On the Philosophical Relevance of Godel's Incompleteness Theorems

Exploring the Foundational Significance of Gödelʼs Incompleteness Theorems

“…The influence of Gödel's incompleteness theorems on philosophy is manifested in the fact that, since their publication, they have triggered extensive and enduring discussions of some philosophical problems related to them such as the nature of logic and mathematics, the nature of mind and machine, the difference between human intelligence and machine intelligence, the limit of machine intelligence, and the relationship among Gödel's theorems, logicism, Hilbert's program, and intuitionism. For the philosophical relevance of Gödel's incompleteness theorems, see Raatikainen (2005). For Gödel's philosophical thought and its influence on the discovery of his incompleteness theorems, see Wang (1990Wang ( , 1997.…”

Saving Liberalism from Itself

“…Now, as I have been arguing throughout the book, it is doubtful whether there really is any tradition-independent way of looking at the world. Reasoning always unfolds within a set of received parameters and, as Gödel's second incompleteness theorem states, even consistent mathematical systems cannot prove their own consistency (Raattkainen 2005). And so, what we call mathematical systems and scientific paradigms might equally be thought of as traditions of what is counted as legitimate knowledge.…”

Mathematical Proofs and Scientific Discovery

“…To see why for Lucas' argument to work one needs to assume that it is absolutely provable that F is consistent, recall that Gödel's second incompleteness theorem has a conditional form (Raatikainen 2005). Indeed, Gödel showed that for any sufficiently strong formal theory F, if F is consistent, a sentence G in the language of F, which is equivalent in F to the sentence expressing the consistency of F, cannot be proved in F. Thus, if F proves only true sentences, i.e.…”

“…This issue deserves a careful examination, since it is related to an important discussion on the philosophical consequences of Gödel's results which took place in the last decades, namely the discussion on whether Gödel's results imply that CTM is untenable (Horsten, Welch 2016;Raatikainen 2005). That discussion is useful to illustrate how even those philosophers who reject the idea that mathematical reasoning can be automated, failed to recognize the inadequacy of the axiomatic view, and so were led astray in their attempts to argue against the computationalist perspective.…”