Trans-finite and the Picturing of Turing Machines

Abstract

The “picturing (tasavvur )” of “Turing machines”, which is “logia (fikriyat )” pertaining to the “descendant (düşkün )”, is essentially developed on the basis of the “circle-free machine” that performs “computation” under “Euclidean geometry” and “arithmetic”. The construction of the “circlefree machine” requires “infinitely often” operations and “infinite” amount of “ink”. As a “picturelessname (suretsiz isim )” pertaining to “logia (fikriyat )”, “infinite” is “relative (izafi )” to the “finite”. The “infinite” however cannot be constructed by starting from the “finite”. By means of “tamganame theographia (tamga-isim theographiası )”, we considered the concept of “trans-finite (sonluötesi)” that is not “relative (izafi )” to the “finite”. By “tamga-name theographia (tamga-isim theographiası )”, we mean “(composed) name writing (müteşekkil isim yazımı )”, by means of “strike (darb )”, in the form of “picture (resim )” within the framework of “substance theographia (cevher theographiası )”, that constructs “trans-finite (sonluötesi )” which is not “relative (izafi )” to the “finite”. Through this approach, we explained that construction of a “(composed) body of trans- finite (inless) length (sonlu-ötesi  (iç’siz ) uzunluk’ta (müteşekkil ) cisim )” belonging to “Euclidean geometry” and “(composed) body of trans-finite (in- less) multitude (sonlu-ötesi  (iç’siz ) çokluk’ta  (müteşekkil ) cisim )” belonging to “arithmetic” is not possible. In this regard, one cannot conceive “infinitely often” operations and “infinite” quantity of ink pertaining to the “circle-free machine”. Hence, in view of “circle-free machine”, one cannot consider “potential-infinitely often (kuvve’de-sonsuz sıklık’ta )” operations and “potentially-infinite (kuvve’de sonsuz )” amount of ink. We pointed out that Cantor’s reasoning, which asserts that “natural numbers” form an “infinite multitute of a countable set (sonsuz çokluk’ta sayılabilir küme )” is essentially “circulus in demonstrando (d.ngüsel gösterim )”. In this respect, Cantor’s “diagonal argument”, which is commonly believed to construct “infinite multitute of an uncountable set (sonsuz çokluk’ta sayılamaz küme )” is invalid. The “picturing (tasavvur )” of “Turing machines” which lacks “essence (asıl )” pertaing to “grounds (zemin )” is therefore “narrative (hikâyat  (historia))” based on “phantasy (tahayyül )”. We briefly stated that to repair this deficiency of “grounds (zemin )” by means of “artificial intelligence” is not possible.