Related papers: Axiomatizing Mathematical Theories: Multiplication
Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit…
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…
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…
We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error value $\bot$ whose main purpose is to always return a value for division. To rings…
We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the…
We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…
The purpose of this essay is to bring out the unique role of Mathematics in providing a base to the diverse sciences which conform to its rigid structure. Of these the physical and economic sciences are so intimately linked with…
The topic of diversity is an interesting subject, both as a purely mathematical concept and also for its applications to important real-life situations. Unfortunately, although the meaning of diversity seems intuitively clear, no precise…
Abstraction logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as…
This paper will develop a single framework for unifying, simplifying and extending our prior results about axiom systems that retain a partial knowledge of their own consistency, via an axiomatic declaration of self-consistency. Its perhaps…
In the paper hereditary classes of ${\rm L}$-structures are studied with language of the form ${{\rm L} = {\rm L_{fin}} \cup {\rm L_\infty}}$, where ${{\rm L_{fin}} = \langle R_1,R_2,\ldots, R_m, = \rangle}$ and ${{\rm L_\infty} = \langle…
Topological statistical theory provides the foundation for a modern mathematical reformulation of classical statistical theory: Structural Statistics emphasizes the structural assumptions that accompany distribution families and the set of…
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the…
Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as "the" axioms of set theory. This suggests that we all carry around with us,…
Signed systems were introduced as a general, syntax-independent framework for paraconsistent reasoning, that is, non-trivialised reasoning from inconsistent information. In this paper, we show how the family of corresponding paraconsistent…
We propose an axiomatic foundation of mathematics based on the finite sequence as the foundational concept, rather than based on logic and set, as in set theory, or based on type as in dependent type theories. Finite sequences lead to a…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and…
Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we…
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation…