The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
On December 26, 1951, Gödel delivered the 25th]. W. Gibbs Lecture at a Meeting of the American Mathematical Association at Brown University. In the lecture, he formulated a disjunctive thesis conceming the limits of mathematical reasoning and the possibility of the existence of mathematical truths that are inaccessible to the human rnind. This thesis, known as Gödel's disjunction, is introduced as a direct consequence of the incompleteness theorems [Gödel 1951, p. 310): Either ... the human mind ( even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems [henceforth, absolutely undecidable mathematical sentences] ... ( where the case that both tenns of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives).That is, either the output of the human mathematical rnind exceeds the output of a Turing machine ( called the anti-mechanist thesis) or there are true mathematical sentences that are undecidable "not just within some particular axiomatic system, but by any mathematical proof the human mind can conceive." The latter are called absolutely undecidable mathematical sentences, i.e. mathematical sentences that cannot be either absolutely proved or refuted [Gödel 1951].According to Gödel, the fact that the disjunctive thesis above holds is a "mathematically established fact" of great philosophical interest which follows from the incompleteness theorems, and as such, it is "entirely independent from the standpoint taken toward the foundation of mathematics" [ Gödel 1951]. Indeed, most commentators agree that Gödel's arguments for this disjunctive thesis are compelling. Since Gödel's disjunction was first formulated in 1951, much effort has gone into finding equally compelling arguments for or against either of the disjuncts. In particular, attemptsGödtl's Disjunction. First Edition. Leon Horsten & Philip Welch ( eds}.
We analyse Kreisel's notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church's thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman.
This chapter first argues that while there are solid objections to be raised to Quine’s view, certain widespread arguments against result from overly crude and uncharitable interpretations of Quine. It then turns to the question of what kind of evidence it would take for a Quinean naturalist to change their mind about certain theses, such as the size of the set theoretic universe. It argues that Quineans might be moved to embrace further set-theoretic ontology in the light of the mathematical utility of large cardinals, and potentially even the ‘multiverse’ position on set theory.
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