Recognizing the limitations of element-wise set-theoretic constructions in accommodating contextual effects in perception, neuroscientists are trying to move beyond elementism. Mathematicians also recognized the shortcomings of 0-dimensional elementism in modelling geometry, which led to the development of higher dimensional algebra. Here we show that higher dimensional algebra meets some of the demands of cognitive neuroscience data and hence may become indispensable in implementing the paradigm shift from acontextual models such as the feature list model of percept and concept to empirically valid models. Drawing upon the experience of higher dimensional algebra, we distil higher dimensional models of percept, concept, emotion, memory, and neural information from experimental data. These studies brought to light a number of parallels between the methods of mathematics and the workings of the brain, which led us to put forward higher dimensional algebra that provides a mathematical model of mathematical practice as a model of the brain.