Editing Argumentation theory (section)

===Mathematical argumentation===
{{Main article|Philosophy of mathematics}}

The basis of mathematical truth has been the subject of long debate. [[Frege]] in particular sought to demonstrate (see Gottlob Frege, [[The Foundations of Arithmetic]], 1884, and ''[[Begriffsschrift]]'', 1879) that arithmetical truths can be derived from purely logical axioms and therefore are, in the end, [[logical truth]]s.<ref>{{cite book|last1=Boolos|first1=George|title=Logic, logic, and logic|date=1999|publisher=Harvard University Press|location=Cambridge, Mass.|isbn=9780674537675|edition=2nd print.|chapter=Chapter 9: Gottlob Frege and the Foundations of Arithmetic}}</ref> The project was developed by [[Bertrand Russell|Russell]] and [[Alfred North Whitehead|Whitehead]] in their ''[[Principia Mathematica]]''. If an argument can be cast in the form of sentences in [[symbolic logic]], then it can be tested by the application of accepted proof procedures. This was carried out for arithmetic using [[Peano axioms]], and the foundation most commonly used for most modern mathematics is [[Zermelo-Fraenkel set theory]], with or without the [[Axiom of Choice]]. Be that as it may, an argument in mathematics, as in any other discipline, can be considered valid only if it can be shown that it cannot have true premises and a false conclusion.