On Saturday, July 27, 2019 at 2:14:26 PM UTC-5, Philip Thrift wrote: > > > > On Saturday, July 27, 2019 at 1:42:43 PM UTC-5, Lawrence Crowell wrote: >> >> On Saturday, July 27, 2019 at 8:38:12 AM UTC-5, John Clark wrote: >>> >>> All that assumes that infinity exists for any meaningful use of the word >>> “exists” and as far as I know nobody has ever found a infinite number of >>> anything. Mathematics can write stories about the infinite in the language >>> of mathematics but are they fiction or nonfiction? >>> >>> John k Clark >>> >>> >> Infinity is not a number in the usual sense, but more a cardinality of a >> set. Infinity has been a source of trouble for some. I work with Hilbert >> spaces that have a form of construction that is finite, but where the >> finite upper limit is not bounded ---- it can always be increased. This is >> because of entropy bounds, such as the Bekenstein bound for black holes and >> Bousso bounds on AdS, that demands a finite state space for local physics. >> George Cantor made some set theoretic sense out of infinities, even a >> hierarchy of them. This avoids some difficulties. However, I think that >> mathematics in general is not as rich if you work exclusively in finitude. >> Fraenkel-Zermelo set theory even has an axiom of infinity. The main point >> is with axiomatic completeness, and mathematics with infinity is more >> complete. >> >> Richard Feynman talked about Greek mathematics, the axiomatic formal >> systems of mathematics, and Babylonian mathematics that is set up for >> practical matters. I have no particular preference for either, and think it >> is interesting to switch hats. >> >> LC >> >> >>> >>> On Sat, Jul 27, 2019 at 7:36 AM Lawrence Crowell < >>> [email protected]> wrote: >>> >>>> On Thursday, July 25, 2019 at 10:02:39 PM UTC-5, Bruce wrote: >>>>> >>>>> On Fri, Jul 26, 2019 at 12:48 PM John Clark <[email protected]> >>>>> wrote: >>>>> >>>>>> When I was younger I read a lot of science fiction, I don't do it so >>>>>> much anymore and technically I didn't do it this time either but I did >>>>>> listen to a audio book called "We Are Legion We Are Bob" it's the first >>>>>> book of the Bobiverse trilogy and I really enjoyed it. You can get a >>>>>> free 5 >>>>>> minute sample of the book here: >>>>>> >>>>>> We Are Legion (We Are Bob): Bobiverse, Book 1 >>>>>> <https://www.amazon.com/We-Are-Legion-Bob-Bobiverse/dp/B01L082SCI/ref=tmm_aud_swatch_0?_encoding=UTF8&qid=&sr=> >>>>>> >>>>>> It tells the story of Bob, a young man who has just sold his >>>>>> software company for a crazy amount of money and decides that after a >>>>>> decade of hard work he's going to spent the rest of his life just >>>>>> goofing >>>>>> off. On a whim he signs with a Cryonics company to have his head frozen >>>>>> after his death and then just hours later while crossing the street >>>>>> to go to a science fiction convention is hit by a car and dies. Five >>>>>> subjective seconds later he wakes up and finds that a century has >>>>>> passed and he's been uploaded into a computer. This is all in the >>>>>> opening chapter. >>>>>> >>>>>> Parts of the story are unrealistic but parts of it are not, I think >>>>>> it was Isaac Asimov who said it's OK for a science fiction writer to >>>>>> violate the known laws of physics but only if he knows he's doing it, >>>>>> and >>>>>> when Dennis Taylor, the creator of Bob universe, does it at one point >>>>>> with >>>>>> faster than light communication it's obvious that he knowns it. And I >>>>>> can't >>>>>> deny it makes for a story that is more fun to read. I have now read >>>>>> (well >>>>>> listened) to all 3 Bob books and, although parts are a little corny and >>>>>> parts a little too Star Trek for my taste, on the whole I greatly >>>>>> enjoyed >>>>>> them all. They're a lot of fun. >>>>>> >>>>>> The only other novel I can think of that treats the subject of >>>>>> uploading with equal intelligence is "The Silicon Man". >>>>>> >>>>>> The Silicon Man by Charles Platt >>>>>> <https://www.amazon.com/Silicon-Man-Cortext-Charles-Platt/dp/1888869143> >>>>>> >>>>>> John K Clark >>>>>> >>>>> >>>>> Consider any of the earlier novels by Greg Egan, the Australian hard >>>>> science fiction write based in Perth, WA: particularly "Permutation City" >>>>> (1994). >>>>> >>>>> Bruce >>>>> >>>> >>>> I had this idea of a science fiction story of where minds are stored in >>>> machines in order to "eternally" punish them. The idea is that if a >>>> million >>>> seconds in the simulated world is a second in the outer world then one can >>>> in effect construct a near version of eternal hell-fire. The setting is a >>>> world governed by complete terror. Then Egan came out with Permutation >>>> city, which explores a similar set of ideas. >>>> >>>> The problem with the idea of putting minds into machines is that >>>> machines can run recursive functions or algorithms, but in a number system >>>> such as Peano's we make the inductive leap that the successor of any >>>> number >>>> can't be the same number or zero in all (infinite number) cases. We can >>>> make an inference from a recursively enumerable set. I would then think >>>> that the idea of putting minds into machines, or robotic consciousness, is >>>> at this time an unknown, maybe an unknowable, proposition. >>>> >>>> LC >>>> >>>> >>>> > > > > If an actual "black hole" (relativistic) computer existed, it would > effectively compute an infinite number of operations in finite time. > > One could have, in effect, an *infinite-time Turing machine*. > > https://arxiv.org/abs/math/9808093 > <https://www.google.com/url?q=https%3A%2F%2Farxiv.org%2Fabs%2Fmath%2F9808093&sa=D&sntz=1&usg=AFQjCNEmfCZe-55X9cBWq8Bcsq11vVN6nA> > > We extend in a natural way the operation of Turing machines to infinite > ordinal time. > > But all this is fiction. >
Black holes are limited to the amount of information they contain by the Bekenstein bound. Also the duration of a black hole is T ~ M^3. So an observer of the outside world would not encounter an infinite Cauchy sequence of null rays at the inner event horizon. The idea of the infinite task or hyper-Turing machine is interesting, but it may only adjust the Chaitin halting probability for any given algorithm closer to 0 or 1, but not absolutely determine halting or non-halting status. LC > > > "In the 1980s, Hartry Field started a project in the philosophy of > mathematics discussing mathematical fictionalism, the doctrine that all > mathematical statements are merely useful fictions, and should not be taken > to be literally true. More precisely, Field holds that the existence of > sets may be denied, in opposition to Quine and Putnam." [Wikipedia] > > @philipthrift > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/e4351752-f43c-41ab-b3db-fae4df0d89de%40googlegroups.com.
