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Mathematics

There are two main approaches to truth in mathematics. They are the model theory of truth and the proof theory of truth.

Historically, with the nineteenth century development of 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 tables.

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 generally assumed to be those statements that are provable in a formal axiomatic system.

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. 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, 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. Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory. 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 intuition, an ability he admitted could be ultimately beyond the scope of a formal theory of logic or mathematics and perhaps best considered in the realm of human 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".