On Sat, Jun 6, 2009 at 4:20 PM, Jesse Mazer <laserma...@hotmail.com> wrote:
> > > > Date: Sat, 6 Jun 2009 21:17:03 +0200 > > > From: tor...@dsv.su.se > > To: everything-list@googlegroups.com > > Subject: Re: The seven step-Mathematical preliminaries > > > > > > Jesse Mazer skrev: > [[[[[[[>> > >> > >>> Date: Sat, 6 Jun 2009 16:48:21 +0200 > >>> From: tor...@dsv.su.se > >>> To: everything-list@googlegroups.com > >>> Subject: Re: The seven step-Mathematical preliminaries > >>> > >>> Jesse Mazer skrev: > >>>> > >>>> Here you're just contradicting yourself. If you say BIGGEST+1 "is then > >>>> a natural number", that just proves that the set N was not in fact the > >>>> set "of all natural numbers". The alternative would be to say > >>>> BIGGEST+1 is *not* a natural number, but then you need to provide a > >>>> definition of "natural number" that would explain why this is the > case. > >>> > >>> It depends upon how you define "natural number". If you define it by: n > >>> is a natural number if and only if n belongs to N, the set of all > >>> natural numbers, then of course BIGGEST+1 is *not* a natural number. In > >>> that case you have to call BIGGEST+1 something else, maybe "unnatural > >>> number". > >> > >> OK, but then you need to define what you mean by "N, the set of all > >> natural numbers". Specifically you need to say what number is > >> "BIGGEST". Is it arbitrary? Can I set BIGGEST = 3, for example? Or do > >> you have some philosophical ideas related to what BIGGEST is, like the > >> number of particles in the universe or the largest number any human > >> can conceptualize? > > > > It is rather the last, the largest number any human can conceptualize. > > More natural numbers are not needed.]]]]]]] > > Why humans, specifically? What if an alien could conceptualize a larger > number? For that matter, since you deny any special role to consciousness, > why should it have anything to do with the conceptualizations of beings with > brains? A volume of space isn't normally said to "conceptualize" the number > of atoms contained in that volume, but why should that number be any less > real than the largest number that's been conceptualized by a biological > brain? > *JohnM:* *Jesse, * *you don't have to go out to 'aliens', just eliminate the format "possible as of 2009". Our un-alien species is well capable of learning (compare to 2000BC) and whatever is restricted today as 'impossible' may be everyday's bread after tomorrow. You are absolutely right - even as of today. * *Especially in your next reply-par below.* > > >> > >> Also, any comment on my point about there being an infinite number of > >> possible propositions about even a finite set, > > > > There is not an infinite number of possible proposition. You can only > > create a finite number of proposition with finite length during your > > lifetime. Just like the number of natural numbers are unlimited but > > finite, so are the possible propositions unlimited but finte. > > But you said earlier that as long as we admit only a finite collection of > numbers, we can prove the "consistency" of mathematics involving only those > numbers. Well, how can we "prove" that? If we only show that all the > propositions we have generated to date are consistent, how do we know the > next proposition we generate won't involve an inconsistency? Presumably you > are implicitly suggesting there should be some upper limit on the number of > propositions about the numbers as well as on the numbers themselves, but if > you define this limit in terms of how many a human could generate in their > lifetime, we get back to problems like what if some other being (genetically > engineered humans, say) would have a longer lifetime, or what if we built a > computer that generated propositions much faster than a human could and > checked their consistency automatically, etc. > > >> or about my question about whether you have any philosophical/logical > >> argument for saying all sets must be finite, > > > > My philosophical argument is about the mening of the word "all". To be > > able to use that word, you must associate it with a value set. > > What's a "value set"? And why do you say we "must" associate it in this > way? Do you have a philosophical argument for this "must", or is it just an > edict that reflects your personal aesthetic preferences? > > Mostly > > that set is "all objects in the universe", and if you stay inside the > > universe, there is no problems. > > *I* certainly don't define numbers in terms of any specific mapping between > numbers and objects in the universe, it seems like a rather strange > notion--shall we have arguments over whether the number 113485 should be > associated with this specific shoelace or this specific kangaroo? One of the > first thing kids learn about number is that if you count some collection of > objects, it doesn't matter what order you count them in, the final number > you get will be the same regardless of the order (i.e. it doesn't matter > which you point to when you say "1" and which you point to when you say "2", > as long as you point to each object exactly once). > > Also, am I understanding correctly in thinking you don't believe there can > be truths about numbers independent of what humans actually know about them > (i.e. there is no truth about the sum of two very large numbers unless some > human has actually calculated that sum at one point)? If in fact you don't > believe there are truths about numbers independent of human thoughts about > them, why do you think there can be truths about the physical universe which > humans don't know about? For example, is there a truth about the surface > topography of some planet that humans have never and will never see up close > or send probes to? In physics most facts about physical systems are > quantitative numerical facts, after all, so if you admit truths about the > surface topography of a planet in another galaxy there's no reason not to > admit truths about the number of atoms in some large volume of space in > another galaxy, even if this number is one no human has ever thought about > specifically. > *JohnM:* *Not about my questioning 'truth' (who's?) but yur basic question was the 'belief' in numbers... Right on. David Bohm regarded numbers as a human invention and after several questioning on this list and others, I did not receive satisfying response why 'numbers' should be the originating 'truth' of the existence? Only consequences of 'it' and mechanisms. * *Would you deny a different mindset in the unlimited? * ** *About the 'physical universe' as truth: I regard it a figment based on the explanations (over millannia) in a steadily enriching epistemic cognitive inventory based on similarly continuously changing explanations of the observations we make in our developing capabilites, 'lately' (?) mostly on a quantified basis (numbers?) building up the edifice of the physical truth. Not vice versa. * > >But as soon you go outside universe, > > you must be carefull with what substitutions you do. If you have "all" > > quantified with all object inside the universe > > But I don't, of course. This is an idiosyncratic way of thinking specific > to you, and you have not given any philosophical justification for the idea > that numbers must be mapped to physical entities. Also, when you say > "universe" are you ruling out a priori any cosmological model which says the > universe is spatially infinite and contains an infinite number of particles? > *JohnM: * *(could we add: "in an infinite number of universes"? I am not for that, my poor imagination does not cover 'infinite' .)* > > > > >> as opposed to it just being a sort of aesthetic preference on your > >> part? Do you think there is anything illogical or incoherent about > >> defining a set in terms of a rule that takes any input and decides > >> whether it's a member of the set or not, such that there may be no > >> upper limit on the number of possible inputs that the rule would > >> define as being members? (such as would be the case for the rule 'n is > >> a natural number if n=1 or if n is equal to some other natural > number+1') > > > > In the last sentence you have an implicite "all": The full sentence > > would be: For all n in the universe hold that n is a natural number if > > n=1 or if n is equal to some other natural number+1. > > I didn't say anything about the universe, I would treat an n as just a > possible symbolic input that could be fed into the algorithm that decides > whether any given string of symbols fits the definition of a natural number, > it doesn't matter if this particular string is ever printed out in the real > physical universe. > *JohnM:* *My ignorance asks for a "definition of natural numbers" beyond an illustration "....2,3,4,.." without combining them to countables. The symbols are meaningful only if we assigne meaning to them. You may call 2 as ///// and 4 as // - the world will not collapse. (I asked Bruno and did not get a satisfying (for me) argument.) **The axioms are drawn to justify our theories about the world we built in our mind - supported by the fragmental and incomplete continuously developing **observations of the day we mustered upon those partially observed phenomena that got to us. (Keeping most of the previous explanations in store). * *A most ingenious edifice humanity erected. * *What does not mean that another 'storytelling' is impossible, as was given e.g. by (then young) Paul Churchland by a differently thinking 'tribe' on "caloric" - a liquid. * *Thanks for starting an expansion upo the 'possible' hoax.* > Jesse > *John M* > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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