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...) — the affirmation of strife
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 affirmation of strife
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". — the affirmation of strife
There is no "person" AFAIK these are called "demonstratives" or something like that. — the affirmation of strife
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.
didn't make sense to me. Humans can use whatever grammar they like, so I'm not sure what you are confused about here.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. — T Clark
parentheses addedThere 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 — the affirmation of strife
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
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. — the affirmation of strife
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. — the affirmation of strife
credibility of mathematics — T Clark
Either the model (physics) is wrong, or the mathematical rules were not followed. — the affirmation of strife
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. — the affirmation of strife
In addition to what StreetlightX said about the "enworlded-ness" of language (arising from the fact that it is invented by humans), — the affirmation of strife
I think some of Turing's fear was justified. — the affirmation of strife
[1]: Is this still controversial? I mean, Einstein called it a language. My first year lecturer did the same. — the affirmation of strife
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.