# RE: The seven step-Mathematical preliminaries

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> Date: Fri, 12 Jun 2009 18:40:14 +0200
> From: tor...@dsv.su.se
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer skrev:
> >
> > > Date: Wed, 10 Jun 2009 09:18:10 +0200
> > > From: tor...@dsv.su.se
> > > Subject: Re: The seven step-Mathematical preliminaries
> > >
> > > Jesse Mazer skrev:
> > >>
> > >>> Date: Tue, 9 Jun 2009 18:38:23 +0200
> > >>> From: tor...@dsv.su.se
> > >>> Subject: Re: The seven step-Mathematical preliminaries
> > >>>
> > >>> For you to be able to use the word "all", you must define the "domain"
> > >>> of that word. If you do not define the domain, then it will be
> > >>> impossible for me and all other humans to understand what you are
> > >>
> > >> OK, so how do you say I should define this type of "universe"? Unless
> > >> you are demanding that I actually give you a list which spells out
> > >> every symbol-string that qualifies as a member, can't I simply provide
> > >> an abstract *rule* that would allow someone to determine in principle
> > >> if a particular symbol-string they are given qualifies? Or do you have
> > >> a third alternative besides spelling out every member or giving an
> > >> abstract rule?
> > >
> > > You have to spell out every member.
> >
> > Where does this "have to" come from? Again, is it something you have a
> > philosophical or logical definition for, or is it just your aesthetic
> > preference?
>
> It is, as I said above, for me and all other humans to understand what
> you are talking about.  It is also for to be able to decide what
> deductions or conclusions or proofs that are legal or illegal.  It has
> nothing to do with my aesthetic preference.

Well, most humans who think about mathematics can understand rule-based
definitions like "0 is a whole number, and N is a whole number if it's equal to
some other whole number plus one"--you seem to be the lone exception. What's
more, I kind of think you're "playing dumb" here, because I bet *you* would
have little problem with a rule-based definition of a finite set that didn't
actually spell out every member, like "0 is a member of the set, and N is in
the set if it's equal to some other member of the set plus one, *unless* the
'other member of the set' is equal to 1,038,712 in which case no members of the
set are larger than that." Here, can't you understand that the set includes
every whole number from 0 to 1,038,712 without my having to write out every
member? And the mental process that allows you to decide whether some string of
symbols (say, 1692) would qualify as a member of that set is exactly the same
as the mental process that would allow you to decide whether some string of
symbols would qualify as a member of the "whole numbers" which have no upper
limit.
As for being "able to decide what deductions or conclusions or proofs that are
legal or illegal", how exactly would writing out all the members of the
"universe" solve that? For example, I actually write out all the numbers from 0
to 1,038,712 and say that they are members of the "universe" I want to talk
about. But if I write out some axioms used to prove various propositions about
these numbers, they are still going to be in the form of general *rules* with
abstract variables like x and y (where these variables stand for arbitrary
numbers in the set), no? Or do you also insist that instead of writing axioms
and making deductions, we also spell out in advance every proposition that
shall be deemed true? In that case there is no room at all for mathematicians
to make "deductions" or write "proofs", all of math would just consist of
looking at the pre-established list of true propositions and checking if the
proposition in question is on there.

> >
> > > Because in a *rule* you are
> > > (implicitely) using this type of "universe", and you will then get a
> > > circular definition.
> >
> > A good rule (as opposed to a 'bad' rule like 'the set of all sets that
> > do not contain themselves') gives a perfectly well-defined criteria
> > for what is contained in the universe, such that no one will ever have
> > cause to be unsure about whether some particular symbol-string they're
> > given at belongs in this universe. It's only "circular" if you say in
> > advance that there is something problematic about rules which define
> > infinite universes, but again this just seems like your aesthetic
> > preference and not something you have given any philosophical/logical
> > justification for.
>
> What do you mean by "some particular symbol-string"?
>
> I suppose that you mean by this is: If you take any particular
> symbol-string from this universe, then no one will ever have cause to be
> unsure about whether this symbol-string belongs in this universe.  So
> you are defining "this universe" by supposing that you have "this
> universe" to start with.  Is that not a typical circular definition?
No, I'm saying "take some particular collection of symbols like
0,1,2,3,4,5,6,7,8,9, then any finite ordered group of them like 21396 is a
valid symbol string, and then you can use your rule to see if it qualifies as a
member of the 'universe' defined by that rule" (for example, if the rule is one
that gives us the set of all odd whole numbers, the symbol-string 21396
wouldn't qualify). Of course I have used the word "any" here, but the mental
process that allows you to judge whether a *particular* symbol string qualifies
is no different than the one that would be involved if I had said something
like "take some particular collection of symbols like 0,1,2,3,4,5,6,7,8,9, then
any ordered group of them *of length 10 or less* like 21396 is a valid symbol
string". Somehow I doubt whether you would have any trouble judging whether a
symbol string I gave you qualified as "valid" according to that criterion, even
though I have not actually written out every valid symbol-string of length 10
or less.
Jesse
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