Related papers: Is mathematics consistent?
We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical set-theoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be…
Math word problem solver requires both precise relation reasoning about quantities in the text and reliable generation for the diverse equation. Current sequence-to-tree or relation extraction methods regard this only from a fixed view,…
We identify points of difference between Invariant Set Theory and standard quantum theory, and show that these lead to noticeable differences in predictions between the two theories. We design a number of experiments to test which of these…
One of the outstanding problems of philosophy of science and mathematics today is whether there is just "one" unique mathematics or the same can be bifurcated into "pure" and "applied" categories. A novel solution for this problem is…
A quotient of a poset $P$ is a partial order obtained on the equivalence classes of an equivalence relation $\theta$ on $P$; $\theta$ is then called a congruence if it satisfies certain conditions, which vary according to different…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…
In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of…
In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are…
In this paper, we propose to study the following maximum ordinal consensus problem: Suppose we are given a metric system (M, X), which contains k metrics M = {\rho_1,..., \rho_k} defined on the same point set X. We aim to find a maximum…
This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the…
Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them…
In this paper we study solution attempts for a problem posed by Ali Enayat: can there be a finitely axiomatized consistent sequential theory that interprets itself plus the (sentential or non-uniform) Tarski biconditionals? We provide a…
We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
Suppose we can choose from a set of linear autonomous systems with bounded process noise, the dynamics of each system are unknown, and we would like to design a stabilizing policy. The underlying question is how to estimate the dynamics of…
Conformal prediction is a statistically rigorous method for quantifying uncertainty in models by having them output sets of predictions, with larger sets indicating more uncertainty. However, prediction sets are not inherently actionable;…
A folk theorem says higher order arithmetic has the proof theoretic strength of set theory with limited power set. This paper makes the theorem precise in terms of several axiom system based on ZF.
Both relativistic mechanics and Newtonian mechanics are based on principles that have ontological implications. We propose a series of formalisms that rigorously define the ontology underlying mechanical theories, in order to clarify and…
This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…
In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA (first-order Peano arithmetic) might…