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"Soft Mathematics" is the title of an article I wrote for Mathematics Awareness Week 96, which takes place April 21-27. The theme for this year is Mathematics and Decision Making. The focus of my article is the blend of mathematics and techniques of the social sciences that is increasingly prevalent in economics, management science, psychology, cognitive science, sociology, linguistics, and the like. The article appears on the Mathematics Awareness Week Web entry, which can be accessed directly from MAA Online. Choose Resources for Mathematics and Decision Making and then pick Original Articles.
By "soft mathematics" I don't mean "applied mathematics." Nor do I mean "mathematical modeling." And I certainly don't mean the use of statistics in the social sciences. Social scientists do indeed make frequent use of statistical techniques to collect the data for their study. There is no question that practically all sciences, natural or social, rely on mathematics in one way or another when it comes to data collection.
Rather, what I mean by the term is a genuine attempt to blend mathematics with other approaches in trying to analyze or describe some phenomenon. The result is certainly not mathematics in the sense I was brought up and trained to understand the word. Just as counterfeit money is not money, so too soft mathematics is not mathematics. Of course, counterfeit money can occasionally make people genuinely rich, and likewise soft mathematics can lead to real (hard) mathematics. (Printing your own money can also, of course, land you in prison, and a lot of what might seem to qualify as soft mathematics is pure junk that by rights ought to land the perpetrators in mathematical prison.)
As an example of soft mathematics, take the following formula from linguistics:
XP --> (SUB) X' YP*
(I said "take it." I did not ask you to understand it.) The symbol X' should be rendered as X with an overbar, a typographical convention that is not universally available on WWW servers and readers. The formula reads: "An X-phrase consists of an optional subject, coupled with an X-bar, coupled with any number of Y-phrases." It is a grammatical rule in what linguists call "X-bar theory", and it is a candidate for one of the rules of language that may be "hard-wired" into the human mind as part of what linguist Noam Chomsky called Universal Grammar.
According to the algebraic rules that go along with this formula, you can replace the symbolic variables X and Y by any of N (for noun), V (for verb), P (for preposition), and A (for adjective) to give a particular rule of grammar. For example,
NP --> (SUB) N' VP*
which says that a noun phrase consists of an optional subject coupled with a noun-bar (a noun is the simplest case) coupled with any number of verb phrases.
To anyone not familiar with contemporary linguistics, this doubtless seems pretty bizarre, and it would involve too great a digression to explain just what is going on. Those sufficiently intrigued should consult linguist Steven Pinker's excellent best-selling book The Language Instinct, published in 1994. My point is simply to indicate that, in order to capture some of the abstract patterns and structures of mind and language, linguists sometimes find it convenient (perhaps necessary) to make use of mathematics, or at least mathematical notation or techniques. The result is not that linguistics becomes part of mathematics, or even a "mathematical science" (in the sense of, say, physics). The aim is not to prove theorems. There might not be any relevant theorems to prove. Ever. There might not even be any deep or revealing definitions. The aim is to do linguistics, to understand language and investigate the way people use language to communicate.
To some mathematicians, the absense of theorems means that what is going on is not only not mathematics, but no self-respecting mathematician should become involved in such a dubious pursuit. I don't agree. Besides the fact that such a view would damn any number of revered mathematicians (for example Leibniz, for his algebra of mental concepts), such an attitude strikes me as dangerously isolationist. There have been a number of infamous occasions where social scientists have made use of mathematical techniques in a highly inappropriate and misleading manner. (Chomsky was not one of them, I hasten to add. He was well-trained in mathematics. He even proved some theorems!) Far better for those of us properly trained in the discipline to offer our services when we are asked (or even if we are not, provided we do it in a collegial fashion, aware that as mathematicians we have only been handed one of many sets of stone tablets, all of which may be valid in their own domains).
To my mind, it is not a cause for dismay that, say, a linguist or a management scientist uses what for me as a professional mathematician is an entirely trivial piece of apparatus. I don't see such use by others as besmirching my discipline. Rather, I view it as an affirmation of the incredible power of mathematics that even its simplest elements can be put to good use by others.
Devlin's Angle is updated on the first of each month.