Re: What is /are The Logic(s) of Life?... and Adjointness is Fundamental in Categories and Topoi of Biological Systems

[Ionel <***>]

Hi, Tim:

Please note and consider three very important points that were also
explained twice before in my postings in this thread on the possible Logics
that we do need to be able to model Life's Complexities:

1. The original (M,R)-system, as printed in 1958, employed just sets;

2. Sets- as defined at that time- are subject to the Axiom of Choice (AC,
cf. also Robert Rosen in  "Essays...", and also widely recognized), and the
Boolean Logic, which is suitable ONLY for automata and machines, i.e.,
simple systems is at the basis of mathematical set theory;

3. Robert Rosen explicitely gives up Sets as UNSUITABLE for modeling
Complex Systems in his "Essays..." book published in 2001, and is in favor
of 'structured' Categories that aren't Sets. Furthermore, "Category Theory
is intrinsically intuitionistic" says explicitely the latest textbook
published in 2004 that I referred you to before.

Last-but-not-least, both Lukasiewicz and Intuitionistic Logics are NOT
'algorithmic' , or recursive, in general, as you assumed in your posting,
and Lukn and Intuitionistic (Heyting) Logics ARE NOT SUBJECT to the Axiom
of Choice; so far these two kinds of Logics are the major break up with 2-
valued logic (Boolean) which served as the basis for sets. This is the
major reason for my posting.

The logic of predicates, or predicative logic, is in essence also Boolean-
based, and as Rashevsky himself showed in several articles published in BMB
in the 50's, it leads to equivalent results to those that are obtained by
Sets and Relations for biological and societal organisms. Hilbert's
predicate theory does not extend to Intuitionistic logic, that you might
call 'impredicative', as double negation is NOT equivalent to affirmation
in Heyting Logics (as it is in Boolean, 2-valued Logic).

Regards,
Ionel



On Mon, 14 Jun 2004 10:47:24 -0400, Tim Gwinn <***> wrote:
(shortened message to essentials).

>Hi Ionel,
>....
The 'logic' for representing these organizations would be the relational
models, such as used for the (M,R)-system model.>>

Please note Point#1 above: which (M,R)-system model version are you
referring to ? The 1958 one in terms of Sets ?, or the 1997/ 2001, and
2004  onein terms of Structured Categories ? , because it does make a very
big difference which one you are referring to.

>>>>>>>
-----------------------------------------

..... I am pretty certain hat neither Lukasiewicz nor Heyting logic nor
some other 'fuzzy' logic will admit such impredicative structures. As best
I understand them all, they remain entirely algorithmic and predicative.>>

I 'm sorry but this is very easily proven to be incorrect; please see
points #2 and #3, above. Current Logics cannot be trifled with by what one
might call 'handwaving' arguments.

______________________________________
>Regards,
>Tim
>

_________________________
>> -----Original Message-----
>> From: ROSEN Forum [mailto:*** Behalf Of I. C.
>> Baianu
>> Sent: Friday, June 11, 2004 2:23 PM
>> To: ***
>> Subject: Re: What is /are The Logic(s) of Life?... and Adjointness is
>> Fundamental in Categories and Topoi of Biological Systems
>>
>>
>> Tim:
>>
>> At this point it is useful to review and expand our discussion on Adjoint
>> Functors in Categories. As you might gather from the two items listed
>> below, in attempting to answer the question "What is Life?" that
>> Schrodinger
>> --one of the great and very famous inventors of Quantum/Wave
('Mechanics')
>> Theory--asked, one has to face also the deeper, implicit question that
>> Robert Rosen provides new answers to in "Life Itself" and his "Essays on
>> Life.":  ------What is /are The Logic(s) of Life? -----. Pairs of inverse
>> Adjoint functors between two Categories of Biological Systems provide, in
>> my opinion one of the very important answers  to the question
>> "What is The Logic of Life?" because adjointness can be employed to
define
>> the fundamental logico-mathematical structures that one can currently
>> employ to generate valid, irreducible models of living organsisms=
>> biological systems. Please note that I have employed the term
>> "irreducible"
>> to denote what until now was called 'non-reductionist', in order to
>> separate Rosennean--or other models following the Robert Rosen-Nicolas
>> Rashevsky Relational Biology paradigm-- from reductionist--linear --
>> machinist--robotic types of models that can be expressed in terms
>> of linear
>> algebra/vector spaces/'linear' maths. One suspects that there is a single
>> correct answer and solution to the question posed by Schrodinger
>> and others:
>> "What is Life", and that the implicit problem/question "What is The Logic
>> of Life?" does have also a UNIQUE solution/answer formulated "The unique
>> Logic of Life is...x " , where 'x' is what we are still searching for.
>> Because we do not know what 'x'--the answer is exactly we have to examine
>> the alternate possibilities that we can reasonably put forward,
>> and thus we
>> need to re-phrase the question as : "What are the possible candidates for
>> 'The Logic of Life'? "-- the title of this post. We do have from recent
>> work on the Foundations of Mathematics Itself--as well as from Robert
>> Rosen's "Life Itself" and "Essays on Life"-- the same 'negative', partial
>> and SURPRISING answer: The Logic of Life and Mathematical Foundations is
>> NOT the Boolean (or chryssipian)logic of 'Yes' or 'No"--'Tertium
>> non datur"-
>> -(The Excluded Third principle of classical logic).  Recent published
work
>> on the Foundations of Mathematics (Lawvere and Rosebrugh,2001: "Sets for
>> Mathematics", avail. for example from www.amazon.com), mostly by
>> F. William
>> ('Bill') Lawvere and Radu Diaconescu, has shown that the axiom of
>> choice --
>> related to the use of "SETS"-- implies Booleaness. Thus, the use
of 'sets'
>> that are subjected to the Axiom of Choice and the chryssipian logic of
The
>> Excluded third--or even categories of sets with this restriction-- as in
>> the original paper of Robert Rosen on the Representation of biological
>> systems in terms of Categories of Sets (published in 1958), limits one to
>> the consideration biological systems models of Boolean
>> Logic-based systems,
>> (i.e., machine-like, robot-like, automata/computer models of living
>> organisms that are LINEAR, DECOMPOSABLE, ANALIZABLE-IN-PARTS, that are
>> identified by Robert Rosen in "Life Itself" as 'REDUCTIONIST'-type
models,
>> because such models attempt -and claim- explicitely the 'reduction' of
>> Biology to Physics, and physical 'mechanism' ONLY). -------On the other
>> hand, the New Mathematical Foundation proposes the use of Intuitionistic
>> Logic and more specifically the use of Heyting Logic Algebras for the
>> Foundations of Mathematics and the concept of a Topos that incorporates
>> both a Category of General Type Structures (not just STRUCTURELESS sets)
>> together with a Heyting Algebra/ 'Subobject classifier". Sets, and
>> Categories of sets, can still be defined of course, as a
>> particular kind of
>> Topos, subject to the Boolean Logic, selected axiomatics, etc. In my own
>> published work in the Bulletin of Mathemical Biophysics (BMB;later re-
>> defined as the 'Bulletin of Mathematical Biology'), as early as 1970 in
>> "Organismic Supercategories: II. On Multistable Systems." ref.II), I have
>> introduced an ETAS axiomatics for models of living organisms in terms of
>> higher-level categories of functional biological system models, as, for
>> example in, categories of categories... (or N-categories/supercategories)
>> of living cell models as a representation of an entire living
>> organism. The
>> ETAS axiomatics I selected was based on F.W. Lawvere's ETAC axioms for
>> Category Theory that avoided employing the concept of set in defining
>> Categories (Baianu, 1970, in BMB, ref.II). Thus, the concepts that I
>> defined in this paper back in 1970 are still current and have passed the
>> test of 34 years, whereas biological modeling in Relational Biology
>> that employ Sets and Boolean Logic as in Nicolas Rashevsky's
>> Organismic Set
>> Theory, and in Robert Rosen's initial work on Categories of Sets for
>> modeling Biological Systems had to be abandoned. This is also explicitely
>> stated by Robert Rosen in his "Essays on Life" (2002) where he selects
>> explicitely Categories of Sructured Objects to represent (M,R)-systems
>> instead of categories of SETS; this essential requirement was pointed out
>> in our earlier papers (Baianu,1970 (BMB); Baianu & Marinescu,1973(RRM-
PA);
>> Baianu, 1980(BMB), Baianu (1983, SIAM--manuscript available with the May
>> and June 2004 updates at this website). Thus the concepts of Categories
>> of Structured Objects and Topos are currently and firmly established to
be
>> essential for the modeling of living organisms in Relational Biology.
>>
>> __________________________________________________________________
>>
>> 1. Adjointness at the Foundation of Current Logics and Mathematics.
>> ---------------------------------------------------------------------
>> The concept of Adjunction/Adjointness and Adjoint Functors is
>> considered to
>> be "the most important concept" ("Categorical Foundations", Pedicchino
and
>> Tholen, Eds. 2004; Cambridge Univ. Press) in the current theory and
>> applications of Categories or Topoi. Without going into a very formal and
>> detailed definition of adjunction/ adjointness, let us consider
>> briefly the
>> fact that Logics other than Boolean, such as Lukasiewicz (N-valued
Logics)
>> and Intuitionistic, Heyting Logic Algebras (that are at the basis of the
>> concept of Topos which is briefly explained at the end of this paragraph)
>> involve a special type of ordered structures, called Lattices.
Definitions
>> of such Logics based on ordered sets abound in the older
>> literature but can
>> be now treated by means of adjunction in the bicategory of
>> ordered objects,
>> Ord(T) in a Topos, T (this is the subject of the first chapter
>> ("I. Ordered
>> Sets via Adjunctions"  by R. J. Wood in Pedicchino and Tholen,2004, pp.5-
>> 48). The concept of Topos extends, and enormously generalizes, the older
>> concept of category that implied the use of sets and classes, and
>> implicitely Bolean Logic for sets. As Saunders MacLane said, one can
think
>> of a Topos as a <generalized--the most general-- form of "Space">.
>>
>> 2.What is /are The Logic(s) of Life?
>>
>> Having chosen Categories of Structured Objects, N-Categories and Topoi
>> as the effective tools for models of living, FUNCTIONAL, organisms in
>> Relational Biology one is still faced with the choice of the appropriate
>> Logic or Logics for such modeling. In the above discussion, the use of
>> Boolean Logic and Sets based on such Logic was rejected as being
>> appropriate
>> only for automata, robots, machines, and all other kind of simple systems
>> that are Linear and that can be 'decomposed' and studied/analyzed part by
>> part, component-by-component without any loss--and then
>> re-assembled or put
>> back together, things that we are unable to do with a living organism
down
>> to the level of its molecules. Within a Topos the most general Logic
>> available is intuitionistic, and more specifically, it involves a Heyting
>> Logic algebra. On the other hand, in a paper published in 1977 in BMB
>> I proposed the use of Lukasiewicz, N-valued Logic and Categories of
>> Lukasiewicz Logic algebras to model Genetic Activities and Nonlinear
>> Dynamicsof Genetic Networks in Living Organisms (Baianu, 1977, BMB; see
>> also my May 2004 posting available at this website). Both Heyting Logic
>> Algebras and Lukasiewicz Logic algebras involve Lattices, that is
>> a certain
>> type of Ordered Objects that establishes a 'Natural' Hierarchy,
>> or 'stratification' of essential objects (see also Hartmann's ontological
>> theory (1936?!), which is currently discusssed and formalized in terms of
>> categories and N-categories-- supercategories-- in recent papers
published
>> in the Axiomathics journal edited by Dr. Roberto Poli). The Category of
>> Lukasiewicz (n-valued Logics) Algebras, LuknCAT is a full
>> subcategory of the
>> Category of Heyting Logic Algebras, HeytingCAT. The functor from
>> LuknCAT to
>> HeytingCAT is both full and faithful! As ordering of objects can be
>> characterized through adjunctions/adjoint functors (as discussed above),
>> -------------------------------------------------------------------------
>> Therefore, one concludes that the Logic of Life INVOLVES fundamental
>> Adjunctions in Ord(T),( with T being a Topos), that can occur in
>> either the
>> category of Lukasiewicz (n-valued)Logics or the Category of
Intuitionistic
>> Heyting Logic Algebras. It would, thus seem also that the Logics
>> of Complex
>> Systems in the Rosennean sense is Lukasiewicz (n-valued), or
>> Heyting Logic , or other related possible Logics that one may consider
>> (notably among them, as an example, the Fuzzy Relational Logic).
>> _________________________________________________________________________
>>
>> Regards,
>>
>> Ionel