Related papers: Hard Proofs and Good Reasons
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
In this paper, we propose standard statistical tools as a solution to commonly highlighted problems in the explainability literature. Indeed, leveraging statistical estimators allows for a proper definition of explanations, enabling…
Is it possible for a large sequence of measurements or observations, which support a hypothesis, to counterintuitively decrease our confidence? Can unanimous support be too good to be true? The assumption of independence is often made in…
The likelihood principle makes strong claims about the nature of statistical evidence but is controversial. Its claims are undermined by the existence of several examples that are assumed to show that it allows, with unity probability,…
"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers…
The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of…
The intuitive reasoning of physicists in conditions of uncertainty is closer to the Bayesian approach than to the frequentist ideas taught at University and which are considered the reference framework for handling statistical problems. The…
The last decade witnessed an ever-increasing stream of successes in Machine Learning (ML). These successes offer clear evidence that ML is bound to become pervasive in a wide range of practical uses, including many that directly affect…
We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric. To represent both aspect of uncertainty, we choose an expected distance function as…
Language technologies that accurately model the dynamics of events must perform commonsense reasoning. Existing work evaluating commonsense reasoning focuses on making inferences about common, everyday situations. To instead investigate the…
Functional languages with strong static type systems have beneficial properties to help ensure program correctness and reliability. Surprisingly, their practical significance in applications is low relative to other languages lacking in…
Beginning with a simple semantics for propositions, based on counting observations, it is shown that probabilistic and fuzzy logic correspond to two different heuristic assumptions regarding the combination of propositions whose evidence…
Data scientists are not mathematicians, but they make heavy use of mathematics in their daily work. While mathematicians can study a mathematical object which is inaccessible to our five senses, data scientists must deal with real-world…
The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable…
Linear logic was conceived in 1987 by Girard and, in contrast to classical logic, restricts the usage of the structural inference rules of weakening and contraction. With this, atoms of the logic are no longer interpreted as truth, but as…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…
A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides…
We believe, in the sense of supporting ideas and considering them correct while dismissing doubts about them. We take sides about ideas and theories as if that was the right thing to do. And yet, from a rational point of view, this type of…
Decisions in organizations are about evaluating alternatives and choosing the one that would best serve organizational goals. To the extent that the evaluation of alternatives could be formulated as a predictive task with appropriate…
A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $\phi$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $\phi$. In…