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J. Alama:
"Reverse-engineering mathematics";
Talk: Seminar Young India Fellows @ Ashoka University, Sonipat, Indien (invited); 2014-10-27.

If I give you a mathematical problem to solve, you're allowed to use
more or less all available means to arrive at a solution. One
combines background knowledge -- sometimes in clever and subtle ways
-- until a solution is found. From an abstract, logical point of
view, we begin with axioms, definitions, hypotheses, lemmas, and
previously established results, until we arrive at an agreed-upon
stopping point: the theorem.

Proceeding in this way, we discover what are known as sufficient
conditions for the theorem (that is, what it takes to derive the
result). What if we go the other way around, deriving axioms from
theorems? In logical terms, we are asking: Can we find the theorem's
necessary conditions? That is, can we find what is true, given that
the theorem is true?

Such an investigation can be understood as a kind of reverse
engineering. But instead of disassembling a physical artifact to
discover how it was made, we are reverse engineering mathematical
facts. One can imagine a detective who treats a mathematical theorem
as a kind of "crime scene" and tries to find out how it all happened.

The distinction between synthesis (putting together) and analysis
(taking apart) is helpful here. Analysis was understood as a distinct
kind of mathematical investigation during the classical Greek era (if
not before, and probably independently in other parts of the world).
In recent years, though, one can turn to automated theorem provers --
computer programs that can search for mathematical structures and
proofs -- to assist with analysis (in the logical or epistemological
sense of the term we are using here). Such programs are indispensable
assistants in this kind of mathematical research because they are
often able to tackle mathematical problems in ways that, for a human
mathematician, would be strange or counterintuitive.

In this talk, I will discuss the idea of reverse-engineering
mathematics and give a glimpse into the fascinating world of current
research in this area.