On the stories of solutions

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equations written on blackboard
A math board. If you ever caught yourself asking someone, in complete confusion over your math work, "When will I ever need this in life?", then this column, by Sharon Spaulding, is surely to have you rethinking how you view the solutions to your math problems. Photo by Gabby K on Pexels.com

If you remember high school algebra, the main goal is to solve for numbers hiding inside equations. In algebra, these numbers are usually disguised as “x”, but later on in math, variables like to be extra mysterious inside equations. Instead of appearing as “x”, a variable might only appear in an equation with a suggestion of how fast it travels — and there may even be more than just one variable. In these situations (partial differential equations), there often ends up being a “family” of solutions that works instead of just a single number. Sometimes we even end up with different solutions depending on where we are in some space where we’re looking for solutions.  

One of my math professors was explaining this for a specific equation this past week. In that situation, there ended up being different solutions depending on where you are in the XY-plane. The way he described it was that in each section there was “a different story” and we had to figure out what the exact “story” was in each one.  

“The way he described it was that in each section there was “a different story” and we had to figure out what the exact “story” was in each one.”

It’s common in math courses to hear various ways of impressing a narrative on abstract concepts with colloquial expressions or informal designations. Different equations that are introduced early on in a proof and then used later on might be referred to as “that guy” or “this guy”. Concepts that eventually come together to prove a theorem are often called “ingredients”. Taken together, a family of solutions creates a “story”.  

various ingredients and frying pan on table
A table of ingredients. Concepts that eventually come together to prove a theorem are often called “ingredients”. Taken together, a family of solutions creates a “story”. Or maybe, even a recipe. Photo by Katerina Holmes on Pexels.com

These terms certainly aren’t ubiquitous. Another math student may or may not have heard these phrases at all depending on whose classes they’ve been in. Yet some forms of expressions like them are used by almost every mathematician I’ve heard lecture or teach, and I know that as a student I use them myself when trying to deconstruct a proof in a textbook or explain a problem while tutoring.  

It’s not so much about creating metaphors in attempts to better visualize concepts. In some situations, at least at my level as a student, trying to visualize something isn’t even all that helpful. I have very little idea about what Borel sets actually look like. It’s not something I could draw or that I have a clear picture of in my head. Yet using certain phrases and terms to narrativize concepts and proofs does have purpose, even if not to provide a way to visualize a problem. Saying that there are different “stories” in each section for a solution to a Cauchy problem as a way to express that the solution differs along its domain didn’t actually alter my understanding of the problem —  but it did do something important. It lit up the idea of “solutions” in my head to mean something more exciting than just a set of equations that I’d solved for. They were stories! A few equations had first appeared on my page as a small set of unobtrusive lines. Then, quite quickly, they began to unravel over the page, expanding and contracting, merging and dividing. Finally they came to rest. Stripped of enigma, they no longer stood at the top of a blank page, isolated — now they told stories.   

“It lit up the idea of “solutions” in my head to mean something more exciting than just a set of equations that I’d solved for. They were stories!”

Outside of math there are other situations we remain distant from — circumstances in worlds not entirely visible to us and which are parts of intricate systems of economic, political, social and vocational connection. It’s impossible to fully understand these situations. It’s impossible to completely understand the full magnitude and scope of how historic and persistent racism permeate our institutions, communities, culture and selves. It’s impossible to completely understand the scale of how our lifestyles disturb natural environments and devastate life beyond our immediate geographies. It’s impossible to completely understand the gravity of living in diverse communities — like UConn — and missing out on connecting with students from different backgrounds, socioeconomic statuses and races with different interests, pursuits, faiths and lifestyles. It’s impossible to completely understand the mind and heart of another human.  

wood people woman girl
A girl laying in the woods, reading a story. As in math, narratives are what tug us forward — reflections on surprising connections, stories from individual people and encounters with art. Photo by Kenyel Nuñez on Pexels.com

Yet in all these situations, limited understanding doesn’t prevent us from responding in meaningful ways. We can learn more. We can incorporate what we know into our own outlooks to creatively and powerfully escape defaults. As in math, narratives are what tug us forward — reflections on surprising connections, stories from individual people and encounters with art. They may not always help us to better understand larger complexities — but they help us to draw nearer to what it is we’re investigating. And as a first step, drawing nearer is perhaps more important.  

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