# RE: The seven step-Mathematical preliminaries

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> Date: Sat, 6 Jun 2009 21:17:03 +0200
> From: tor...@dsv.su.se
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer skrev:
>>
>>
>>> Date: Sat, 6 Jun 2009 16:48:21 +0200
>>> From: tor...@dsv.su.se
>>> 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.
>> 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
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.
>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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