The study of speech-gesture-thought co-production serves multiple purposes, providing deep insight into many areas of investigation that go from psycholinguistics and cognitive linguistics to discourse analysis, and to the neuroscience of motor action and language. In this piece, building on my previous work (with George Lakoff) on the cognitive science of mathematics (Lakoff & Núñez,1997, 2000; Nunez & Lakoff 1998, 2005), I focus on the study of gesture production in order to address the question of the nature of mathematics and its foundations. I analyze (gestural) convergent evidence of the psychological reality of fundamental conceptual metaphors that we claim make infinitesimal calculus possible (in particular, what concerns limits and continuity). These conceptual metaphors, which are analyzed in detail in Where mathematics comes from (Lakoff & Núñez, 2000), emerge from fundamental mechanisms of everyday human imagination, language, and cognition, and structure the inferential organization of mathematical concepts and ideas. In this chapter, I show how the study of the gesture production of professional mathematicians turns out to be crucial in characterizing (in real-time) fundamental metaphorical contents that, while making the very mathematical ideas possible, are not captured by the standard well-accepted formalisms that are taken to "define" what mathematical concepts really are. (Rafael Núñez)