FOM: "mistaken-proof" theory?
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Fri Sep 18 01:56:50 EDT 1998
Harvey raises the spectre of incorrect proofs of theorems finding
their way into the literature, and the errors in the proof going
unnoticed (at least, for a considerable time). He thinks that
mathematicians will as much as confess to the likelihood of such
happenings, by not being prepared in general to wager all their
personal assets on the correctness of a result claimed to be proved,
even when that claim is accepted according to the prevailing exacting
standards of appraisal within the research community.
I am here not so much interested in the decision-theoretic and
financial aspects of Harvey's speculations (thought they might turn
out to be very rich and interesting), as in possible epistemological
consequences of such situations.
Suppose the claim mistakenly taken to be proved is P. I shall call P
a 'blooper'. Let's take the matter first from the standpoint of a
classical realist. Let's also suppose that P is in some branch of
mathematics like number theory or real analysis, where there is
supposed to be an intended model, and where the point of one's
investigations is to discover whether claims are *true* or *false*
outright. (Not just: true-in-M but false-in-N, but rather:
true-in-the-intended-model/false-in-the-intended-model.) The
classical realist countenances the following distinct possibilities:
T1. P is true, but there is no proof of P; that is, there is not even
a *possible* proof of P, in any conceivable extension of our axioms
and methods of proof that we would accept if we got to consider them
and their consequences. (We somehow have to rule out the silly case of
adopting P as a new axiom.)
T2. P is true, and there is a proof of P; but such a proof is not
constructible within the system of axioms and methods of proof that we
currently accept. It requires adoption of some as-yet-unconsidered
axioms and/or methods of proof that we have not yet considered. (Same
caveat as above.)
T3. P is true, and there is a proof of P in the system of axioms and
methods of proof that we currently accept. (The only trouble is, the
'proof' of P that we have at present is not a legitimate proof.)
F1. P is false, but there is no refutation of P; that is, there is not
even a *possible* refutation of P, in any conceivable extension of our
axioms and methods of proof that we would accept if we got to consider
them and their consequences. (We somehow have to rule out the silly
case of adopting not-P as a new axiom.)
F2. P is false, and there is a refutation of P; but such a refutation
is not constructible within the system of axioms and methods of proof
that we currently accept. It requires adoption of some
as-yet-unconsidered axioms and/or methods of proof that we have not
yet considered. (Same caveat as in F1.)
F3. P is false, and there is a refutation of P in the system of axioms
and methods of proof that we currently accept.
Let's call the 1-type cases 'transcendent'; the 2-type cases
'eventual'; and the 3-type cases 'available'. (Note that I have not
complicated matters further by distinguishing between 'feasible' and
'non-feasible' proofs and refutations.) I want to raise the question:
What can we say about the case of a mistakenly 'proved' proposition P
in each of the situations:
T-transcendent;
T-eventual;
T-available;
F-transcendent;
F-eventual;
F-available ... ?
Question: What well-known bloopers P fall into which of these
categories?
I would imagine that F-available bloopers might be revealed sooner
than bloopers of the other kinds because mathematicians will assume
them (thinking they have been proved) and then deduce patently false
statements from them. Unwilling to believe that they have thereby
(correctly) proved the inconsistency of their axiomatic foundations,
they will 'smoke out' the error in the alleged proof of the blooper,
because they will be motivated now to scrutinize it really closely.
Question: are there any famous examples of this?
But what about the other extreme---a T-transcendent blooper? After
all, it is (ex hypothesi) *true* in the intended structure. So,
relying on it to prove further results won't lead one to any
contradiction. It's just that one will be in the grip of a mistaken
impression that one has certain knowledge (both of the blooper, and of
any result deduced by using it as a premiss). Bloopers of this kind
might go unnoticed (at least, as *not really proven*), because one
will (ex hypothesi) never be prodded into hypercritical scrutiny of
its 'proof'. But, interestingly, if the error in the 'proof' is
discovered (say, by a keen graduate student working his/her way into
the literature) that will not be enough for us to know that the claim
is a *T-transcendent* blooper. All we shall know is that it's a
blooper. In fact, by definition, we could *never* know that a given
blooper is T-transcendent. We could never know its truth value, or
that it is transcendent!
Similarly, it follows from the very definition of 'F-transcendent
blooper' that we could never know its truth value, or that it is
transcendent! Suppose P is an F-transcendent blooper. This means that
(unless we spot the error in the 'proof') we shall never deduce
anything untoward by assuming P. Now, how can an apple be bad without
*coming to be seen as* affecting, if not the whole barrel, then at
least some other parts of it? Any result depending crucially on P as a
premiss would, itself, have to be F-transcendent. Ouch! We would have
been diverted into an undetectably-in-vast-error body of propositions,
taking them as having been proved, hence true, while being massively
mistaken. They would all be false, and transcendently so. Our *only*
way of forswearing our mistaken meanderings in The False would be to
discover the error in the 'proof' of P! Ironically, even then we
would not be able to know that it was F-transcendent progeny of the F-
transcendent blooper that we were mistakenly embracing as true. For
the blooper P's very F-transcendence will never be known.
The epistemological status of a blooper (once we have found the error
in its 'proof') with regard to the realist's *non*-transcendent
categories can in general be difficult to determine. That's because
the *genuine* proofs (refutations), though they exist, might remain
undiscovered. Once they are discovered, however, the category of the
blooper will have been established.
Question: Is there evidence of any kind as to a blooper's
category---once the error(s) in its 'proof' have been
discovered---that could fall short of a genuine proof or refutation of
it? Would a blooper of any category other than T-transcendent
introduce any weird wrinkles in the web of mathematical belief, which
would not themselves provide a definitive proof or refutation of it,
but enable us to finger the blooper as, potentially, a blooper?
So far I have been thinking along lines suggested by the classical
realist's outlook. Let's now try an intuitionistic outlook instead.
The intuitionist will recognize only the four categories
T-eventual;
T-available;
F-transcendent;
F-eventual
as possibilities in principle for any given blooper. The absence of
the transcendent categories is very interesting---especially the
absence of the category 'T-transcendent'. There is an important
contrast to be made out. The classical realist carries on from a
T-transcendent blooper (whose status *as a blooper* has not been
discovered) confident that he is still discovering *truths* (since the
blooper is, ex hypothesi, true; and classical proof methods preserve
truth). By his own lights, the classical realist will have discovered
only truths via the blooper; but, once it's discovered to be a
blooper, he will not be able to claim certain knowledge of any of the
truths that he deduced essentially via the blooper. The intuitionist,
however, cannot even countenance the possibility of "accidentally
getting at further truths" via a blooper. This is because, for the
intuitionist, the error in the (indirect) 'proof' of the blooper will
presumably have blocked his having an effective method for finding a
direct proof (= canonical warrant) for the blooper. The intuitionist
cannot, as the classicist by contrast can, make sense of the
possibility that he might actually have got at certain truths via the
blooper, but been deprived only of his certainty in those results upon
discovering the error in the 'proof' of the blooper. (I use "direct"
v. "indirect" in the proof-theoretic sense of Prawitz and Dummett.)
Question: do these considerations justify one in thinking that an
intuitionist, perhaps even more than the classical mathematician,
should be much more concerned that proofs be correct? Perhaps
not---because the classicist has so much more to fear from having been
misled by an F-transcendent blooper.
Further question: Do we know anything about characteristic features or
the structure of incorrect proofs of P that would enable us to
*induce*, with better-than-random results, from the 'proof' (now
appreciated as incorrect) to the truth-value eventually shown to be
the correct truth-value for P? Do bloopers in twentieth-century
mathematics, when their category is finally determined, tend to be
T-bloopers rather than F-bloopers?
What kinds of logical mistakes are most frequently committed by good
research mathematicians when they 'prove' bloopers? Something as crass
as affirming the consequent? Or a quantifier switch (from "for all x,
there exists y..." to "there exists y such that for all x...")? Or
missing a lurking case in a proof by cases? Or forgetting that an
operation is not commutative? Or forgetting that a relation is not
transitive/symmetric/... etc.? Is it more often a case of smuggling
in an unjustified further assumption, or of making logical misuse of
permitted assumptions?
Can every proof of a T-blooper P be transformed into a mathematically
interesting, non-trivial proof of P from the original permitted
assumptions *plus* some extra assumption(s) which one can see, in
hindsight, how to *read off* from the original mistake in the 'proof'?
If not "every" proof of a T-blooper, then how about "almost every"?
"most"? "a significant few"? "none"?
There seems to be a prospect here of what might deserve the title
"mistaken-proof theory".
Neil Tennant
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