Related papers: Constructive Mathematics
Technology is currently ubiquitous and is also part of the educational system at all levels. It started with communication technology systems, and later continued with digital competence. Nowadays, although these previous concepts are still…
Unlike mathematics, in which the notion of truth might be abstract, in physics, the emphasis must be placed on algorithmic procedures for obtaining numerical results subject to the experimental verifiability. For, a physical science is…
This paper documents and reviews the state of the art concerning computational models of construction grammar learning. It brings together prior work on the computational learning of form-meaning pairings, which has so far been studied in…
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…
We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching…
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an…
Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an…
This paper proposes critical pragmatism as a philosophy of sustainable mathematics education to bridge the gap between critical theory and the existing patchwork implementations. Combining existential sustainability as a holistic concept…
A discursive, non-technical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the…
We illustrate the concept of mathematical proof.
The multi-layered nature of societal developments poses a challenge in the teaching and learning of history of mathematics. As an attempt to tackle this challenge we experimented with the use of Knowledge-Maps. The focus of our interest…
It is rare to succeed in getting mathematics into ordinary conversation without meeting all kinds of reservations. In order to raise public awareness of mathematics effectively, it is necessary to modify such attitudes. In this paper, we…
Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight…
We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a…
This essay examines the relationship between artificial intelligence and the historical evolution of modern Mathematics. Rather than viewing AI as an external rupture, we argue that its effectiveness reveals a structural tendency already…
There is no mysterious link between mathematics and physics, because both of them are human inventions designed to study the world.
The aim of this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole.