Related papers: Euclid After Computer Proof-checking
There are growing uncertainties surrounding the classical model of computation established by G\"odel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those…
The recently initiated approach called computability logic is a formal theory of interactive computation. See a comprehensive online source on the subject at http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a soundness…
The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic and to ludics, which has been recently…
Mechanized theorem proving is becoming the basis of reliable systems programming and rigorous mathematics. Despite decades of progress in proof automation, writing mechanized proofs still requires engineers' expertise and remains labor…
The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows…
"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…
The ubiquitous presence of computer simulations in all kinds of research areas evidence their role as the new driving force for the advancement of science and engineering research. Nothing seems to escape the image of success that computer…
Mathematics has become inescapable in modern, digitized societies: there is hardly any area of life left that isn't affected by it, and we as mathematicians play a central role in this. Our actions affect what others, in particular our…
Eudaemonics is the study of the nature, causes, and conditions of human well-being. According to the ethical theory of eudaemonia, reaping satisfaction and fulfillment from life is not only a desirable end, but a moral responsibility.…
In this paper, we analyze axiomatic issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms changed the algorithmic universe, making it open and allowing…
Euclid is expected to establish new state-of-the-art constraints on extensions beyond the standard LCDM cosmological model by measuring the positions and shapes of billions of galaxies. Specifically, its goal is to shed light on the nature…
We explore the relationship between Brouwer's intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called "The contradictority of elementary geometry". In that paper, he showed that a certain classical…
It is nowadays common to consider that proof must be part of the learning of mathematics from Kindergarten to University1. As it is easy to observe, looking back to the history of mathematical curricula, this has not always been the case…
In this article we present and describe a notion of "logical perfection". We extract the notion of "perfection" from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver…
The Euclid mission aims to measure the positions, shapes, and redshifts of over a billion galaxies to provide unprecedented constraints on the nature of dark matter and dark energy. Achieving this goal requires a continuous reassessment of…
Philosophy of science attempts to describe all parts of the scientific process in a general way in order to facilitate the description, execution and improvements of this process. So far, all proposed philosophies have only covered existing…
The essay consists of three parts. In the first part, it is explained how theory of algorithms and computations evaluates the contemporary situation with computers and global networks. In the second part, it is demonstrated what new…
This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions.…
When working on intelligent tutor systems designed for mathematics education and its specificities, an interesting objective is to provide relevant help to the students by anticipating their next steps. This can only be done by knowing,…