We define emergence algebraically in the context of discrete dynamical systems modeled as transformation semigroups. Emergence happens when a quotient structure (coarse-grained dynamics) is not a substructure of the original system. We survey small groups to show that algebraic emergence is neither ubiquitous nor rare. Then, we describe connections with hierarchical decompositions and explore some of the philosophical implications of the algebraic constraints.

The game of Go was the last great challenge for artificial intelligence in abstract board games. AlphaGo was the first system to reach supremacy, and subsequent implementations further improved the state of the art. As in chess, the fall of the human world champion did not lead to the end of the game. Now, we have renewed interest in the game due to new questions that emerged in this development. How far are we from perfect play? Can humans catch up? (...) How compressible is Go knowledge? What is the computational complexity of a perfect player? How much energy is really needed to play the game optimally? Here, we investigate these and related questions with respect to the special properties of Go (meaningful draws and extreme combinatorial complexity). Since traditional board games have an important role in human culture, our analysis is relevant in a broader context. What happens in the game world could forecast our relationship with AI entities, their explainability, and usefulness. (shrink)

Computational implementations are special relations between what is computed and what computes it. Though the word “isomorphism” appears in philosophical discussions about the nature of implementations, it is used only metaphorically. Here we discuss computation in the precise language of abstract algebra. The capability of emulating computers is the defining property of computers. Such a chain of emulation is ultimately grounded in an algebraic object, a full transformation semigroup. Mathematically, emulation is defined by structure preserving maps (morphisms) between semigroups. These (...) are systematic, very special relationships, crucial for defining implementation. In contrast, interpretations are general functions with no morphic properties. They can be used to derive semantic content from computations. Hierarchical structure imposed on a computational structure plays a similar semantic role. Beyond bringing precision into the investigation, the algebraic approach also sheds light on the interplay between time and computation. (shrink)

The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to knowledge by (...) making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative. (shrink)