> What is really meant by a "PLAUSIBILITY argument"? What is the difference > between "use reasoning to develop plausible arguments" and "use deductive > reasoning to construct valid arguments"? The authors say that an argument is > plausible if "it makes common sense and is mathematically correct." > According to the dictionary, something is plausible if it is believable, > apparently valid, or acceptable, if it has the appearance of truth. Perhaps > someone with special interest/expertise in proof could elaborate?
What, indeed, is really meant by the phrase "plausiblity argument"? We can see just how thorny this question really is by recasting it: What is a proof? I know of no really satisfying answer to the latter question--though, like most mathematicians, I believe that I know one when I see one.
Solow [_How to Read and Do Proofs_, 2nd Ed., Wiley, 1990, pg. 3] wrote "... a mathematical proof is a convincing argument expressed in the language of mathematics," while Smith, Eggen, & St. Andre [_A Transition to Advanced Mathematics_, 3rd Ed., Brooks/Cole, 1990, pg. 26] wrote "A proof...is a logically valid deduction of a theorem, from axioms or the theorem's premises, and may use previously proved theorems." Greenberg [_Euclidean and Non-Euclidean Geometries_, 2nd Ed., Freeman, 1980, pg. 33] wrote "...a proof is a list of statements, together with a justification for each statement, ending up with the conclusion desired. ... *Only six types of justification are allowed.*..." (Author's emphasis.)
In the next-to-last paragraph of the introduction to their monumental _Principia Mathematica_, Whitehead & Russell wrote a sentence that has since been paraphrased and applied to "proof" by many subsequent authors: "Most mathematical investigation is concerned not with the analysis of the complete process of reasoning, but with the presentation of such an abstract of the proof as is sufficient to convince a properly instructed mind." They went on to say, at the end of the same paragraph: "In this connection it may be remembered that the investigations of Weierstrass and others of the same school have shown that, even in the common topics of mathematical thought, much more detail is necessary than previous generations of mathematicians had anticipated."
Evidently, previous generations of mathematicians, who knew a proof when they saw one, had been mistaken. Maybe I am, too. Standards of proof are relative.
So what are they relative to? It's here that I think Whitehead and Russell hit the nail on the head: Standards of proof vary with the propriety of the instruction that the apprehending mind has received. Every age answers differently the question "What is a properly instructed mind?" (Deep epistemological questions lurk here. I am too cowardly to deal with them, but not cowardly enough to refuse to admit it.)
What has this to do with plausibility arguments? I think we use that phrase of arguments that our own "properly instructed" minds find unacceptable or incomplete, but that we think our students' "improperly instructed" minds will find convincing. (This is not a reflection on the propriety of the instruction we've provided our students, but on its incompleteness at the time these issues arise.)
The very most formal of proofs are written in symbolic logic and are almost incomprehensible to most readers--even highly trained and successful research mathematicians. A few (relatively speaking) such proofs can be found in such fields as formal verification of computer programs. Nobody has ever read them. Not even their authors.
But the rest of us, as experts, commonly accept plausibility arguments written for experts. They are, as Whitehead and Russell said, abstracts of the formal proofs, and they're as incomprehensible to our students as the really formal stuff is to us. Paradoxically, the higher the level of mathematics, the less formal the proofs. That's because, at those higher levels, the author can expect the reader to supply more of that niggling detail students find so hard.
What's more, the author writing at those higher levels knows not just that the reader can supply the detail, but that the reader will know what detail must be supplied. This directly addresses the issue of plausibility. The self-same incomplete argument, constructed diabolically enough, could convince--that is, might be plausible to--both novice and expert. That's because the novice doesn't know the details aren't there and the expert not only knows they aren't but also knows how to put them there. (I give arguments like this in my calculus courses all the time--gave one Thursday, in fact. And again today.)
There is danger here. Lebesgue pointed it out long ago when he reflected that mathematics teachers often take verbal precautions that the student is certain to misconstrue. I used to do that in the past, but I've given it up because it's dishonest. Now I make it a point to default on detail instead of on meaning. I want to mean what I say, say what I mean, and have students hear both what I say and what I mean. In fact, I want them to know that I'm trying to have what I say and what I mean be the same thing because I want them to aim for that goal too.
The crux of the matter is this: A good teacher can be precise without being pedantic. Issues that an expert-in-training (EIT) will think important will seem pedantic to both the novice and the real expert. Precision demands those issues be dealt with; the EIT is learning this and must find them and deal with them publicly--not only to convince herself that she can but to convince her teachers as well. The expert knows about the issues and deals with them privately. (Dealing with issues privately: One of my favorite teachers taught me that it is impolite to compute in public.) The novice doesn't yet know about the issues--let alone understand them or know how to deal with them.
In point of fact, almost all "proofs" are really plausibility arguments. Good teaching is the art of casting our plausibility arguments at a level that convinces the EIT but keeps him stretching for that conviction, teasing him deeper into more arcane issues, forcing him gently but inexorably toward the expert level.