The Mathematician as Mathematics : Theories of Computation in Light of Wittgenstein's Thought

• Theories of computation are considered in light of Wittgenstein's thought. The dissertation is divided into three parts.The first part presents the philosophical approach taken in the dissertation. The approach takes Wittgenstein's philosophy, in particular as it has been understood by Rush Rhees, as a model. Emphasis is put on the difference between technical work and philosophical work. Philosophical work starts in the lifeworld and is akin to anthropology. The descriptions needed in this work can be given by considering our language use and by drawing up objects of comparison. It is nonetheless important not to confuse philosophical problems with problems of a purely intellectual character, since problems in philosophy can be described as bearing similarities to moral problems.In the second part, which is the main part of the dissertation, the classical theories of computation are considered against the background of the first part. The chapter starts by briefly tracing the historical development of the theories back to Hilbert. It is noted that Hilbert was concerned with epistemological questions. After this, it is investigated how well these epistemological questions were dealt with in the theories of computation that followed. In particular, work by Gödel, Church, and Turing is considered. Topics that are discussed are the very idea of mathematically analysing mathematical activity itself, the concept of finite procedure, what is natural in mathematics, and the idea that computations should be described as transitions between states.The third part is about a more recent theory of computation: abstract state machines. It is shown that many of the aspects of the classical theories that were problematic from an epistemological perspective are carried over to the abstract state machines. This includes the notion of state. Other topics are the idea of axiomatising computability, and the use of abstract state machines for proving Church's thesis.

Ämnesord

HUMANIORA  -- Filosofi, etik och religion -- Filosofi (hsv//swe)

HUMANITIES  -- Philosophy, Ethics and Religion -- Philosophy (hsv//eng)

Nyckelord

theories of computation

philosophy of mathematics

Wittgenstein

Publikations- och innehållstyp

vet (ämneskategori)

dok (ämneskategori)