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Mathematics is becoming increasingly specialized, divided into a vast and growing number of subfields. While this division of cognitive labor has important benefits, it also has a significant drawback: it can sometimes impede mathematical progress by making it difficult for mathematicians to make connections to subfields other than their own. Mathematicians can address this by making their own subfield more accessible to researchers working in other areas. One way they can do this is by engaging in exposition, as I illustrate with the User's Guide Project in algebraic topology. However, the current reward structure of mathematics does not appropriately credit mathematicians who make their subfields more accessible via exposition. I thus conclude that the reward structure of mathematics should be changed to more highly value such work, with changes being adopted at the level of departments, professional societies and funding agencies. [ABSTRACT FROM AUTHOR]

The problem of 'applicability' of mathematics to modern physical sciences has been labeled as an 'unreasonably effective' and unexplainable 'miracle' by prominent physicists such as Eugene Wigner and Paul Dirac. Philosophers of science from contending traditions have also contributed to the debates at large. While some have tried to trivialize the problem, others, in particular those with a phenomenological orientation, have attempted to eliminate this riddle, claiming that the two domains are 'identical'. This paper, while challenging some of these interpretations, proposes a solution from a dialectical materialist position. Drawing on Evald Ilyenkov's concept of the 'ideal' as the objectivized form of human activity in social nature, it is suggested that the two realms are necessarily interconnected as schemata of human activity which aim for a highly quantified conceptualization of reality. [ABSTRACT FROM AUTHOR]

Despite societal expectations that graduate students in science, technology, engineering, and mathematics (STEM) disciplines integrate broader impacts into their professional activities, few of them actually receive any formal training in this area. This paper describes a graduate seminar entitled “Broader Impacts of Natural Sciences on Society” taught at the University of Florida in 2006 and 2008. In addition to course goals, recruitment, expectations, format, content, and outcomes, this paper describes challenges and recommendations for others who might want to teach a similar course. [ABSTRACT FROM AUTHOR]