Related papers: A cute proof that makes $e$ natural
An inductive inference system for proving validity of formulas in the initial algebra $T_{\mathcal{E}}$ of an order-sorted equational theory $\mathcal{E}$ is presented. It has 20 inference rules, but only 9 of them require user interaction;…
In this paper we study the property of normality of a number in base 2. A simple rule that associates a vector to a number is presented and the property of normality is stated for the vector associated to the number. The problem of testing…
In this paper we shall relate computational complexity to the principle of natural selection. We shall do this by giving a philosophical account of complexity versus universality. It seems sustainable to equate universal systems to complex…
It has long been known that scientific output proceeds on an exponential increase, or more properly, a logistic growth curve. The interplay between effort and discovery is clear, and the nature of the functional form has been thought to be…
This work presents a formalization of analogy on numbers that relies on generalized means. It is motivated by recent advances in artificial intelligence and applications of machine learning, where the notion of analogy is used to infer…
In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This…
For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness.…
The paper discusses relationships between aesthetics theory and mathematical models of mind. Mathematical theory describes abilities for concepts, emotions, instincts, imagination, adaptation, learning, cognition, language, approximate…
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…
As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a…
Ramsey Theorem [6] for pairs is intuitionistically but not classically provable: it is equivalent to a subclassical principle [2]. In this note we show that Ramsey may be restated in an intuitionistically provable form, which is informative…
The basic system E of dyadic deontic logic proposed by {\AA}qvist offers a simple solution to contrary-to-duty paradoxes and allows to represent norms with exceptions. We investigate E from a proof-theoretical viewpoint. We propose a…
Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a…
The ever-growing complexity of mathematical proofs makes their manual verification by mathematicians very cognitively demanding. Autoformalization seeks to address this by translating proofs written in natural language into a formal…
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching. In this paper, we will investigate the proof-theoretic strength of this result and related theorems. We show that a…
We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q >…
We outline an unified introduction to the evolution equations of classical and quantum systems intended for a high school students audience. The attempt consists in circumventing the lack of mathematical knowledge with the use of simplified…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
We derive an interesting congruence relation motivated by an Indian Olympiad problem. We give three different proofs of the theorem and mention a few interesting related results.
The Logic of Approximate Entailment (LAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds…