Mathematics in the Digital Age: The Case of Simulation-Based Proofs

Moritz Lucius Sümmermann,Benjamin Rott,Daniel Sommerhoff

doi:10.1007/s40753-020-00125-6

Abstract

Digital transformation has made possible the implementation of environments in which mathematics can be experienced in interplay with the computer. Examples are dynamic geometry environments or interactive computational environments, for example GeoGebra or Jupyter Notebook, respectively. We argue that a new possibility to construct and experience proofs arises alongside this development, as it enables the construction of environments capable of not only showing predefined animations, but actually allowing user interaction with mathematical objects and in this way supporting the construction of proofs. We precisely define such environments and call them “mathematical simulations.” Following a theoretical dissection of possible user interaction with these mathematical simulations, we categorize them in relation to other environments supporting the construction of mathematical proofs along the dimensions of “interactivity” and “formality.” Furthermore, we give an analysis of the functions of proofs that can be satisfied by simulation-based proofs. Finally, we provide examples of simulation-based proofs in Ariadne, a mathematical simulation for topology. The results of the analysis show that simulation-based proofs can in theory yield most functions of traditional symbolic proofs, showing promise for the consideration of simulation-based proofs as an alternative form of proof, as well as their use in this regard in education as well as in research. While a theoretical analysis can provide arguments for the possible functions of proof, they can fulfil their actual use and, in particular, their acceptance is of course subject to the sociomathematical norms of the respective communities and will be decided in the future.

Highlights

Proofs can come in many forms, ranging from a system of logical deductions done in a formalistic symbolic way, as in the Principia Mathematica (Whitehead and Russell 1910), over plain text argumentations supplemented by formulas, which are the standard form in many research papers and textbooks, to so-called “Proofs Without Words” given by an image only (Nelsen 1993)
This deeply rooted view on such technology as being there for exploration and conjecturing, and on proofs as being formal in their representation, led to attempts on doing proofs by integrating such technology and formal proofs in digital environments. This manifests, for example in, the implementation of proof assistants in dynamic geometry environments (Albano et al 2019; Nam 2012; Kovacs 2015; Miyazaki et al 2017; Hanna et al 2019). We argue that these limitations and separation in informal/exploration – formal/proving is mostly due to the nature of geometrical constructions and the role of Dynamic Geometry Environment (DGE) software in proof and technology related activities
To emphasize that these limitations are not inherent to digital environments in general, but to the practice of use of those from geometry, we present several proofs2 done in a different type of software, a Dynamic Topology Environment called ARIADNE (Summermann 2019a)

Summary

Introduction

Proofs can come in many forms, ranging from a system of logical deductions done in a formalistic symbolic way, as in the Principia Mathematica (Whitehead and Russell 1910), over plain text argumentations supplemented by formulas, which are the standard form in many research papers and textbooks, to so-called “Proofs Without Words” given by an image only (Nelsen 1993).Computers have added to this variety by giving rise to computable proofs, i.e. proofs that can be checked by a computer (Voevodsky 2015), or even proofs executed by a program such as the much debated (Tymoczko 1979) proof of the Four-Color Theorem (Appel and Haken 1977), too long to ever be completely reviewed by a human.The rise of computers and with it digital transformation of all aspects of human activities has made environments possible that enable to do mathematics in an interactive way. We argue that these limitations and separation in informal/exploration – formal/proving is mostly due to the nature of geometrical constructions and the role of Dynamic Geometry Environment (DGE) software in proof and technology related activities. These simple examples do not represent proving situations, but suffice to showcase the different possible errors of user and software on the mathematical content.

Results

Conclusion