Kurt Gödel y San Agustín : platonismo matemático y la existencia de Dios

Resumen: El trabajo explica el siguiente razonamiento: si las verdades matemáticas no son creación humana, entonces necesariamente hay que aceptar la existencia de un Intelecto Eterno que sea su sustento metafísico. La premisa es defendida a partir del descubrimiento gödeliano de la imposibilidad de formalizar la totalidad de las matemáticas en un sistema deductivo formal. A partir de allí se muestra la necesidad de admitir una existencia eidética objetiva de las “entidades matemáticas”. Luego, tomando un argumento de San Agustín en De libero arbitrio, se llega a la necesidad de la existencia de Dios como Intelecto que sostiene el ser de las Ideas.

Lacan, Gödel, a ciência e a verdade

Jaques Lacan, the thinker who proposes a return to the fundamentals of psychoanalysis in Freud states that the math would face as a privileged way of transmission of knowledge by the science. Although he was a follower of the mathematization of nature as the foundation of modern science, for him this principle does not imply eliminating the subject that produces it. That would be equivalent to saying that there can not be a language, whatever, even the math, that may "erases" the subject assumption in science. In the text The science and the truth we will try to introduce the idea, not so simple, by the way, the truth as the cause. Citing the framework of the causes in Aristotle, Lacan will speak of a homology between the truth as formal cause, in the case of science, and the truth as material cause, on the side of psychoanalysis. Among its aims with this text, he wants to establish that the unconscious of the subject would be none other than the subject of science. The famous incompleteness theorems of logical-mathematical Kurt Gödel enter here as a chapter of this issue. Recognized as true watershed, these theorems have to be remembered as revealing even outside the mathematical environment, and Lacan himself is not indifferent to this. He makes mention of Gödel's name and draws some observations apparently modest support for his own theory. Since some technical sophisticated knowledges awaits the reader who intends understand this supposed corroboration that Gödel provides to psychoanalysis, introduce the student of Lacan in the use he makes of the incompleteness theorems is the objective of this work. In The science and the truth, which fits us to locate the name of Gödel, one must question how seize such an idea without incurring the extrapolation and abuse of mathematical knowledge, almost trivial in this case. Thus, this paper aims to introduce the reader to the reasoning behind the theorems of Gödel, acquaint him about the Lacan’s mathematical claims, and indicate how to proceed using this implicit math in the text The science and the truth.

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.

Portrait of Kurt Lichtenstein with his wife Portraits Couples

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Kurt Hirschfeld seated in his study Portraits Men

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Kurt Hirschfeld (left) speaking with Thomas Mann Portraits Groups Men

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Godshaw (Gottschalk) children Hal, Elisabeth, Kurt, Walter, Ursula, and Freddy.

Front row from left to right: Walter, Ursula, Freddy; top row from left to right: Hal, Elisabeth, Kurt

Studio Portrait of Elizabeth, Kurt and Hal Godshaw (Gottschalk).

From left to right: Elizabeth, Kurt, Hal Godshaw

Kurt and Hal Gottschalk exercising with a medicine ball in a reproduction of an illustration for the book "Kinder-Turnen in Haus" by Fritz Strube. Sports

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Kurt Godshaw (Gottschalk) as a teenager lying in a meadow; Hannover, Germany.

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Wedding portrait of Kurt Godshaw (Gottschalk) and Edith Godshaw nee Osterer; Quito, Ecuador. Portraits Couples

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Kurt and Edith Godshaw (Gottschalk) nee Osterer during a visit with Maren Matthiesen; Switzerland.

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Freddy, Ursula, Elizabeth, Hal, Walter, and Kurt Gottschalk sharing a bed with their parents.

From left to right: Fred Gottschalk, grandma, Ursula Gottschalk, Elizabeth Gottschalk, grandpa, Hal Gottschalk, Walter Gottschalk, and Kurt Gottschalk

Walter, Freddy, Kurt and Elizabeth Gottschalk with their mother Therese Gottschalk nee Molling, and the maid on an easter egg hunt; Hannover, Germany.

From left to right: Walter Gottschalk, Therese Gottschalk nee Molling, Freddy Gottschalk, the maid, Kurt Gottschalk, and Elizabeth Gottschalk

Henny Molling with Therese, Hal, Kurt and Elizabeth Gottschalk at the swim club; Hannover, Germany.

From left to right: Therese Gottschalk, Hal Gottschalk, Kurt Gottschalk, Henny Molling; on the ground: Elizabeth Gottschalk