Long Chen

The syntactic interpretation of mathematics (SIM) refers to the view that interprets mathematical propositions as true expressions solely by virtue of the syntactical conventions governing the use of those expressions and not by corresponding to objective mathematical facts. This view, exemplified by the Vienna Circle and particularly Rudolf Carnap, is the target of Gödel's sustained critique and his defence of mathematical realism and intuition. I analyse Gödel's central arguments against SIM: the arbitrariness and asymmetry arguments, which challenge the notion that mathematics is contentless, and the circularity argument, which concerns the role of consistency proofs. I also examine two contemporary defences of Carnap: one appealing to dependable rather than provable consistency, and another recasting Carnap's project as transforming justificatory questions into pragmatic choices of linguistic frameworks. I contend that both defences fail to fulfil the original philosophical ambitions of the syntactic view, and that this failure indirectly bolsters the case for mathematical realism.

Despite his unreserved appreciation of Turing’s analysis for being a “precise and unquestionably adequate definition” of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a “philosophical error” in Turing’s argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing’s work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing’s response to the “mathematical objection” based on Gödel’s incompleteness theorem and Gödel’s own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no “error” on Turing’s side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together.