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...
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.
Hey, thanks for the shoutout! Yes, I do appeal to Lawvere's theorem in my work, and link it to the notion that many of the properties of quantum mechanics become derivable once you have a fundamental limit to the information that can be obtained about any given physical system.
But I've recently become aware that effectively, Paul Benioff beat me to the punch here, since he showed that if you formalize the mathematics of quantum mechanics within a given model of ZFC, then a string of measurement outcomes will in general not be part of this model (https://doi.org/10.1063/1.522953, https://doi.org/10.1063/1.522954), which directly constitutes an undecidability result. Unfortunately, his work hasn't really gotten the attention that (I think) it deserves, probably because of the (to most physicists) unfamiliar language of model theory and forcing. I hope to eventually untangle this well enough to try and connect it explicitly to my work.
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.
Let me get this right. It essentially points to the fact that, whenever something points to itself, there is an invariant point which cannot be pointed to (the fixed point). This means you can never fully point to yourself.
This means everything that tries to describe itself will always contain something that it cannot describe, something it cannot explain - a true statement that cannot be proven from within the thing (Gödel).
This feels irreducible, like there is something we cannot explain no matter what we do. People might interpret all sorts of things into it, invent names, identify it with the self. One might say: See, there it is, right there is the self - right there where I cannot point. It feels like a center. Like you are an unexplainable paradox at the center of yourself. (I love this kind of stuff.)
Here is how you solve the paradox: Stop pointing.
That's it. When you point, then you are creating a loop. Every loop has a fixed point. But the point isn't a thing. It's the loop that is the thing that comes into existence. At the invariance there is pure symmetry, it has no properties other than not having properties. It's the loop that is creating an asymmetry between the loop and the fixed point. When the loop disappears, then only symmetry is left. But it is impossible to talk about, because every way to talk about it is indexing, creating a loop. Impossible to think about. It is undefinable, inevitable, cannot be named, not a thing, nothing.
When nothing points to itself, it is creating something. Existence in its simplest form is a recursion, a loop. Yet, existence is completely born out of non-existence. Nothingness pointing to itself is still nothingness, or not?
Either you say they are the same, then there is nothing to talk about (pun intended).
Or you say they are not the same. But you can only claim that when ignoring the possibility that they are.
These things are only different because you forget that they are the same. Existence is created through ignorance. By letting go of ignorance you see things how they really are: empty of inherent existence.
Existence implies nothingness and nothingness implies existence. When you identify with the fixed point, you are identifying with nothingness, but the process of identification already creates somethingness. When you point to yourself, you are already taking an outside view. By looking at something you are creating a boundary between it and you, subject and object. You are creating the illusion of your self by drawing boundaries. When you stop drawing boundaries "you" merge back into the universe. You become one with everything.
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.
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.
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).
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.
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.
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.
Hey, thanks for the shoutout! Yes, I do appeal to Lawvere's theorem in my work, and link it to the notion that many of the properties of quantum mechanics become derivable once you have a fundamental limit to the information that can be obtained about any given physical system.
But I've recently become aware that effectively, Paul Benioff beat me to the punch here, since he showed that if you formalize the mathematics of quantum mechanics within a given model of ZFC, then a string of measurement outcomes will in general not be part of this model (https://doi.org/10.1063/1.522953, https://doi.org/10.1063/1.522954), which directly constitutes an undecidability result. Unfortunately, his work hasn't really gotten the attention that (I think) it deserves, probably because of the (to most physicists) unfamiliar language of model theory and forcing. I hope to eventually untangle this well enough to try and connect it explicitly to my work.
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.
Read the Laws of Form by g spencer brown, check it out.
Read the Laws of Form by g spencer brown, check it out
Let me get this right. It essentially points to the fact that, whenever something points to itself, there is an invariant point which cannot be pointed to (the fixed point). This means you can never fully point to yourself.
This means everything that tries to describe itself will always contain something that it cannot describe, something it cannot explain - a true statement that cannot be proven from within the thing (Gödel).
This feels irreducible, like there is something we cannot explain no matter what we do. People might interpret all sorts of things into it, invent names, identify it with the self. One might say: See, there it is, right there is the self - right there where I cannot point. It feels like a center. Like you are an unexplainable paradox at the center of yourself. (I love this kind of stuff.)
Here is how you solve the paradox: Stop pointing.
That's it. When you point, then you are creating a loop. Every loop has a fixed point. But the point isn't a thing. It's the loop that is the thing that comes into existence. At the invariance there is pure symmetry, it has no properties other than not having properties. It's the loop that is creating an asymmetry between the loop and the fixed point. When the loop disappears, then only symmetry is left. But it is impossible to talk about, because every way to talk about it is indexing, creating a loop. Impossible to think about. It is undefinable, inevitable, cannot be named, not a thing, nothing.
When nothing points to itself, it is creating something. Existence in its simplest form is a recursion, a loop. Yet, existence is completely born out of non-existence. Nothingness pointing to itself is still nothingness, or not?
Either you say they are the same, then there is nothing to talk about (pun intended).
Or you say they are not the same. But you can only claim that when ignoring the possibility that they are.
These things are only different because you forget that they are the same. Existence is created through ignorance. By letting go of ignorance you see things how they really are: empty of inherent existence.
Existence implies nothingness and nothingness implies existence. When you identify with the fixed point, you are identifying with nothingness, but the process of identification already creates somethingness. When you point to yourself, you are already taking an outside view. By looking at something you are creating a boundary between it and you, subject and object. You are creating the illusion of your self by drawing boundaries. When you stop drawing boundaries "you" merge back into the universe. You become one with everything.
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.
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)
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).
"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.