By: Tony Kulesa
Recently passed away last month is the storied mathematician Alexander Grothendieck, who, in between his early quest to personally assassinate Hitler and his later (asc)descension into anarchist hermitdom, amassed a Christ-like following of mathematicians to rebuild entire fields of mathematics into a single theory of sublime generality, starting with the definition of a point.
Steven Landsburg gives a fitting layman’s description of Grothendieck’s profoundly powerful approach to mathematics:
“Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.
Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.
It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.
In other words, [Grothendieck’s] philosophy was this: If a phenomenon seems hard to explain, it’s because you haven’t fully understood how general it is. Once you figure out how general it is, the explanation will stare you in the face.”
To me, this analogy begs the question, what would have happened had the Isaac Newton of the apple-falling legend been one of today’s molecular biologists? Rather than writing down a new theory of gravity, would he have been looking for the genetic mutations that lead to the apple’s fall from its tree branch (armed with a tree farm, apple collecting robots, and a MiSeq, all on the NIH’s tab)?
What, as Landsburg puts it, is staring us in the face if we could just figure out how to ask the right question? In simple concepts like the definition of a point, Grothendieck saw the potential to build entirely new ways of thinking about geometry that turned long outstanding problems into obvious truths. What would a person like Grothendieck make of the basic axioms from which we build biology? Perhaps there are alternative ways of thinking about the basic building blocks, a gene, an enzyme, an organism, a species, that freshly reinterpret their functions in a way that would reinvent our field.
There are likely many reasons why Grothendieck was able to do what he did for mathematics, but many that knew him suggest the most significant to be his courage. Not only is it tremendously risky to spend years trying to reinvent existing fields, but its also lonely, and surely he would not have achieved anywhere near as much without the support of his peers. There are some biologists that come to mind, indeed even in our own department, who have voiced or even dedicated bodies of work to theoretically “out-there” ideas, and are widely dismissed by their colleagues. In light of Grothendieck’s triumph in mathematics, maybe we shouldn’t so quickly dismiss wild theories, but embrace and encourage them.
Steven Landsburg’s piece on Grothendieck: