Related papers: Euclid After Computer Proof-checking
In this short essay, we show how computer experiments, and especially visualization, allowed for the investigation and discovery of phenomena which would have passed unnoticed. We shall also highlight the importance of interactivity between…
The Euclid satellite will provide data on the clustering of galaxies and on the distortion of their measured shapes, which can be used to constrain and test the cosmological model. However, the increase in precision places strong…
This paper reports on empirical work aimed at comparing evidential reasoning techniques. While there is prima facie evidence for some conclusions, this i6 work in progress; the present focus is methodology, with the goal that subsequent…
We introduce Voevodsky's univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-L\"of type theory. Agda allows us to write mathematical definitions,…
Computation, the use of a computer to solve, simulate, or visualize a physical problem, has revolutionized how physics research is done. Computation is used widely to model systems, to simulate experiments, and to analyze data. Yet, in most…
Part 1 : For more than two millennia, ever since Euclid's geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time,…
We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We…
We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
We introduce a logic for reasoning about evidence that essentially views evidence as a function from prior beliefs (before making an observation) to posterior beliefs (after making the observation). We provide a sound and complete…
This article examines two approaches to verification, one based on using a logic for expressing properties of a system, and one based on showing the system equivalent to a simpler system that obviously has whatever property is of interest.…
Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and…
We discuss models of computing that are beyond classical. The primary motivation is to unearth the cause of nonclassical advantages in computation. Completeness results from computational complexity theory lead to the identification of very…
Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn't produce a manuscript with their…
The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for meta-theoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
Valid empirical studies build confidence in scientific findings. Fortunately, it is now common for software engineering researchers to consider threats to validity when designing their studies and to discuss them as part of their…
Mathematics has many useful properties for developing of complex software systems. One is that it can exactly describe a physical situation of the object or outcome of an action. Mathematics support abstraction and this is an excellent…
Are we smarter now than Socrates was in his time? Society as a whole certainly enjoys a higher degree of education, but humans as a species probably don't get intrinsically smarter with time. Our knowledge base, however, continues to grow…
We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code…
Procedural computer languages have long been used in many aspects of mathematics pedagogy. In this work, we examine the use of Prolog, a declarative language for the same purpose. We find the facts+rules aspect of Prolog to be a novel…