Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

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.

Tarskin totuus : Totuuden määrittelemisen taustaehtoja ja seurauksia

Tutkielma esittelee Alfred Tarskin formalisoitujen kielien totuusmääritelmän filosofisia ja loogisia taustaehtoja ja seurauksia. Tutkimuksen keskeisimmät menetelmät ovat filosofinen analyysi ja matemaattinen logiikka. Keinona on myös aatehistoriallinen tutkimus. Keskeisin tutkimuksen kohde on Alfred Tarskin artikkeli "Der Wahrheitsbegriff in den formalisierten Sprachen", 1935 (julkaistu käännöksenä teoksessa Logic, Semantics, Metamathematics, Papers from 1923 to 1938, Clarendon Press, Oxford, 1956). Tarskin alkuperäinen totuusmääritelmä formalisoiduille kielille ja totuuden määrittelemättömyystulos ovat esitetty nykyaikaisella notaatiolla. Analyyttisen ja historiallisen lähestymisen kautta Tarskin totuusmääritelmän ja malliteorian suhteita selvennetään absoluuttisen ja suhteellisen totuuskäsityksen avulla. Ajatusta formaalisista kielistä ennalta annetusti tulkittuina kielinä esitellään ja sen historiallisia taustoja selvitetään. Tämän idean olennaisuus todetaan Tarskin 1930-luvun totuusmääritelmän muodostumiselle. Eräiden käännöksen käsitteeseen liittyvien epäselvyyksien esiin nostaminen ja aksiomaattisten totuusteorioiden ja totuusmääritelmän suhteen esitteleminen ovat osa tutkielman loogista osuutta. Näiden lisäksi tutkielma esittelee S. Shapiron ja J. Ketlandin totuusteoreettisen deationismin kritiikkiä, jossa käytetään Tarskin totuusmääritelmää vastaavaa totuusteoreettista aksiomatisointia.

Freddy Godshaw (Gottschalk) with his friend Peter Zander celebrating their shared birthday; Cellerina, Switzerland.

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Studio portrait of Liselotte "Lotte" Stern nee Wallerstein.

Photograph probably taken shortly before her marriage to Alfred Stern in 1928.

Shimon Gottschalk with his wife Jo and family on his 70th birthday; Florida Portraits Family

Front row left to right: Tikkun, Back row: the twins, age 8, Shimon's wife Jo,, right: Ariela, age 28, the eldest

Liederkranz trip to Hundseck Portraits Groups

Liederkranzausflug nach Hundseck; Standing l-r; Sidi Ullmann, Ehepaar, Alfred und Toni Hirschland, Willi Ullmann, Lisel Suess, Lene Hesse-Sinzheimer, Julius Suess, ?, Hermann Boehm, Eugen Hirsch, Dr. Carl Kahn, Frau von Michael Rothschild (Ella?), Max Sinzheimer, Grete Hirschland and Alfred Liebmann ; Seated l-r: Adele Boehm, Michael Rothschild, rosel Suess, Louis Kahn (Junior), Ehepaar Dr. Gustav Hecht, Paula Hirsch and Marianne Kahn

Studio portrait of Liselotte "Lotte" Stern nee Wallerstein.

Photograph probably taken shortly before her marriage to Alfred Stern in 1928.

Studio portrait of Adolf Rothschild and Sophie Rothschild nee Cohn.

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Studio portrait of the six daughters of Solomon and Emilie Kohnberger: (l-r) Laura, Irene, Johanna, Melanie, Hermine and Martha.

Laura (1878-1917) died during a flu epidemic. She married Sigmund Stiassny and had two children, Lisbeth (Gersuny) (1900-1986) and Walter (1902-1912) who died of a ruptured appendix.

Schoenewald family portrait; Bad Nauheim, Germany.

Includes Rosi Schoenewald (sister of Dora), Dora Schoenewald (wife of Alfred Apfel and mother of Hannah Busoni), Henni Apfel, Tilly Schoenewald, Luise Schoenewald, Max Schoenewald (brother of Dora), Felix Schoenewald (brother of Dora), Samuel Schoenewald (brother of Dora), and Gerhardt Schoenewald

Studio portrait of Joseph and Meta Gruenstein Portraits Couples

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Studio portrait of Meti and Meta Gruenspecht Portraits Women

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Studio portrait of Max Gruenspecht and Regina Gruenspecht nee Epstein Portraits Couples

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Dieter, Herbert, Meta, Josef Gruenstein with Max Gruenspecht and Regina Gruenspecht née Epstein in front of the synagogue in Schwarza, Thuringia, Germany

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