Q&A: A Synthetic Proposition in Mathematics
A Synthetic Proposition in Mathematics
Question
Have a good week, Rabbi,
Recently I heard about the continuum hypothesis in mathematics, and I heard that it was proved that it cannot be proven false and also cannot be proven true. Does that mean there can be synthetic propositions in mathematics? That is, propositions about which there is not, and cannot be, certainty, but which still have some truth value?
Best regards,
Answer
Yes. This is a famous example of Gödel’s incompleteness theorem in mathematics (the weaker version, I think). But there is no place for syntheticity in mathematics. Whatever is synthetic is not mathematical. Once they proved that both possibilities are consistent, it stops being a conjecture and becomes a matter of free choice. The choice itself is not a mathematical act. Gödel showed that there is also strong incompleteness (where a statement is true but cannot be proved), and that is closer to what you are looking for. But there too, the statement is true because it was proved, except that the proof is carried out outside the axiomatic system in question.