The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

the modernity of the meraner lehrplan for teaching geometry today in grades 10 11 exploiting the power of dynamic geometry systems

In 1905, at the meeting of the Deutschen Mathematiker-Vereinigung in Meran, the "Meraner Lehrplan," a mathematics syllabus, was proposed. This document, which contains many of Felix Klein's ideas on teaching geometry in school, proposed approaching geometry via intuitive geometry, which is the ability to see in space, in order to provide elements for both interpreting the real world and developing logical skills (see Treutlein in Der geometrische anschauungsunterricht als unterstufe eines zweistufigen geometrischen unterrichtes an unseren hoheren schulen. Teubner, Leipzig/Berlin 1911). Klein's ideas still hold today: An intuitive approach to geometry can be facilitated using information technology. Some activities related to a space geometry approach based on the analogy among figures and on the use of a dynamic geometry system will be presented in this chapter

Semantic closure demonstrated by the evolution of a universal constructor architecture in an artificial chemistry

We present a novel stringmol-based artificial chemistry system modelled on the universal constructor architecture (UCA) first explored by von Neumann. In a UCA, machines interact with an abstract description of themselves to replicate by copying the abstract description and constructing the machines that the abstract description encodes. DNA-based replication follows this architecture, with DNA being the abstract description, the polymerase being the copier, and the ribosome being the principal machine in expressing what is encoded on the DNA. This architecture is semantically closed as the machine that defines what the abstract description means is itself encoded on that abstract description. We present a series of experiments with the stringmol UCA that show the evolution of the meaning of genomic material, allowing the concept of semantic closure and transitions between semantically closed states to be elucidated in the light of concrete examples. We present results where, for the first time in an in silico system, simultaneous evolution of the genomic material, copier and constructor of a UCA, giving rise to viable offspring

Revisiting mu-puzzle. A case study in finite countermodels verification

© 2018, Springer Nature Switzerland AG. In this paper we consider well-known MU puzzle from Goedel, Escher, Bach: An Eternal Golden Braid book by D. Hofstadter, as an infinite state safety verification problem for string rewriting systems. We demonstrate fully automated solution using finite countermodels method (FCM). We highlight advantages of FCM method and compare it with alternatives methods using regular invariants