Assumptions, in FotCM discourse, are ideas, value judgements or unquestioned premises which color one's thinking, generally without one's own awareness.
Mental activity needs to be based on some sort of model or context. We could call such a thing a world view, working model or working hypothesis, depending on the application. Thoughts, value judgements or interpretations of reality do not exist without assumption, nor can they exist in a vacuum. In this sense thinking is never free of 'assumptions' which form a sort of implicit context to anything that can be thought.
The term "assumption", however, here implies that the context is not consciously recognized as such and thus is assumed to be valid or applicable to the question at hand a priori, without review, implicitly. This is how belief systems, prejudices and false information can color thinking and lead to results which seem logical and self-consistent but are in fact opposed to external reality.
Hasty or habitual thinking, the opposite of "thinking with a hammer", often involves unconscious assumptions. This is necessary for efficiency in situations requiring rapid response but detrimental otherwise. Assumptions are also linked to anticipation.