Generalized Logic
From a New Zealand Mathematics Journal.
(This might come in handy for a math midterm.)
THE USES OF FALLACY
In the last hundred
years or so, mathematics has undergone a tremendous growth in size and
complexity and subtlety. This growth
has given rise to a demand for more flexible methods of proving theorems than
the laborious, difficult, pedantic, “rigorous” methods previously in
favor. This demand has been met by what
is now a well-developed branch of mathematics known as Generalized Logic. I don’t want to develop the theory of
Generalized Logic in detail, but I must introduce some necessary terms. In Classical Logic, a Theorem consists of a
True Statement for which there exists a Classical Proof. In Generalized Logic, we relax both of these
restrictions: a Generalized Theorem
consists of a Statement for which there exists a Generalized Proof. I think that the meaning of these terms
should be sufficiently clear without the need for elaborate definitions.
The applications of
Generalized Proofs will be obvious.
Professional authors of textbooks use them freely, especially when
proving mathematical results in Physics texts.
Teachers and lecturers find that the use of Generalized Proofs enables
them to make complex ideas readily accessible to students at an elementary
level (without the necessity for the tutor to understand them himself). Research workers in a hurry to claim
priority for a new result, or who lack the time and inclination to be pedantic,
find Generalized Proofs useful in writing papers. In this application, Generalized Proofs have the further
advantage that the result is not required to be true, thus eliminating a
tiresome (and now superfluous) restriction on the growth of mathematics.
I want now to consider
some of the proof techniques that Generalized Logic has made available. I will be concerned mostly with the ways in
which these methods can be applied in lecture courses – they require only
trivial modifications to be used in text books and research papers.
The reductio
methods are particularly worthy of note.
There are, as everyone knows, two reductio methods
available: reductio ad nauseam
and reductio ad erratum. Both
methods begin in the same way: the mathematician denies the result he is trying
to prove, and writes down all the consequences of this denial that he can think
of. The methods are most effective if
these consequences are written down at random, preferably in odd vacant corners
of the blackboard.
Although the methods
begin in the same way, their aims are completely different. In reductio ad nauseam the lecturer’s
aim is to get everyone in the class asleep and not taking notes. (The latter is a much stronger
condition.) The lecturer then has only
to clean the blackboard and announce, “Thus we arrive at a contradiction, and
the result is established.” There is no
need to shout this – it is the signal for which everyone’s subconscious has
been waiting. The entire class will
awaken, stretch, and decide to get the last part of the proof from someone
else. If everyone had stopped taking
notes, therefore, there is no “someone else” and the result is established.
In reductio ad
erratum the aim is more subtle. If
the working is complicated and pointless enough, an error is bound to
occur. The first few such mistakes may
well be picked up by an attentive class, but sooner or later one will get
through. For a while, this error will
lie dormant, buried deep in the working, but eventually it will come to the surface
and announce its presence by contradicting something that has gone before. The theorem is then proved.
It should be noted that
in reductio ad erratum the lecturer need not be aware of this random
error or of the use he has made of it.
The best practitioners of this method can produce deep and subtle errors
within two or three lines and surface them within minutes, all by an
instinctive process of which they are never aware. The subconscious artistry displayed by a really virtuoso master
to a connoisseur who knows what to look for can be breathtaking.
There is a whole class
of methods which can be applied when a lecturer can get from his premisses P to
a statement A, and from another statement B to the desired conclusion C, but he
cannot bridge the gap from A to B. A
number of techniques are available to the aggressive lecturer in this
emergency. He can write down A, and
without any hesitation put “therefore B.”
If the theorem is dull enough, it is unlikely that anyone will question
the “therefore.” This is the method of
Proof by Omission, and is remarkably easy to get away with (sorry, “remarkably
easy to apply with success”).
Alternatively, there is
the Proof by Misdirection, where some statement that looks rather like “A,
therefore B” is proved. A good bet is
to prove the converse “B, therefore A”:
this will always satisfy a first-year class. The Proof by Misdirection has a countably infinite analogue, if
the lecturer is not pressed for time, in the method of Proof by Convergent
Irrelevancies.
Proof by Definition can
sometimes be used: the lecturer defines
a set S of whatever entities he is considering for which B is true, and
announces that in future he will be concerned only with members of S. Even an honors class will probably take this
at face value, without inquiring whether the set S might not be empty.
Proof by Assertion is
unanswerable. If some vague waffle
about why B is true does not satisfy the class, the lecturer simply says, “This
point should be intuitively obvious. I’ve
explained it as clearly as I can. If
you still cannot see it, you will just have to think very carefully about it
yourselves, and then you will see how trivial and obvious it is.”
The hallmark of a Proof
by Admission of Ignorance is the statement, “None of the text-books makes this
point clear. The result is certainly
true, but I don’t know why. We shall
just have to accept it as it stands.”
This otherwise satisfactory method has the potential disadvantage that
somebody in the class may know why the result is true (or, worse, know why it
is false) and be prepared to say so.
A Proof by Non-Existent
Reference will silence all but the most determined troublemaker. “You will find a proof of this given in
Copson on page 445” which is in the middle of the index. An important variant of this technique can
be used by lecturers in pairs. Dr.
Jones assumes a result which Professor Smith will be proving later in the year
– but Professor Smith, finding himself short of time, omits that theorem, since
the class has already done it with Dr Jones...
Proof by Physical
Reasoning provides uniqueness theorems for many difficult systems of
differential equations, but it has other important applications besides. The cosine formula for a triangle, for
example, can be obtained by considering the equilibrium of a mechanical
system. (Physicists then reverse the
procedure, obtaining the conditions for equilibrium of the system from the
cosine rule rather than from experiment.)
The ultimate and
irrefutable standby, of course, is the self-explanatory technique of Proof by
Assignment. In a textbook, this can be
recognized by the typical expressions “It can readily be shown that...” or “We
leave as a trivial exercise for the reader the proof that...” (The words “readily” and “trivial” are an
essential part of the technique.)
An obvious and fruitful
ploy when confronted with the difficult problem of showing that B follows from
A is the Delayed Lemma. “We assert as a
lemma, the proof of which we postpone...”
This is by no means idle procrastination: there are two possible denouements. In the first place, the lemma actually may be proved later on,
using the original theorem in the argument.
This Proof by Circular Cross-Reference has an obvious inductive
generalization to chains of three or more theorems, and some very elegant
results arise when this chain of interdependent theorems becomes infinite.
The other possible fate
of a Delayed Lemma is the Proof by Infinite Neglect, in which the lecture
course terminates before the lemma has been proved. The lemma, and the theorem of which it is a part, naturally will
be assumed without comment in future courses.
A very subtle method of
proving a theorem is the method of Proof by Osmosis. Here the theorem is never stated, and no hint of its proof is
given, but by the end of the course, it is tacitly assumed to be known. The theorem floats about in the air during
the entire course, and the mechanism by which the class absorbs it is the
well-known biological phenomenon of osmosis.
A method of proof, which is regrettably little used in undergraduate
mathematics, is the Proof by Aesthetics (“This result is too beautiful to be
false”). Physicists will be aware that
Dirac uses this method to establish the validity of several of his theories,
the evidence for which is otherwise fairly slender. His remark “It is more important to have beauty in one’s
equations than to have them fit experiment” [1] has achieved a certain
fame.
I want to discuss
finally the Proof by Oral Tradition.
This method gives rise to the celebrated Folk Theorems, of which
Fermat’s Last Theorem is an imperfect example.
The classical type exists only as a footnote in a textbook, to the
effect that it can be proved (see unpublished lecture notes of the late
Professor Green) that... Reference to
the late Professor Green’s lecture notes reveals that he had never actually
seen the proof, but had been assured of its validity by in a personal
communication, since destroyed, from the great Sir Ernest White. If one could still track it back form here,
one would find that Sir Ernest heard of it over coffee one morning from one of
his research students, who had seen a proof of the result, in Swedish, in the
first issue of a mathematical magazine which never produced a second issue and
is not available in the libraries. And
so on. Not very surprisingly, it is
common for the contents of a Folk Theorem to change dramatically as its history
is investigated.
I have made no mention
of Special Methods such as division by zero, taking wrong square roots,
manipulating divergent series and so forth.
These methods, while very powerful, are adequately described in the
standard literature. Nor have I
discussed the little-known Fundamental Theorem of All Mathematics, which states
that every number is zero (and whose proof will give the interested reader many
hours of enjoyment, and excellent practice in the use of the methods outlined
above). However, it will have become
apparent what riches there are in the study of Generalized Logic, and I appeal
to Mathematics Departments to institute formal courses in this discipline. This should be done, preferably at
undergraduate level, so that those who go to teaching with only a Bachelor’s
degree should be familiar with the subject.
It is certain that in future nobody will be able to claim a mathematical
education without a firm grounding in at least the practical applications of
Generalized Logic.
[1] PAM Dirac, The Evolution of the Physicist’s
Picture of Nature
Scientific American, May 1963, p
47.