On Saturday, July 18, 2015 at 4:35:06 AM UTC+10, Bruno Marchal wrote: > > > On 17 Jul 2015, at 06:21, Pierz wrote: > > > > On Thursday, July 16, 2015 at 7:07:50 PM UTC+10, Bruno Marchal wrote: >> >> >> On 15 Jul 2015, at 20:54, John Mikes wrote: >> >> I think JC resoinded to Brent: >> >> *"I don't have a visceral grasp of the true immensity of infinity. Do >> you?" * >> >> I wonder if 'immensity' means - B I G - ? in which case I cannot refrain >> from thinking about the* infinite SMALL* as well. >> >> >> The infinitely small is infinitely large, in the relative way. >> >> And in string theory I believe there are two solutions to the size of the > universe relative to the Planck length - either we are much, much bigger or > we are much, much smaller. Once the universe collapses below the size of > the Planck length, it becomes mathematically identical to expansion, so > collapse becomes expansion and vice versa. All very weird. Reminds me of a > nightmare my brother used to complain about as a kid which he called > "macro-micro", in which a boulder would become immense, and at the same > time as it expanded to infinity, it would become infinitesimal, like the > cube which can be seen in one of two perspectives. He found this > unpleasant, even terrifying. > > > I can understand. I made similar nightmare in my youth, notably involving > infinitely complex infinite knots. >
You're still having that nightmare. But now it's called your job. > > > > At any rate, from the point of view of this question, infinitesimal is as > good as infinite, since computationally it involves the same problem of > unending iteration. > > > Sure, same difficulty. Now, is not like consciousness? I mean, it is easy > to do a machine which can grasp many notion of infinite and reason with > them. The mystery is more in the apparent instantaneous qualitative > understanding of infinity that we seem to have, when we say that we > understand that a program like "10-got-10" will never stop. Simple machine > can prove that, and know that in the Theaetetus' sense, but how could they > feel that in a finite time? That suggests to me that consciousness is more > on the side of truth (p) than representation ([]p) in the Theaetetus > definition of knower []p & p. That would confirm the filter of > consciousness that brain and memories would impose in the relative way. It > is very counter-intuitive though, but *that* fact can be intutited when > familiar with machine's self-reference. > I'll take your word for it! You're right that the real point I am making is with respect to the qualia. You can make a machine manipulate the symbol of infinity, and call that "reasoning about infinity", as JC does above, but that's a little bit the Chinese Room. Now the Chinese Room may not be the greatest philosophical argument, but it does make *some *sense. A parrot can proclaim, "To be or not to be, that is the question!" but that doesn't make it an existential philosopher. What I am sure of is that my Macbook Pro, which is a damned complicated piece of computational hardware, does not understand infinity, and with all my programming skills I have no way of making it. It seems to me to reside in the domain of the unprovable mathematical intuition. > > Bruno > > > > > > >> The Mandelbrot set illustrates this well: >> >> https://www.youtube.com/watch?v=bo-MB1QPZ7E >> >> The more you zoom, the more the mini-mandelbrot set are small, with ever >> bigger filaments around them. >> >> Perhaps it is even clearer in this zoom where we go near 8 mini-brots: >> >> https://www.youtube.com/watch?v=nkNNrfZz7dg >> >> >> Just like I may think for 'eternal' as >> being momentary and timeless. >> >> >> >> Sometimes we can distinguish eternity from timelessness, depending on the >> context. >> >> >> >> We like to imagine meanings for concepts as >> we like. >> >> >> That is why we have to be careful to not introduce wishful thinking in >> the picture. >> We must be able to not deny the logical consequences of our beliefs. >> >> Bruno >> >> >> >> >> >> >> >> JM >> >> On Wed, Jul 15, 2015 at 4:26 AM, Bruno Marchal <[email protected]> wrote: >> >>> >>> On 14 Jul 2015, at 20:25, [email protected] wrote: >>> >>> >>>> >>>> >>>> On 07/14/15, John Clark wrote: >>>> >>>> >>>> >>>> >>>> On Tuesday, July 14, 2015 , Brent wrote: >>>> >>>> >>>> >>>> > Just ask yourself how you grasp the notion of infinity. >>>> >>>> >>>> I don't have a visceral grasp of the true immensity of infinity. Do >>>> you? >>>> >>>> >>>> No, I don't, which was more or less my point. What we think of as our >>>> "grasp of infinity" is an ability to consistently manipulate and use some >>>> symbol that just means "bigger than anything else we're concerned with". >>>> In mathematics it mostly comes up in proofs by induction. There's an >>>> interesting book available online, >>>> http://www.cs.utexas.edu/users/moore/publications/moore-wirth-2014a.pdf >>>> which describes one somewhat successful effort to have a computer do >>>> automatic proof by induction; which is what I would regard as one kind of >>>> 'grasping infinity'. >>>> >>> >>> Another one is a theorem prover for a formal and effective (the theorems >>> are recursively enumerable) set theory. It has the axiom that there is an >>> infinite set, but soon or later the theorem prover will prove Cantor >>> theorem that all sets have a smaller cardinal than their power set. This is >>> proved by diagonalization instead of induction. In fact diagonalization is >>> very often effective or computable, and that is why machine can be aware of >>> their own limitation. >>> >>> Bruno >>> >>> >>> >>> >>> >>>> Brent >>>> On Mon, Jul 13, 2015 at 9:26 PM, Pierz <[email protected]> wrote: >>>> >>>> >>>> >>>> >>>> > Sure. It's a concept even very young children can understand >>>> >>>> >>>> >>>> Have you actually tried this experiment? I think if you ask a very >>>> young child for the largest number there is he will say something like a >>>> million zillion, if you counter with a million zillion +1 he will look >>>> puzzled for a second and then with a note of triumph in his voice will say >>>> a million zillion +2 and it will take some time to convince him that still >>>> isn't quite right. >>>> >>>> > Computers just iterate until told or forced to stop, they cannot >>>> reason about their own iterative processes. >>>> >>>> >>>> Actually they can. >>>> The computer program Mathematica >>>> uses iteration to calculate the numerical value of PI, if you tell it >>>> to calculate the first 500 digits to the right of the decimal point it can >>>> do it in about half a second, if you tell it to calculate the first >>>> 10,000 digits to the right of the decimal point it can do it in about >>>> 3 second >>>> s, but if you ask it to calculate an infinite number of digits to the >>>> right of the decimal point it won't even start the iteration procedure, >>>> instead it will tell you that is an impossible task and you're being a >>>> idiot for asking it to do such a thing. Well OK,... the program is more >>>> polite than that and its language more diplomatic but I have a hunch >>>> that's >>>> what it's thinking. >>>> >>>> >>>> > infinity and zero are about equally easy mathematical concepts to >>>> grasp - historically both appeared in Indian mathematics around the same >>>> time. >>>> >>>> >>>> And yet the idea that there was more than one sort of infinity and >>>> some infinite things were bigger than others wasn't >>>> discovered until about 1880, not because the proof was so technically >>>> difficult it isn't (the ancient Greeks could have discovered it), but >>>> because before Georg Cantor nobody had even tried; before Cantor everybody >>>> thought it was obvious that nothing could be larger than infinity and that >>>> was that. Everybody thought they understood infinity but they did not. >>>> >>>> >>>> John K Clark >>>> >>>> >>>> >>>> >>>> >>>> -- >>>> >>>> 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 post to this group, send email to [email protected]. >>>> >>>> Visit this group at http://groups.google.com/group/everything-list. >>>> >>>> For more options, visit https://groups.google.com/d/optout. >>>> >>>> >>>> -- >>>> 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 post to this group, send email to [email protected]. >>>> Visit this group at http://groups.google.com/group/everything-list. >>>> For more options, visit https://groups.google.com/d/optout. >>>> >>> >>> http://iridia.ulb.ac.be/~marchal/ >>> >>> >>> >>> >>> -- >>> 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 post to this group, send email to [email protected]. >>> Visit this group at http://groups.google.com/group/everything-list. >>> For more options, visit https://groups.google.com/d/optout. >>> >> >> >> -- >> 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 post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> http://iridia.ulb.ac.be/~marchal/ >> >> >> >> > -- > 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] <javascript:>. > To post to this group, send email to [email protected] > <javascript:>. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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