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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.

Although the formalizations of computability provided in the 1930s have proven to be equivalent, two different accounts of computability may be distinguished regarding computability as an epistemic concept. While computability, according to the epistemic account, should be based on epistemic constraints related to the capacities of human computers, the non-epistemic account considers computability as based on manipulations of symbols that require no human capacities other than the capacity of manipulating symbols according to a set of rules. In this paper, I shall evaluate, both from a logical and physical point of view, whether computability should be regarded as an epistemic concept, i.e., whether epistemic constraints should be added on (physical) computability for considering functions as (physically) computable. Specifically, I shall argue that the introduction of epistemic constraints have deep implications for the set of computable functions, for the logical and physical Church-Turing thesis—cornerstones of logical and physical computability respectively—might turn out to be false according to which epistemic constraints are accepted.

The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators.