"... the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there..."
The quote above, from a lecture given in 1907, refers to a division of mathematical labor that is still being discussed today. Freeman Dyson's Birds and Frogs and Timothy Gowers's The Two Cultures of Mathematics are two recent articles elaborate on the rift that Keyser pointed out: that there are two distinctly different approaches to doing mathematics - an approach from above, favored by the bird-like theoreticians, and an approach from below, taken by frog-like problem solvers.
For non-mathematicians, the Gowers and Dyson articles provide windows into the usually opaque world of how mathematicians work and how they think about what they do. That mathematicians have differing views on what doing mathematics means would surprise many; that some view their work in almost heroic terms (Dyson's 'birds' and Gowers's 'theory builders') would surprise even more. The orientation that Dyson describes as 'frog' and that Gowers describes as 'problem solver' (categories into which they fit themselves, respectively) would probably be close to what the general public sees as the mathematical mind-set, although many would still be surprised that these problem solving frogs are attacking genuinely new problems.
Often when discussing how mathematicians view their work, we attempt to sort them by their 'philosophy of mathematics' - do they see themselves as neo-platonists, as formalists, as constuctivists? And, related to this, do they see themselves as discovering or inventing? Both the Dyson and Gowers articles avoid this over-used characterization of mathematicians via philosophy. Much more productively, both authors choose to focus on how mathematicians actually work, and on what they actually do. Focusing on actual mathematical practice and on the culture that grows up around it, as these articles do, provides much more insight into how mathematicians view their work and understand each other.
Through their articles, Dyson and Gowers are striving for a reconciliation between the two camps of mathematicians that they describe. The idea that mathematicians may fail to value the contributions of collegues who approach their work differently may come as another surprise for many non-mathematicians - the unfortunate reality is that, as in all fields, there are plenty of hierarchies and cliques in the world of mathematics. Gowers notes that areas of mathematics that are associated more with 'problem solving' (like combinatorics) receive less prestige than those that are perceived to be theoretically richer, and that these perceptions are often based on misconception. (The disdain goes both ways, of course - category theory is still labeled as 'abstract nonsense,' a label only sometimes jokingly applied.) Both authors end on notes that call out for a greater awareness and sympathy for how the two spiecies, or cultures, of mathematicians can work together so that mathematics can advance collectively.
