Related papers: Current Mathematics Appears to Be Inconsistent
We present an overview of how certain computational tools currently interact with mathematical practice, and reflect on the implications for research mathematics in the short to medium term, as the field navigates the emerging age of AI and…
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…
We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.
We explore the rational, formal and non-formal criteria of consistency, non-triviality and redundancy in the mathematical research now a days. We develop a paradigmatic discussion by analysing the different conceptions of those criteria,…
In this paper we investigate the problem of which Lie algebras appear as the derived algebra of a Lie algebra. We present new results that further develop this study and address two questions raised in a paper concerned with the…
Development of several alternative mathematical models for the biological system in question and discrimination between such models using experimental data is the best way to robust conclusions. Models which challenge existing theories are…
The values of $\alpha_s$ determined from low- and high-energy measurements are in irreconcilable contradiction with each other. The current status of the problem is critically reviewed. Consequences of the $\alpha_s$ contradiction, in…
A perplexing problem in understanding physical reality is why the universe seems comprehensible, and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather…
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from…
Research in foundations of physics is stagnant, as claimed by many scientists during the last years. I suggest that one reason might be the reification of language, particularly of mathematics, which favour the search for answers to wrong…
We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the…
We prove some constructive results that on first and maybe even on second glance seem impossible.
Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than…
The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a…
In the paper [Hong-Shi Zong, Wei-Min Sun, Phys. Lett. B 640 (2006) 196], the authors claim that our proof of the inconsistency of the ladder approximation to QCD [Phys. Lett. B 611 (2005) 129] was incorrect. However, their claim is based on…
Predictive models are being increasingly used to support consequential decision making at the individual level in contexts such as pretrial bail and loan approval. As a result, there is increasing social and legal pressure to provide…
The success of present and future cosmological studies is tied to the ability to detect discrepancies in complex data sets within the framework of a cosmological model. Tensions caused by the presence of unknown systematic effects need to…
A working mathematician's summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the August 2025 version, with a completely revised section on vanishing cycles.
Two conjectures recently proposed by one of the authors are disproved
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…