MATHEMATICAL ORGANISMS
F. G. ASENJO
1. PURPOSE. This paper presents an axiomatic theory of organisms which is in partial contrast to axiomatic set theory. Herewith, are two of the distinctive characteristics. (A few words on
over-all motivation are offered in the final remarks). The principal objective of the axiom of foundation in axiomatic set theory
is to eliminate formulas of the form xex. The reason, so it is said,
is that such formulas serve no useful purpose. This is an atomistic prejudice. One o f the properties o f organisms is that they
have overlapping extensions whic h in many cases include for
each organism the organism itself as an additional member. I n
physiology the division between part and whole, member and
aggregate, is not so easy to draw as in macroscopic physics —
organisms are not simply built out of elements. The axiom system
presented in the next section excludes the axiom of foundation
as well as any of its alternative forms.
The axiom o f extensionality is a mathematical convenience
derived fr om our understanding o f macroscopic physics. But
again, in physiology organisms can be coextensive and at the
same time distinguishable from one another. There are already
nonextensional set theories in the literature: for example, Fraenkel-Mostowski's system, originally devised t o prove the independence of the'axiom of choice, in which so-called "atoms" are
coexistensive and distinguishable. Also, Paul J. Cohen's generic
sets for the proof of the independence of the axiom of choice
in more general set-theoretic settings, although not atoms in the
Fraenkel-Mostowski sense, share some of the characteristics of
nonextensionality in the relationships between the generic sets
and the forcing conditions that determine them. I n the next
section some mathematical organisms are subjected to the axiom
of extensionality, while others are not.
302
F
.
G. ASENIO
2. MATHEMATICAL ORGANISMS. We shall describe a first-order
theory MO. MO has a finite number of predicate letters = , a, a
o
l
tion
, kletters. We shall use capital letters X
and
l
aX, Y, Z, t o represent arbitrary variables. Intuitively, a,
, Xn a
relations,
2 d k according to the terms related. XaY w ill be read " X
stands
in Y",
, a aa
s Xa,Y (I = 1, k ) will be read " X i-stands in Y".
r interpretations
v In
af the
i a b l e we
s have in mind the variables take organisms as
values.
We
define
an organism to be an organ if and
i e
only i f t i t stands i n itself, otherwise organisms which are not
n are called proper organisms.
organs
i h
t o
Definition
I . org(X) for XaX.
e u
Definition
n g
2. Org(X) for —org(X).
h
u
t introduce small letters x
Let
m us
variables
i b o for organs, and boldface small letters x
special,
for propel organisms (x , y , z ,
i , ex f.restricted
, ,
a variables
s
represent
arbitrary
organs,
and
x , y , z , r e p r e s e n t arbitrary
, sx r pa e
c i a l ,
proper
organisms).
In
other
words,
(x,)A(x,) stands for (X)(org
2 r o el s t r i c t
(X)D
A(X)),
i.e.,
A
holds
for
all
organs;
(Ex,)A(x,) stands for
t
, ef d
a
s
(EX)(org(X)8c.A(X)),
i.e., A holds f or some organ; ( E x
i e
istands rfor (EX)(Org(X)8LA(X)), i.e., A holds for some proper
n
organism.
)A(xn
,)
d now present the first proper axioms of MO , the axioms
We
i a the relative symmetry of a and the internal relatedness
concerning
of proper
v t organisms.
i i
Al.
d xv aX=Xax .
u e
A2.
a xl ax & yax D xay.
l y
Notice
c a that the extension o f each organ includes — apart
fromo the
s organ itself (Def. I ) - all the proper organisms in
which ithe organ stands ( Al) , plus all the organs of these proper
n
organisms
(A2).
s n
t t
a e
n r
t n
s a
, l
b o
M AT H EM AT I C AL O R G AN I SM S
3
0
3
A3. ( E X ) ( Y )
-Definition
Y o X 3. (Y)—(Ya0).
( N u l
Theorem
I . Org(0).
l
O r g
A
a l and
n i A2 obstruct the usefulness o f a pairing axiom for
organs as well as the usefulness of an algebra of organs similar
m
to sthe algebra
of sets. Given our lack of definite motivation for
A
x
selecting any specific axiom system for organs, and in order to
i
o freedom of choice, the existence of organs and proper
preserve
m
) w ill be provided here by proper axioms chosen more
organisms
.
to keep
various possibilities open than t o formalize regular,
specific relationships between organs and proper organisms —
relationships whose regularity would make them more interesting
from a mathematical viewpoint. As an example of a particular
finite system of organisms that shows the possibilities we have
in mind, let us consider the following axiom system M O
a
1
b,
i, wm h , e nr are
e individual constants for proper organisms.
,
b SI. a
b
i
, S2.
ia a
m
a b m, a m (i 1 , k
St.
t (i
t( i+ 1. a
St
, =
St
j)= . + 2. a
p kSt
lo
1 b+ 3.
( ja
„ iSt
-, c" +. =4." b
io
q )o
kSt
1
( jboa.
2 n+ ,5.
+
6.
a .St
2 c . ) doa.
3
=
o
r St
.) + ,7. aob.
1
. + )8. b o n .
3
e St
. + 9. e o n .
i St
n St + 10. pap.
d St + 11. a
i 1
Here, between curved brackets, are the extensions (as far as a
v aq
is concerned)
of a few organisms from MO, (the bracket notation
i 2
d .
u
a
l
c
o
n
304
F
.
G. ASENIO
does not designate an univocal function for organs — see next
section).
a
{a
l
a
: l
a
3
a ,
b
l
:m a
(Same extension for c
cf, k
k
,
:
{
k 3
lP
2
a
a i
ch:
. )
:P
,
3
h
, ,
: {a
c}a
b
1
b a
d
1.i l l
,
.
ka
, (a
,d
, .
4
2,b a l
lCOLONIES, SPECIES AND 1-11(PERSPECIES. We define
3. FAMILIES,
,
1
k
,a
,
1 b organism
a proper
to be a family o f organs i f and only i f it
.a
1
c
1 ,2 ,
,
a are called colonies.
families
of organs
- .1
,
1 e
a
,
s 3
b l
3
i
6.cFam(x) for x o
1
t Definition
,
. .
,
l
,c
a Definition
7.l Col(x) for —Fam(x).
1
,
n xb
l .
3
We
define
a
ccolony to be a proper colony if and only i f it
d ,
2-stands
2
i
n
its
kelf, otherwise colonies whic h ar e n o t proper
s n
c
colonies
are called
2-species.
,
i 1l1
.
i
n .,
Definition 8., col(X) for Col(X) & X a
i 3
2
9.c2-Spec(X) f or Col(X) &— col(X).
t Definition
,
X.
l
s c
Inductively, we
. define an i-species (2 t o be an (i + 1)e ,
proper species if and only if it (i + I)-stands in itself, otherwise
l
i-species
which are not proper species are called (i + 1)-species.
f
, Definition 10. ( i + I)-spec(X) for i-Spec(X) & XCIi +
o 1X.
t Definition 11. ( i + 1)-Spec(X) for i-Spec(X) &—(i + 1)-spec
(X). (Col(X) and ( i + 1)-Spec(X) are, of course, abbreviations
h
e
r
w
i
s
e
p
MATHEMATI CAL ORGANISMS
3
0
5
for O rg(X) & — Fam(X) and i-Spec(X) &—(i + 1)-spec(X) r espectively.)
The following three axioms are the axioms of extensionality.
A4a. x = x.
A4b. x = y ( X ) ( X o x X o y ) & ( X ) ( X o
& & ( X) ( Xo
l
xAzle.X Xko = Y D(org(X) & org(Y)) V (Fam(X) & Fam(Y)) V
x X o & col(Y)) V ( k - s p e c ( X ) & k-spec(Y)) V (kl
(col(X)
y ) kSpec(X) & k-Spec(Y)).
y ) .
Notice that equality o f organs is indiv idual identity — for
organs, some extension does not guarantee equality.
Definition 12. X c Y for (Z)(ZOX D ZoY) & ( Z ) ( Z o
& &(Z)(Zo
i
X
D Z 13.
a kX Y for X c Y & — X = Y. ( P r o p e r InDefinition
X
DZOk Y)
l
clusion).
.
Y )
Notice that unequal
( I norgans
c l w it h the same extension ar e
properly included uin sonei another.
o n
Theorem 2. M O) is. a first-order theory with equality.
Theorem 3. org(X) & X = Y D org(Y).
Thorem 4. Org(X) & X = Y O r g( Y) .
Theorem 5. x o y D —3c = y .
The following are axioms for proper organisms which establish
for them properties that are to a certain extent similar to those
of sets.
A5a. (x)(y)(Ez)(u)(Fam(x)D(Fam(y) ( F a m ( z ) & (Fam(u)
( uoz u=x Vu=y ) ) ) ) ) .
A517. (x)(y)(Ez)(u)(col(x) D (col(y) ( C01( Z ) 8L(C01(111)
A5c.
( 1 1 0 1 ZE
(x)(y)(Ez)(u)(i-spec(x)
- - 1 1 ' X V
(i-spec(u)
I I =
(uo
y
)
)
)
j
) ) .
z
=
7
u
=
y
u
0
)
/
)
D (i-SpeC(y) D (i-spec(z) &
306
F
.
G. ASENJ O
These are the pairing axioms. Notice that the " pair " determined by two proper organisms of the same k ind s a y , two
proper colonies — may include an unlimited number of organisms of other kinds — organs, families of organs and various
kinds o f hyperspecies. Ax iom systems can easily be given i n
which it is not possible to separate set-theoretically organisms
of one kind from organisms of other kinds. Still, this does not
preclude the definition of ordered "pairs " of proper organisms
of the same k ind in the usual manner. But first, the definition
for unordered "pairs".
Definition 14. Fam(x) & Fam(y) & (z)(Fam(z)D(zo[x,y]
z = x V z = y)))V .
✓ (col(x) & col(y) & ( z ) ( c o l( z ) D ( z o
l= y))) V
lx , y 1
z
=
✓ (k-spec(x) & k-spec(y) & (z)(k-spec(z)D(za[x,y]
xz = x V
z
Vz =y )))V
✓ (k-Spec(x) & k-Spec(y) &—
=- 0) .
0
& —y
0
84x,y1
Notice that the value of the function [x ,y ] is unique for any
given pair of non-null proper organisms of the same kind (A4b),
althought Ix, y l does not necessarily indicate the extension of the
pairing organism thus formed.
Definition 1 5 . [ x l = [x,x1.
Theorem 6. (x )(y )([x l = [ y l D x = y ).
Definition 16. ‹ x , y > =
,
[x,y1] •
Theorem 7. ( x ) ( y ) ( u) ( v ) ( <x ,y > = < U N >
= v).
X
u & y
Definition 16 can be extended inductively in the usual manner
to ordered n-tuples, for which we shall use the standard notation
<x
i Additional axioms similar to S t
, introduce
further relationships between different kinds of orgax S t The
nisms.
+ choice
1 1is, again, more heuristic than systematic (r„s,
„ a r e
> n
o
w
. i
n
o
r
d
e
r
t
o
MA THE MA TI CA L ORGANISMS
3
0
7
are individual constants for organs, r,s are individual constants
for proper organisms).
St+ 12. Fam(a) & Fam(b) &
SI + 13. a,a
St+
14. a
i
St
+
o
r &/5.
St
c l +a /6.
r r) o
S
t
+
17.
so
2
.i
ls . + 18.
St
& c
19. l a b ] = r .
iSt+
c o l
.St+
a
. 20. [r,slon.
(St s+ 21
)
.
2
•
& Fam(n).
r.
With these new axioms the extension of each a
with
i ir sandi s.n The
c r extension
e a s e of
d each c, is increased with r. The
"pair" la b ] is a proper colony whose extension is ( a, It, a
s, c
l
l now complete M O with an axiom schema similar to the
, We
general
, class existence theorem in Gbdel's axiomatic set theory.
(LikeCthe theorem, the axiom schema can be obtained from a
finiteknumber of proper axioms.)
3
A6. Let c f( x
}
occur
l among x
.
iup, to xk-proper
„ , species are quantified. We c all such a w f preE Th e n
dicative.
, •Y • • ,
t
x A6a.
„ , , ,(Ez)(x1)(x2)• • • (x0(—k-Spec(x1) D ( k -S p e C(X )
c
Y ) i •( D ( ••••• k - S p e c ( x „ ) < X I , X
.
• b • •iYIt , • - Yni) ) • •• )•
> O Z
(
X
I
,
, e
A6b.N(Ez)(x1)(x2)...(x„)(--k-Spec(x
•
X
n
i a
,
1.
D k - Spec ( x „ ) D ( < x
„ w
,
Yi,
) D
(l•••, Ym))•••).
k
a fn
, „ . ,
x
d w
Boolean
operations
are
S P e C„ ( >X a2 ,) z now definable for proper organisms.
i hParadoxescarep avoided
( x with the following line of reasoning.
nLeto if be —xl ax or — x c i
w
i s
h
,
ix e( 1c- < , „ i - < , k ) .
hT v h a e
n
o( r x i )
n( a
b— k lS l p e e c
y( s x )
o
r
308
F
.
G. ASEN)0
(xo
Spec(z).
i
z Definition 17. Given any predicative w f (j,(x
=
Yin),
t we shall use • i i p ( x l • •••• x , Yi, •••,Yn) to denote
xthe, proper
x „ organism
.
Y o f all tuples < x
o
such
that
—k-Spec(x,)
& & —k-Spec(x„).
i 1
,, , x „ >
18. V stands for i i ( - xs Definition
a t i s f y i n g
,
)c Theorem(8.
p
—k-Spec(x)
,
D XGV X01 \IV ...VXCrkV.
&
) k -Spec (x )
=
- 9. k-Spec(V).
x
)
.
, Theorem
w
h 4. THE CATEGORY OF MATHEMATICAL ORGANISMS. Org a n isms
can be be described as forming a modified category system in the
i
following way. The category o f nzathematical organisms is an
c
organism A, not a k-Species, together with an organism C that
his a disjoint union o f the form C = U [X,Y] for all ( X,Y) in
lA X A. Each IX,Y1 is not an organ, nor is it a k-Species, and can
ebe void. When it is not void it is composed of a single organism,
anamely, the correspondence composed o f all pairs (U,V) such
dthat U o
swhere
the latter are any of the predicate letters a, c r
i
addition,
for each triple ( X X ,Z) o f members o f A we are to
tkX
from [Y,Z1 to [X,Y], eventually void. The image
ohave
.a n a
I function
n
of
this
function
is denoted by [Y,Z l • 1X,Y1 and called the comcd
position
o
f
[Y,Z
] by [X,Y]. Composition functions satisfy two
oV o
conditions.
ni
tY ,
(i) Whenever compositions make sense,
ra
(1Z,Wl • [Y,Z]) • [X,Y] = [Z,14
an
,
d
(ii)
For
A not
we have an element 1, in
] • each
( [ XY in
, Z
1 a k-Species
•
is
C, namely
EX,X],
such
that
[X,X1
•
[Y,X]
= [ Y, X] and [ X, Y] •
[ X , Y
cu
IX,X1 ]= )[ X, Y] whenever the composition makes sense.
tc
•
ih Given
a member IX,Yj in C, X is called the domain and Y the
otcodomain o f [X,Y) . Since C is composed o f single organisms,
nh
a
nt
dU
ia
m
l
p,
lV
MATHEMATICAL ORGANISMS
3
0
9
A is somewhat similar to what in category theory is called an
ordered class. Also, organisms in A can be called objects while
organisms in C can be called correspondences. ("Morphism", a
truncation of homomorphism, suggests too stronlgy a condition
of single-valuedness that is not intended.)
The elimination o f indiv idual objects (k-Species always excluded, of course) is now possible in a manner similar to the socalled non-objective approach in category theory. Objects can
be defined in terms o f correspondences and retrieved through
appropriate identities.
Saunders Mac Lane considers Paul J . Cohen's proof o f the
independence of the continuum hypothesis a startling indication
that current axiomatic set theories do not appropriately describe
the entities a mathematician needs to match his intuitive idea of
set as it derives from mathematical usage. MacLane is at present
attempting (and he is not alone in this) to reconstruct mathematically useful fragments of set theory from the standpoint of pure
category theory ("categories without sets"). This is indeed a
considerable project and it is s till too early to predict where it
may lead. However, it is perhaps to the point to remark that
categories, given the great generality with which they are now
defined, allow all kinds of unexpected structures with or without
slight modifications — and mathematical organisms are a case
in point. I t seems as if category theoreticians, in their desire for
a better set theory (or something that replaces it), may be getting
more than they bargained for — which might be all to the good.
5. FINAL REMARKS. Although this is not the place to elaborate
on biological motivations, nevertheless a few comments are indispensable. Our logic is the logic of solids, said Bergson, which
is why the application of logical concepts to biology is always
ridden with innumerable exceptions and qualifications, and worst
of all, why i t fails t o convey tw o important biological facts
(i) Although organs have simple location anatomically conceived,
physiologically they influence one another to such a degree that
they must be thought o f as having the same extension as the
whole organism and often a greater one; further, the general
state of the organism constitutes an additional term in the phy-
310
F
.
G. ASENJO
sociological composition of each organ. (ii) Environment and the
hierarchy o f life are intrinsic to each organism, and although
organisms associate i n colonies, colonies i n species, etc., this
does not in any way imply that organisms are built with completely separated individuals or any other elementary abstractions. Indeed, set-theoretic conceptions and hierarchies have a
tendency t o hide and b lu r biological significance. T hey are
suitable to describe collections of macroscopic solids but not life.
We need a logic of fluids.
In this logic of fluids a certain amount of ambiguity must find
formal expression, which requires beginning with what we may
call a relativization o f the indiv idual. ( I n biology one mus t
distinguish an organ without completely separating it in a settheoretic sense, lest the organism vanish altogether). This relativization is of course implic it in M O and M O
different,
1
approach to such relativization see [21.) It is our belief
that
. ( only
F o through
r
a na relativization
o t h e r of
, the individual can we give
to networks of relations the primary descriptive role that they
should have, a role that is now overshadowed by excessive emphasis on final components. Ultimately, one should be able to
describe organisms as pure relations, that is, as systems of correspondences in which events are mere changes in the system.
University of Pittsburgh, Penna.
F
.
G. ASENIO
REFERENCES
[1] E. G. ASENIO, El Todo y las Partes: Estudios de Ont ologie Farinai.
Madrid: Edit orial Mart inez de Murguia, 1962.
[2] E. G. ASENIO, Theory o f multiplicities, Logique et Analyse, Vol.
VI I 1 (1965), pp. 105-110. (Mis print : O n page 106, line 7 f rom bottom.
1p-distinguishable should be u-distinguishable).
[3] Ludwig v on BERTALANITY, Problems of Life. London: C. A . Watts
Co., 1952.
[4] Kurt GOLDSTEIN, Der Auf bau des Organismus. The Hague: Ma r t i
nus• Nijhof f , 1934. (The Organism. New York: American Book Co., 1938).