FOM: Intuition, Logic, and Induction.
Alexander Zenkin
alexzen at com2com.ru
Tue Mar 9 20:39:56 EST 1999
Charles Silver (in particular) wrote (RE: FOM: Intuition, Logic, and
Induction. Tue, 9 Mar 1999 10:18:18) to Paul Prueitt:
> I ... did not understand Alexander Zenkin's EA Theorem.
> I still do not understand his points, which you suggest are
> complex and not amenable to the simple kind of answer that I requested.
Dear Charlie,
Sincerely sorry for some misunderstanding! But that was not a trivial
misunderstanding: it showed very clearly how strongly a pure logic and a pure
mathematics are connected in real (scientific!) life with our (scientific!)
psychology, our (scientific!) intuition, and our (scientific!) super-inductive
subconsciousness : -).
Indeed, in my message (Subject: FOM: Intuition, Logic, and Induction. Date:
Sat, 06 Mar 1999 14:54:47) I gave the EA-Theorem formulation in the following
form.
EA-THEOREM. IF there exists at least one natural number, say n*, such
that a predicate Q(n*) holds, THEN for all natural number n > n* a predicate
P(n) is true, or, in a short symbolic form:
_E n* Q(n*) == > _A n>n* P(n). (1*)
Here, P and Q=f(P) are some number-theoretical properties (predicates) , and
symbols "_E" and "_A" simply replace here the usual mathematical words "there
exists" and " for all", correspondingly.
So,
1) any mathematician, seeing the equality Q=f(P), takes, naturally, the
simplest particular case, Q=P, and obtains:
_E n* Q(n*) == > _A n>n* Q(n). (1*)
and, on the spot, he/she presents the killing example: Q(n)="n is a prime
number", since Q(2) is true, then all natural number > 1 are prime. That
mathematician says: "It is a full stupidity". Of course, such the
mathematician will be absolutely right.
The author's psychology: today, I know only four different type of
EA-Theorems (only two are mine); every type is based on an invention, on a
discovery of a unique logical and mathematical connection between two
DIFFERENT unique mathematical properties (predicates) P and Q produced by P,
i.e. P=/=Q. In my message just that unique logical and mathematical
connection is denoted for short by Q=f(P). Why "for short"? - Because the
Super-Induction account occupies about six hours in my lectures course : -)
So, the notation Q=f(P) is not a common mathematical function, and therefore,
in order to avoid misunderstandins, I should have written explicitely the
condition P=/=Q. My psychology did not allow me to foresee such the possible
case.
2) Any logician, seeing the (1*), can say: "an AUTHENTIC inference of a
COMMON statement _A n>n* P(n) from a SINGLE statement _E n* Q(n*) is
impossible, because such the inference contradicts to J.S.Mill's inductive
logic, to all our scientific experience, and to our scientific intuition!"
The author's argumentation: my scientific experience and my scientific
intuition, for a long time, too were hindering me to understand that
EA-Theorems are a new type of logical and mathematical authentic reasonings,
but, on the other hand, the MATHEMATICAL EA-Theorems are proved by rigorous
MATHEMATICAL methods, and it is not admissible to dispute against mathematics.
Ultimately, being a mathematician, I put absolute trust in MATHEMATICS : -).
But there are two arguments that, I hope, can shake the natural distrust of
logicians: firstly, all EA-Theorems statements refer (today) to the natural
numbers (or to the natural indexes of objects), but the series of natural
numbers, thanks to "if n then n+1", is a very specific, strong coherent
structure (it is not "a J.S.Mill's set of particular (frequently, - quite
accidental) facts") which is adapted very well for a realization of "chain
reactions", and, secondly, such the predicate, P, which such the predicate, Q,
can be invented for, so that the corresponding EA-Theorem (1*) could be
proved, is very rare event in modern mathematics: repeat, there are only four
such the events. It needs to study their properties much better in order to
understand why they are in existence.
At last, the notation (1*) does not contain explicit logical and
mathematical definitions of concrete predicates P and Q. In oder to any
mathematician and logician will believe that the notation (1*), taking into
account its "impossible" form, is not an anti-Sokal-like mystification, he/she
must personally to test a real EA-Theorem and its real mathematical proof.
Unfortunately, our "linear" texts are adapted for mathematical language not
so well. I am afraid that even the real H.E.Richert's EA-Theorem formulation
which is contained in the end of my previous message {Subject: FOM: Intuition,
Logic, and Induction. Date: Sat, 06 Mar 1999 14:54:47} can be read not
adequately by different browsers (I see that my own FOM-reply is quite
deformed).
Therefore, I can suppose only the references to some EA-Theorems with
their proofs (in English).
1. H.E.Richert's EA-Theorem, W.Sierpinski, Elementary Theory of Numbers. -
Warszaw, 1964, Chapter III. Prime numbers, pp. 143-144.
2. A.A.Zenkin, 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 Akademii Nauk, Vol 354, No. 5, 1997, pp. 587 -
589.
3. A.A.Zenkin, Waring's problem: g(1,4) = 21 for fourth powers of positive
integers.- An International Journal "Computers and Mathematics with
Applications", Vol.17, No. 11, pp. 1503 - 1506, 1989.
In my previous message, I described the main stages of the both methods:
the Complete Mathematical Induction and the Super-Induction. Here I would like
to add the following.
1) In the both methods, there is a threshold number n*. In the Complete
Mathematical Induction the number n*, usually, is equal to 0 or 1 that,
usually, is defined by a trivial non-feasibility of the predicate P (a
devision by zero, an imaginarity and so on). In the Super-Induction metod, the
threshold number n* can have any finite values (of course, in different
problems).
2) In the Super-Induction method, we can invent the Q, and we can prove the
corresponding EA-Theorem (1*), but, untill we will find a number n* (a trigger
of the Super-Induction) such that Q(n*) holds, our "proof" proves nothing,
because the EA-Theorem itself does not guarantees the existence of the
threshold numbers n*. As is known, the last problem simply does not appear in
a framework of the Complete Mathematical Induction method.
The following example elucidates the said.
Consider the old expression n^2+n+41 and the predicate P(n) = "n^2+n+41 is
a composite number". Since P(n) is false for all n=0,1,2,...,39, but P(40) is
true, nobody will wish to use here the common Complete Mathematical Induction.
But the Super-Induction, generally speaking, can be used here under the
following two conditions.
1) If for the given predicate P(n), we will be able (today) TO INVENT a
predicate Q(n), which, of course, will be depended upon P (and, possibly, upon
other parameters of the problem), and such Q(n) that the EA-Theorem (1*) could
be proved. If so, then
2) We must find a natural number n*.
After that only, we can state that for all n>n* the expression n^2+n+41
defines only composite numbers. Of course, if a joker already did not proved
some earlier that a quantity of prime numbers generated by the expression
n^2+n+41 is infinite : -).
> On Tue, 9 Mar 1999, Paul Prueitt wrote to
> "'Charles Silver'" <csilver at sophia.smith.edu>:
> Your question is a good question and should be given clarity at the
> minimal complexity, so that we can all share in the insight of super
> induction. For example, how does Robert's epistemology of social
> science see the difference between classical mathematical induction and
> the super induction of Zenkin. My point was that there is a
> neuropsychological (and quantum field theory) basis for grounding any
> representation of anything that can be associated with the term
> "induction". and I asked Alex to address this in as simple a fashion as
> possible.
Dear Paul,
you absolutely right as to the intention to "the minimal complexity": you have
guessed my constant intention to a maximal clarity at "the minimal
complexity". Above, I have tryed to do that. I hope the problem became some
more clear, but as Sokrat (?) said "I know that I know nothing"! The Charles
Silver's and yours questions touched upon some very deep problems, connected
with Intuition, Logic, and Induction, and especially in connexion with the
FOM-discussion on Visual Proofs. The question is such. For number-theory
problems solution by means of the Super-Induction method, I use my Cognitive
Computer Visualization System DSNT, i.e., I simply visualize an intitial
segment of the natural numbers series with the predicates P and Q given on the
set. Of course, the corresponding (color-musical) 2D-images are constructed by
a computer. So, I can now to find the threshold number n* not only visually,
but also hearing the corresponding mathematical music. It is not a mystic, it
was done many years ago. But a very interesting question (of an
epistemological-social sense : -) arises: is a physical fact of my hearing the
number n* a legitimate argument for the authentic truth of the common
mathematical statement _A n>n* P(n)? : -) The same question refers to the
case (if the segment is very large) when I stare at not a static picture, but
at, say, a 30-min color-musical cartoon? : -)
In one word, I agree with you that very interesting and quite unexpected
"neuropsychological (and quantum field theory), and especially educational
investigations might be done here. All the more, as you certainly keep in
mind, Karl Pribram and I discovered very unexpected similarity between his
experimental pictures (if my memory serves correctly) of active points of
cerebral cortex, and my color-musical pythograms of Euler's Theorems on sums
of two squares of non-negative integers. That time, we agreed that such the
coincidence might be not accidental. Alas, as far back as 1995, at IEEE'95,
Monterey.
So, dear Charles and Paul, once more thanks very much for your deep
questions, remarks, and desires. I would be very glad to answer all possible
questions for clarification of the Super-Induction logical Nature.
Sincerely yours,
A Z
**************************************************
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computer Center
of the Russian Academy of Sciences.
e-mail: alexzen at com2com.ru
WEB-Site http://www.com2com.ru/alexzen
**************************************************
"Infinitum Actu Non Datur" - Aristotle.
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