Ok, I can see now that, careful as we have been, there has been an ambiguity in our conversation regarding the meaning of the word “proof”. Two possible meanings:
a valid and convincing argument
a structure in a formal system which is equivalent to a valid and convincing argument (if one assumes the truth of the axioms and the validity of the rules in that formal system).
I have sometimes been using one of those meanings and sometimes the other.
I suspect the best thing to do with this conversation is to simply shut it down. But before doing so, I’d like to thank you for making this posting. I just recently stumbled upon Nelson’s work on “internal set theory” and non-standard analysis. Good stuff! I’m less impressed with ultrafinitism - I’m more of a intuitionist or constructivist—but it certainly is fascinating.
Ok, I can see now that, careful as we have been, there has been an ambiguity in our conversation regarding the meaning of the word “proof”. Two possible meanings:
a valid and convincing argument
a structure in a formal system which is equivalent to a valid and convincing argument (if one assumes the truth of the axioms and the validity of the rules in that formal system).
I have sometimes been using one of those meanings and sometimes the other.
I suspect the best thing to do with this conversation is to simply shut it down. But before doing so, I’d like to thank you for making this posting. I just recently stumbled upon Nelson’s work on “internal set theory” and non-standard analysis. Good stuff! I’m less impressed with ultrafinitism - I’m more of a intuitionist or constructivist—but it certainly is fascinating.
Nice.
Up to you, I’m having a great time.