Research Interests

Text Box:

Research Interests

Text Box: Mathematical reasoning fascinates me. However, my research interests have narrowed over the years. Initially, I had focused both on mathematical statements and on statements about mathematics. Proofs and solutions, that is, had occupied my attention as much as their implications on logic and reasoning. Such implications remain a diversion because of my obsession with mathematical hegemony, the extent to which the potency of mathematical reasoning extends beyond the technical. The boundary beyond which mathematical reasoning loses its effectiveness has an allure that I cannot deny. Nonetheless, lately I have been limiting my researches to mathematics-proper. Given a particular mathematical analysis—proof or solution—I look closely at the mathematician’s choice of analytic strategies. My approach is as much a mathematical undertaking as it is meta-mathematical. For, an analytic device does not present as a choice unless there are alternative ways of establishing the proof or solution. As such, it is imperative that I have a fundamental understanding of the mathematics at hand—a basic understanding will not suffice. With the mathematical landscape carefully surveyed, I endeavor to discover the mathematician’s motive for choosing specific analytic strategies over others. Is it, for example, simply a matter of expedience? Is the mathematician unable to proceed in any other way? On the one hand, expedience alone is an unlikely motive. Expedience cannot occur in vacuo any more than Ockham’s Razor can deliver residuum from naught. But, qua communication, a mathematical analysis enables the mathematician to mean more than s/he says. Maybe the mathematician chooses analytic devices that promote the primacy of constructivism (cf. LEJ Brouwer and Errett Bishop) over formalism or logicism or some-other-ism. Or, perhaps the mathematician seeks to subordinate p-adic analysis to topology. On the other hand, the vast majority of results in the refereed journals are (dare I say) unremarkable enough that an author never strays so far from familiar territory—if at its fringes—that s/he cannot discern more than one path to take. The kinds of questions I examine include the likes of Why has the mathematician proven a given number-theoretic result in terms of algebraic topology? Why has the mathematician chosen finite Einstellung over tertium non datur or reductio ad absurdum? Why has the mathematician argued here by counterexample and there a fortiori? I have found abundant vistas at the boundaries of sub-disciplines such as topology/analysis or algebra/number theory or the like. Of course, I cannot see beyond the horizon delineated by my own mathematical aptitude and proclivities. Fortunately, the horizon continually expands, revealing more than enough for me to explore.