with coding emerging as a particularly successful application. This progress has naturally motivated efforts to extend these…
In many real-world situations, there is often not enough information to know that a certain strategy will succeed in achieving the goal, but there is a good reason to believe that it will. The paper introduces the term ``doxastic'' for such…
Humans engage in informal debates on a daily basis. By expressing their opinions and ideas in an argumentative fashion, they are able to gain a deeper understanding of a given problem and in some cases, find the best possible course of…
The problem of belief tracking in the presence of stochastic actions and observations is pervasive and yet computationally intractable. In this work we show however that probabilistic beliefs can be maintained in factored form exactly and…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
Inference is the process of using facts we know to learn about facts we do not know. A theory of inference gives assumptions necessary to get from the former to the latter, along with a definition for and summary of the resulting…
It is a widespread belief that results like G\"odel's incompleteness theorems or the intrinsic randomness of quantum mechanics represent fundamental limitations to humanity's strive for scientific knowledge. As the argument goes, there are…
As scientists we like to think that modern societies and their members base their views, opinions and behaviour on scientific facts. This is not necessarily the case, even though we are all (over-) exposed to information flow through…
This is an explanation and defense of "mathematical conceptualism" for a general mathematical and philosophical audience. I make a case that it is cogent, rigorous, attractive, and better suited to ordinary mathematical practice than all…
Formally verifying properties of software code has been a highly desirable task, especially with the emergence of LLM-generated code. In the same vein, they provide an interesting avenue for the exploration of formal verification and…