Jesse Mazer wrote:
> > Date: Sat, 6 Jun 2009 21:17:03 +0200
> > From:
> > To:
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> >
> > Jesse Mazer skrev:
> >>
> >>
> >>> Date: Sat, 6 Jun 2009 16:48:21 +0200
> >>> From:
> >>> To:
> >>> 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?
> >>
> >> 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.

In fact we don't know "truths" about the surface topography of planets, 
including the Earth.  Our knowledge is in terms of models, like WGS84, 
or topographic maps.  We have confidence in these as approximations in 
some degree, but not in the same way as the "truths" we know about the 
mathematical models.  For example we can calculate the path distance 
between two points on the WGS84 ovoid and we know that value, but it is 
only approximately equal to the value we would measure by surveying or 
by GPS.  When we use infinities and infinitesimals it greatly simplifies 
our mathematical models, but it probably doesn't correspond to anything 
in the world.

> >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?
> >
> >> 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.
> Jesse
> >

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