Related papers: Unity and Disunity in Mathematics
A universal Turing machine is a powerful concept - a single device can compute any function that is computable. A universal spin model, similarly, is a class of physical systems whose low energy behavior simulates that of any spin system.…
This article presents an overview of the idea that "information compression by multiple alignment, unification and search" (ICMAUS) may serve as a unifying principle in computing (including mathematics and logic) and in such aspects of…
Statistics experiences a storm around the perceived misuse and possible abuse of its methods in the context of the so-called reproducibility crisis. The methods and styles of quantification practiced in mathematical modelling rarely make it…
The nature of the existence, revealed through Human cognitive system, has been evolving since the development of the languages. Part of such revelations were the geometrical forms and the numbers, whose beauty and order, wondrous and…
This is a survey of the diversity of problems in additive number theory. Equity requires the consideration of less currently popular problems, and suggests their inclusion in the additive canon. Of particular interest are problems about the…
The information technology explosion has dramatically increased the application of new mathematical ideas and has led to an increasing use of mathematics across a wide range of fields that have been traditionally labeled "pure" or…
The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a…
Objects are a centerpiece of the mathematical realm and our interaction with and reasoning about it, just as they are of the physical one (if not more). And humans' mathematical reasoning must ultimately be grounded in our general…
The main aim of this paper is to make a remark about the relation between (i) dualities between theories, as `duality' is understood in physics and (ii) equivalence of theories, as `equivalence' is understood in logic and philosophy. The…
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
We contemplate the notion of ambiguity in mathematical discourse. We consider a general method of resolving ambiguity and semantic options for sustaining a resolution. The general discussion is applied to the case of `fraction' which is…
A few remarks on how mathematics quests for freedom.
A new mathematics, the constructive one, characterizes a singular limit as undecidable. Hence, a singular limit between two theories actually represents a difference between two different kinds of mathematics. This particular situation…
The quantities of dimension one - known as the dimensionless quantities - are widely used in physics. However, the debate about some dimensionless units is still open. The paper brings new interrelated arguments that lead to the conclusion…
What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the…
Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note we consider the analytic properties of diversities, in particular the generalizations of uniform…
Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice…
Programming with logic for sophisticated applications must deal with recursion and negation, which together have created significant challenges in logic, leading to many different, conflicting semantics of rules. This paper describes a…
We reveal a contradiction in measure-theoretic probability. The contradiction is an "equation" $1/2 = 0$ with its two sides representing probabilities. Unlike known paradoxes in mathematics, the revealed contradiction cannot be explained…
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…