Date: January 16, 2026
Gödel poses an interesting argument against mathematical conventionalism, which he characterizes as follows:
Gödel identifies the syntactical view with three assertions. First, mathematical intuition can be replaced by conventions about the use of symbols and their application. Second, “there do not exist any mathematical objects or facts,” and therefore mathematical propositions are void of content. And third, the syntactical conception defined by these two assertions is compatible with strict empiricism.
The argument runs as follows:
According to Gödel, a rule about the truth of sentences can be called syntactical only if it is clear from its formulation, or if it somehow can be known beforehand, that it does not imply the truth or falsehood of any ‘factual sentence’ or ‘proposition expressing an empirical fact’. But, so the argument continued, this requirement would be met only if the rule of syntax is consistent, since otherwise the rule would imply all sentences, including the factual ones. Therefore, by Gödel’s second theorem, the mathematics not captured by the rule in question must be invoked in order to legitimize the rule, and thereby the claim that mathematics is solely a result of syntactical rules is contradicted.
One would not necessarily need to justify the consistency of the rule of syntax mathematically. Instead, one could appeal to intuition or empirical induction. The case is similar to, e.g., the question of the consistency of ZFC. Gödel mentions that this is a move one can take.
But now it turns out that for proving the consistency of mathematics an intuition of the same power is needed as for deducing the truth of the mathematical axioms, at least in some interpretation. In particular the abstract mathematical concepts, such as “infinite set,” “function,” etc., cannot be proved consistent without again using abstract concepts, i.e., such as are not merely ascertainable properties or relations of finite combinations of symbols. So, while it was the primary purpose of the syntactical conception to justify the use of these problematic concepts by interpreting them syntactically, it turns out that quite on the contrary, abstract concepts are necessary in order to justify the syntactical rules (as admissible or consistent)…the fact is that, in whatever manner syntactical rules are formulated, the power and usefulness of the mathematics resulting is proportional to the power of the mathematical intuition necessary for their proof of admissibility. This phenomenon might be called “the non-eliminability of the content of mathematics by the syntactical interpretation.”
References
- Panu Raattkainen, On the Philosophical Relevance of Gödel’s Incompleteness Theorems, pp. 513-534. https://shs.cairn.info/journal-revue-internationale-de-philosophie-2005-4-page-513
- Kurt Gödel, Is Mathematics a Syntax of Language?
- https://plato.stanford.edu/entries/goedel/content-mathematics.html