David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Dissertation, University of Helsinki (2009)
One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.
|Keywords||Gödel Tarski Semantical argument Mathematical truth|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Neil Tennant (2002). Deflationism and the Gödel Phenomena. Mind 111 (443):551-582.
M. Redhead (2004). Mathematics and the Mind. British Journal for the Philosophy of Science 55 (4):731-737.
Ryan Christensen (2011). Theories and Theories of Truth. Metaphysica 12 (1):31-43.
Jeffrey Ketland (1999). Deflationism and Tarski's Paradise. Mind 108 (429):69-94.
Peter Milne (1997). Tarski on Truth and its Definition. In Timothy Childers, Petr Kolft & Vladimir Svoboda (eds.), Logica '96: Proceedings of the 10th International Symposium. Filosofia. 198-210.
Bo Mou (2001). The Enumerative Character of Tarski's Definition of Truth and its General Character in a Tarskian System. Synthese 126 (1-2):91 - 121.
Jean Fichot (2003). Truth, Proofs and Functions. Synthese 137 (1-2):43 - 58.
Luis Fernández Moreno (2001). Tarskian Truth and the Correspondence Theory. Synthese 126 (1-2):123 - 147.
Aladdin M. Yaqub (1993). The Liar Speaks the Truth: A Defense of the Revision Theory of Truth. Oup Usa.
Mario Gómez-torrente (1998). Logical Truth and Tarskian Logical Truth. Synthese 117 (3):375-408.
Aladdin Mahmūd Yaqūb (1993). The Liar Speaks the Truth: A Defense of the Revision Theory of Truth. Oxford University Press.
Dale Jacquette (2010). Circularity or Lacunae in Tarski's Truth-Schemata. Journal of Logic, Language and Information 19 (3):315-326.
Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
Gregor Damschen (2011). Questioning Gödel's Ontological Proof: Is Truth Positive? European Journal for Philosophy of Religion 3 (1):161-169.
Volker Halbach (1995). Tarski Hierarchies. Erkenntnis 43 (3):339 - 367.
Added to index2012-03-19
Total downloads55 ( #33,521 of 1,413,138 )
Recent downloads (6 months)12 ( #17,931 of 1,413,138 )
How can I increase my downloads?