mathematical language, legal language?

A friend of mine made the recent comment (on someone else's private journal; I won't quote it here in detail) that a "pure" subject is suspect to her, and she believes that all fields of study are linked together in intricate ways. Her opinion make sense ..

.. but I wonder whether "pure mathematics" falls into that category or not. Most of the time? Some of the time?

Off and on I've discussed matters with scientists (or rather, grad students in the hards sciences) and on less occasions, folks in the social sciences and humanities. As pure mathematicians we get our share of flak for not doing much interdisciplinary work. Maybe they think that we mathematicians are arrogant, that we don't need them, but they need us. Maybe I'm imagining things.

There are the occasional hotspots in research: dynamical systems are useful for scientific models and more recently, computational biology. Algebra and combinatorics become useful in encryption and search algorithms. The nascent theory of wavelets has grown in parallel to needs of image and signal processing. I'm confident I could find more examples of this sort.

However, I believe that most of pure mathematics is not immediately applicable and to the rest of the world, "current worthless." Maybe different areas will have untold worldly uses in the future, but maybe they won't. I get the impression that past and present administrations of American government fund mathematical research because of hope towards the former, not because of any noble purpose on their part.

If our mathematics helps the rest of the world, fine; if not, then as Archimedes said, don't disturb our circles.

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Me, I like to think of mathematics as logic, or even philosophy, applied to very specific situations, often to geometry or the structure and interaction of numbers and their abstractions. It's a convenient way of saying that it's the study of abstract objects using abstract tools.

Many modern problems have a tendency towards abstraction and take a long time to ponder; perhaps this is the fault of logical formalism and rigor, which adds apparent volume and mass to a little monster, so we fear it more and make much of it.

If you ask a mathematician about a problem, (s)he may tell you the full regalia of the issue and possibly give a few definitions. If you ask a mathematician about his/her ideas, then more likely than not, you'll get a concrete example and possibly heuristical reasons and intuition in the form of pictures and diagrams.

So why the logical formalism? Why the rigor and fearsome-looking definitions? Why a formulaic outline when assessing and/or solving a problem whose words are hard to understand?

It's the price we pay for consistency and accuracy, and it plays a similar role as legal language. In some ways, legal documents and mathematical papers are similar. For example, the first section is full of definitions. In a union contract, how do you define "Employee?" and on some apartment leases, what exactly is a "tenant?"

If you believe your method is correct and it accurately addresses the problem at hand, then you must "fill out the legal paperwork" .. in short, write out precisely what you mean in a consistent way that can be checked logically.

On the other hand, when reading mathematical work it takes a careful mind, often a mathematician's, to decipher the full strength (or weakness) of a new theorem. Is it argued correctly, any logical mistakes? Does it address even the pathological cases, if the definition allows such possibilities? For example, it's a bit like hiring a lawyer for when you're closing on a house: the last thing you want is to hear that sinkholes aren't covered by a "good condition" policy.

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Perhaps a problem with mathematics and interdisciplinary work is, oddly enough, a language barrier. Another problem is that mathematicians are attached to their own problems, just like any other type of academian. Combining the two, sometimes it seems like a one-sided relationship with other disciplines.

A mathematician might be helpful to a think tank of physicists or economists or biologists, but only after (s)he processes the situation and question in a way (s)he can understand, and then asks enough questions so that (s)he understands what can be done and what cannot: maybe it's an issue of conservation of matter, or that negative profit is bad. If this is successful, then (s)he might suggest an idea, and with any luck (s)he can explain the rationale apart from any tendency towards excessive terminology. Maybe it helps, who knows?

I remain wary of a think tank of mathematicians discussing a common (open) problem to a physicist, economist, or biologist. It may take too long to explain, and even with a good dose of patience, it may be too many new concepts to juggle -- and this is not an insult to non-mathematicians, but a matter of technical details. It's a lot to ask someone to think about something away from their area of expertise, and with many issues to consider.

However, I do see the value in mathematicians asking non-mathematicians how they think about their own problems, because it could help the mathematician learn how to think about the mathematical problem in a new way. It could be as simple as analogy: the notion of extremal length, also called modulus of curve families, is mysterious unless you describe it in terms of electro-magnetism and current through a condenser. It doesn't proves your theorems, but it helps you write your proofs.

But there remains the problem: how did we know to think of extremal length in terms of electromagnetic current? Someone else has to answer that one, though my guess is complicated and needs some explanation [1].

I don't claim to know the answers. If non-mathematicians had a tried-and-true method that could help mathematicians with their own research problems, then great! Sign me up! But every time I explain what I'm studying to the non-mathematician, we mutually give up and it leads to little added progress, though by explaining I might understand the issues better.

When from experience others are of little help, we tend to ask less often and continue on by ourselves. After a while you forget to ask, and expect confusion.

What can I say?

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[1] I think it has to do with the relation of extremal length and a notion called "capacity" which, physically speaking, describes the least kinetic energy under a given arrangement of position. If this relationship is true, then there may be an optimal arrangement of the magnetic field induced by the current, and the resulting field lines may give some comparison, as curves, to the given curve family.