Fundamental tensions exist between formal-logical approaches and cognitive approaches to linguistic meaning. The divergence arises from the fundamental differences in nature and form between formal/mathematical structures of natural language meaning and their cognitive representations. While the former are abstract and logical categories of representations, the latter are ultimately embodied and grounded in sensory-motor systems of the brain. This article aims to motivate a unifying theory/formalism of linguistic meaning from a general biologically integrative perspective in the context of current theorizing in linguistics, neurobiology and cognitive sciences on human language meaning within which two divergent approaches for the mathematical and cognitive aspects of linguistic meaning exist. The tensions can be somewhat neutralized if formal-mathematical structures and cognitive representations of natural language meaning can be shown to have representational duality and unity in brain dynamics. This work shows a broad outline of one, if not the only one, path toward this vision. © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.