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

[Ionel <***>]

Hi, Tim:

I am answering your specific questions one-by-one as we go along with the
quotes.

On Tue, 15 Jun 2004 13:19:40 -0400, Tim Gwinn <***> wrote:

>Hi Ionel,
>A couple of comments/questions interposed below.
>Regards,
>Tim

RE:

>> 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;
>
________________________________________________________________________
-------Question from TG: By this, are you asserting that Rosen's original
(M,R)-model, utilizing the category Ens, is invalid? Or only that Ens (or
Set, for that matter) is inadequate in the role of your RealCAT?
__________________________________________________________________

-------Answer from ICB: The point that Robert makes in his "Essays on
Life..." , printed in 2001-- with which I totally agree-- (as I proposed
something similar in 1970 in my BMB paper "Organismic Supercategories: II.
On Multistable Systems."), is that the Boolean Logic and the Axiom of Choice
limit severely both the entailment and the realizations of (M,R)-system
models, and therefore a 'structured' category , or even better now, a Topos,
is much more appropriate from the Logical standpoint for modeling Life,
cells, organisms, complex systems, etc.

Therefore, 'Yes' , Ens is NOW unsuitable FROM A LOGICAL, and also
ontological, STANDPOINT not only for 'my' choices of RealCAT, BUT also for
Robert Rosen's last choice--as he says it himself in the "Essays on Life".

>
>> 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.

>
>TG: Can you give me the page reference for that? I can't seem to recall
that.>>
------------------------------------------------------------

ICB's Response: I will provide you with a precise page number, but it would
be far better... that you find it... because you'd have then also the
benefit of Robert's original thinking behind this important choice.
-----------------------------------------------------------------
>> 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.

 TG: "I will have to read more, but I think I disagree at least on the
'algorithmic' aspect. From what I read, "intuitionistic" logics are
entirely constructivist in nature: in order for any statement to be true in
an intuitionist logic, a proof - an algorithmic chain of statements - must
be able to be constructed (and similarly, if a statement is to be proven
false, a proof for it must be constructible)."

ICBs Response: ...'Constructed' would be much more accurate, rather
than 'constructible'!  I don't think constructivists like to use the
term 'constructible', because they'd be then prone to ask "constructible by
whom?"

ICBs Response:

A. 'Constructivist' in Logic and Fundamental Mathematics is not limited to
either recursive, or algorithmic, or finitary proofs; it is exactly the
opposite. This is why Topoi are presently the most general 'spaces' known
to man, and they are the ones with the fewest logical constraints or
restrictions. A simple example of the  non-algorithmic character of
Intuitionistic, Heyting Logics is that Non(Non X) is different from X
(that is, not equal to X), whereas ANY algorithmic, or recursive, logic
requires that Non(Non X)= X , that is the twice repeated 'Non's cancel each
other out, as in our everyday 'correct' grammar, but not as in the
grammatically incorrect slang , such as in: "they don't know nothing!"--
which the user intends to mean a strong negation, and not an affirmation--
as it is in classical, chryssippian/Boolean logic. So people using such
slang may be using intuitionistic logic too!

B. Currently, famous logicians also say emphatically that
only 'constructivist' logic provides for--- 'true entailment'--------------
, and that the 'other' Logics, such as Boolean or predicative logic,
finitary, etc., ----do NOT allow for 'true' Entailment----which is also
Robert Rosen's main point and is well-made in his book on "Essays on
Life..."

C. Please also see Robert' substantial discussion in his "Essays..." book
about Hilbert's program to provide a foundation for all mathematics on just
Boolean, predicate logic and sets, or predicates, that according to Robert--
and all the mainstream mathematicians today-- failed to achieve. Robert
Rosen also gives in the "Essays..." the very good reasons for such a
failure of the famous Hilbert program of 'reducing' mathematics to
predicative, Boolean, or finitary, Logics.

I also wrote previously: "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' because double negation is NOT
equivalent to affirmation in Heyting Logics (as it is in Boolean, 2-valued
Logic)."

So, I hope that you'll agree that it does make sense to ask again and
again: "What is/are the Logics-- (appropriate to modeling)-- of Life ?" ,
as I did in my previous posting.

The famous Erwin Schrodinger was asking it differently: "What is <(the
quantum theory that describes)> Life? with the corner brackets added by me
to expalin his implicit question which is contained in his 1945 boolet.
This does not, however, mean that,
either : A. Schrodinger found/wrote it in his booklet as a good, 'real'
answer,

or:      B. That there is a 'good' answer to the question he asked ('What
is --<the quantum theory that would best explain/describe>-- Life ?')
either in terms of his day, or the present day, quantum 'mechanics'
theories, even though there have been many sophisticated claims to that
effect, such as Bohm's implicit or holistic, own version of quantum theory.
Bohm's book on Quantum Theory makes very interesting reading though, and in
many ways, he and Robert have some 'common ground', so to speak, even
though they also have a substantial difference of opinion about the
usefulness of QT in understanding Life -- as Judith Rosen made it amply
clear in several of her recent postings (please see her postings on this
subject). Bohm still believes that a quantum theory along his line of
thinking may explain eventually everything--perhaps even Life Itself-- and
obviously Robert Rosen did come to the opposite conclusion. Then, again if
Howard's arguments are right--one may need different complementary--
'inconsistent' views/models of reality, as approximations of that reality
in order to obtain a 'complete'- or closer to a 'complete'- modeling of
such a complex reality. After all, such complex reality is NOT restricted
to our own Boolean, finitary, or predicative logic, that needs always be
logically consistent when a finite model is proposed. Again, according
to intuitionistic logic, the 'inconsistent' (or complementary) models (of
reality) don't cancel each other out because (Non(Non X)) is different from
just "X" in intuitionistic logics! I'd also readily agree that not everyone
is gonna like, or accept, 'inconsistent',or complementary, models in
Science, and they'd be in need of being reminded of Godel's proof and
quantum theory' s  wave and particle duality --that was introduced by Louis
de Broglie (NP)-- and that Albert Einstein himself was supportive of--which
are two 'inconsistent' models of the same reality of just ONE microphysical
real 'thing', such as 'the' electron,'the' proton, just one neutron, etc.

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
>> >
>>
>> _________________________