On Sat, Jun 6, 2009 at 4:20 PM, Jesse Mazer <lasermazer.domain.name.hidden> wrote:
>
>
> > Date: Sat, 06 Jun 2009 21:17:03 +0200
>
> > From: torgny.domain.name.hidden
> > To: everything-list.domain.name.hidden
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> >
> > Jesse Mazer skrev:
> [[[[[[[>>
> >>
> >>> Date: Sat, 6 Jun 2009 16:48:21 +0200
> >>> From: torgny.domain.name.hidden
> >>> To: everything-list.domain.name.hidden
> >>> 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 02 as
///// and 04 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*
>
> >
>
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Received on Sun Jun 07 2009 - 11:38:08 PDT