Editing Truth (section)

===Mathematics===
{{anchor|Truth_in_mathematics}} 
{{Main|Model theory|Proof theory}}

There are two main approaches to truth in mathematics. They are the ''[[model theory|model theory of truth]]'' and the ''[[proof theory|proof theory of truth]]''.<ref>Penelope Maddy; ''Realism in Mathematics''; Series: Clarendon Paperbacks; Paperback: 216 pages; Publisher: Oxford University Press, US (1992); 978-0-19-824035-8.</ref>

Historically, with the nineteenth century development of [[Boolean algebra (logic)|Boolean algebra]], mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In [[propositional logic]], these symbols can be manipulated according to a set of [[axioms]] and [[rules of inference]], often given in the form of [[truth table]]s.

In addition, from at least the time of [[Hilbert's program]] at the turn of the twentieth century to the proof of [[Gödel's incompleteness theorems]] and the development of the [[Church–Turing thesis]] in the early part of that century, true statements in mathematics were [[logical positivism|generally assumed]] to be those statements that are provable in a formal axiomatic system.<ref>Elliott Mendelson; ''Introduction to Mathematical Logic''; Series: Discrete Mathematics and Its Applications; Hardcover: 469 pages; Publisher: Chapman and Hall/CRC; 5 edition (August 11, 2009); 978-1-58488-876-5.</ref>

The works of [[Kurt Gödel]], [[Alan Turing]], and others shook this assumption, with the development of statements that are true but cannot be proven within the system.<ref>''See, e.g.,'' Chaitin, Gregory L., ''The Limits of Mathematics'' (1997) esp. 89 ''ff''.</ref> Two examples of the latter can be found in [[Hilbert's problems]]. Work on [[Hilbert's 10th problem]] led in the late twentieth century to the construction of specific [[Diophantine equations]] for which it is undecidable whether they have a solution,<ref>M. Davis. "Hilbert's Tenth Problem is Unsolvable." ''American Mathematical Monthly'' 80, pp. 233–269, 1973</ref> or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, [[Hilbert's first problem]] was on the [[continuum hypothesis]].<ref>Yandell, Benjamin H.. ''The Honors Class. Hilbert's Problems and Their Solvers'' (2002).</ref> Gödel and [[Paul Cohen (mathematician)|Paul Cohen]] showed that this hypothesis cannot be proved or disproved using the standard [[axiom]]s of [[set theory]].<ref>Chaitin, Gregory L., ''The Limits of Mathematics'' (1997) 1–28, 89 ''ff''.</ref> In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.

Gödel thought that the ability to perceive the truth of a mathematical or logical proposition is a matter of [[logical intuition|intuition]], an ability he admitted could be ultimately beyond the scope of a formal theory of logic or mathematics<ref>{{cite web| last=Ravitch| first=Harold| title=On Gödel's Philosophy of Mathematics| year=1998| url=http://www.friesian.com/goedel/chap-2.htm| access-date=2018-05-25| archive-date=2018-02-28| archive-url=https://web.archive.org/web/20180228005628/http://friesian.com/goedel/chap-2.htm| url-status=live}}</ref><ref>{{cite web| last=Solomon| first=Martin| title=On Kurt Gödel's Philosophy of Mathematics| year=1998| url=http://calculemus.org/lect/07logika/godel-solomon.html| access-date=2018-05-25| archive-date=2016-03-04| archive-url=https://web.archive.org/web/20160304030146/http://www.calculemus.org/lect/07logika/godel-solomon.html| url-status=live}}</ref> and perhaps best considered in the realm of human [[comprehension (logic)|comprehension]] and communication. But he commented, "The more I think about language, the more it amazes me that people ever understand each other at all".<ref>{{cite book| last=Wang| first=Hao| title=A Logical Journey: From Gödel to Philosophy| publisher=The MIT Press| year=1997| url=https://books.google.com/books?isbn=0262261251}} (A discussion of Gödel's views on [[logical intuition]] is woven throughout the book; the quote appears on page 75.)</ref>