On Tue, May 22, 2012 at 11:08 AM, Stephen P. King <stephe...@charter.net>wrote:

> On 5/22/2012 10:56 AM, Joseph Knight wrote: > > > > On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net>wrote: > >> On 5/21/2012 6:26 PM, Russell Standish wrote: >> >> snip >> >> Hi Russell, >> >> I once thought that consistency, in mathematics, was the indication >> of existence but situations like this make that idea a point of >> contention... CH and AoC <http://en.wikipedia.org/wiki/Axiom_of_choice>are >> two axioms associated with ZF set theory that have lead some people >> (including me) to consider a wider interpretation of mathematics. What if >> all possible consistent mathematical theories must somehow exist? >> > > Joel David Hamkins introduced the "set-theoretic multiverse" idea > (link<http://arxiv.org/abs/1108.4223>). > The abstract reads: > > "The multiverse view in set theory, introduced and argued for in this > article, is the view that there are many distinct concepts of set, each > instantiated in a corresponding set-theoretic universe. The universe view, > in contrast, asserts that there is an absolute background set concept, with > a corresponding absolute set-theoretic universe in which every > set-theoretic question has a definite answer. The multiverse position, I > argue, explains our experience with the enormous diversity of set-theoretic > possibilities, a phenomenon that challenges the universe view. In > particular, I argue that the continuum hypothesis is settled on the > multiverse view by our extensive knowledge about how it behaves in the > multiverse, and as a result it can no longer be settled in the manner > formerly hoped for." > > > Hi Joseph, > > Thank you for this comment and link! Do you think that there is a > possibility of an "invariance theory", like Special relativity but for > mathematics, at the end of this chain of reasoning? > I am doubtful, simply because, for example, the Continuum Hypothesis and its negation are both consistent with ZF set theory. Ditto for the axiom of choice, of course. I find it fascinating that, at this level of the foundations of mathematics, mathematics becomes almost an intuitive science. Questions are asked such as: *Ought *the axiom of choice be true? Are its consequences in line with how we intuit sets to behave? This is the intersection of minds and mathematics. > My thinking is that any form of consciousness or theory of knowledge has > to assume that there is something meaningful to the idea that knowledge > implies agency <http://en.wikipedia.org/wiki/Agency_%28philosophy%29> and > intention <http://plato.stanford.edu/entries/intention/>... > > > > >> >> >> Its one reason why Bruno would like to restrict ontology to machines, >> or at most integers - echoing Kronecker's quotable "God made the >> integers, all else is the work of man". >> >> >> >> >> I understand that, but this choice to restrict makes Bruno's >> Idealism even more perplexing to me; how is it that the Integers are given >> such special status, especially when we cast aside all possibility (within >> our ontology) of the "reality" of the physical world? Without the physical >> world to act as a "selection" mechanism for what is "Real", why the bias >> for integers? This has been a question that I have tried to get answered to >> no avail. >> > > I think Bruno gives such high status to the natural numbers because they > are perhaps the least-doubt-able mathematical entities there are. The very > fact that talks of a "set-theoretic multiverse" exist makes one ask, how > real are sets? Do set theories tell us more about our minds than they do > about the mathematical world? (Obviously, as David Lewis pointed out, you > need something like a set theory in order to do mathematics at all, and as > Russell says, for the average mathematician it really doesn't matter.) > > > My skeptisism centers on the ambiguity of the metric that defines "the > least-doubt-able mathematical entities there are". > I understand. At the end of the day, it may be up to the individual to decide what is doubt-able and what is not. > We operate as if there is a clear domain of meaning to this phrase and yet > are free to range outside it at will without self-contradiction. Set > theory, whether implicit of explicitly acknowledged seems to be a > requirement for communication of the 1st person content. Is it necessary > for consciousness itself? Might consciousness, boiled down to its essence, > be the act of making a distinction itself? > This is an extremely interesting line of thought. Sets do seem to be necessary for the communication of mathematical ideas, maybe even the communication of ideas period. I will have to give this more thought. > > > > Also: *No one here has questioned the reality of the physical world. *Should > I append this statement to every email until you stop countering it? > > > I frankly have to explicitly mention this because the "reality of the > physical world" is, in fact, being questioned by many posters on this list. > Only its status as fundamental is being questioned, to my knowledge. There are a couple of posters whose messages I ignore, however, so I could be missing something. > That you would write this remark is puzzling to me. I think that I can > safely assume that you have read Bruno's papers... Maybe the problem is > that I fail to see how reducing the physical world to the epiphenomena of > numbers does not also remove its "reality". > It's "real" because I see it, I interact with it. It's not fake, whatever that could possibly mean. It's just made epiphenomenal by COMP. > > > > > >> >> >> >> This is the origin of Bruno's claim that COMP entails that physics is >> not computable, a corrolory of which is that Digital Physics is >> refuted (since DP=>COMP). >> >> >> Does the symbol "=>" mean "implies"? I get confused ... >> >> >> Yes, that is the usual meaning. It can also be written (DP or not COMP). >> >> >> "=>" = "or not"] >> > > Actually "a implies b" is defined as "not a or b". > > > Thank you for this clarification! Would you care to elaborate on this > definition? > We understand A to imply B. If A is true, then B should be. If B is false, then A better be too. If A is false, then we don't really care about B. This is the standard definition of "implies" throughout mathematics -- as a definition, in terms of "not" and "or". > > -- > Onward! > > Stephen > > "Nature, to be commanded, must be obeyed." > ~ Francis Bacon > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-list@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > -- Joseph Knight -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.