We demonstrate the utility of a new methodological tool, neural-network word embedding models, for large-scale text analysis, revealing how these models produce richer insights into cultural associations and categories than possible with prior methods. Word embeddings represent semantic relations between words as geometric relationships between vectors in a high-dimensional space, operationalizing a relational model of meaning consistent with contemporary theories of identity and culture. We show that dimensions induced by word differences (e.g. man - woman, rich - poor, black - white, liberal - conservative) in these vector spaces closely correspond to dimensions of cultural meaning, and the projection of words onto these dimensions reflects widely shared cultural connotations when compared to surveyed responses and labeled historical data. We pilot a method for testing the stability of these associations, then demonstrate applications of word embeddings for macro-cultural investigation with a longitudinal analysis of the coevolution of gender and class associations in the United States over the 20th century and a comparative analysis of historic distinctions between markers of gender and class in the U.S. and Britain. We argue that the success of these high-dimensional models motivates a move towards "high-dimensional theorizing" of meanings, identities and cultural processes.