why we do mathematics

A colleague recently asked, in the course of an unrelated conversation, for my views on why society should fund pure mathematics.  Particularly during hard economics times, it’s a reasonable question to ask, why do we as a society fund the things we fund.  On the other hand, it is a question about which I have a strong vested interest and a significant bias.

She asked me as I was preparing to talk to a group of pre-university students, giving them reasons that they should consider studying mathematics at university, and the combination of the 2 questions got me thinking.

I am by inclination and training a pure mathematician.  I will admit to having dabbled in a few bits of more applied areas of mathematics, but I like geometry and I like studying geometry for its own sake.  I enjoy groups and like studying groups for their own sake.  I enjoy the abstraction and I love the art of being a mathematician. 

My first answer is the same as it would be for most academic endeavours, the subjects covered by the departments that make up universities.  It seems bizarre to me that as a society, we would ever be willing  to say, we can stop now because we’ve learned enough. It seems bizarre to me that we as a society would ever be willing to say, we have a sufficient understanding of the the inner workings of the physical realm and the realm of ideas, of the human mind and our place in the wider world, that we don’t need to learn any more.

We will never come to a point where we know everything, where we’ve learned enough.  Part of this is developing our understanding, and part of this is creating art.  There is art in exploring ideas, navigating through possibilities, making connections between hitherto unrelated ideas and concepts.

But beyond this, for mathematics of all stripes and types, there are other reasons.

Mathematicians predict the future.  We construct models that allow us to extrapolate from the past and the present, based on the information we have and based on our understanding of the mechanisms of the world, and make reasonable guesses of what the world might be like in days, or weeks, or months.  We bear witness to this every day, when we hear the weather report and learn what the future might hold for us.

Mathematicians reconstruct the past.  My favorite example here concerns phylogenetic trees.  We start with evolution and natural selection.  If we make the assumption that life began once on earth, or at least that it is only the descendants of one beginning of life, which I naively believe is the more reasonable assumption, then life is a tree.  Where the branches split from one another mark where one species gives birth to two, and so on, and so on.

Viewing things in the present day, we don’t know where are the branches in this tree.  But if we know far apart the current species are (say, in how different their DNA is one from another), then we can reconstruct this tree, and in doing so, we can reconstruct the history of all life.

Mathematicians explore the unseen.  I like the Radon transform as an example of this.  A CAT scan works by shooting beams of radiation through an object with variable density.  Some highly sophisticated mathematics allows us to reconstruct the density of the object from the absorption of radiation along each of these beams.

Mathematicians are explorers.  Our explorations link the inner world of the mind, with its notions of beauty and structure and order, with the outer world in which we live, allowing us to create models to understand the rhythms of the world.  We forge unexpected connections, and even the most abstract and obtuse piece of mathematics has the possibility of shedding light on a poorly understood (or even well understood) corner of our world. 

And that I think is sufficient reason.

~ by Jim Anderson on 1 May 2016.

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