Hard Analysis vs. Soft Analysis

Terence Tao, the UCLA Math Professor and Fields Medalist, has a blog. In a recent posting, he discussed the differences and similarities between hard analysis and soft analysis. I will not re-iterate what Dr Tao writes on his page since that would just be ridiculous. However, I would like to state how this is related to the titular mathematical behavioral sciences. Tao notes that analysts generally specialize in either hard or soft analysis.

At first glance, the two types of analysis look very different; they deal with different types of objects, ask different types of questions, and seem to use different techniques in their proofs. They even use^[2] different axioms of mathematics; the axiom of infinity, the axiom of choice, and the Dedekind completeness axiom for the real numbers are often invoked in soft analysis, but rarely in hard analysis. (As a consequence, there are occasionally some finitary results that can be proven easily by soft analysis but are in fact impossible to prove via hard analysis methods; the Paris-Harrington theorem gives a famous example.) Because of all these differences, it is common for analysts to specialise in only one of the two types of analysis. For instance, as a general rule (and with notable exceptions), discrete mathematicians, computer scientists, real-variable harmonic analysts, and analytic number theorists tend to rely on “hard analysis” tools, whereas functional analysts, operator algebraists, abstract harmonic analysts, and ergodic theorists tend to rely on “soft analysis” tools. (PDE is an interesting intermediate case in which both types of analysis are popular and useful, though many practitioners of PDE still prefer to primarily use just one of the two types.

Now one reason the mathematical behavioral sciences are so difficult is that one often has to use the tools of both soft and hard analysis. For example, in my work on researching the formal properties of models, I use hard analysis (e.g., mathematical logic and algorithms) and I use the tools of soft analysis when I investigate the dynamics of a system (e.g., dynamics and ergodic theory).

Thus, this posting by Tao ends up being an excellent exposition on how a mathematician can move from one domain of mathematics to another by going through a particular example, viz. the (finite) convergence principle. Being able to do this is important for many mathematicians, but this sort of situation arises for us in the mathematical behavioral sciences all of the time.