From the view of a pure mathematician, those working in pure mathematics produce pure knowledge. Whether used or not, it has a great elegance and value in and of itself. Those in applied mathematics simply pick up what has been done and use it in designing or building things. Number theory is often used to illustrate this, where work done decades ago in pure mathematics is now central to encryption.
However, the relationship between pure and applied mathematics is a bit more complicated. New insights drawn from applications have been at the root of interesting new areas and questions in pure mathematics. Fourier analysis, sparse matrix computation, and graph theory all demonstrate this.
Some have argued that, whether pure or applied, mathematics is not really needed by the average person. Alfred North Whitehead, a Harvard mathematician and philosopher, once stated, “ideas [from mathematics] are of highly specialized application, and rarely influence thought.” In other words, mathematics is a specialized skill, but not a liberal art.
Using examples from n-dimensional linear algebra, I will show why I believe the areas of pure and applied mathematics are deeply tied, and that this field does indeed influence thought in areas like understanding relationships and political discourse.