FOM: About some epochal FOM-jokes.
Alexander Zenkin
alexzen at com2com.ru
Tue Aug 11 16:29:16 EDT 1998
Neil Tennant {of the Date: Sun, 2 Aug 1998} and Stephen G Simpson {of
the Mon, 3 Aug 1998} offered the lists of the epochal items concerning
the FOM-area and call to extend the list by related items. As is known,
the outstanding number theorist G.H.Hardy stated: "┘there is no
permanent place in the world for ugly mathematics." I believe a bore
gives birth the ugly mathematics. Therefore I offer the following item:
epochal FOM-jokes.
In what follows, three of such , in my opinion, epochal jokes are
presented.
1) How to generalize the classical Aristotel▓s modus ponens rule?
As is known, the modus ponens rule is as follows: A, A ==> B |- B,
i.e., if the premises A and A ==> B are authentic, then B is authentic
too.
From Aristotel▓s time, all classical logicians and natural scientists
believe (almost at the genetic level) that here the premise A must be a
common statement, and the consequence B may be only a less common
statement, or a partial statement, or even a single one.
As is known, the modern META-Mathematics states that the modus ponens
rule (the postulate of the propositional calculus by S.Kleene) is used
correctly under the sole condition: the formulas A and A ==> B must be
deducible from a given axiom system. So, the modern META-Mathematics
does not formally forbid such the interpretation of the modus ponens
rule where the consequence B is a common statement and the premise A is
a less common one.
But the question arises: does the last ⌠impossible■ situation exist,
say, in Mathematics in general?
The following little known Lemma of the German mathematician
H.E.Richert says ⌠Yes■ {see W.Sierpinski, Elementary Theory of Numbers.
- Warszaw, 1964, Chapter III. Prime numbers, pp. 143-144. Below, I give
it in some inessentially changed notation}.
H.E.Richert▓s LEMMA (1949). Let { m1, m2, ┘} be an infinite
sequence of natural numbers such that for a certain natural number k the
inequality
(18) m_(i+1) < = 2 m_i for all i > k,
holds and let s0 >= m_(k+1) be a natural number.
In such the case,
IF there exist an integer n* > = 0 such that each of the numbers
(21) n*+1, n*+2, ┘, n*+s0
is the sum of different terms of the sequence { m1, m2, ┘, mk },
THEN every natural number n > n* is the sum of different terms of the
sequence {m1, m2, ┘ }.
Define the following two number-theoretical predicates:
P(n) = "n is the sum of different terms of the sequence { m1, m2, ┘ }",
and
Q(n) = Q1(n)&Q2, where Q1(n) = " n+j is the sum of different terms of
the sequence { m1, m2, ┘, mk } for every j=1,2,..., s0 " and Q2 = "s0
>= m_(k+1)",
Then we can rewrite H.E.Richert's Lemma in the following form.
H.E.Richert▓s LEMMA. Let { m1, m2, ┘ } be an infinite sequence of
natural numbers such that for a certain natural number k the inequality
(18) m_(i+1) <= 2 m_i for i > k
holds and let s0>= m_(k+1) be a natural number.
In such the case,
IF there exists a number n* >= 0 such that Q(n*) is true,
THEN for any n > n* P(n) is valid,
or, in the short symbolic notation,
_En*Q(n*) ==>
_An>n*P(n), (1)
{Remark: here and further: ⌠_E■ = ⌠Exists■, ⌠_A■ = ⌠for any■ and ⌠==>■ =
⌠IF ┘, THEN ┘■}.
So, the MATHEMATICAL PROOF ITSELF of the H.E.Richert Lemma is the
authentic inference of the common statement _An>n*P(n) from the single
statement _En*Q(n*) ! But the Lemma itself does not prove the truth of
the common statement _An>n*P(n). To do that, it is necessary to prove
the truth of the single (!) statement _En*Q(n*) out of the Lemma. And in
order to do the last, it is sufficient to find even one a number n* for
which mathematical predicate Q(n*) is true. Only from the proven truth
of the single statement _En*Q(n*) and the proven truth of the
conditional statement (1), it follows, by the modus ponens rule, that
the common statement _An>n*P(n) is the authentic truth.
Using his Lemma, H.E.Richert proved many interesting theorems on the
representability of natural numbers by sums of different prime numbers.
As I know today, the H.E.Richert Lemma is the first example in all
History of Mathematics and Logic, which generalizes the classical modus
ponens rule on the case with a single premise A and a common consequence
B.
I am sure that the H.E.Richert Lemma is really an epochal
achievement in classical Mathematics and classical Logic from the
ancient epoch.
2) How to disprove the classical J.S.Mill▓s inductive Logic Paradigm?
As is known, the main J.S.Mill paradigm of the Classical Inductive
Logic says: a common statement, say, B may be inferred from a partial
(and even single) statement, say, A, but such the common statement B
will always be only a plausible, probable, but never authentic
hypothesis.
More than 150 years, all classical inductive logic and all natural
science are based on that paradigm as if it went without saying.
As said above, the H.E.Richert Lemma states that, in Mathematics,
there exist such the Theorems that are authentic inferences ⌠from a
single statement to a reliable common one■, i.e., disprove an absolute
character of J.S.Mill▓s paradigm of classical induction.
It is really an epochal result too.
3) How to generalize the classical B.Pascal Method of the Complete
Mathematical Induction (CMI)?
As is well known, the classical B.Pascal CMI-method for a given
mathematical predicate P(n) is formulated as follows:
[P(n*)&[_An>=n*[P(n) ==> P(n+1)]]] ==> _An>=n*P(n). (2)
As far back as 1978-1979, I discovered, independently of H.E.Richert,
and proved two different classes of theorems
{see, for example,
[4] Zenkin A.A., Superinduction: A New Method For Proving General
Mathematical Statements With A Computer. - Doklady Mathematics, Vol.55,
No.3, pp. 410-413 (1997). Translated from Doklady Alademii Nauk, Vol
354, No. 5, 1997, pp. 587 - 589.
[6]. Zenkin A.A., Generalized Waring's Problem: An Estimation Of
G(m,r)-Function Via g(m-1,r)-Function By Means Of The Superinduction
Method. - Doklady Mathematics, vol 56, No. 1, pp. 597-600 (1997).
Translated from Doklady Alademii Nauk, Vol 355, No. 6, pp. 727 -
730.(1997). }
of the same form:
_En*Q(n*) ==>
_An>n*P(n), (1)
Just the fact that I have two different types of such the theorems
permitted me to understand and to formulate the Super-Induction (SI)
method which is a common method for proving such unusual mathematical
Theorems and which I have said about earlier, at the FOM (Febriary-98).
Remember some about the SI-method.
Let we wish to prove a common mathematical statement, _An>=1P(n).
Then we invent (contrive, think out, devise, ⌠spin out of thin air■)
some new mathematical predicate Q = f(P) such that a) the conditional
mathematical statement of the form _En*Q(n*) ==> _An>n*P(n) has a
mathematical sense, and b) the mathematical statement (1) can be proven.
If such the proof was a success (but of course, it does not always
work), then it remains only to find a natural number n* (even one) such
that Q(n*) is true (it is also not guaranteed). If these a) and b) were
successes, then we have proved our common statement either in the
traditional form _An>=1P(n) or in the some non-traditional form:
_An>=1P(n) except for n (- Ne, where the finite exceptional set Ne =
{1<=n<=n* : not-P(n)}.
I wish to emphasize that the choice of Q = f(P) is quite arbitrary
and must satisfy only the conditions a) and b). Therefore, we can
assume, for example, that, for a given P(n), predicate Q(n*) =
P(n*)&[_An>=n*[P(n) = => P(n+1)]]. Substituting that value of Q(n*) in
(1) of the SI-method, we get the following conditional statement:
_En* [P(n*)&[_An>=n*[P(n) ==> P(n+1)]]] ==> _An>=n*P(n), (3)
that is we get the CMI-Method of B.Pascal in the almost classical form
(2).
I must emphasize that (3) is not a formal attachment of the ⌠existence■
quantifier to the ⌠formula■ (2): the expressions (1), (2) and (3) are
not formulas of any formal calculus, they are brief symbolic notations
of the corresponding theorems of the classical mathematics.
So, the classical B.Pascal CMI-Method is a special case of the
SI-Method.
As is known, the famous META-mathematical generalization of the
classical B.Pascal CMI-Method, - the without doubt epochal transfinite
induction method (up to Cantor▓s e0 = ⌠Epsilon_Zero■), - is the main
tool for proving the consistency of the classical Peano arithmetic. I am
sure that if META-mathematicians will ever generalize the classical
SI-Method, they will be able to arrive the first unattainable ordinal
number, if, of course, they will be able to find a suitable ⌠akupuncture
point■ n* in a sufficiently long transfinite series.
Thus, the German mathematician H.E.Richert turned out a very fine
joker: it took the entire second-half of the XX Century in order to
understand that his single mathematical Lemma generalizes at once
Classical Aristotel▓s Logic (to the case of the modus ponens rule with a
single premise and a common consequence), Classical J.S.Mill▓s inductive
Logic (to the case of the authentic induction ⌠from a single to a
common■), and Classical B.Pascal▓s CMI-Method (to the SI-method).
I am sure that such the H.E.Richert▓s joke may be entirely assessed as
the epochal event.
Best Regards,
Alexander Zenkin
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