Hi Eric, My undergraduate adviser wrote a book on constructive analysis. An Amazon review is quoted below. It seems like it wasn't so short or pleasant:
*Foundations of Constructive Analysis* * A Brilliant Book* By Frank Cannonito - July 16, 2013 *Amazon Verified Purchase* Errett Bishop was my friend and colleague and we had many discussions about this book and its subject matter. It is a difficult book because the way of thinking about the subject is unfamiliar to classically trained mathematicians, and this was a disappointment for Bishop. But in it Bishop found how to give, for example, a constructive proof of the Riemann Mapping Theorem - something which Goedel told Hilbert would not be possible (despite Ostrowskii's contemporary proof which was constructive except for the last step which was hanging by a hair). There is much more in this remarkable book and we are fortunate that Ishi Press International has reprinted it (with a New Forward by Michael Beeson). Highly recommended but difficult. Sent from my Verizon Nexus 6 4G LTE Phone (505) 670-9918 Right. I thought the point was that you can have propositions that are "true" in the sense of being consistent within the system, but not provable by constructions defined within the system. But all this, too relies heavily on what you consider to constitute truth value for propositions (some acceptance criterion more liberal than strict constructivism). Also, the incompleteness theorems are a particular property of the indexing of the integers, and their maps to proofs. I believe there are no counterpart problems within the reals, because the cardinality mismatch is not the same. A book on this that I have liked is Torkel Franzen's relatively short and pleasant survey: http://www.amazon.com/G%C3%B6dels-Theorem-Incomplete-Guide-Abuse/dp/1568812388 <http://www.amazon.com/Gödels-Theorem-Incomplete-Guide-Abuse/dp/1568812388> If there are any here who don't like non-constructive notions of truth, there is recent work to find out how much of mathematics can be built only from constructive arguments (I think I have this right). Perhaps we have discussed it before on this list (getting old and dotty), but a wikipedia summary is here: https://en.wikipedia.org/wiki/Univalent_foundations and the group's webpage is here https://www.math.ias.edu/sp/univalent/goals All best, Eric On Dec 28, 2015, at 1:33 PM, Grant Holland wrote: Oh yes, it need not be neither. It just can't be both! Grant Sent from my iPhone On Dec 28, 2015, at 3:29 PM, Grant Holland <[email protected]> wrote: Glen, Eric, If "reality" is complete, must not then (assuming that it is at least as complex as arithmetic), aka Godel, it be also inconsistent? Grant Sent from my iPhone On Dec 28, 2015, at 11:23 AM, glen <[email protected]> wrote: On 12/28/2015 06:30 AM, David Eric Smith wrote: A language that is not even internally consistent presumably has no hope of having an empirically valid semantics, since evidently the universe "is" something, and there is no semantic notion of ambiguity of its "being/not-being" some definite thing, structurally analogous to an inconsistent language's being able to arrive at a contradiction by taking two paths to answer a single proposition. It's not clear to me that the presumption is trustworthy. Isn't it possible that what is (reality) does not obey some of the structure we rely on for asserting consistency (or completeness)? In other words, perhaps reality is inconsistent. Hence, the only language that will be valid, will be an inconsistent language. Of course, that doesn't imply that just any old inconsistency will be tolerated. Perhaps reality is only inconsistent in very particular ways and any language that we expect to validate must be 1) inconsistent in all those real ways and 2) in only those real ways. Further of course, inconsistency is a bit like paradox in that, once you identify an inconsistency very precisely, you may be able to define a new language that eliminates it. ... which brings us beyond the (mere) points of higher order logics and iterative constructions, to the core idea of context-sensitive construction. There is no Grand Unifying Anything except the imperative to approach Grand Unified Things. And this targets Patrick's argument against the idealists (e.g. libertarians and marxists). The only reliable ideal is the creation and commitment to ideals. Each particular ideal is (will be) eventually destroyed. But for whatever reason, we seem to always create and commit ourselves to ideals. Old people tend to surrender over time and build huge hairballs of bandaged ideals all glued together with spit and bailing wire. Any serious conversation with an old person is an attempt to navigate the topology of their iteratively constructed, stigmergic, hairball of broken ideals ... and if that old person is open-minded, such conversations lead to new kinks and tortuous folds ... which is why old people make the best story tellers. But I can't help wondering why music is dominated by the young. [sigh] -- -- ⊥ glen ⊥ ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
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