Editing Self-reference (section)

==In logic, mathematics and computing ==
In classical [[philosophy]], [[paradoxes]] were created by self-referential concepts such as the [[omnipotence paradox]] of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The [[Epimenides paradox]], 'All Cretans are liars' when uttered by an ancient Greek Cretan was one of the first recorded versions. Contemporary philosophy sometimes employs the same technique to demonstrate that a supposed concept is meaningless or ill-defined.<ref>{{cite book| url = https://plato.stanford.edu/entries/liar-paradox/| title = ''Liar Paradox''| year = 2020| publisher = Metaphysics Research Lab, Stanford University}}</ref>

In [[mathematics]] and [[computability theory]], self-reference (also known as [[impredicativity]]) is the key concept in proving limitations of many systems. [[Gödel's incompleteness theorems|Gödel's theorem]] uses it to show that no formal [[Consistency|consistent]] system of mathematics can ever contain all possible mathematical truths, because it cannot prove some truths about its own structure. [[The halting problem]] equivalent, in computation theory, shows that there is always some task that a computer cannot perform, namely reasoning about itself. These proofs relate to a long tradition of mathematical paradoxes such as [[Russell's paradox]] and [[Berry's paradox]], and ultimately to classical philosophical paradoxes.

In [[game theory]], undefined behaviors can occur where two players must model each other's mental states and behaviors, leading to infinite regress.

In [[computer programming]], self-reference occurs in [[Reflection (computer science)|reflection]], where a program can read or modify its own instructions like any other data.<ref>{{cite web|last1=Malenfant|first1=J.|last2=Demers|first2=F-N|title=A Tutorial on Behavioral Reflection and its Implementation|url=http://www2.parc.com/csl/groups/sda/projects/reflection96/docs/malenfant/malenfant.pdf|publisher=PARC|archive-url=https://web.archive.org/web/20170821214626/http://www2.parc.com/csl/groups/sda/projects/reflection96/docs/malenfant/malenfant.pdf|access-date=17 May 2015|archive-date=21 August 2017}}</ref> Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in [[recursion]] (related to the mathematical [[recurrence relation]]) in [[functional programming]], where a code structure refers back to itself during computation.<ref name="Drucker2008">{{cite book |last=Drucker |first=Thomas |title=Perspectives on the History of Mathematical Logic |url=https://books.google.com/books?id=R70M4zsVgREC&pg=PA110 |date=4 January 2008 |publisher=Springer Science & Business Media |isbn=978-0-8176-4768-1 |page=110}}</ref> 'Taming' self-reference from potentially paradoxical concepts into well-behaved recursions has been one of the great successes of [[computer science]], and is now used routinely in, for example, writing [[compilers]] using the 'meta-language' [[ML (programming language)|ML]]. Using a compiler to compile itself is known as [[bootstrapping (compilers)|bootstrapping]]. [[Self-modifying code]] is possible to write (programs which operate on themselves), both with [[Assembly language|assembler]] and with functional languages such as [[Lisp (programming language)|Lisp]], but is generally discouraged in real-world programming. Computing hardware makes fundamental use of self-reference in [[Flip-flop (electronics)|flip-flops]], the basic units of digital memory, which convert potentially paradoxical logical self-relations into memory by expanding their terms over time. Thinking in terms of self-reference is a pervasive part of programmer culture, with many programs and acronyms named self-referentially as a form of humor, such as [[GNU]] ('GNU's not Unix') and [[Pine (email client)|PINE]] ('Pine is not Elm'). The [[GNU Hurd]] is named for a pair of mutually self-referential acronyms.

[[Tupper's self-referential formula]] is a mathematical curiosity which plots an image of its own formula.