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Peiyuan Zhu (Nov 27 2022 at 01:51):How to understand that this says "In the case of formal systems this means that one is working entirely on the symbolic side, quite a different matter from the biology where there is no intrinsic symbolism, only our external descriptions of processes in such terms."
Peiyuan Zhu (Nov 27 2022 at 01:51):was found in this paper https://arxiv.org/pdf/1512.04325.pdf
Peiyuan Zhu (Nov 27 2022 at 01:52):I'm not sure what he means by "intrinsic symbolism" and "external description of processes"
Peiyuan Zhu (Nov 27 2022 at 02:23):An overall reading of the paper: is he suggesting an analogy between quantum logic and DNA replication? Is this process realizable in classical computer? In the paper Kauffman cited Laws of Form of George Spencer-Brown. In what way is what's proposed in the paper different from Spencer-Brown's calculus?
Peiyuan Zhu (Nov 27 2022 at 02:42):
"This is no accident, since topology is indeed a controlled study of cycles and circularities in primarily geometrical systems." I'm not sure what does this mean.
Peiyuan Zhu (Nov 27 2022 at 04:33):"For example, Heinz von Forerster points out that the sentence “I am the observed relation between myself and observing myself.” defines the concept of “I” as an eigenform of the transformation" also not sure if I really understand this. Is there a real-life example of relation between myself and observing myself? What is a relation? Is it a mathematical construction? ?
John Baez (Nov 27 2022 at 09:50):My advice: use double dollars, not single dollars, around math expressions.
Peiyuan Zhu (Nov 27 2022 at 10:41):It sounds like “reflexive domain” is just another word for dynamical system and “eigenform” is just another word for fixed point, am I right?
Morgan Rogers (he/him) (Nov 27 2022 at 10:53):Peiyuan Zhu said:
I'm not sure what he means by "intrinsic symbolism" and "external description of processes"
"Symbols" are the formal objects we use to represent objects, such as names of things or variables in equations or formulae. When we work in logic, we directly manipulate symbols. We can do this to reason about biological systems, for example. In the biological system itself, however, there are no formal symbols being manipulated (or so this author claims).
Morgan Rogers (he/him) (Nov 27 2022 at 10:54):Peiyuan Zhu said:
"This is no accident, since topology is indeed a controlled study of cycles and circularities in primarily geometrical systems." I'm not sure what does this mean.
This quote is quite funny. The author says that "we avoid cycles in mathematics" and then goes on to talk about an area of mathematics where cycles are the objects of study...
Morgan Rogers (he/him) (Nov 27 2022 at 11:01):Peiyuan Zhu said:
It sounds like “reflexive domain” is just another word for dynamical system and “eigenform” is just another word for fixed point, am I right?
There are some red flags for me in this text, but I'll try to answer anyway. "Eigenform" only makes sense for linear operators, and means a value for which applying the operation gives a multiple of that value (linearity ensures that this property will moreover be shared by the multiples of the value). The author really wants to talk about fixed points, which make sense more broadly.
I don't see "reflexive domain" in the part of the text that you quoted.
Peiyuan Zhu (Nov 27 2022 at 11:54):Morgan Rogers (he/him) said:
Peiyuan Zhu said:
It sounds like “reflexive domain” is just another word for dynamical system and “eigenform” is just another word for fixed point, am I right?
There are some red flags for me in this text, but I'll try to answer anyway. "Eigenform" only makes sense for linear operators, and means a value for which applying the operation gives a multiple of that value (linearity ensures that this property will moreover be shared by the multiples of the value). The author really wants to talk about fixed points, which make sense more broadly.
I don't see "reflexive domain" in the part of the text that you quoted.
image.png
On reflexive domain
Peiyuan Zhu (Nov 27 2022 at 11:55):Morgan Rogers (he/him) said:
Peiyuan Zhu said:
"This is no accident, since topology is indeed a controlled study of cycles and circularities in primarily geometrical systems." I'm not sure what does this mean.
This quote is quite funny. The author says that "we avoid cycles in mathematics" and then goes on to talk about an area of mathematics where cycles are the objects of study...
I think he intend to separate topology & knots from most of other mathematics. But how topology deal with circularity is what I'm not sure about.
Peiyuan Zhu (Nov 27 2022 at 11:57):Morgan Rogers (he/him) said:
Peiyuan Zhu said:
I'm not sure what he means by "intrinsic symbolism" and "external description of processes"
"Symbols" are the formal objects we use to represent objects, such as names of things or variables in equations or formulae. When we work in logic, we directly manipulate symbols. We can do this to reason about biological systems, for example. In the biological system itself, however, there are no formal symbols being manipulated (or so this author claims).
But I don't see why lambda calculus being extrinsic (is he contrasting object-oriented programming as intrinsic?). Isn't lambda calculus a kind of formal system?
Morgan Rogers (he/him) (Nov 27 2022 at 15:32):Yes, lambda calculus is a formal system. So its constituents are symbols, whereas that is not the case (or at least not obviously the case) for biological systems.
Morgan Rogers (he/him) (Nov 27 2022 at 15:34):A reflexive domain is a structure required to interpret lambda calculus, one with an isomorphism/equivalence of types .
Jorge Soto-Andrade (Nov 27 2022 at 15:53):Morgan Rogers (he/him) said:
Peiyuan Zhu said:
It sounds like “reflexive domain” is just another word for dynamical system and “eigenform” is just another word for fixed point, am I right?
There are some red flags for me in this text, but I'll try to answer anyway. "Eigenform" only makes sense for linear operators, and means a value for which applying the operation gives a multiple of that value (linearity ensures that this property will moreover be shared by the multiples of the value). The author really wants to talk about fixed points, which make sense more broadly.
I don't see "reflexive domain" in the part of the text that you quoted.
Hi Peiyuan Zhu, Regarding reflexive domains and fixed points, you may want to have a look at
J. Soto-Andrade, F. J. Varela, "Self reference and fixed points", Acta Appl. Math., 2(1984), 1-19.
https://www.researchgate.net/publication/225997258_Self-reference_and_fixed_points_A_discussion_and_an_extension_of_Lawvere's_Theorem
For a rather friendly survey on self-reference and its many avatars, see:
J. Soto-Andrade, S. Jaramillo, C. Gutiérrez, J.-C. Letelier (2011), Ouroboros avatars: a mathematical exploration of self-reference and metabolic closure, in Advances in Artificial Life,
ECAL 2011: Proceedings of the Eleventh European Conference on the Synthesis and Simulation
of Living Systems, T. Lenaerts, M. Giacobini, H. Bersini, P. Bourgine, M. Dorigo & R. Doursat
(Eds.), The MIT Press, Cambridge MA, p. 763-770.
https://www.researchgate.net/publication/220039783_Ouroboros_avatar_A_mathematical_Exploration_of_Self-reference_and_Metabolic_Closure