Date: Mar 12, 1997 1:29 AM Author: Lou Talman Subject: Plausibility Arguments Ron Ward asks:

> 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.

--Lou Talman