FOM: Reverse Mathematics and GAPs_in_THE_CONTINUUM.

[Alexander Zenkin]

At 10:31 AM 8/6/99 +0300, Karlis Podnieks FOM-wrote (among other
things):
>Hence, reverse mathematics seems to be a normal mathematical approach
to >some problems.

I agree with Karlis Podnieks.
In the science, there are two main processes of cognition:
A) to create a theory describing a set of given facts (in particular, to
construct a (minimal) axioms system which produces a set of given
statements, as today the reverse mathematics does it);
B) to produce (to deduce, to calculate, etc.) a set of new (theoretical)
facts from the given theory (in particular, to deduce a set of (formal)
statements from the given axioms system, as the modern meta-mathematics
does it ).
 In reality, we have the following uninterrupted dialectical process:
Facts = > Theory = > New Facts = > New Theory = > New Facts = > and so
on.
 Such the process is a usual way of a ScienceTheory development in
mathematics.
 We need in a New Theory every time when the existing (Old) Theory
becomes unsatisfactory in either certain sense.
I believe the appearance itself of the Harvey Friedman's and Stephen
Simpson's reverse mathematics is evidence of a dissatisfaction with the
modern meta-mathematics, since the reverse mathematics is an opposition
not to the really working, even pure, mathematics, but to the modern
axiomatic proof theory, i.e., just to the meta-mathematics. Therefore, I
consider the reverse mathematics as an opportune, useful and perspective
scientific direction.
 The famous Hilbert's program has failed not only thanks to the known
Geodel's theorems, but first of all because that program was trying to
formalize all the mathematics by means of the banishment of the
semantics of that mathematics from its (of the mathematics) formal
descriptions (recall the famous Hilbert's "semantic isomorphism" between
the words "point, line, plane" and the words "table, chair, beer mug").
But to rob the Mathematics of its semantics is much more dangerous than
"to ban boxers to use their fists". And I hope that new versions of a
formal proof theory-XXI will less negligently treat and more effectively
utilize that mathematics semantics. For example, as the modern
mathematics (mathematical modelling as a whole, Victor K.Finn's
quasi-axiomatic systems, and so on) is able to do that.

At 10:47 8/6/99, Martin Davis FOM-responded to Karlis Podnieks  (among
other things):

>To quote Jacobi: "Man muss immer umkehren". That is: One must always
invert.
> Perhaps a light touch will dispell some of the spurious melodrama seen
on the list these days!

 That short quotation defines the epistemological backgraound of the
situation best of all.
 I think the question (possibly, in its more wide aspect) is about the
most general "technologies" of the human-being's cognition (not only
about the mathematical one).
 As is known, a human-being cognizes any unknown things by virtue of
known things, viz
A) either by an analogy - a kind of a "short-range" cognition: a known
property of one thing is attributed to other ("resembling") thing;
B) or by an opposition (inversion) - a kind of a "long-range" cognition:
a negation (frequently, not very strong one) of a known property of one
thing is attributed to other ("opposite") thing.

 Of course, not all is so simple here from the point of view of the
formalization of these cognitive processes. Firstly, much dependes from
a choice of a concrete property (in the both cases A and B), and,
secondly, - from a "degree" of the inversion of a choiced property (in
the case B).
 I would like to illustrate that "technologies" by the two, quite
unusual examples.

 FIRST EXAMPLE. Consider the property (operation) "multiplication". What
a property (operation) will be an inverted one? - Obviously, the
property (operation) "division" from the common arithmetics point of
view.
 As is known, the Eratosthenes' sieve may be called as the
MULTIPLICATIVE sieve since the corresponding algorithm allows to divide
the set of all natural numbers into two subsets: the set of prime
numbers and the set of all possible PRODUCTs of prime numbers. That is
such the partition is defined by the property (operation)
"multiplication". Now, whether anybody can imagine a like algorithm
defined by the property (operation) "division"? - I think, it will be
not easy, if any.
 But if we replace the property (operation) "multiplication" by its more
soft opposition (the case B), viz by the property (operation)
"addition", we obtain an ADDITIVE Eratosthenes' sieve with the same,
almost "word-for-word" algorithm (the case A) allowing to divide the set
of all natural numbers into two subsets: a set of so-called ADDITIVE
PRIME numbers and a set of all possible SUMs of elements from some given
basic subset of natural numbers.
 As it was shown, the main problems of Classical Additive Number Theory
(for example, such as Fermate's, Goldbach's, Waring's problems, etc.)
can be formulated in a framework of the ADDITIVE Eratosthenes' sieve. A
lot of new number-theoretical results were really obtained by means of
that ADDITIVE Eratosthenes' sieve basing on the Super-Induction method
and EA-Theorems. {More details and references are presented in my
[HM]-message:
 Subject: [HM] ADDITIVE VERSION OF ERATOSTHENES' SIEVE AND EA-THEOREMS.
     Date: Fri, 30 Jul 1999 11:57:59 +0400
Reply-To: historia-matematica at chasque.apc.org}.
 So, this example shows that, using quite not close analogies (the case
A) and quite soft oppositions (the case B) to known mathematical
statements, we can produce really new and very not-trivial mathematical
results.

 SECOND EXAMPLE. Consider the notion of the real number (geometrical
point), say, of the segment [0,1]. What a notion will be an inverted
one? - Probably, somebody will name the complex number notion. It is so.
But it turns out that there is the following, quite unexpected
opposition, inversion (the case B) to the real number (geometrical
point) notion.

 However, first of all, I would like to thank John Conway (20 July),
Udai Venedem (21, 23 July), and James A. Landau (27 July) answered
kindly my [HM]-request (19 July) as to the subject: "an indeterminacy of
rational numbers representation", and to make some remarks.
 If my memory serves me well, N.Bourbaki said that Mathematics is a
science about (formal) notations in the sense that if a thing has not
(can't have) a notation then such a thing does not exist from the
Mathematics point of view. Anyway, from the FORMAL Mathematics point of
view. Therefore, I don't deem that "to think of numbers in terms of
their notations is our thought-habit" only, as John Conway wrote. I
believe that it is one of the important demand of modern formal
Mathematics (and, of course, modern meta-Mathematics and Mathematical
Logic): there is not a MATHEMATICAL object if it hasn't any notation.
 I don't agree with "a Platonic point of view, where *the nature of
numbers precedes the nature of everything that has a nature* (as says
Ibn 'Arabi)" (see Udai Venedem's messages), - though all depends, of
course, on the interpretation of the first occurence of the term
"nature" in Ibn 'Arabi's citation, - but I agree with Udai Venedem in
that "lines on a stick" for a shepherd, - and also fingers and stones
for more ancient human-beings calculations, - were just the notations of
numbers (maybe, some more primitive and material notations than the
formal symbolic notations of the modern meta-Mathematics), since were
such the lines, fingers, and stones absent, ancient people could speak
about anything except for the numbers and calculations.

 James A. Landau writes (among other things):

 Zeno's "Dichotomy" Paradox can be expressed as follows:  (using binary
fractions) Achilles cannot reach the end of a field one stadium long
because first he must reach the halfway point (0.1 stadium), then the
halfway point  of the remaining distance (0.11 stadium), and so on.
 One of SEVERAL possible ways of interpreting the Dichotomy Paradox is
that Zeno is arguing that the number 0.11111... does NOT equal 1.0.

 Alexander Zenkin (AZ):
 I agree with James A. Landau and Zeno but under the two following
conditions: 1) if that "Dichotomy" is the POTENTIALLY infinite PROCESS
since only in that case 0.11111... =/= 1.0, and 2) if Achilles realizes
that POTENTIALLY infinite process not by means of his foots, but by
means of his, and Zeno's, head, i.e., in their imagination only. : -)

 Further James A. Landau writes:
 Now take an arbitrary real number from (0, 1).  What is the probability
that its decimal expansion will repeat with a period of one (as does
1/30 =  .0333333...)  Start with any decimal digit.  The odds that the
next digit of the arbitrary number will be the same is 1/10.  The odds
that the next two  will be the same is 1/10x1/10.  The odds that all
remaining digits will be the same is the limit as n goes to infinity of
(1/10) to the n-th, which is  zero.  This is true no matter at which
decimal position you start.  Hence the  probability is zero that this
arbitrary real number has a repeating decimal expansion of length one.
Similarly for length 2, 3, etc.  Hence the  probability is zero that an
arbitrary real number is rational, from which we can deduce that the
real numbers are uncountable.

 AZ:
 I believe that James A. Landau proved much more, viz that the rational
number, say 1/30 =  .0333333... ,  simply does not exist in the Nature,
as well as all other rational numbers though (of course, from the
Probability Theory point of view only). : -)

 Now, as to my question to [HM]-list (on the Date Mon, 19 Jul 1999
23:36:01) about "an indeterminacy of rational numbers representation".
 I believe that the AGREEMENT that the two DIFFERENT binary notations
0. a1 a2 . . . ak 1 0 0 0 . . . ,
and
0. a1 a2 . . . ak 0 1 1 1 . . . ,
where all ai ,  i = 1, 2, ..., k,  are "0" or "1",
represent THE SAME rational number (here - proper fraction) could appear
only after the appearance of the infinite binary (decimal etc.) notation
of real numbers and of the notion of the accuracy of calculations since
the two DIFFERENT notations, for example, 1.0000 and 0.1111 define the
same number only "within the accuracy of", say 2^(-3).

 Now, some words about the case where such the AGREEMENT does not work.

 Consider the well-known INFINITE binary graph - the tree, say T, with
the root V:

                        V
                   /             \
              0                     1
          /       \              /       \
        0         1             0         1
      /  \      /  \         /   \     /    \
   0     1    0    1        0     1     0     1
  / \   / \  / \  / \      / \   / \   / \    / \
 .   .   .   .   .   .   .   .   .   .   .   .   .

 As is known, the tree T is a graphic representation of the set, say X,
of all real numbers (or, that is the same, of all geometrical points) of
the segment [0,1], in the sense that there is a 1-1-correspondence, say
PSI, between a set of all infinite binary paths on the tree, T, of the
form:

(1) V a1 a2 a3 . . . an . . . , where for all i>=1 [[ai =0]OR[ai =1]],

and the set, X.

 It is obiously, that only one path (1) PSI-corresponds to a given real
number (geometrical point) x from [0,1], and only one real number
(geometrical point) x from [0,1] PSI-corresponds to a given path (1).
Otherwise the tree T would have circles that conflicts with the
definition itself of the tree notion of the modern Graph Theory.
 Further, as it is easy to see, the two different infinite RATIONAL
paths

V 0 1 1 1 . . .
and
V 1 0 0 0 . . .

are possessed of the unique property that THERE ARE NOT OTHER PATHS
BETWEEN THEM, i.e., these two paths define two different geometrical
points (rational numbers), say r1 and r2, of the segment [0,1] which
there are not other real numbers (points) between. Thus, these two
rational paths define a GAP on the segment [0,1] between the different
rational numbers (points) r1 and r2. Obviously, that the GAP's "length"
is equal to zero.
 I think this FACT is "of clear and obvious <mathematical and
philosophical> GENERAL INTELLECTUAL INTEREST " and is a unique "NATURAL
EXPERIMENT" which the Nature realized in a human-being mind. : -)
 Since EVERY vertex of the tree T generates at least one GAP, and
different vertexes generate different GAPS, we have the

THEOREM 1. The set, say G, of all GAPS on the segment [0,1]  is an
INFINITE set.

 Let now x1 and x2 be two ARBITRARY different real numbers (geometrical
points) of the segment [0,1], and the FINITE binary sequence,

V a1 a2 a3  . . . an,

is a common part of the two paths PSI-corresponding to x1 and x2. As is
easy to see, the two infinite RATIONAL paths,

(1a) V a1 a2 a3 . . . an 1 0 0 0 . . .
and
(1b) V a1 a2 a3 . . . an 0 1 1 1 . . .

define a GAP between x1 and x2. Call such the gap as a RATIONAL GAP.

So, according to the Classical Mathematical Analysis, we have the

THEOREM 2. The infinite set of all RATIONAL GAPS is an ORDERED,
CONTINUAL, EVERYWHERE DENSE set on the CONTINUAL segment [0,1]. The set
is ORDERED by the same way as the set of real numbers.

Consider any IRRATIONAL  number (point), say x, belonging to [0,1]:

(2) x = 0. a1 a2 a3 . . . an . . .  .

For any index n>=1 we have a RATIONAL gap, say gn, defined by the pair
of rational numbers (1a) and (1b).
So, we obtain a FUNDAMENTAL (in Cauchy'es, Weierstrass', and Dedekind's
sense ) sequence of the RATIONAL gaps,

(3) g1, g2, g3, . . . , gn, . . . ,

which defines (converges to) an IRRATIONAL GAP, say gx, connected with
(and generated by) the given IRRATIONAL number x:

  lim gn = gx,  by n == > oo.

Thus, we have proved

THEOREM 3. The cardinality (power) of the complete, ordered, infinite
set, G, of all gaps in the segment [0,1] is equal to the CONTINUUM
power.

 Now, consider the Classical THEORY OF (REAL) NUMBERS by Cauchy,
Weierstrass, and Dedekind. As it is easy to see, substituting the term
"real number" for the term "real gap" and using the above results
concerning the properties of that gaps, we obtain a THEORY OF GAPS that
is the ("word-for-word") DUAL one to the CLASSICAL THEORY OF REAL
NUMBERS.

 So, we have

THEOREM 4. The segment [0,1] contains A CONTINUAL SET, G, OF REAL GAPS,
and THE THEORY OF GAPS is reciprocal (dual) to THE CLASSICAL THEORY OF
REAL NUMBERS.

I believe deeply that such the THEORY OF REAL GAPS as a dual INVERSION
(the case B) of the CLASSICAL THEORY OF REAL NUMBERS will considerably
extend our knowledge of the true nature of the mathematical, physical,
and philosophical CONTINUUM CONCEPTION itself.

"He that hath an ear, let him hear what the Spirit saith ┘ "    : -)

AZ

############################################
  Prof. Alexander A. Zenkin,
  Doctor of Physical and Mathematical Sciences,
  Leading Research Scientist of the Computer Center
  of the Russian Academy of Sciences,
  Member of the Philosophical Society of the Russia,
  Full-Member of the Creative Unity of the Russia Painters.
  e-mail: alexzen at com2com.ru
  WEB-Site   http://www.com2com.ru/alexzen
  "Artistic "PI"-Number Gallery":
  http://www.com2com.ru/alexzen/galery/Galery.html
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  "Infinitum Actu Non Datur" - Aristotle.
  "Drawing is a very useful tool against the uncertainty of words" -
Leibniz.