What is the role of language in cognition? Do we think with words, or do we use words to communicate made-up decisions? The paper briefly reviews ideas in this area since 1950s. Then we discuss mechanisms of cognition, recent neuroscience experiments, and corresponding mathematical models. These models are interpreted in terms of a biological drive for cognition. Based on the Grossberg-Levine theory of drives and emotions, we identify specific emotions associated with the need for cognition. We demonstrate an engineering application of the developed technique, which significantly improves detection of patterns in noise over the previous state-of-the-art. The developed mathematical models are extended toward language. Then we consider possible brain-mind mechanisms of interaction between language and cognition. A mathematical analysis imposes restrictions on possible mechanisms. The proposed model resolves some long-standing language-cognition issues: how the mind learns correct associations between words and objects among an astronomical number of possible associations; why kids can talk about almost everything, but cannot act like adults, what exactly are the brain-mind differences; why animals do not talk and think like people. Recent brain imaging experiments indicate support for the proposed model. We discuss future theoretical and experimental research.