I got so excited that I just bought your book "Lectures on the Philosophy of Mathematics". I have two Brazilian books on mathematical philosophy, but they don't have as many topics as yours, besides, I found your way of narrating much more friendly and exciting.
Could the properties of the natural numbers be the properties shared among all the interpretations? I just think that there does not have to be one correct interpretation. The plurality is conceptually intriguing. Also, the idea of interpreting a theory in a background theory sounds like an embedding, and there are generally many ways to embed an object into an ambient space.
Yes, indeed, we have many interpretations of the natural numbers in diverse foundational systems, and this was a major theme of my essay. Interpretations are rather like embeddings, but with a definability aspect, since the embedding itself must be definable in the system in which the interpretation is made. One problematic issue with your proposal to understand the arithmetic properties as what is true in common in all interpretations, however, is that this could cause us to think that some arithmetically expressible properties are neither true nor false—the common theory, after all, is simply not a complete theory. For example, the usual interpretation of arithmetic in ZFC set theory proves Goodstein's theorem, an arithmetic assertion, but this is not provable for other interpretations. Do we want to hold that Goodstein's theorem is somehow indeterminate? This will be the situation if we follow your proposal. Meanwhile, many mathematicians expect that there is a fact of the matter about arithmetic truths—should we expect that every arithmetic question should be either determinately true or determinately false?
I got so excited that I just bought your book "Lectures on the Philosophy of Mathematics". I have two Brazilian books on mathematical philosophy, but they don't have as many topics as yours, besides, I found your way of narrating much more friendly and exciting.
I am so glad to hear it!
Could the properties of the natural numbers be the properties shared among all the interpretations? I just think that there does not have to be one correct interpretation. The plurality is conceptually intriguing. Also, the idea of interpreting a theory in a background theory sounds like an embedding, and there are generally many ways to embed an object into an ambient space.
Yes, indeed, we have many interpretations of the natural numbers in diverse foundational systems, and this was a major theme of my essay. Interpretations are rather like embeddings, but with a definability aspect, since the embedding itself must be definable in the system in which the interpretation is made. One problematic issue with your proposal to understand the arithmetic properties as what is true in common in all interpretations, however, is that this could cause us to think that some arithmetically expressible properties are neither true nor false—the common theory, after all, is simply not a complete theory. For example, the usual interpretation of arithmetic in ZFC set theory proves Goodstein's theorem, an arithmetic assertion, but this is not provable for other interpretations. Do we want to hold that Goodstein's theorem is somehow indeterminate? This will be the situation if we follow your proposal. Meanwhile, many mathematicians expect that there is a fact of the matter about arithmetic truths—should we expect that every arithmetic question should be either determinately true or determinately false?