From the earliest days, mathematics has typically received a hallowed treatment from those who have studied it. Plato, who studied a special set of five three-dimensional shapes now known as the Platonic solids, wrote that one of these, the dodecahedron, was used by the gods “for arranging the constellations on the whole of heaven.” This sort of mythos has continued through the years — a modern example comes from the prolific mathematician Paul Erdős, who would refer to “The Book,” a collection known only to God that contained the most elegant proofs of mathematical theorems.
As this column has tried to evince over the years, math does seem to have an inherent beauty, with many mathematical ideas either possessing an elegant structure or an unreasonable knack for describing the complex dynamics of our lived experience. Hence, I think this has led to the relatively common conception that math exists, and new contributions to a mathematical field are discovered by researchers. This notion of discovery contrasts with views more often associated with engineering, where practitioners invent technologies to address a given need.
I’m not the first person to bring up this question of whether we should always think of new math as discovered or invented—for some exciting discussions, one can listen to Grant Sanderson or Neil deGrasse Tyson—but I do have some general thoughts I’d like to share on this debate.
Mainly, I think having a flexible notion between discovery and invention is the best way to view the work of mathematical research. On the one hand, mathematicians cannot discover a new equation or theorem that fails to hold in certain cases, much like a scientist cannot discover some new law unsupported by existing knowledge or the data found in an experiment. The existing mathematical theory guides researchers to results that are indeed worthy of discovery; past work can be used to show whether the new conclusion is true.
But all this past work is ultimately in the realm of people, who have a key role to play in questioning new mathematical discoveries and judging their worth. This process is key to the advancement of any field of study, but comes with its flaws. Humans are biased in ways we often do not realize; research can be presented in obscure ways that lead to an under—or over—appreciation of the work in question, and there’s a limited amount of resources to fully support every proposed research project.
In this more uncertain reality, mathematics can take on the flavor of inventors pitching their inventions — “I am presenting this new idea I developed: here’s how and why it works.” I don’t think this is a bad thing at all — I find research conferences and workshops, where these exact pitches are made, to provide a wealth of incredibly exciting and fulfilling moments. But I feel this concept is underacknowledged, in that we are tempted to treat math as coming down from on high. Instead we should see it as another set of ideas to be questioned, explored, and challenged in order to make it better.
As you all continue in your semesters of coursework and/or research, you may come across math and think things like “this is too much; I’m never going to understand” or “it surely must be right; I mean, it’s math and there’s always a right answer.” I’ve thought these things too, I think, because I may see myself as too much a discoverer of math, rather than an inventor of new ideas. But balancing these two notions helps to keep an open, inquisitive mind, while duly deferring to past works or current experts when necessary. This is much easier said than done, but I believe that the moments of greatest fulfillment come when reaching that effective interplay of active inventiveness and guided discovery.
