While agreeing wholeheartedly with Robert, I would like to point a finger at what I think of as the "monolithic mathematics mob", MMM. These are the people who treat mathematics as a single theory. Carl Hewitt, with whom I shared an admin at MIT for a decade, has a proof of the consistency of mathematics based on that premise at https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/edit?usp=sharing along with a little more that so far I've been unable to pin down. But I rather suspect that pretty much everyone who finds G??del's second incompleteness theorem paradoxical shares Hewitt's view of mathematics as a single theory. I find the following difficulties with the MMM view. 1. You can't have a theory without a language. What is the language of mathematics? Judging by appearances it would seem to be a living thing that grows in different directions following the many varied and evolving interests of mathematicians, pure, applied, Arcturan, or whatever. This leads to: Principle 1. There will never come a time when mathematicians have settled on the language of mathematics. 2. In the unlikely event that Principle 1 is violated, namely by collecting every mathematical symbol that will ever be needed in mathematics into a single language possessed of a single consistent theory T, there is no reason to expect any such thing to be recursively enumerable. With the requisite assumptions this is G??del's first rather than his second incompleteness theorem. But this raises the imponderable question of whether mathematics is what mathematicians know, or what they could ever know (by enumeration of theorems), or the aforementioned theory T, which they can never know at any future time t even as all the consequences-in-principle of whatever finite axiomatization of T they have agreed on by time t. Which leads to: Principle 2. There will never come a time when mathematicians have settled on what constitutes mathematics. G??del's first incompleteness theorem suffices for Principle 2. Principle 1 is even more elementary. Much as I appreciate G??del's second incompleteness theorem, it seems to me that his first is all that's needed to answer those who find the second paradoxical. Vaughan Pratt On 7/29/2015 6:56 AM, Robert Dawson wrote: ...
However, the loop is not closed, and cannot be. There are questions which are legitimate parts of mathematics/logic that cannot be answered internally. I'm not talking about Goedel incompleteness here (though one might), but about why we do what we do. If we want to say what constitutes mathematics worth doing - to say why Fermat's Last Theorem or the Riemann Hypothesis are more important that the (3n+1) problem or finding palindromic sequences in the decimal expansion of pi - we cannot do this by calculation and proof. This is an example of a place where philosophy of mathematics can have a genuine connection.
-Robert Dawson
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