actual vs potential ∞
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I know I've posted this before. I don't remember it getting any traction with y'all. But it's relevant to my struggles with beliefs in potential vs actual infinity: Belief in the Sinularity is Fideistic https://link.springer.com/chapter/10.1007%2F978-3-642-32560-1_19 Not unrelated, I've often been a fan of trying identify *where* an argument goes wrong. And because this post mentions not only 1/0, but Isabelle, Coq [⛧], Idris, and Agda, I figured it might be a good follow-up to our modeling discussion on Friday, including my predisposition against upper ontologies. 1/0 = 0 https://www.hillelwayne.com/post/divide-by-zero/ Here's the (really uninformative!) SMMRY L7: https://smmry.com/https://www.hillelwayne.com/post/divide-by-zero/#&SM_LENGTH=7 > Since 1 0, there is no multiplicative inverse of 0⁻. Okay, now we can talk about division in the reals. > > So what's -1 * π? How do you sum up something times? While it would be nice if division didn't have any "Oddness" to it, we can't guarantee that without kneecapping mathematics. > > We'll define division as follows: IF b = 0 THEN a/b = 1 ELSE a/b = a * b⁻. > > Doing so is mathematically consistent, because under this definition of division you can't take 1/0 = 1 and prove something false. > > The problem is in step: our division theorem is only valid for c 0, so you can't go from 1/0 * 0 to 1 * 0/0. The "Denominator is nonzero" clause prevents us from taking our definition and reaching this contradiction. > > Under this definition of division step in the counterargument above is now valid: we can say that 1/0 * 0 = 1 * 0/0. However, in step we say that 0/0 = 1. > > Ab = cb => a = c but with division by zero we have 1 * 0 = 2 * 0 => 1 = 2. [⛧] I decided awhile back to focus on Coq because it seems to have libraries of theorems for a large body of standard math. But still NOT having explored it much, yet learning some meta-stuff surrounding the domain(s), I'm really leaning toward Isabelle. I suppose, in the end, I won't learn to use any of it, except to pretend like I know what I'm talking about down at the pub. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
uǝʃƃ ⊥ glen
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My opinion. 1/0 is undefined. Depending on the context you can define it in a way that's useful in that context. To say that lim(1/x) as x ->0 = infinity means precisely: For any r in R, however large, there exists an x in R such that 1/x > r. Frank --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Mon, Aug 3, 2020, 11:03 AM uǝlƃ ↙↙↙ <[hidden email]> wrote:
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In reply to this post by gepr
To be a little clearer on my hand-wringing, here is a section where Bringsjord et al argue that belief in the Singularity is not rational:
> A > (P1) There will be AI (created by HI). > (P2) If there is AI, there will be AI+ (created by AI). > (P3) If there is AI+, there will be AI++ (created by AI+). > ) > There will be AI++ (= S will occur). > [...] > Our certainty in > the lack of certainty here can be established by showing, formally, that the denial > of (P1) is consistent, since if not-(P1) is consistent, it follows that (P1) doesn’t > follow from any of the axioms of classical logic and mathematics (for example, > from a standard axiomatic set theory, such as ZF). How then do we show that not- > (P1) is consistent? We derive it from a set of premises which are themselves > consistent. To do this, suppose that human persons are information-processing > machines more powerful than standard Turing machines, for instance the infinite- > time Turing machines specified and explored by Hamkins and Lewis (2000), that > AI (as referred to in A) is based on standard Turing-level information processing, > and that the process of creating the artificial intelligent machines is itself at the > level of Turing-computable functions. Under these jointly consistent mathematical > suppositions, it can be easily proved that AI can never reach the level of human > persons (and motivated readers with a modicum of understanding of the mathe- > matics of computer science are encouraged to carry out the proof). So, we know > that (P1) isn’t certain. Note the "for instance" of the ∞ time Turing machines, which itself seems to refer to a stable output in the long run that is taken as a non-halting output ... maybe kindasorta like the decimal format of 1/7 ... or Nick's conception of reality 8^D. I keep thinking, with no decision in sight so far, that Wolpert and Benford's attempt to resolve Roko's Basilisk is related, that there's some underlying set-up that makes the whole controversy dissolve. You'll note the higher-order nature of AI+ and AI++. And if there are some higher-order operators that simply don't operate over potential infinities, what are they? And can we simply define our way out of it, as in defining 1/0 ≡ 0? On 8/3/20 10:02 AM, uǝlƃ ↙↙↙ wrote: > > I know I've posted this before. I don't remember it getting any traction with y'all. But it's relevant to my struggles with beliefs in potential vs actual infinity: > > Belief in the Sinularity is Fideistic > https://link.springer.com/chapter/10.1007%2F978-3-642-32560-1_19 > > Not unrelated, I've often been a fan of trying identify *where* an argument goes wrong. And because this post mentions not only 1/0, but Isabelle, Coq, Idris, and Agda, I figured it might be a good follow-up to our modeling discussion on Friday, including my predisposition against upper ontologies. > > 1/0 = 0 > https://www.hillelwayne.com/post/divide-by-zero/ -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
uǝʃƃ ⊥ glen
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In reply to this post by gepr
In 2011, my buddy Ralf offered me a summer *artist in residence* in Eugene
Oregon. We attended the 10th Annual Oregon Programming Languages Summer School[⏧], where a few days were spent in a giant lecture hall full of mostly young men fiddling with Coq. One night, he and I even ran into Benjamin Pierce riding his bicycle back from the conference! Agda has always seemed promising to me, however, its dependence on Emacs has remained a deterrent for me. [⏧] https://www.cs.uoregon.edu/research/summerschool/summer11/curriculum.html -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by Frank Wimberly-2
I might modify this slightly to For any r in R, however large, there exists x in R, and epsilon > 0 in R such that 1/x > r for x < epsilon. I'm not sure that makes a difference but it may make it clearer. On Mon, Aug 3, 2020 at 11:14 AM Frank Wimberly <[hidden email]> wrote:
Frank Wimberly
140 Calle Ojo Feliz Santa Fe, NM 87505 505 670-9918 - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by gepr
This is where folk tales are wonderful. Out of all the complex clutter of daily life among all the different people, they recognize a big question and put a marker on it by wrapping it in a small story or metaphor, which turns out to have staying power as a meme, because it resonated with what really was a big question.
Are the Celts (or even more specifically, the Irish?) the only ethnicity that had a specific meme equivalent to a pot of gold at the end of the rainbow? Or did it convergently evolve in several cultures? > On Aug 4, 2020, at 2:02 AM, uǝlƃ ↙↙↙ <[hidden email]> wrote: > > > I know I've posted this before. I don't remember it getting any traction with y'all. But it's relevant to my struggles with beliefs in potential vs actual infinity: > > Belief in the Sinularity is Fideistic > https://link.springer.com/chapter/10.1007%2F978-3-642-32560-1_19 > > Not unrelated, I've often been a fan of trying identify *where* an argument goes wrong. And because this post mentions not only 1/0, but Isabelle, Coq [⛧], Idris, and Agda, I figured it might be a good follow-up to our modeling discussion on Friday, including my predisposition against upper ontologies. > > 1/0 = 0 > https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fwww.hillelwayne.com%2fpost%2fdivide-by-zero%2f&c=E,1,JxsXytueRHA0GCfR0UOa_3uDRb1upQSgWOk-Xn9W0El902gHmLp9YG0abXsverWIfnV9N-7WHZnF5x4UojpbFdwztwOiAwefuhlrHNfbWDzzwCA,&typo=1 > > Here's the (really uninformative!) SMMRY L7: > https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fsmmry.com%2fhttps%3a%2f%2fwww.hillelwayne.com%2fpost%2fdivide-by-zero%2f%23%26SM_LENGTH%3d7&c=E,1,ojb4vUe5aPs24YNNGQSrZVPwWP0D69QletaevbLEpj0OxdCjjavwpY9GtAJu5a2Mc1d5Sv4p18nm2y0FjBAFLDAm8jUY5swj5w4XCY72UHrz&typo=1 >> Since 1 0, there is no multiplicative inverse of 0⁻. Okay, now we can talk about division in the reals. >> >> So what's -1 * π? How do you sum up something times? While it would be nice if division didn't have any "Oddness" to it, we can't guarantee that without kneecapping mathematics. >> >> We'll define division as follows: IF b = 0 THEN a/b = 1 ELSE a/b = a * b⁻. >> >> Doing so is mathematically consistent, because under this definition of division you can't take 1/0 = 1 and prove something false. >> >> The problem is in step: our division theorem is only valid for c 0, so you can't go from 1/0 * 0 to 1 * 0/0. The "Denominator is nonzero" clause prevents us from taking our definition and reaching this contradiction. >> >> Under this definition of division step in the counterargument above is now valid: we can say that 1/0 * 0 = 1 * 0/0. However, in step we say that 0/0 = 1. >> >> Ab = cb => a = c but with division by zero we have 1 * 0 = 2 * 0 => 1 = 2. > > > > [⛧] I decided awhile back to focus on Coq because it seems to have libraries of theorems for a large body of standard math. But still NOT having explored it much, yet learning some meta-stuff surrounding the domain(s), I'm really leaning toward Isabelle. I suppose, in the end, I won't learn to use any of it, except to pretend like I know what I'm talking about down at the pub. > > -- > ↙↙↙ uǝlƃ > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe https://linkprotect.cudasvc.com/url?a=http%3a%2f%2fredfish.com%2fmailman%2flistinfo%2ffriam_redfish.com&c=E,1,UTc_8AxdA9Q4p6xHPwfg1ZGYkn1L6Rshm3PZVQLgSnP-lfe22-94Fxyz_-jF-b4_qqvtfHMdPvw7zxkaWIFajGc_LfeQOy_wJ-qz0_KI0A,,&typo=1 > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC https://linkprotect.cudasvc.com/url?a=http%3a%2f%2ffriam-comic.blogspot.com%2f&c=E,1,XICmjtgygQaXCjBo6Rp4fi9GaQGLm50k3Y9rWCzegK2dveyCNd7uQIgSA_zsoIT6rpmgYujmIv6_r-JZ2va6n6-XalhE8VU7SzLyuhEMRHc5HZtCw08,&typo=1 - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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Yes! And playing that same note (along with cargo cults, mnemonics, and the specialness/detail-preservation of narrativity), I committed to posting that I was wrong and Jon's *epiphenomena* are appropriately named (based primarily on the oracle-sort idea (I like "oracle" better than "key") [⛧]. But near the end, I asked EricC whether or not evolutionary biologists have a typical way of speaking about contingency/ancillary/contextual causation as opposed to, for lack of a better word, driven causation. I think I asked that as a result of Jon's suggestion that *mystery* isn't fundamental, here. I've forgotten how EricC actually responded. But while responding, the concepts of "critical path", polyphenism, and robustness was what came to mind. And now I think I'm wrong about being wrong. [⛤]
It often seems that such folk tales exhibit some universality (e.g. virgin births or Jungian archetypes). But it's difficult for someone like me to a) guess at their cross-culture applicability and b) guess at which contingent causes have to be dragged along as the narrative moves from one detail-rich context to another ... like so many privileged post-yuppies saying Namaste after their Hot Yoga. [⛧] That idea being to shuffle a list, you place the items to be shuffled into a key-value map where the keys are drawn from a [pseudo-]random or arbitrary number source, then extract the ordered values and toss the keys. There's structure there, but it would have to be *reconstructed*. [⛤] My dad used to be accused of never admitting he was wrong. So, he often told the minimalist-for-him joke: "I was wrong once. Then it turned out I was right." On 8/3/20 2:44 PM, David Eric Smith wrote: > This is where folk tales are wonderful. Out of all the complex clutter of daily life among all the different people, they recognize a big question and put a marker on it by wrapping it in a small story or metaphor, which turns out to have staying power as a meme, because it resonated with what really was a big question. > > Are the Celts (or even more specifically, the Irish?) the only ethnicity that had a specific meme equivalent to a pot of gold at the end of the rainbow? Or did it convergently evolve in several cultures? -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
uǝʃƃ ⊥ glen
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