[Fis] FIS discussions. Beyond Equality in Math. Implication

Thu Oct 31 15:14:29 CET 2019

Dear All,

This note contains some very useful notions and references. It seems to me
that there are very familiar ones which have not received discussion here:
one is the obvious greater than > or less than <. The sign for non-equality
is simply the negation of = and doesn’t bring much new.

The relation that might be much more important is implication c. Its use is
standard in logic, for ‘material’ implication, but I think this can be
extended to a concept of implication as a process of causation in ‘real
matter’. Any takers? Does it correspond to Pedro’s First Hypothesis, since
“novel whole is formed” can be read “new entity emerges”? Can this sign be
used to imply distributions of qualities in real, non-number systems?

Best,

Joseph

  _____  

From: Fis [mailto:fis-bounces at listas.unizar.es] On Behalf Of Pedro C.
Marijuan
Sent: jeudi, 31 octobre 2019 14:06
To: fis at listas.unizar.es
Subject: Re: [Fis] FIS discussions. Units

Dear List,

Some brief responses to Jerry and Karl.

Q4: Does an informed path exist which logically organizes the inanimate into
the animate?

Q5: What are relationships between the inanimate objects and the animate
objects?

Hypothesis: If two independent forms (parts) are copulated (linked,
conjoined, connected, bound) together, a novel interdependent informed whole
is formed. 

Hypothesis: A set of atomic numbers can be composed into an animate object
by copulating the set of parts into a natural sort or kind (an organized
whole). 

Pedro: Do these assertions add any light  to you critical quation about
possible relationships between units,  the animate and the inanimate?  Is
any simpler scientific mathematics possible?

Thanks for the abstraction effort, Jerry. Your whole questions set is a very
good discussion guide although enormously difficult to be answered, at least
in the biological realm. Even for a very simple cell, eg the prokaryote
(bacteria), the way its components are coupled and the relationship they
keep with their environment has not been properly put in informational terms
yet, as far as I know. A couple of years ago I made a pretty complete
catalogue of the "signaling parts" of E. coli, and the result was surprising
for me (see Marijuan et al., BioSystems, 2017). In a few words, "nothing was
eaten that had not been previously recognized by some signaling apparatus".
It is literal, for in the order of 200 'receptors' of all sort could check
for 300 or more different types of 'food' molecules. Putting in another way,
the "energy flow" and the "information flow" of the living cell are
completely interrelated. And the result of their 'logical' coupling is the
systematic emergence of a life cycle that includes reproduction --Spinoza's
principle of conatus. What kind of elegant informational/logical synthesis
could be made (beyond the ensuing Darwinian Dogma)? 

Responding to Karl, I was surprised to find, some posts ago, a critique of
the equality sign. His idea, well argued from his multidimensional
partitions argument (equality hides from view the many possible variable
distributions of qualities inside the number's sumands), has been
coincidentally developed by other mathematicians in a different field:
"infinite categories". See the abstract below, (courtesy of Malcolm Dean). 

With Category Theory, Mathematics Escapes From Equality
Two monumental works have led many mathematicians to avoid the equal sign.
Their goal: Rebuild the foundations of the discipline upon the looser
relationship of “equivalence.” The process has not always gone smoothly.

Kevin Hartnett, Senior Writer

https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from
-equality-20191010/
<https://www.quantamagazine.org/with-category-theory-mathematics-escapes-fro
m-equality-20191010/?utm_source=Quanta+Magazine&utm_campaign=388bce3947-RSS_
Daily_Mathematics&utm_medium=email&utm_term=0_f0cb61321c-388bce3947-38991267
7&mc_cid=388bce3947&mc_eid=9dd29ead65> 

Quanta Magazine, 10 October 2019

The equal sign is the bedrock of mathematics. It seems to make an entirely
fundamental and uncontroversial statement: These things are exactly the
same.

But there is a growing community of mathematicians who regard the equal sign
as math’s original error. They see it as a veneer that hides important
complexities in the way quantities are related — complexities that could
unlock solutions to an enormous number of problems. They want to reformulate
mathematics in the looser language of equivalence. “We came up with this
notion of equality,” said Jonathan Campbell
<http://www.jonathanacampbell.com/>  of Duke University. “It should have
been equivalence all along.” The most prominent figure in this community is
Jacob Lurie <https://www.ias.edu/scholars/lurie> . In July, Lurie, 41, left
his tenured post at Harvard University for a faculty position at the
Institute for Advanced Study in Princeton, New Jersey, home to many of the
most revered mathematicians in the world. Lurie’s ideas are sweeping on a
scale rarely seen in any field. Through his books, which span thousands of
dense, technical pages, he has constructed a strikingly different way to
understand some of the most essential concepts in math by moving beyond the
equal sign. “I just think he felt this was the correct way to think about
mathematics,” said Michael Hopkins <http://www.math.harvard.edu/~mjh/> , a
mathematician at Harvard and Lurie’s graduate school adviser. Lurie
published his first book,
<https://press.princeton.edu/books/paperback/9780691140490/higher-topos-theo
ry-am-170> Higher Topos Theory, in 2009. The 944-page volume serves as a
manual for how to interpret established areas of mathematics in the new
language of “infinity categories.” In the years since, Lurie’s ideas have
moved into an increasingly wide range of mathematical disciplines. Many
mathematicians view them as indispensable to the future of the field. “No
one goes back once they’ve learned infinity categories,” said John
<https://sites.math.northwestern.edu/~jnkf/>  Francis of Northwestern
University.

So... very good point by Karl! Could new mathematical ideas provide the
bio-mathematical (informational) synthesis needed?

Best wishes to all,

--Pedro

-- 
-------------------------------------------------
Pedro C. Marijuán
Grupo de Bioinformación / Bioinformation Group

pcmarijuan.iacs at aragon.es
http://sites.google.com/site/pedrocmarijuan/
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