BioSystems 42 (1997) 191-205
The topology of perceptive functions as a corollary of the
theorem of existence in closed spaces
M. Bounias
a*b,c**,A. Bonaly d
a UniversitC d’Avignon, BiomathPmatique et Toxicologic, Fact& des Sciences, F 84000 Avignon, France
b The Alexandria Institute-New York, Hastings-on-Hudson, NY 10 706, USA
’ INRA: D&pt. de Phytopharmacie and bcotoxicologie, F 78026 Versailles, France
’ MathPmatiques, Universitt Paris X, Paris, France
Abstract
The capability for a system of perceiving both outer objects and an inner self are two fundamental features of
abstract mathematical objects endowed with the properties of topologically closed sets. Such structures exist upon
intersection of topological spaces owning different dimensions. Then, the theorem of Jordan-Veblen
provides their
capability of being observable, while the theorems of Brouwer and of Banach-Caccioppoli
provide two kinds of fixed
points which account for the properties of so-called right and left brain functions. Fixed points account for the
biological ‘self, and the system provides theoretical justification for the existence of brain structure/function
relationships, including memory, emotion, and respective characteristics of right and left hemispheres. Hence, an
abstract topological reasoning based on set properties, provides evidence that the observer’s function directly infers
from the phenomenon of existence and that it belongs to the same mathematical system as the property of being
observable. Order relations are raised from equivalence relations by Poincare groups, upon mappings on the sets of
functions and related homotopic transformations in sequences of intersections. Therefore, time is a construction of
abstract brain functions, and a living organism just fills the system with appropriate molecular structures. 0 1997
Elsevier Science Ireland Ltd.
Keywords: Physical existence; Closed sets; Equivalence classes; Order relations; Homotopic
Fixed points; Biological self; Brain structure/function
relationships
groups; Poincare groups;
Matter is real because it is an expression of the mind
1. Introduction
(Marcel
One of the most fundamental
question still
arising in modern cosmology (Magnon, 1996), is
the identification of ‘reality’ with ‘existence’, and
Proust,
1918)
the relationship
* Corresponding author. Fax: + 33 04 90711476.
0303-2647!97/$17.00
0 1997 Elsevier
PII SO303-2647(97)01706-l
Science Ireland
Ltd. All rights
reserved.
of existence,
if any,
with
some
192
M. Bounias, A.
Bonaly/BioSystems 42 (1997) 191-205
origin which has been called a ‘beginning’ or a
‘creation’ by philosophers, and ‘initial conditions’
by physicists, particularly in the Newtonian view.
In fact, as early as half a millenary B.C., Protagoras of Abdera claimed that “Man is the measure
of all things, those which are in that they are, and
those which are not, in that they are not” (Protagoras of Abdera, 480-410 B.C.). However, this
hits the paradox that what is not perceived does
not exist or is false. Plato improved the system by
introducing the fact that Reality can be reached
by intelligence through reasoning (Plato, 428-348
B.C.). The former was thus finally led to reject
mathematics, while the latter emphasized their
formal interest for the geometrical description of
universe (Taton, 1957).
Much later, during the XVIIth century, Leibnitz, one of the fathers of modern differential
calculus, reflected about the possibility of an abstract reasoning system in which the whole of the
truth could be contained (Leibnitz, 1666, 1684).
Much later, he hypothesized the model of the
monades (Leibnitz, 1714), providing some features of a complete construction of a physical
world from fundamental components. This posed
the problem in terms of causality, which may
reach an extreme stage with the so-called ‘anthropic principle’ (Barrows and Tipler, 1986)
which is not so far from a religious concept.
However, more than half a century was needed
before Kant expressed the dichotomic suggestion
that on one hand both space and time are the
product of human intuition, without link with any
independent properties of objects, and that on the
other hand, the abstract reasoning needs to be
supported by experimental observation, otherwise
it would fall into contradictions that he called
‘antinomies’ (Kant, 1781). These views are also
paradoxical, since part of the universe may not be
observable (see the concept of ‘foamy topologies’
in Magnon, 1996), so that no support could be
found for some part of reality.
At present, theoretical physicists still are facing
the problems of the reality of time and the confrontation between relativity and quantum physics
lets unsolved the question of causality.
The existence of any entity has been claimed as
depending on its interactions with others (Leclerc,
1972; Earley, 1992), although consideration for
the ‘observer’ was reported as “excessively anthropocentric and needlessly restrictive” (Earley,
1986). To which extent existence can be asserted
independently of a phenomenon of perception or
a construction of human’s mind, remained a
largely debated and till recently unsolved problem
(d’Espagnat, 1985), although “Humans are not
apart from the natural world, but rather are parts
of nature, and of large-scale natural systems such
as planetary ecology” (Earley, 1991). Obviously,
there remains a strong need for a clearer approach
of the relationships between abstract and physical
concepts. This is what drove us to reformulate the
problem in terms of the transition from mathematical to physical spaces.
In previous works (Bounias and Bonaly, 1994;
Bonaly and Bounias, 1995), we have developed in
terms of mathematical topology a theory of existence that we intended to build as much independently as possible from our own human
subjectivity. In other words, we aimed at starting
a description of universe through the identification of mathematical conditions which would acfor the phenomenon
of existence,
count
independently of any observation. One basic remark, first conjectured by Bonaly (1994: personal
communication),
is that since physical objects
should be in some way perceivable, they must
belong to the class of closed sets.
Our demonstration of necessary and sufficient
conditions for existence of closed spaces (the ‘theorem of existence’), first needed intersections of
parts of a topological set endowed with different,
up to four topological dimensions, whatever the
nature of their elements. Further, we demonstrated that such an abstract space could exist as
a corollary of the existence of the empty set as the
primary axiom (Bounias and Bonaly, 1996). At
this stage, a logical continuum was emerging from
an abstract concept to a physical one. However,
there remained to understand how could living
systems be eventually characterized from and
classified among other physical objects.
In other words, instead of a ‘theory of anything’, we were in search for particular corollaries
of a ‘theory of something’. One key principle is
that in closed objects, there exist several kinds of
M. Bounias, A. Bonaly/BioSystems
fixed points linked either to nonmetric topological
properties or to metric properties connected to the
sets of functions available in the primary set.
Since a nxed point provides any given closed with
correspondences between a full set of other closed
objects and one single point of this closed, this
property lets emerge a mathematical support for a
natural concept of the ‘self.
We now intend to examine what are the fundamental properties of correspondences
between
closed sets, which could actually involve a phenomenon of the same kind as that which we
would call ‘perception’, in a biological meaning.
Furthermore, provided this goal is achieved, it
would become possible to examine to which extent the abstract concept is reflected in biological
systems. In this respect, one should keep in mind
that the phenomenon called perception, in the
physiological sense, results from physical and
chemical interactions of waves and molecules with
more complex molecular structures. For example,
in biological systems, the information from outside signals reaches sensory cells, which contain
chemical receptors (chemoceptors).
The latter
control
the excitability
of specific organs
(Slepecky and Ulfendahl, 1993) which is communicated to sensory neurons, via neuromediators
(Ault and Hildebrand, 1994). The relevant information is then carried, under hormonal control
(Kern et al., 1994), from thalamic areas to the
brain sensory cortex (Fox and Zahs, 1994) and
further processed upon interactive neuro-hormonal communication of other cortex areas with
hippocampus, amygdala nuclei, and various other
brain structures, including the putamen, the caudate nucleus, and the colliculus, of which some
are used for coordination tasks.
Considering the result of this cascade of operations, it could be suggested that what physicists
are doing is to use similar neuronal networks to
elaborate some theories finally dealing with the
physics of themselves: in other words, there is
little doubt that the biological brain has been, in
some sort, self-organized. In effect, its construction has been performed through the evolutionary
improvements of self-generated metabolic systems
(Buvet, 1974; Bounias, 1990) which have been
further rearranged in increasingly complex organisms (Norel, 1984; Mayr et al,, 1978).
193
42 (1997) 191-205
However, nothing is ascertained concerning either the real origins or even the basic reasons for
such an apparently intricate organization. In addition, in their usual meaning, the concept of
‘origins’ and ‘evolution’, and generally speaking,
the notions of ‘life’, are not separate from the
concept of ‘time’ (Norel, 1984), although this is
not justified by any theoretical support.
The present work will now strengthen our previous demonstrations
that ‘existence’ does not
primarily involve ‘time’, and that ‘origins’ does
not imply a ‘beginning’ but rather the logics of a
minimal system: namely the properties of an abstract topological set. Furthermore,
it will be
shown that the identification of fundamental
brain functions, as a software, does not necessarily need to be supported by the previous development of any envelope called a ‘biological system’,
as a hardware, but instead emerges as a primary
concept.
2. Preliminaries
2.1. Mathematical
words and symbols.
Conjunction: a A b means: a and b (e.g.: propositions simultaneously true).
Disjunction: proposition a v b is true if and
only if (noted iff) at least one of either 1zor b is
true.
Belonging: ZE W: z is a member of W.
Empty set : 0 the set that contains no member.
Implications:
a =>b (a implies b); a-b
(a implies b and b implies a), with a and b propositions, properties, relations or functions. This is
also called ‘logical equivalence’.
Inclusion: XC Y: X included in Y: all members
of X are members of Y. Xc Y: X can be identical
to Y. X is said a part or a subset of Y.
Intersection:
En F contains all members belonging to both of sets E and F. Also noted n {E,
F}, i.e.: intersection of the members of the pair
{W1.
Reunion: A u B contains all members belonging
to both of sets A and B.
Negation of a proposition: 7A means: not A, or:
A is not true.
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M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
Relations: xRy means: R is a relation connecting x to y. An equivalence relation is reflexive
(xRx), transitive (xRy and yZ?z implies XRZ) and
symmetric (xRy_yRx),
while an order relation is
not symmetric.
Symbols of the composition of functions or mappings: fig means function f composed with func-
tion g (also represented by 0 and other symbols,
such as T, 0, . ..).
2.2. Mathematical glossary: definitions and
concepts
A topology, or topological structure on a set E,
is a set of parts of E, noted 0, owning the
following three properties: (i) the reunion of every
family of members of Q belongs to a. (ii) the
intersection of any finite family of members of CD
belongs to 0. (iii) the filled part and the empty
part of E belong to @ (Chambadal, 1981)
An open part, or open is any part of E belonging to CD.
its
An open interval does not contain
boundaries.
An open ball is the set of all points whose
distance from a point is lower than a given value.
A closed ball is the set of all points whose
distances to a point is lower than or equal to a
given distance.
A neighbourhood (V,) of a point or member or
set x is a set containing an open containing x.
The adherence or closure of a set is the smallest
closed containing this set (Choquet, 1984).
A filter on a set E is a family F of nonempty
parts of E such that: (i) F is stable through finite
intersections, and (ii) F is stable through oversetting, i.e. VMEF, VAc E, AIM*AEF.
Correspondences, connecting two members of
one or several sets are expressed as functions and
mappings. Group properties emerge from the considered functions (Godbillon, 1971).
A separated space in the Hausdorff sense is
such that for any two members (x, y) of a set (E),
(x ZY)J(~X)~(~,)
= @.
The homotopy relation, involving pathways as
continued mappings of segment Z= [0, l] in set
(E), is an equivalence relation in the set of pathways joining (x) to (y) (Godbillon, 1971).
Morphisms, in the sense of homomorphisms,
are defined on sets (E), (F), provided with combination rules (T) and (I). A mapping (f) is a
morphism of (E,T) on (FJ),
if and only if:
(Chambadal,
\J(x, y)~E$(f(xTy) =f(x)lfO)
1981).
If additionally function (f) is bijective, the
corresponding mapping becomes an isomorphism.
A homeomorphism of a topological set (E) on a
topological set (F) provides a bijection (one to
one and onto mapping) of the set of opens of (E)
on the set of opens of (fl. Homeomorphism
therefore denotes isomorphism on sets.
A relation or function (R) is said collectivizing
relatively to element (x) if there exists a set (E)
such that R(x) axe(E).
A retract (or retraction) will be considered as a
collectivizing mapping relatively to a part of the
definition set identical to the set of values.
A constant mapping applies to a mapping of a
set (E) in a set (E) in such a way that for any pair
(x, y) of members of (E):
f(x) =“KY) = co*
A fixed point of a function u> is the element
(x0) such that: f(xo) = x0. This is considered apart
from the identity function (h), such that for any
element (x), /r(x) = x.
The set of equivalence classes for an equivalence relation (R) on a set (E) is the quotient set of
(E) by (R) or: (E/R). When in addition a group
(G) is defined, a quotient set on (G) by (R) is a
quotient group. The equivalence class of the neutral element of (G) is a subgroup of (G) noted
(G’) said distinguished (normal).
Hence, given a morphism (f) on group (G) in
group (H), the reciprocal image of the neutral
(identity) element of (H) by (f) is also a distinguished subgroup called the kernel:Ker(f).
Further details can be found in Schwartz
(1991), Choquet (1984), James and James (1992).
2.3. Basic theorems
Two related theorems and a third one will be
used in particular.
First, the theorem of Brouwer states that in a
closed (O), or a compact convex normed space,
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M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
there exists a fixed point for a mapping (0 x 0)
of (0) into itself.
Then, the theorem of Banach-Caccioppoli
states that in a non-empty complete metric space
(M), given a contraction (T> of (M x M), for any
(x&(M), the sequence {(x0), T(x,), T(T(x,)), . ..>
converges to a limit x* which is the fixed point of
(7): T(x*) =x*.
Lastly, the theorem of Jordan-Veblen says that
two points one inside a closed and the other in the
complement can be joined by a pathway owning a
non empty intersection with the frontier. In contrast, the intersection with the frontier of two
points both belonging to either the interior or
to the complement of the closed is empty (James
and James, 1992; Godbillon, 1971; Chambadal,
1981).
3. Mathematical results
defined from a property of members of the sets,
and from union and intersection operations,
(James and James, 1992), no outer intervention is
necessary and the primary set (E) intrinsically
gives rise to the tribe of metric spaces.
It is not our aim, here, to consider the whole of
the countable unions of closed (F,) or intersections of open (Gg) of the o-algebra of subsets.
Instead, we will consider some particular elements
of the following tribes:
W(T) = n (A”,} 3 W(T) 3 0
including that (Choquet,
n(i+
m)
@i
=)
l(Q),
(l-4)
1984):
U-5)
(@)I
Then, upon consideration of a dimension, that
is some way of characterization of the occupation
of space (Tricot, 1993), if (m) and (n) are the
respective dimensions of subspaces W,,, and W,,,
the following theorem holds (Bounias and Bonaly,
1994):
3.1. Space structuration
m#n=z-@(W,)ncD(W,)=O(W,nW,)
The only primary condition we have previously
accepted as an unescapable one is that a primary
open set (E) exists (Bounias and Bonaly, 1994;
Bonaly and Bounias, 1995). The first corollary is
the existence of parts of (E) (which confers to (E)
the status of a space), and of the set P(E) of these
parts. Still intrinsically, P(E) allows the emergence of combination rules, in particular union
and intersection. Let (Q) denote an open and (0)
a closed, in generic terms, for completing what
represents a ‘mathematical system’ (James and
James, 1992). Then:
This restricted case provides connected subspaces, while W(T) is separated in the Hausdorff
sense.
For our purpose, it is sufficient that Eq. (l-4) to
Eq. (l-6) state for the natural existence (in the
mathematical sense) of closed structures from a
primary open set, while the complementaries
CO( W,), remain open.
n
bJP,JQ1= I@,,,(@I,.1
(l-6)
3.2. Topology in one intersection
Let (Si) and
(S,) two intersections
of the
( W, n W,) kind. We will first consider section (Si)
U-1)
Two tribes (T) of closed can be considered on
the basis of closed and the complementaries (C)
of open:
U,.(O),.EO(A)
(l-2)
u ,.C( @),.= A”,,
(l-3)
An algebra of Bore1 sets is defined on these
structures, and provided a measure is applied,
metric spaces are allowed to emerge. Since, at
least in the Caratheodory
sense, a measure is
alone, with some objects represented by sets (Ai),
(Bi), (Ci). Let
bers of the sets (Fig. 1).
Theorem 1 “The functions among closed sets of
a section define equivalence classes which are
represented by the groups of Poincare of the
section.”
Proof: upon continuity in (S,), there exists a
path (here, a piecewise smooth curve) joining
point (a,) to points (xi) or (z,), denoted cp(0, E) =
q(a, xi), with EE[O,11, such that v(O) = ai and
q(l) =x,.
(ai)E(Ai),
(xi)E(Bi),
CzijECCi)
mm-
196
M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
Fig. 1. Schematic representation of the interactions of three closed sets (A, B, C) imbedded in two intersections (Si and S,) of a space
W,, with a space (W,), m > n. The denomination of points (a, x, z: members of sets A, B, C) and of correspondences is as in text.
(P’), (f), and (H) are functions, (7’) and (r) are homotopic transformations. The Brouwer’s fixed points are denoted (a& the
Banach-Caccioppoli fixed points are denoted ( # a), and the Jordan-Veblen’s points are denoted &,).
we will define as the Jordan’s
if and only if
the interacting sets are closed:
Accordingly,
points the following intersections,
~(ai,xi)n(A)=k&and
~(aeXi)E(B)=Xie
(2-l)
Similarly, we can denote two other paths and
their Jordan’s points:
q(ai, zi) n (A) = &),
and p(+ xi) n(B) = gin
(2-2)
V(G zi) n (B) = ti&
and 40(x, zi) n (Cl = nix
(2-3)
Members xi and yi are connected by a correspondence (here bijective) associated to a mapping Fi of (B) on (C):
E;:x$+*F&)
= zi
(2-4)
The immediate corollary is that function (Fi)
owns an homotopic correspondent, as function Hi
of (A x A), on the condition that (A) is closed.
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M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
H,:(u~)HH,@,)
= (&)
(2-5)
This latter is an endomorphism, in contrast
with the former case corresponding to an isomorphism. The homotopic deformation will be noted
(T,), such that:
T(a, 0) = Hi(a), and T(a, 1) = Fj(xi)
(2-6)
Now, if (A) is a closed, there exists a fixed point
(ao) for any function v) defined in (A), such that
f(a,) = (a”). Hence:
T(aO, O) e
Hi(aO)
=
(2-7)
(uO)
A fixed point therefore coincides for both the
homotopic function and any function (f, in (A x
A). It can be represented by a part P,(A@P(A),
the set of parts of (A).
Groups of homotopy are defined for all the
paths joining, for instance, (ai) to F(xJ, as well as
(Hi) to (Fi) and also for the loops of a path (2):
(/i) = Cp
t”i3
F(xi))
u
P
C”i3
zi>
(2-8)
The quotient set of these groups therefore represents an equivalence class including the Poincare group of (S,), which completes the proof.
3.3. Space organization between intersections
Closed objects are contained in each of the
following intersections, called sections (S,) with
respect to their properties as Poincare sections in
an embedded space (Bonaly and Bounias, 1995).
{Sm,n}i= f IV, n W,,} u Fj, noted(&)
(3-l)
where Fi is now the set of functions available on
the elements of kind (e,) of section (S,), and 3
m, n, m >rz, i.e. m =4, n = 3 for our observable
universe (Bonaly and Bounias, 1995), in consistency with Eq. (l-6). Another set of functions will
therefore define another section. Since upon continuity in W, a path is defined between two members ei and ej of sections Si and Sj, members of
sets which are closed in each section will be
however arcwise connected between Si and 5” In
a previous work (Bonaly and Bounias, 1995) we
have therefore proposed the following notation
for the conjecture of existence, that we will now
call the ‘something function’:
M = f-Jp,, + 1.n...)w-u~cY
(3-2)
with {n + 1, n }, n > 0 the infimum of the set of
dimensions of P(W), (I) a correspondence operation and ‘Avcly’ = arcwise connectivity.
Physical information is carried by order relation, such as irreversibility of phenomena. It is
worth recalling, in this respect, that even in classical thermodynamics, the second law can be expressed in terms of gain and loss of information,
with crucial role played by the surroundings of a
biological system (Welch, 1993b).
Generally speaking, functions or mappings belong to the class of equivalence relations, and do
not imply a causality, while in contrast, order
relations may involve a causality. It is therefore of
major importance to examine the nature of
changes occurring in the relations connecting
closed objects, as for instance those contained in
one section and those contained in another section. We will thus prove the following proposition.
Lemma 2. “Upon mapping from one section to
another, any alteration of information is expressed as a partial transfer of equivalence relations into order relations”.
Proof: each section is defined by the corresponding set of functions (F), and an ordered
sequence of sections (Sj) appears if:
3{S,,,,1, eE(W,):F,=
F[Fde)l
(3-3)
Let (e) and (c) denote closed and open members or element families belonging to the same
class of sets (Y) = {(A), (B), (C), . . .}. Then:
V(e, c)&F),
vVe,cE(Si)2
(e, c)~(w,)
ve,cE:(wm)
(3-4a)
(3-4b)
Even if space (S,) is Hausdorff-separate,
its
embedding space (W,) may not be necessarily
such, with respect to structures in (S,), since the
case for non-empty intersections between two sections has not yet been examined. In contrast,
spaces W(T) could be strictly Hausdorff-separate
if, for any two sections (Si) and (S’): (S,) n (S’) =
@, (i Zj).
Let Iii the function that connects the corresponding two sets of functions by some operation
(0):
M. Bounias, A.
198
l-i
=
l-j=
Body /BioSystems 42 (1997) 191-205
{F,(e)OF,(C)l
(3-5)
{~(F,(e))OJYl;;,(C))I
(3-6)
The mapping F,: F,HF,, Vee( IV,) involves an
homotopic correspondence if the mappings Gij:
eit+ej = G(ei) are bijective, that is the correspondence between sets Yiwq is homeomorphic. In
such cases, one can write:
G(e,) bijectiveor,
j - Ti,j
of (A) in the collection of sets in (S,), (Sj), etc.
may finally constitute one single group of Poincare of (S): rci(S, a,), on the condition that a
symmetric element exists with respect to a given
combination rule, and fixed points and invariants
as neutral elements. Provided this conditions is
completely or at least partially fulfilled (right
hand or left hand symmetry defining semigroups), some properties finally emerge. Let:
(3-7)
However, Fij does not necessarily involve the
latter condition, and members (e,) and (ei) can be
represented by subsets with a cardinal Curd(e) >
1, and Carti
# Curti(
Hence, onto (surjective) and one to one (injective) mappings can
occur as well.
In these cases, the previous groups of homotopy will not necessarily be defined, and equivalence classes will be at least partially replaced by
order relations. The lemma is therefore proved.
MO, zi) = U%
3.4. Fixed points and invariants of
The group n,(S, a,) therefore provides a junction between equivalence and order relations
through the set of all sections (S,), that is in (IV,)
upon sequencing by (Ti,j). The theorem is thus
proved.
Proposition 4. “A second kind fixed point is
defined in a sequence of Poincare sections of
closed spaces.”
When at least set (A) is a metric space endowed
with the properties of a Banach space, a contraction Ki of (A x A) can be defined. Then, for any
point or member (a&(A), the Banach-Caccioppoli
theorem
states
that
the
sequence
{a,, K(a,), K(K(a,)), . . .} converges to a unique
element ( # a,)E(A) which is the fixed point of the
contraction function, that is: K( # a) = ( # a).
This allows appear ‘a new invariant in (A) upon
ordering of sequences from (Si) to (Sj). It is
noteworthy, for further considerations, that this
property of ( # a) holds provided a distance and a
norm are defined in (A), a condition not required
for (a,), although generalizable to non-metric
spaces (Bounias and Bonaly, 1995).
Corollaries 4. Properties of the involvement of
point ( # a) in homotopic correspondences and
mappings of set (A) with elements of C,(A) by
(Ti,j) will be matter of further consideration.
correspondences
Theorem 3. “The correspondence joining equivalence relations to order relations in sequences of
sections (Si) is provided by the groups of Poincart
of the sections.”
Proof. Consider the fixed point (ao)E {A} as the
origin of the mappings from {A} to any (B) or
(C) sets. Since Fi(ao) = a,,, one may write for the
homotopic transformations:
Ti,j(a,, a)(a~[O,
ll):F~(aoJ~~(atq)
l$(a,) = FIFi(aO)] = Fi(ao) = (a,), etc.
(4-l)
(4-2)
so that:
c(aO) = &(a& and (aoJ = (aoi)
(4-3)
Any F(a,) is also a fixed point of the set (F) of
functions, and the morphisms involving (aJ may
have no kernel, since the reciprocal mapping of
(ao) covers the whole set of functions Fi, Fj, G, Gj,
so far without demonstrated need for an identity
function.
Considering the related homotopic correspondences, the paths (A) or the loops (6) joining (aJ
to any member (e)E {C,(A)}, the complementary
F(xi))
&(ziy a,) = MF’ - ‘(xi), ao)
(4-4)
(4-5)
Then, with respect to relations Eq. (3-5) Eq.
(3-6), Eq. (3-7), the loops denoted s(a,) originating in (ao) include the homotopic mappings, and
are of the following type:
(4-6)
199
M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
4.1: The fixed point ( # aa) does not provide K
with the property of an identity function. Indeed,
considering the fixed point (uO) of the closed set
(A), one can write K(a,) = a,. However, for two
consecutive members in the sequence of contractions, K(a,) # a,.
4.2: Only the morphisms on (i) either K(a,), or
(ii) the whole sequence K(K(K(K. ..))), fulfil the
condition for the reciprocal mapping to be considered as a kernel. The first has previously been
questioned as such, and the second therefore constitutes the only remaining possibility. Since the
set of point ( # a,, # ub, . . . } represents a surjective (onto) mapping for all sequences of combined
functions K(a,), K(a,), K(a.,,), this set of functions might own some properties of a kernel.
4.3: Then, it depends on whether ( # a,), ( # a,,),
( # a,,,), are identical or not, that the second kind
fixed point of the section is a set or a unique
point.
4.4: Furthermore, a sequence Fij,...(u) involving
a Lipschitzian factor Aij between successive sections would introduce a convergence limit for
some part of the system after a finite number of
sections. Topologically, this convergence limit (L)
is contained in a neighbourhood of ( # a), i.e.
LE V( #a), and may therefore not be an endpoint. These questions are under current consideration.
4. Discussion
The present study has pointed out a set of
mathematically supported conditions associated
to the existence of correspondences
between
closed topological sets, themselves accounting for
the existence of physical objects. We will now
examine some implications of these results, with
respect to the phenomenon of Life, which we do
not aim to define a priori, since the definition will,
in fact, infer from a more complete development
of the mathematical description of the ‘topological world’.
4.1. Time: a secondary parameter
Order relations necessarily emerge on the ba-
sis of a sequencing of sets of functions (F(F(. . .F
and non-linear phenomena are therefore
to be subsequently expected. Since all kinds
of ‘onto’ mappings can be involved, the organizational
system will be able to identify
ordered changes of shapes from one section
(Si) to another (noted 5”). The existence of
any set of functions of the F(x) type warrants
such a sequencing, so that (Si) will be considered either ‘preceding’ or ‘following’ (Si) in
terms of ‘chronology’. The existence of such a
sequence is mathematically
associated with a
set of correspondence
properties which nearly
define a ‘perception’ phenomenon, regardless of
what we would consider a living or not living
system.
It is noteworthy that according to mathematical
considerations, time represents the counterpart of
what we have identified as a loss of information,
or a change in filtering, from fine to rough topologies. Provided this perceptive function can be
traduced in terms of living (and eventually human) properties, and the system would finally
define what is considered as ‘time’ in human
perception.
Since in a complex interacting system, a human aims to be considered at least sometimes
as an observer, although he can be also observed, let us now examine some arguments
allowing biological components to be classified
in the manifold of topological properties, and
particularly
in the class of sets Y = {(A),
G(A)).
(x)))),
4.2. The biological self
In set (A), the Jordan’s points must be connected to the interior by some specific functional
path with reciprocal property. The former two
conditions are fulfilled via sensory systems, involving receptor structures connected to sensory
neurons. Reciprocal property is achieved when
the fixed point (a& is involved, as for instance for
any composite function (0) giving a loop:
~(%I, a,,)
_ o G!,> %I)= %)
= a0
(5-l)
200
hf. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
Invariant properties are linked to the existence
of two kinds of fixed point. The first one, (a,), and
the second, (family # a,) connect all various Jordan’s Points or sets to one single image or image
set.
This property therefore provides support to the
concept that a biological ‘self should be composed of several elements.
We are now currently investigating on constant
mappings in the system, that is points (x), b)
such that I;(x) = Fb), and also on retraction
properties. The biological self will be finally considered as contained in the supremum of the
collectivizing
functions
on
a single
set:
((a,), u ( # a,)) *
It is interesting to note that a particular type of
mapping involved in the perceptive functions connects (A) to (A).
V(a # a,) E {A >,3F(a):~[~(~)l
f F(a)
(5-2)
Hence, any perceptive organism can also perceive some elements of itself, while it cannot for
some others, since K(u,) # (a,).
Considering Jordan-Veblen’s
points, it is important to note that a link is required between the
topography of those from the observed objects
and those from the observer. In effect, recent
works have provided evidence for the importance
of the topographical configuration of neurons in
relation with their function (Lewin et al., 1994).
These various considerations allow set (A) to be
denoted as an ‘observer’, in the anthropogenic
sense, although it is worthy to note that no ‘anthropic principle’ (see Merleau-Ponty, 1984) has
been needed, so far.
It should be recalled that our perceived spacetime (W,,,) represents an optimum case, with respect to mathematical properties, that is also to
‘organizational’ availabilities (Bonaly and Bounias, 1995). However, the topology of spaces endowed with dimensions higher than 3 exhibits
quite different features (Berger, 1990), so that the
present findings cannot be directly generalized to
higher dimensioned hyperspaces.
4.4. Theoretical perceptive functions and the
organs of the bruin
4.3. The mathematical bruin
Since two kinds of ‘self functions are available,
in case of redundancy in a biological structure,
the combination of least action principle with
entropic distribution would lead each redundant
part to take in charge only one of these two
functions.
The Banach-Caccioppoli
principle additionally
connects the second kind invariant by (Tij) to the
acquisition of a distance, that is, in some way, the
perception of a metrics. Concerning the first kind
invariant, subsets {A,} c {A}, are fixed upon
transformations by functions (F) in the ordered
sequence of sections (S,), in contrast with subsets
{Ab} c (A) of elements involved in successive
mappings, eventually non-homotopic, by (TJ.
In a biological system, members of {A,} and of
{ # A,) are the most closely connected with the
neuronal network, nearly invariant, although
u ( # a,) may exhibit some ageing-like transformations. In contrast, the others, denoted (A,},
will be those involved in turnover functions, that
is metabolic structures and functions.
4.4.1. Identljicution of equivalence relations
A sub-system is required in set (A) for identification of the equivalence classes. Since receptors
of excitatory transmission are contained in the
thalamus of vertebrates brain (Wang et al., 1994)
this organ might exhibit some of the relevant
properties.
4.4.2. IdentiJicution of order relations
Such relations should be identified as a restriction from equivalence ones. It is noteworthy that
80% of the cerebral cortex cells contain NMDA
receptors as also encountered in thalamus, and
that these receptor functions exhibit modulation
at various stages of cortical processing (Conti et
al., 1994), indicating higher ‘filtering’ capabilities.
The transition function from equivalence to order
relations could thus exhibit some consistency with
cortex capabilities.
4.4.3. Emotional component
Confrontation between equivalence and order
relations need to be operated by a mediator sub-
M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
system. Upon extension of the involved homotopic deformations to the endomorphisms, this
subsystem additionally
accounts for so-called
emotional functions: let Hi a function of (A x
A x ... x A)(, times) composed with a sequencing
component Hij. Then:
(5-3)
In the biological brain, there exists such a physiological subsystem connected to both the thalamus and the cortex, that is the amygdala nuclei.
The involvement of this organ in the Poincare
group functions could be strengthened by many
recent findings supporting the hypothesis that the
role of the amygdala is closely related to those
types of behavioral responses which involve emotional feelings (Davis et al., 1994). When this
system is extended to two sets of the (A) type, the
resulting effects involve mutual perception.
It can be now precised that the first of the two
components of the ‘self function is oriented towards emotional aspects (system based on a,),
while the second one (system based on #a) involves the definition of a metrics. This fairly
clearly explains the widely accepted knowledge on
the distribution of functions respectively between
the so-called ‘right brain’ and ‘left brain’. However no such separation is formerly needed, and
the system could work using a single organ rather
than a paired one.
4.4.4. Memory
Short-term memorization
functions of perceived outer phenomena are first justified by the
endomorphism Hi&, Q,
which also accounts
for functions (Fi) upon homotopy. Hence, longterm memory (noted ‘Z’) will be represented by
the closure of combinations (0) of functions (a
‘loop of functions’):
c =Fio~o...oF,oFF;
(5-4)
Bonaly (1994) has recently depicted this system
as a fractal structure represented by neuronal
chaining. This is consistent with the basically nonlinear structure of events considered through the
sequence of Poincart sections. Irrespective of
work currently in progress in this area of research,
a confrontation of relations Eq. (5-3) and Eq.
201
(5-4) raises the hypothesis of a link of memory
functions with emotional properties.
These theoretical considerations
quite well
agree with biological observations.
In effect,
declarative and spatial memory are first largely
associated to hippocampal functions, in addition
to prelimbic cortex contribution (Birto, 1992).
Then, some data on adrenergic modulation (Jaffard et al., 1992) do suggest a possibility of link
with the emotional functions, in consistency with
the recent finding of specific forms of learning in
amygdala (Burns et al., 1995).
4.4.5. On l$e paradigms
The problem of a correct definition of life still
remains unsolved. Recently, Rebek (1994) emphasized the replication phenomenon, with error-mediated evolution eliciting higher reproductive
capabilities of organisms. However, this aspect
denotes just a means by which life continues and
expands.
An interesting former study of Lofgren (1968)
dealt with an alternative meaning of the reproduction concept. The author pointed out that an
observed object can be understood within the
limits of which part of it can be reproduced.
Therefore, the unsolved question remains the definition of the reference for checking the correctness of understanding, even through complete
reproduction,
since this raises the question of
whether life is able of understanding itself.
In contrast, Polanyi (1968) considers the importance of the shapes rather than of the nature of
the substance, and notes that living beings could
transcend the laws of chemistry and physics
through morphological features, which impose
new limits to the laws of nature, with respect to
inanimate objects. At each upper stage of organization, new properties are gained, which are not
shared by lower stages (see also Conrad, 1977) up
to the phenomenon of consciousness.
Such a concept is not necessarily characteristic
of life, since it is contained in the topological
properties of the whole universe involving a hierarchy of filters applying to basic set properties,
and bounded by ultrafilters.
The other problem raised above, concerning
self-understanding, has been considered by Varela
202
M. Bounias, A. Bonaly /BioSystems
(1975), as the self-reference, or self-information
problem. The author points out the essential role
of the observer, which classifies the observed objects in belonging or not to itself. However, the
observer is not topologically distinct from the
observed object, and the question therefore just
reduces to identification of classes of perceptive
functions. In addition, self-entry of information in
one’s own indicative space falls within the properties of fractal objects, which is emerging as a
nearly universal property (Le MChautC, 1990).
The problem of self-references also appears, although indirectly, in Lijfgren (1968), when the
author recalls the case of previously pointed paradoxes of functions that could be their own arguments or range. This is also included in
topological properties with stable parts, fixed
points of functions and retractions, that is mappings which, upon composition with a function in
a set, allow this function to correspond with the
identity mapping on this set.
4.4.6. From physics to biology
Welch (1993a) wrote that “Contemporary biologists, unlike physicists though, are not accustomed to thinking of ‘geometry’ as a deterministic
dynamical element in the process of the living
state”. The author noted that present-day theoretical physics is essentially based on dynamical
space-time symmetries and invariance relations,
both inherent to the geometrical field concept. He
therefore proposes to develop a theoretical biology based on the transposition of the same concepts to living systems. In another paper, Welch
(1993b) also notes the prominent role of electric
fields in the processing of reactions seen as the
computational
phase of a molecular machine
working between input and output of information. The enzymes would serve as the local ‘fieldtransducers’, and the biological ‘gauge bosons’
would be represented by the hydrogen-bond
phonons. Welch settles a (quasi) Lagrangian
equation for the dissipation function accounting
for diffusion in steady states situations, and
derives a geodesic equation, both built by using
analogy with the equivalent forms in physics. The
author finds the analogical construction advantageous, as “a key element in the construction of
42 (1997) 191-205
theories of natural phenomena” (Welch, 1992).
Previously, the author rightly noted that the geometrical complexity of the cellular structure contributes to make biochemical processes in vivo
irreducible to their chemical expression in vitro
(Welch, 1987, 1989). Again emerges, here, the
hierarchical network corresponding to the topological existence of ordered sets of filters of various roughness. Here, Welch wrote, the invariance
principle would lie in the fact that “the same type
of relational forces govern the spatiotemporal arrangement of the basic entities at all levels-at
the sociological, the multicellular, the subcellular.” Indeed, this reminds self-similar properties
of fractal systems, that the author relates to relational invariance, while speaking of the second
law of thermodynamics (Welch, 1993b).
This approach may well prove fruitful, and
allow some progress in describing living systems.
However, the arising question is to which extent
symmetry and invariance paradigms really are
needed and even justified, keeping in mind that
so-called ‘laws of physics’ are not much more
than repeatable observations fitting a timely accepted paradigm, rather than absolute proofs.
A physical-like theory may unfortunately hit
the same snags as does theoretical physics. In
particular, the violation of parity at low scales,
conflicting with large scale observations
(see
Magnon, 1996 for review), could also be transposed to the living world, since biological scales
range from nanometres (within cell organelles) to
thousands of kilometres (for population clusters)
(Welch, 1987). On one hand, the invariance
paradigm needs a suitable measure to be available, and present metrics remain inappropriate for
unification. On the other hand, symmetrical properties must be quoted with respect to a reference
frame external to the system. When topological
shapes are considered in themselves, there is neither symmetry from inside of them nor need of
invariance. On one hand, the object topology is
primary over its coordinates, and the latter even
vanish in the embedding n-spaces (n > 3). On the
other hand, fractal structures have no such kinds
of symmetries, while conservation applies to internal properties upon changes of scale.
M. Bounias, A. Bonaly / BioSystems 42 (1997) 191-205
Our theory copes with these points. First, the
fractal self-similarity involves homotopic transformations, which we have shown to be important
features of perceptive functions, and therefore of
world’s natural description. Thus, the question of
invariants emerges again, here, as the conservation of an internal topology. Our work in progress also deals with this question, since a whole
manifold of fixed points and stable parts infer
from specific topological properties of closed sets
(Sharma, Sahu, Bounias and Bonaly, 1997; Singh,
Bounias and Bonaly, 1996, in preparation).
There is substantial hope that this part of our
theory finally meets with the justification of exisof relativistic
physics
tence of invariants
(Ashtekar and Magnon-Ashtekar,
1979), which
would therefore provide Welch’s propositions
with further consistency.
5. Conclusion
The topological approach demonstrates that at
least some of the brain structure/functions
relationships are closely justified by the major features that can be mathematically derived from the
theoretical study of a purely abstract system endowed with the topological properties of some
correspondences. The latter are oriented to objects outside and inside the observer and may
therefore account for the sensory perceptions and
to the observer’s self, without formal need for the
previous building of an organic envelope provided
with the characteristics usually attributed to a
living organism. Instead, the concept of time
emerges as a corollary of these mathematical features, and the concept of a ‘chronological origin’
is just the apparent consequence of the identification of timeless order relations. In addition, our
work in progress on fixed points in both metric
and nonmetric settlement is currently providing
justification for unicity of the self.
Hence, we have shown that a purely abstract
reasoning, starting from the axiomatic existence
of the empty set, leads to the inference of perceptive functions which include some characteristics
that can be considered as properly representing
the self of a living being. This just needs the
203
presence of appropriate molecular support for the
various subsystems, and finally, it might well appear that if actually, as conjectured by Polanyi
(1968): “consciousness is a principle that fundamentally transcends not only physics and chemistry but also mechanistic principles of living
beings,” this is because it is a universal topological property, not necessarily strictly restricted to
living beings.
Facing the problem of whether or not the reality (if any) is created by the observer has become
a classic situation in quantum mechanics (Pagels,
1985) but not in biology, even when biologists
have been tempted to transfer to biological systems some essential concepts (like ‘fields’) and
operators (like ‘Lagrangians’) usually handled in
physics.
Although such an operation was set in appropriate terms by some researchers
(Welch,
1993a,b), no great progress seems to have been
done, so far: this could come from the fact that
the concept of ‘field’ may well represent in physics
what ‘vitalism’ is in biology: a word posed on an
observation. It may be that the question was
worthy to be asked in the reverse way, that is
without postulating first that life is a real phenomenon and without using this observation as a
major element of the reasoning.
This is the alternative way we have chosen to
follow, and the present stage of our work essentially shows that perceptive functions are logically
deriving from physical existence, that is the respective properties of both the observer and the
observed objects are identically inherent to the
topology of closed spaces. All members of the
involved subsystems are therefore direct corollaries of the theorem of existence involving a
primary open set endowed with a structure, originating in the empty set without need for a ‘Creation Act’. The question of minimal life now
raises in rather unexpected terms, since less difference appears than formerly expected between living and inanimate matter. Both of the latter
fundamentally share similar topological properties, while life could be characterized by the level
of equipment of physical objects with sensory
structures, providing physical support to mathematical paths theoretically involved in all closed
objects.
204
M. Bounias, A. Bonaly/BioSystems
Acknowledgements
The authors wish to express their thanks to Dr
S. Santoli
for stimulating
discussions
on
‘minimum life’ and for valuable suggestions in the
presentation of this paper. They also acknowledge
with thanks the invitation by theoretical physicist
Prof Anne Magnon for a seminar lecture on this
topics at the Department of Mathematics of the
Blaise Pascal University of Clermont-Ferrand
(France).
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