This is the title of a discussion about self-reference

I’d be interested in hearing about situations where self-referential ideas actually contribute rather than obscure.

Descartes' "Cogito, ergo sum" was intended to be an irrefutable argument from undeniable premises. Descartes could not doubt the fact that he thought. [...] The reason is that Descartes' act of doubting itself requires thinking [...]. Basically, Descartes' famous dictum is shorthand for something more like: I am thinking this thought, and this I cannot doubt because my doubting requires my thought.

Professor Hill denounced the judge who had harassed her.

The law school professor who had worked for him denounced Judge Thomas.

The law school professor who had worked for him denounced the judge who had harassed her.

As the article alludes to at the end, things get even more interesting when thinking about self-modification of programs or self-specialising compilers (I've lost a bookmark to an interesting and not too technical blog post about this, maybe I can find it again...)

In terms of mathematics, the book "Vicious Circles" by John Barwise and Lawrence Moss seems to be a good reference for what they call "hyperset" theory, an extension of set theory that allows for self-referencing and circularity. I haven't read much, and it's very dense. Working understanding of set theory required. I wonder if there are any mathematicians here that could break it down for us.

The tongue in cheek title of our book is intended to suggest that circularity
has an undeservedly bad reputation in philosophical circles. On the other hand,
we certainly do not think that every proposal or argument using circularity
bears close scrutiny. For example, one of the morals of our resolution of
the Hypergame Paradox is that certain kinds of circular definitions really are
incoherent.

I'll need to look into it more to give better examples of "useful self-reference".

"This" is not second person.

There is no "person" AFAIK these are called "demonstratives" or something like that.

There is a set R which consists of all and only non-reflexive sets:
R = {x | x is non-reflexive}
But then we see that R belongs to R iff R is non-reflexive, which holds iff R does not belong to R. Hence either assumption, that R belongs to or R does not belong to R leads to a contradiction.

Suppose we have some set b and form the Russell set using b as a universe.
That is, let R_b, = {c ? b | c is non-reflexive}
There is nothing paradoxical about R_b - The reasoning that seemed to give rise to paradox only tells us that R_b ? b. In other words, the Russell construction gives us a way to take any set b whatsoever and generate a new set not in b.

Why are self-referential sentences like the liar sentence (3) only in the 2nd person while we humans can do the same in one additional way viz. in the 1st person?

The liar's paradox seems like a little joke that people have decided to take seriously. I can't see how it gives any insight into meaning or truth, as some propose.

There is no single first-order formula that serves to define the truth of all sentences of first-order logic in the universe (of sets).

There is no single first-order formula that serves to define the truth of all sentences of first-order logic in the universe (of sets).
parentheses added

Descartes' "Cogito, ergo sum" was intended to be an irrefutable argument from undeniable premises. Descartes could not doubt the fact that he thought. [...] The reason is that Descartes' act of doubting itself requires thinking [...]. Basically, Descartes' famous dictum is shorthand for something more like: I am thinking this thought, and this I cannot doubt because my doubting requires my thought

Cool what you did with the title.

There is a set R which consists of all and only non-reflexive sets:
R = {x | x is non-reflexive}
But then we see that R belongs to R iff R is non-reflexive, which holds iff R does not belong to R. Hence either assumption, that R belongs to or R does not belong to R leads to a contradiction.

So, it looks like the value of the liar's paradox or Russel's paradox etc. comes from the insight into how we can or can not formulate truth.

credibility of mathematics

Either the model (physics) is wrong, or the mathematical rules were not followed.

The problem: what should we do if we are presented with contradictory mathematical rules. For the language analogy, this is like finding a contradiction in your Japanese grammar book. On page 24 it tells you to say X in situation Y, but on page 135 (it's not an easy language, you understand) it instructs you to say the opposite i.e. (not X) in situation Y. Solution: buy a new grammar book.

In addition to what StreetlightX said about the "enworlded-ness" of language (arising from the fact that it is invented by humans),

I think some of Turing's fear was justified.

[1]: Is this still controversial? I mean, Einstein called it a language. My first year lecturer did the same.

Liar's paradoxes show us that certain assumptions we make lead to illogical conclusions. That's incredibly important, because what if you are making those assumptions in arguments that are not liar's paradoxes?

So if the sentence is false, its true, and if its true, its false. We definitely have a contradiction.

We realize we've said nonsense by being too implicit. That's the lesson we can glean. Just because we can say or posit an idea in language, doesn't mean it makes sense. You've previously posted the question, "What is metaphysics?" Many times people use metaphysics to disguise liars paradoxes. Terms that are ambiguous are great ways to hide nonsense terms and conclusions within them. If you can pick them out, you can ask for clarification.

Solving the liar's paradox can give us a tool to solve other nonsense points while keeping within the spirit of the discussion.

Liar's paradoxes are a great teaching tool about the ambiguity of language, but also about seeing through the intentionality of a person's argument.

I don't think this analogy applies. Seems like with the Russel paradox, we start with what appear to be consistent rules and get contradictory results.

what if you are making those assumptions in arguments that are not liar's paradoxes

The liar's paradox only shows up when we are talking about sentences that we would never use in normal speech

Is this the issue, that mathematicians and logicians don't believe math was invented by humans? That they think it is intrinsic to the world?

I don't get it.

There are certainly people who believe that the Russell paradox says something profound about math and logic.

You are right: there is only a danger if this paradox within set theory has an effect within the practical mathematics (which I suggested would necessarily always be detectable, but maybe not trivially apparent). I don't have an example to hand, although they might be found in e.g. differential geometry (foundation for General Relativity) or, where this all came to light, in computability theory (foundation for, well, computers).

math paradoxes we're talking about are trivial

It has magnitude and direction? Cool, so what's a direction?

So, my impression is that most self-reference is useless.

Have you revised this view?

So, my impression is that most self-reference is useless.

As to the usefulness of self-reference, it was pointed out that it is pivotal to iteration. Any iterative procedure by definition calls itself. Now that's indispensable in coding, but it also leads to many a curiosity. So for example, this beast:

Douglas Hofstadter made use of iteration in his discussion of consciousness, a notion that has not dissipated over the years. Chaos theory in general relies on iteration.

Self-referentiality points to our tendency to conflate the thing with our thoughts about said thing.

Also, more generally, it points to the possibility of saying one thing and meaning two things.

I don't think iteration and self-reference are the same thing. I'm not sure of that.

Confusing "the moon" with the moon doesn't strike me as a self-reference issue.

Also, more generally, it points to the possibility of saying one thing and meaning two things.
— baker

I don't understand what you mean.

BTW perturbative quantum field theory was recently put on pretty firm mathematical footing (see Perturbative Algebraic Quantum Field Theory by Kasia Rejzner). This uses Greens functions which are calculated recursively (i.e. G = f[G]).

I saw a nice self-referencing puzzle the other day.
Question: If you pick an answer at random, what are the chances that the percentage written in the pick is equal to the chance of picking that percentage?
There were four answers given from which you could pick at random. One said 50%. One said 25%. One said 60%. And another one said 25%. Altogether there were four answers from which a random choice would be made.

The problem is also known as "confusing the map for the territory".

Are all iterative processes self-referential? Maybe someone else brought this up previously. Is that the same kind of self-reference we're talking about?

All recursive ones processes are, and calculation of the Greens function is recursive. But no, not all iterative ones.

Percentage = 0. Right?

can find non-trivial self-referential paradoxes, such that they could arise from seemingly well-founded frameworks. I'm no longer sure that it is even possible,

# Iterative

iterative_factorial( x ):

  x_factorial = 1

  while x > 1:

      x_factorial *= x

      x -= 1

  return x_factorial



# Recursive

recursive_factorial( x ):

  return 1 if x == 0 else (x * recursive_factorial(x-1))

# This function prints the path to the exit, if there is one.

navigate( maze, path_taken ):



  current_position = get_position(maze, path_taken)

  past_positions = [get_position(maze, path_taken[:n]) for n in len(path_taken)]

  if current_position not in past_positions:



    if AT_EXIT: 

     print( path_taken )

     exit_program()



    else:

     if CAN_GO_STRAIGHT:

         navigate(maze, path_taken + [ STRAIGHT ])

     if CAN_GO_LEFT: 

         navigate(maze, path_taken + [ LEFT ])

     if CAN_GO_RIGHT: 

        navigate(maze, path_taken + [ RIGHT ])

There are certainly people who believe that the Russell paradox says something profound about math and logic.

but it’s clear it’s neither profound nor important.