Abstract: Number concepts are the basic building blocks of numerical thought. For example, the thought that there are two Elmos on the table, or that two is greater than one, or that Mary is contestant number two, all plausibly feature a certain number concept as a constituent, namely TWO. But what are number concepts, exactly? For example, what is the content of TWO, and how do we come to possess that concept? There is a very large empirical literature dedicated to answering these and related questions. What's more, some of that research is of immediate relevance to certain influential programs within the philosophy of mathematics. Conversely, the empirical literature is rife with distinctively philosophical assumptions regarding, for example, the metaphysics of natrual numbers and the relationship between the contents of number concepts and foundational characterizations of natural numbers. Despite these interesting points of contact, there is currently no comprehensive survey of the number cognition literature which critically assess these various philosophical commitments. The primary purpose of this monograph is to provide such a survey, one which is hopefully helpful to both philosophers and number cognition researchers alike. We develop a rough taxonomy of the most influential views within number concept research (what we call "the Mainstream"), summarize and critically assess those views, and isolate and critically evaluate certain commonplace philosophical assumptions underlying them. We also sketch an important, heretofore unrecognized problem resulting from attempting to bring into alignment two sciences which are especially well suited to specify the contents of number concepts: developmental psychology and linguistic semantics. It would appear that these two sciences imply inconsistent theses regarding when certain number concepts are acquired, thus leading to what we call the Developmental Puzzle.
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