Thank you for mentioning Safal. Safal reminds me of myself. When I was a kid >35 yrs ago I was interested in both Lawvere (via Goldblatt) and horizontal gene transfer due to em radiation...
Your statement "The act of letting an object feed itself into its own description is generated by this theorem" is a useful step closer to the difference between 'self-referential' and 'self-referral' --- a key issue in the Vedic account of higher states of consciousness. But it (as well as 'fixing a point') is within the 'object-referral' or 'self-referential' experience (subject/object independence, 'I-it' duality) of the ordinary waking state of consciousness. It does not yet reflect appreciation of the 'direct' experience of transcending all mental activity to consciousness itself -- 'self-referral' beyond language and the intellect. The 'I' of the 'self' remains the individual 'I' or' 'self', which when transcended is unified with the universal 'I' or 'Self' (one in One). That union of individual and universal -- first 'directly experienced' in transcendental consciousness (turiya, 4th state of consciousness, samadhi) is permanently established in the 5th state (cosmic consciousness, Kaivalya, turiyatit chetana). In the 7th state of consciousness (unity consciousness, Brahman consciousness, Brahma chetana), all relative diversity of the phenomenal universe (maya, mithya) is 'directly' known to be one's universal Self ("I am Totality", 'aham brahmasmi). It is from that ultimate unified perspective that Vedanta and the Brahma Sutras (e.g.) describe the 'infinite self-referral dynamics' of the unified field of consciousness -- in which all relative diversity is instantiated.
Again, please consider the paper "A Psychological Critique of Mathematicians", and also "Ignorance and Enlightenment: What's the Difference?"(both at ResearchGate.net) for clarification of these issues referred to in Godel, Turing, the 'No Supervenience Theorem', etc. related to the distinction between 'self-referential' and 'self-referral' mentioned above -- which seems rarely understood, apparently due to no 'direct experience' of transcendental consciousness.
At first I wrote: ""is it a stretch to say "there is *one* theorem that underlies all of these"."" but on second thoughts Lawvere's result really is a unifying thread.
Gödel’s theorem certainly relies on the ability to construct self-referential statements using a diagonal construction, so it leans on Lawvere (before Lawvere). But unique prime factorization is also absolutely critical for Gödel. Turing requires at least a mental (gedanken) construction of a machine. (An idealized "procedure".) So there is something unique about all the self-referential results, but I think your essay is right about this underlying common theme in many so-called self-referential "paradoxes".
There are other circular logic types of paradox that do not use diagonalization or fixed points:
* Grelling’s Paradox — arises from categorization and definition rather than a fixed-point structure.
* Quine's Quinning? — Self-reference but not diagonalization? A think linguistics is too weak to have a diagonalization construction. Though I could be wrong.
* Yablo’s Paradox — avoids direct diagonalization by not having a single statement referring to itself, but still collapses into an infinite regress of contradictions.
* Various self-Referential Probability Paradoxes e.g., "The probability of this statement being true is less than 1/2." This does not follow a clear diagonalization path but introduces paradox through the probabilistic nature of its self-reference.
* Singular Limits — the Urn Game. The Grim Reaper paradox. (Not really self-referential perhaps. More in the class of, "things to be careful about if your are an Infinite entity." 🤣)
However, in the spirit, Lawvere does seem a major universal aspect of most of these self-referential paradoxes.
These are papers written by physicist and philosopher Jochen Szangolies, he argue there, that fixed point theorem is very connected to foundations of quantum physics. I am not educated enough to follow his arguments but people from FQXI liked it a lot.
I'd bet though that an epistemic horizon is not a *fundamental* principle. We only need non-trivial spacetime topology. This is because the only effective epistemic horizon needed for QM is a practical one, a measurement limit horizon, and geometrodynamics already has this (hence also any theory or model that has a similar limit). Barandes' PSQM has such an horizon too.
I propose a No Direct Self-Reference Theorem: no statement can meaningfully and non-trivially refer to itself by itself. More strictly, no direct self-reference is possible. Even when we make statements that purport to be directly self-referring, we are already relying on external terms to mediate this relation.
The theorem can be proven by substituting any nominally ‘self-referring’ statement for the term in the statement that ‘refers’ to the statement, which evidently result in infinite regress (an absent object). Such statements are therefore essentially incomplete, and never make a proper sentence.
From this I concluded that self-reference is necessarily mediated, and the kind of self-reference that we call ‘subjectivity’ or ‘reflexive consciousness’ is intrinsically, structurally socially mediated.
Hi Curt! I was referring to the fact that concepts can't contain reality. It is a Buddhist precept. I've read a few sutras (Mahayana sutras) . I don't remember exact page. But it is a well known fact about buddhism as philosophy that dismisses rational thought as means of reaching reality. And I mean the reality of consciousness. Exactly like the theorems demonstrations in your article. There is a chapter in Godel Escher Bach that exposes the same buddhist perspective. Douglas Hofstadter is a student of Zen Buddhism but his scientific training makes him struggle with the conclusion of Godel's incompleteness. He quips in several parts of the book how a self referent system can't escape the language in witch is written. That is exactly the problem! Consciousness is not a level of language. For Hofstadter the levels of physical reality quantum , chemical, bio - cellular , neuro systems they all are systems like logic. One system constructed upon the other. They are separated every level having new meanings. Macro-languages though constructed on micro-languages can have different different meaning. I hope I don't irritate you by stressing that reality is not this objective nice mathematical object that can be bagged into a well ordered countable set. Though mathematics invented ever new number systems to precisely quantify it. So you see I may be a dilettante but I am a dilettante with a very broad self made education. The truth pops up everywhere like in the history and meaning of this chinese character 我. I - the real I , I mean is a fight with the world that tries to measure with nefarious purpose. So I is a hand with a weapon in protection of self. What else is the struggle between the two programs in the halting problem?
I hope I made myself clear. No teacher of mine ever gave me that. Why else so much formalism. I've explained these complicated theories to my grandma and she understood it. You know what she said? ''That's what they do in life to you''
I'm far from your swiftness of devouring math and physics but I have a steady tortoise pace. I barely understand sometimes but reflection is a good advisor. I think more than 100 years of formalization and obfuscation can be rendered eventually in normal everyday language. That's why science is in shambles today: obfuscation stemming from the conflict of reality.
Yanofsky's paper "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points" shows that we can cover this material using sets and functions -- we don't really need category theory. Also, the applications are more interesting than the theorem (the theorem is pretty easy).
How are all these theorems something new when Buddhist philosophical tradition concluded that reality isn't captured in concepts? Mathematicians construct new concepts with new names structures upon structures like a veritable Qlippoth shells that hide reality.
Abstract this abstract that-"to make abstraction of'' means to ignore. Strange isn't it to call Mathematics ignorance? It only shows that the domain of objectivity is delimitated by something else. “perfect self-awareness” is not achievable with understanding. Something else is needed. Raymond Tallis said something like that - I've seen a 'short' on youtube.
“there’s a statement you can’t tame.” The Billion $ question is why do you need to tame it? :)
Each of the theorems is something new. Lawvere only underpins the structure that is common in each, but cannot alone be used to derive the other results. There must also be a Buddhist - or other tradition - parable or aphorism for this — that adding a little to what already exists can be yuuuge! In mind right now I have the "Less is More" essay and Carl Bender's lectures which are delightful in showing an innocent touch to a differential equation can turn it from analytically sovlable to not even sovlable using Taylor series.
They are new in the sense that they show the same thing from different perspectives. But they all speak about the same thing. As for your reference about Taylor series I wouldn't know. My mathematical education was a trauma made by inhumane teachers.
Every notion I would have stopped to ask more about it's origin and construction was treated as self-understood. To quote from the books I have : ''it is easy to see'' ''it is easy to derive'' ''it is elementary. Do it as an exercise!'' Fortunately Principia Mathematica does not skip logical steps. I understood more from this book and I developed a instinct to reduce higher mathematical language to low level language. Some notions remain though hidden to me until someone unclenches his teeth and loosens a definition in simple words. I've become like an acid vat that dissolves higher language.
It all revolves around x ∈ x or x ∉ x ~ to be or not to be. I mean the paradoxes and math itself.
The last part of you comment which I saw in my Inbox seemed to be removed. Bijou is not my username. It's the one on my birth certificate. I am not familiar with the face dancers in _Dune_.
As for the rest of your comment, nice comments but... Forrest/Trees (?) 🤣
I think to claim they show "the same thing from different perspectives" is a very narrow read. The theorems all show an application of one idea (Lawvere's or diagonalization) in totally different contexts.
You can take the Zen-like idea "All is One" to far too big an extreme. It's more allegorical, not literal. You can also say with a serious face "The One is Many". IN fact I have, agreeing with G. Cantor and R. Rucker. Again, it's more poetry than logic.
''Forrest/Trees'' it is precisely why science today is stopping. The precise logical notions are not that 🤣 . Narrow indeed not to see the polysemy of reality.
"''Forrest/Trees'' it is precisely why science today is stopping. "
However, there is nothing in what I wrote to imply I am failing to recognize the polysemy of reality. The issue is not that non-scientific traditions are incomplete, quite the opposite. The issue is that they are always inexact since they go beyond logic and can admit inconsistency. So you can use scriptures to say something (maybe even useful) about anything. It will not mean you've divined the Truth. Unless you are a god.
There is a huge difference, an infinite gulf. Logic is exact and consistent, but incomplete. Poetry can be complete (has a thing to say about anything), but inexact and inconsistent. They are complimentary.
It's delightful to be able to look at the incompleteness theorem as a fixed point result. I guess fixed point theorems are a blessing and a curse. With them we prove the existence of solutions to many interesting mathematical problems, and with them we also prove the existence of paradoxes that show the holes in our systems of axioms.
I am a starting writer at the intersection of literature, philosophy (of mind and mathematics), and would love your feedback! https://substack.com/home/post/p-156101031
Have you read "Laws of Form (LAW)", by g spencer brown? Worth reading and studying. Using LAW and my Pseudo Infinity set (i.e. the finite set of different pictures that a given digital camera can take); I have been exploring the set of 0 dimensional mathematical pixels (as form) and missing pixels (as emptiness) with and without properties. Assuming CH, and the axiom of choice; we get 2^aleph1 = aleph1^aleph1 = aleph2 This set contains from finite to infinite, possible mathematical/physics/thought/etc. constructions.
Perhaps then this could support RQM (Rovelli)? "Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world" (Rovelli).
In other words, you only know yourself through relationships?
In consciousness you only know yourself through yourself.
Relationalism taken to the extreme is garbage (imho). It is infantile (imho) avoidance of a fundamental duality: self and other. If you want to go down the road of "all is relational" and deny the existence of unique souls, then just be warned, it can be used against you in extremist politics and economics. ("I'm not robbing you, I'm taking from us to give to us.")
Note that Dualism or something like it (Pluralism, many 'uniquenesses', pluralist ontologies) do not preclude relationalism. All things can be related by some intermediations. That's basically the whole lot of the Standard Model of particle physics.
So fundamental physics gives us pretty good working model for dualism, and is totally compatilble with relationalism too, just not absurd extreme relationalism where the relation is both object and arrow.
Thank you for mentioning Safal. Safal reminds me of myself. When I was a kid >35 yrs ago I was interested in both Lawvere (via Goldblatt) and horizontal gene transfer due to em radiation...
I only understand my future self once the present has passed.
Your statement "The act of letting an object feed itself into its own description is generated by this theorem" is a useful step closer to the difference between 'self-referential' and 'self-referral' --- a key issue in the Vedic account of higher states of consciousness. But it (as well as 'fixing a point') is within the 'object-referral' or 'self-referential' experience (subject/object independence, 'I-it' duality) of the ordinary waking state of consciousness. It does not yet reflect appreciation of the 'direct' experience of transcending all mental activity to consciousness itself -- 'self-referral' beyond language and the intellect. The 'I' of the 'self' remains the individual 'I' or' 'self', which when transcended is unified with the universal 'I' or 'Self' (one in One). That union of individual and universal -- first 'directly experienced' in transcendental consciousness (turiya, 4th state of consciousness, samadhi) is permanently established in the 5th state (cosmic consciousness, Kaivalya, turiyatit chetana). In the 7th state of consciousness (unity consciousness, Brahman consciousness, Brahma chetana), all relative diversity of the phenomenal universe (maya, mithya) is 'directly' known to be one's universal Self ("I am Totality", 'aham brahmasmi). It is from that ultimate unified perspective that Vedanta and the Brahma Sutras (e.g.) describe the 'infinite self-referral dynamics' of the unified field of consciousness -- in which all relative diversity is instantiated.
Again, please consider the paper "A Psychological Critique of Mathematicians", and also "Ignorance and Enlightenment: What's the Difference?"(both at ResearchGate.net) for clarification of these issues referred to in Godel, Turing, the 'No Supervenience Theorem', etc. related to the distinction between 'self-referential' and 'self-referral' mentioned above -- which seems rarely understood, apparently due to no 'direct experience' of transcendental consciousness.
Thanks for your wonderful work.
RW Boyer (Bob)
At first I wrote: ""is it a stretch to say "there is *one* theorem that underlies all of these"."" but on second thoughts Lawvere's result really is a unifying thread.
Gödel’s theorem certainly relies on the ability to construct self-referential statements using a diagonal construction, so it leans on Lawvere (before Lawvere). But unique prime factorization is also absolutely critical for Gödel. Turing requires at least a mental (gedanken) construction of a machine. (An idealized "procedure".) So there is something unique about all the self-referential results, but I think your essay is right about this underlying common theme in many so-called self-referential "paradoxes".
There are other circular logic types of paradox that do not use diagonalization or fixed points:
* Grelling’s Paradox — arises from categorization and definition rather than a fixed-point structure.
* Quine's Quinning? — Self-reference but not diagonalization? A think linguistics is too weak to have a diagonalization construction. Though I could be wrong.
* Yablo’s Paradox — avoids direct diagonalization by not having a single statement referring to itself, but still collapses into an infinite regress of contradictions.
* Various self-Referential Probability Paradoxes e.g., "The probability of this statement being true is less than 1/2." This does not follow a clear diagonalization path but introduces paradox through the probabilistic nature of its self-reference.
* Singular Limits — the Urn Game. The Grim Reaper paradox. (Not really self-referential perhaps. More in the class of, "things to be careful about if your are an Infinite entity." 🤣)
However, in the spirit, Lawvere does seem a major universal aspect of most of these self-referential paradoxes.
https://arxiv.org/abs/1805.10668
https://arxiv.org/abs/2007.14909
These are papers written by physicist and philosopher Jochen Szangolies, he argue there, that fixed point theorem is very connected to foundations of quantum physics. I am not educated enough to follow his arguments but people from FQXI liked it a lot.
Nice find.
I'd bet though that an epistemic horizon is not a *fundamental* principle. We only need non-trivial spacetime topology. This is because the only effective epistemic horizon needed for QM is a practical one, a measurement limit horizon, and geometrodynamics already has this (hence also any theory or model that has a similar limit). Barandes' PSQM has such an horizon too.
I propose a No Direct Self-Reference Theorem: no statement can meaningfully and non-trivially refer to itself by itself. More strictly, no direct self-reference is possible. Even when we make statements that purport to be directly self-referring, we are already relying on external terms to mediate this relation.
The theorem can be proven by substituting any nominally ‘self-referring’ statement for the term in the statement that ‘refers’ to the statement, which evidently result in infinite regress (an absent object). Such statements are therefore essentially incomplete, and never make a proper sentence.
From this I concluded that self-reference is necessarily mediated, and the kind of self-reference that we call ‘subjectivity’ or ‘reflexive consciousness’ is intrinsically, structurally socially mediated.
Hi Curt! I was referring to the fact that concepts can't contain reality. It is a Buddhist precept. I've read a few sutras (Mahayana sutras) . I don't remember exact page. But it is a well known fact about buddhism as philosophy that dismisses rational thought as means of reaching reality. And I mean the reality of consciousness. Exactly like the theorems demonstrations in your article. There is a chapter in Godel Escher Bach that exposes the same buddhist perspective. Douglas Hofstadter is a student of Zen Buddhism but his scientific training makes him struggle with the conclusion of Godel's incompleteness. He quips in several parts of the book how a self referent system can't escape the language in witch is written. That is exactly the problem! Consciousness is not a level of language. For Hofstadter the levels of physical reality quantum , chemical, bio - cellular , neuro systems they all are systems like logic. One system constructed upon the other. They are separated every level having new meanings. Macro-languages though constructed on micro-languages can have different different meaning. I hope I don't irritate you by stressing that reality is not this objective nice mathematical object that can be bagged into a well ordered countable set. Though mathematics invented ever new number systems to precisely quantify it. So you see I may be a dilettante but I am a dilettante with a very broad self made education. The truth pops up everywhere like in the history and meaning of this chinese character 我. I - the real I , I mean is a fight with the world that tries to measure with nefarious purpose. So I is a hand with a weapon in protection of self. What else is the struggle between the two programs in the halting problem?
I hope I made myself clear. No teacher of mine ever gave me that. Why else so much formalism. I've explained these complicated theories to my grandma and she understood it. You know what she said? ''That's what they do in life to you''
I'm far from your swiftness of devouring math and physics but I have a steady tortoise pace. I barely understand sometimes but reflection is a good advisor. I think more than 100 years of formalization and obfuscation can be rendered eventually in normal everyday language. That's why science is in shambles today: obfuscation stemming from the conflict of reality.
Yanofsky's paper "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points" shows that we can cover this material using sets and functions -- we don't really need category theory. Also, the applications are more interesting than the theorem (the theorem is pretty easy).
How are all these theorems something new when Buddhist philosophical tradition concluded that reality isn't captured in concepts? Mathematicians construct new concepts with new names structures upon structures like a veritable Qlippoth shells that hide reality.
Abstract this abstract that-"to make abstraction of'' means to ignore. Strange isn't it to call Mathematics ignorance? It only shows that the domain of objectivity is delimitated by something else. “perfect self-awareness” is not achievable with understanding. Something else is needed. Raymond Tallis said something like that - I've seen a 'short' on youtube.
“there’s a statement you can’t tame.” The Billion $ question is why do you need to tame it? :)
我 = I 手 =hand + 戈 =a kind of war axe with three blades (https://en.wiktionary.org/wiki/%E6%88%91)
Each of the theorems is something new. Lawvere only underpins the structure that is common in each, but cannot alone be used to derive the other results. There must also be a Buddhist - or other tradition - parable or aphorism for this — that adding a little to what already exists can be yuuuge! In mind right now I have the "Less is More" essay and Carl Bender's lectures which are delightful in showing an innocent touch to a differential equation can turn it from analytically sovlable to not even sovlable using Taylor series.
They are new in the sense that they show the same thing from different perspectives. But they all speak about the same thing. As for your reference about Taylor series I wouldn't know. My mathematical education was a trauma made by inhumane teachers.
Every notion I would have stopped to ask more about it's origin and construction was treated as self-understood. To quote from the books I have : ''it is easy to see'' ''it is easy to derive'' ''it is elementary. Do it as an exercise!'' Fortunately Principia Mathematica does not skip logical steps. I understood more from this book and I developed a instinct to reduce higher mathematical language to low level language. Some notions remain though hidden to me until someone unclenches his teeth and loosens a definition in simple words. I've become like an acid vat that dissolves higher language.
It all revolves around x ∈ x or x ∉ x ~ to be or not to be. I mean the paradoxes and math itself.
The last part of you comment which I saw in my Inbox seemed to be removed. Bijou is not my username. It's the one on my birth certificate. I am not familiar with the face dancers in _Dune_.
As for the rest of your comment, nice comments but... Forrest/Trees (?) 🤣
I think to claim they show "the same thing from different perspectives" is a very narrow read. The theorems all show an application of one idea (Lawvere's or diagonalization) in totally different contexts.
You can take the Zen-like idea "All is One" to far too big an extreme. It's more allegorical, not literal. You can also say with a serious face "The One is Many". IN fact I have, agreeing with G. Cantor and R. Rucker. Again, it's more poetry than logic.
''Forrest/Trees'' it is precisely why science today is stopping. The precise logical notions are not that 🤣 . Narrow indeed not to see the polysemy of reality.
I might not disagree with this:
"''Forrest/Trees'' it is precisely why science today is stopping. "
However, there is nothing in what I wrote to imply I am failing to recognize the polysemy of reality. The issue is not that non-scientific traditions are incomplete, quite the opposite. The issue is that they are always inexact since they go beyond logic and can admit inconsistency. So you can use scriptures to say something (maybe even useful) about anything. It will not mean you've divined the Truth. Unless you are a god.
False memory sorry. It was Bijaz :
https://dune.fandom.com/wiki/Bijaz
Reality does no difference between poetry and logic. They are the same reality. ''One is all and all is one''
https://www.youtube.com/watch?v=9yHiQGWDEc0
There is a huge difference, an infinite gulf. Logic is exact and consistent, but incomplete. Poetry can be complete (has a thing to say about anything), but inexact and inconsistent. They are complimentary.
"The Gulf of America" Mr. Bijaz
Which specific scripture / verse / etc. are you referring to that is like Lawvere's theorem?
"Thanks Curt for these interesting notes on The Mathematics of Self..."
(It doesn't seem my comments got through the first time, so I'm trying again.)
Please consider my recent paper that discusses core issues from
Goedel (Incompleteness Theorem), Turing, and importantly, the recent
'No Supervenience Theorem' from CM Reason and K Shah. It also
distinguishes 'object- referral',' self-referential', self-reflection, and introspection
from 'self-referral'. It is available full-text/open-access on ResearchGate.net,
"A Psychological Critique of Mathematicians". I think you'd find it interesting
and practically useful.
Bob (RW Boyer)
It's delightful to be able to look at the incompleteness theorem as a fixed point result. I guess fixed point theorems are a blessing and a curse. With them we prove the existence of solutions to many interesting mathematical problems, and with them we also prove the existence of paradoxes that show the holes in our systems of axioms.
I've begun writing a bit about St. Augustine's understanding of self-knowledge from the trinitarian perspective https://substack.com/@keithcannon/note/c-90964877
I intuit that reality is a Gordian knot tied around a paradox.
I'd like to understand this description better. Do you have any recommendations for introducing one's self to set theory and category theory?
I am a starting writer at the intersection of literature, philosophy (of mind and mathematics), and would love your feedback! https://substack.com/home/post/p-156101031
Have you read "Laws of Form (LAW)", by g spencer brown? Worth reading and studying. Using LAW and my Pseudo Infinity set (i.e. the finite set of different pictures that a given digital camera can take); I have been exploring the set of 0 dimensional mathematical pixels (as form) and missing pixels (as emptiness) with and without properties. Assuming CH, and the axiom of choice; we get 2^aleph1 = aleph1^aleph1 = aleph2 This set contains from finite to infinite, possible mathematical/physics/thought/etc. constructions.
Perhaps then this could support RQM (Rovelli)? "Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world" (Rovelli).
In other words, you only know yourself through relationships?
In QM you only know others through measurement.
In consciousness you only know yourself through yourself.
Relationalism taken to the extreme is garbage (imho). It is infantile (imho) avoidance of a fundamental duality: self and other. If you want to go down the road of "all is relational" and deny the existence of unique souls, then just be warned, it can be used against you in extremist politics and economics. ("I'm not robbing you, I'm taking from us to give to us.")
Note that Dualism or something like it (Pluralism, many 'uniquenesses', pluralist ontologies) do not preclude relationalism. All things can be related by some intermediations. That's basically the whole lot of the Standard Model of particle physics.
So fundamental physics gives us pretty good working model for dualism, and is totally compatilble with relationalism too, just not absurd extreme relationalism where the relation is both object and arrow.
Or, perhaps, you can only know yourself through others? :)
Yes, that's what is implied here.