Devlin's Angle

October 2009

Soft Mathematics

In my book Goodbye Descartes: The End of Logic and the Search for a New Cosmology of the Mind (John Wiley, 1997), I introduced the term "soft mathematics" to refer to the use of mathematical ideas and ways of thinking in domains that are inherently non mathematical, having observed instances in several disciplines, among them linguistics, psychology, sociology, economics, political science, management science, and intelligence analysis. (You also see instances, generally fictional though sometimes based on fact, in the CBS television crime series Numb3rs.)

I looked at this phenomenon again in my recent article "What will count as mathematics in 2100?" in the MAA Spectrum book Proof and Other Dilemmas: Mathematics and Philosophy, edited by Bonnie Gold and Roger Simons (MAA, 2008, pp.291-311).

So much of my research over the past twenty years has been spent working with experts from other disciplines, in particular in linguistics and in the intelligence analysis community, that I long ago forgot that such uses of some of the notions of mathematics, along with an analytic or design approach based on mathematics, can seem alien to the mathematician. Becoming an expert in a particular discipline (I would like to say "being indoctrinated", but some might read into that phrase a negative connotation that I do not imply), involves learning to see things in a particular way, and in the process we can "forget" (or never learn) how to see things in other ways.

This tendency to see things in ways we have become familiar with - and to be unable to see distinctions that others take for granted - was brought home to me recently in an email from a group of students in Singapore who were working through Goodbye Descartes and were unable to see why I introduced the term "soft mathematics" for what they saw as plain old "mathematical modeling."

There is a distinction, and to my eyes it is a big one. But the exchange made it clear that what was clear to me was not necessarily clear to others. (I was reminded of Richard Dawkins, who when asked why so many of his books seemed to be just variations of his original The Selfish Gene, replied that he kept saying the same thing in different ways because people so often failed to understand the points he was trying to make.)

In mathematical modeling, you create a definite model - an equation or system of equations, a set of inequalities, of whatever - that is intended to capture key behavioral features of some worldly domain, and then you do mathematics - of the traditional sort. It looks like mathematics, it smells like mathematics, and by golly it is mathematics. The results may be applied to the world insofar as the model actually does capture the intended features of the world.

Soft mathematics is quite different. There is little or nothing that looks like, or is, traditional mathematics. There may not even be any mathematical symbols tossed around - though in many cases there are. Soft mathematics is not mathematics as that discipline is generally thought of, and it remains an open question whether at some time in the future our conception of what constitutes mathematics will change to incorporate such activities. (I address that question in my MAA Spectrum article referred to above.)

What is clear, however, is that the mathematical way of thinking is such a powerful one that, when applied in a soft manner, it has on occasion led to considerable advances in our understanding of various phenomena in the messy, and decidedly non-mathematical social realm of people. One of the best examples I have come across was in the field of linguistics. I describe this example at some length in Chapter 9 of Goodbye Descartes, from which I have abridged the following shorter account.

Grice's maxims

In a lecture given at Harvard University in 1967 - and subsequently published under the title Logic and Conversation - the logician H. P. (Paul) Grice described a "logic" of everyday conversations, the structure that any conversation must have in order to be successful, regardless of its topic and purpose. He did so by formulating a set of "maxims" (his term) that participants in a conversation implicitly follow. It was a brilliant attempt to apply a mathematical approach to the structure of conversation, very much in the spirit of Euclid's formulation of axioms for plane geometry.

Grice began his analysis by observing that a conversation is a cooperative act, which the two participants enter into with a purpose. He tried to encapsulate the cooperative nature of conversation by what he called the Cooperative Principle:

Make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged.
His next step was to derive more specific principles - his maxims - from the Cooperative Principle, by examining consequences under four different headings: quantity, quality, relation, and manner. He illustrated these four categories by means of non-linguistic analogies:
Quantity. If you are assisting a friend to repair his car, your contribution should be neither more nor less than is required; for example, if your friend needs four screws at a particular moment, she expects you to hand her four, not two or six.

Quality. If you and a friend are making a cake, your contributions to this joint activity should be genuine and not spurious. If your friend says he needs the sugar, he does not expect you to hand him the salt.

Relation. Staying with the cake making scenario, your contribution at each stage should be appropriate to the immediate needs of the activity; for example, if your friend is mixing the ingredients, he does not expect to be handed a novel to read, even if it is a novel he would, at some other time, desire to read.

Manner. Whatever joint activity you are engaged in with a friend, your partner will expect you to make it clear what contribution you are making, and to execute your contribution with reasonable dispatch.

In terms of conversation, the category of quantity relates to the amount of information the speaker should provide. In this category, Grice formulated two maxims:
  • Make your contribution as informative as is required.
  • Do not make your contribution more informative than is required.

    Under the category of quality, Grice listed three maxims, the second two being refinements of the first:

  • Try to make your contribution one that is true.
  • Do not say what you believe to be false.
  • Do not say that for which you lack adequate evidence.

    Under the category relation, Grice gave just one maxim:

  • Be relevant.

    Finally, under the category of manner, Grice listed five maxims, a general one followed by four refinements:

  • Be perspicuous.
  • Avoid obscurity of expression.
  • Avoid ambiguity.
  • Be brief.
  • Be orderly.

    As Grice observed, his maxims are not laws that have to be followed. In that respect they are not like mathematical axioms. If you want to perform an arithmetical calculation in a proper manner, you have to obey the rules of arithmetic (even if you are not consciously aware of so doing). But it is possible to engage in a genuine and meaningful conversation and yet fail to observe one or more of the maxims Grice listed. The maxims seem more a matter of an obligation of some kind. In Grice's own words:

    "I would like to be able to think of the standard type of conversational practice not merely as something which all or most do in fact follow, but as something which it is reasonable for us to follow, which we should not abandon." [Emphasis as in the original.]
    Clearly, Grice's maxims fall under my notion of soft mathematics. Grice made successful use of his maxims in analyzing a widespread conversational phenomenon he called "conversational implicature". This is when a person says one thing and means something other than the literal meaning. For example, suppose Naomi says to Melissa, "I am cold" after Melissa has just entered the room and left the door wide open. Literally, Naomi has simply informed Melissa of her body temperature. But what she means - or what she probably means - is "Please close the door." Naomi's words do not actually say this; rather it is implicated by her words. Grice used the word "implicate" rather than "imply" for such cases since Naomi's words certainly do not imply the "close the door" meaning in any logical sense. Assuming Melissa understands Naomi's remark as a request to close the door, she does so because of cultural knowledge, not logic.

    Conversational implicatures

    Conversational implicatures are ubiquitous in our everyday use of language. They can be intended by the speaker, or can be made by the listener. Grice used his maxims to analyze the phenomenon. Let's take a look at his analysis.

    Suppose Mark meets Naomi and says,"How is the car your brother lent you?" Naomi replies, "Well, it hasn't broken down so far."

    Mark's question seems straightforward enough. What about Naomi's reply? Assuming both Mark and Naomi are obeying Grice's Cooperative Principle, that is to say, they are engaged in a genuine attempt to have a conversation, and not trying to mislead each other, what are we to make of Naomi's words? Presumably Naomi is implying, in a roundabout way, that she does not expect her brother's car to be in good order. She is implicating this unspoken meaning. Most people in Mark's position would probably take Naomi's reply that way. But what is the logic behind this particular use of language? After all, Naomi certainly does not come out and say "My brother's car is likely to be unreliable."

    In terms of the maxims, here is a Gricean analysis of the Mark and Naomi example. On hearing Naomi's reply, Mark could reason as follows:

    1. Naomi's remark appears to violate the maxim "Be perspicuous."
    2. On the other hand, I have no reason to suppose she is opting out of the Cooperative Principle.
    3. Given the circumstances, I can regard the irrelevance of Naomi's remark as appropriate if, and only if, I suppose she thinks her brother's car would be likely to break down.
    4. Naomi knows I am capable of working out that last step.
    5. Thus Naomi is implicating that her brother's car would be likely to break down.

    Of course, few if any of us would actually go through such a reasoning process. But that is not the point. In a similar vein, people rarely consult the axioms of logic when putting forward a logical argument, but that does not prevent a logician from analyzing their argument and checking to see if is valid by seeing if it accords with the rules of logic.

    Though scientists like to understand how something "really" is, they often settle for a plausible explanation of the phenomenon that fits the known facts. In explanations of human activities such as reasoning and conversing, one way to see if a particular explanation fits the facts is to see if it would provide a reasonable response to a challenge of "How did you reach that conclusion?" In the case of Mark and Naomi's conversation, imagine a bystander asks Mark what he understood by Naomi's concluding remark, and to explain how he reached that conclusion. Most people in Mark's position would probably respond with an explanation something like the one just given, though perhaps much shorter and, unless they knew about Grice's maxims, not using his technical terminology.

    Though Grice makes no claim that people have any conscious awareness of his maxims, his discussion of conversational implicature establishes a strong case that the maxims capture part of the abstract structure of conversation. They do after all enable the linguist to provide satisfactory, after-the-event explanations of a variety of conversational gambits.

    According to Grice, a participant in a conversation, say Bill in conversation with Doris, may fail to fulfill a maxim in various ways, including the following.

    (1) Bill may quietly and unostentatiously violate a maxim. In some cases, Bill will thereby mislead Doris.

    (2) Bill may opt out from the operation both of the maxim and the Cooperative Principle, making it plain that he is unwilling to cooperate in the way the maxim requires. For example, he might say, "I cannot say more. My lips are sealed."

    (3) Bill may be faced with a clash. For example, he may find it impossible to satisfy both the quantity maxim "Be as informative as required" and the quality maxim "Have adequate evidence for what you say."

    (4) Bill may flout or blatantly fail to fulfill a maxim. Assuming that Bill could satisfy the maxim without violating another maxim, that he is not opting out, and that his failure to satisfy the maxim is so blatant that it is clear he is not trying to mislead, then Doris has to find a way to reconcile what Bill actually says with the assumption that he is observing the Cooperative Principle.

    Case (4) is the one that Grice suggests most typically gives rise to a conversational implicature.

    Let's take a look at some more examples of everyday conversational implicatures.

    For some implicatures, no maxim is violated. For example, suppose Roger drives up to a policewoman and says, "I'm almost out of gas," and the policewoman replies, "There's a gas station around the corner." By the maxim "Be relevant," Roger can infer that the gas station is open.

    In contrast to the gas station scenario, the next example involves apparent violation of the "Be relevant" maxim in a very clear way in order to produce the intended implicature.

    Arthur says, "Bill doesn't seem to have a girl friend these days." Susan replies, "He has been spending a lot of time in Denver lately." Susan's response will violate the "Be relevant" maxim unless she intends her reply to implicate the fact that Bill has, or at least she suspects that he has, a girlfriend in Denver, and she wants her remark to suggest that that is the reason for his frequent visits there.

    For another kind of example, suppose Greg has been telling Melissa of his intention to visit Europe, and has mentioned that he would like to visit her friend Yannis. He asks, "Where does Yannis live?" and Melissa replies, "Somewhere in Greece." Clearly, Greg was asking for the name of the location where Yannis lives, in order to see if it would be possible to visit him. Hence Melissa's reply violates the quantity maxim "Make your contribution as informative as is required." Assuming that Melissa is not violating the Cooperative Principle, the conclusion Greg can draw is that Melissa violates the quantity maxim because to say more would require that she violates the quality maxim "Do not say that for which you lack adequate evidence." In other words, Greg concludes that Melissa does not know the city or town where Yannis lives. Indeed, assuming Melissa is being as informative as she can, Greg may conclude that Melissa cannot be more specific than she has.

    People sometimes flout maxims in order to achieve by implicature an information exchange they would, for some reason, prefer not to state explicitly. For example, suppose Professor Alice Smith is writing a testimonial for her linguistics student Mark Jones, who is seeking an appointment at a university. She writes a letter in which she praises Jones's well groomed appearance, his punctuality, his handwriting, and his prowess at tennis, but does not say anything about his ability as a student. Clearly, Professor Smith is flouting the quantity maxim "Make your contribution as informative as is required." The implicature is that Professor Smith has nothing good to say about Jones's academic ability, but is reluctant to put her opinion in writing.

    Irony is often achieved by a violation of the quality maxim "Do not say what you believe to be false." For example, suppose Jane has been telling Richard how badly her friend Sally had let her down, and Richard comments, "Well, Sally certainly is a great friend." The implicature is that Sally is a very poor friend.

    Metaphor is another linguistic affect that may be achieved by flouting the same quality maxim. For example, if Tom says to his wife, "You are the cream in my coffee," the implicature is that Tom thinks his wife is the completion to his life.

    Violation of the quality maxim "Do not say what you believe to be false" may also be used to achieve the effect of understatement. An example of this is where Barbara and George have had an enormous fight, in which Barbara ended up flinging crockery all over the kitchen, and the next morning Barbara approaches George and says, "I was a bit annoyed last night." The implicature is that Barbara was, as George knows full well, thundering mad. In this case, George probably takes her words as an acknowledgement of, or even an apology for, her behavior.

    So far, none of the examples have involved the maxims of manner. Here are three that do.

    Parents of young children sometimes flout the manner maxim "Avoid obscurity of expression" in order to communicate with each other in a manner that their children cannot comprehend, saying things like "Did you pick up the you-know-what on your way home?"

    Politicians sometimes try to violate the "Avoid ambiguity" manner maxim in order to mislead their audience.

    Neither of the above two examples results in an implicature. However, suppose John says to Sally, "Mary produced a series of sounds on the piano that sounded like 'Home on the Range'." This violates the manner maxim "Be brief," and the implicature is clearly that Mary's piano playing was not very good.

    The above examples illustrate that way a person can make (implicit) use of Grice's maxims to convey a meaning other than the literal meaning of the words actually spoken. The maxims provide some of the "logic" of conversation, just as Aristotle's syllogisms provided some of the "logic" of reasoning.

    Is Grice's work mathematics? Clearly not. (Hence his work does not constitute mathematical modeling.) Is it inspired by, and modeled on, the mathematical approach. You bet your life it is.

    NOTE: Grice's original essay is widely available on the Web as a PDF file.

    Devlin's Angle is updated at the beginning of each month. Devlin's most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.
    Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition.