On Aug 1, 5:24 am, "Stephen P. King" <stephe...@charter.net> wrote:
> On 7/31/2011 7:40 PM, Pzomby wrote:
> > The following quote is from the book What is Mathematics Really? by
> > Reuben Hersh
> > 0 (zero) is particularly nice. It is the class of sets equivalent
> > to the set of all objects unequal to themselves! No object is unequal
> > to itself, so 0 is the class of all empty sets. But all empty sets
> > have the same members .none! So they re not merely equivalent to each
> > other they are all the same set. There s only one empty set! (A set
> > is characterized by its membership list. There s no way to tell one
> > empty membership list from another. Therefore all empty sets are the
> > same thing!)
> > Once I have the empty sets, I can use a trick of Von Neumann as an
> > alternative way to construct the number 1. Consider the class of all
> > empty sets. This class has exactly one member: the unique empty set.
> > It s a singleton. Out of nothing I have made a singleton set a
> > canonical representative for the cardinal number 1. 1 is the class
> > of all singletons all sets but with a single element. To avoid
> > circularity: 1 is the class of all sets equivalent to the set whose
> > only element is the empty set. Continuing, you get pairs, triplets,
> > and so on. Von Neumann recursively constructs the whole set of
> > natural numbers out of sets of nothing.
> > .The idea of set any collection of distinct objects was so simple and
> > fundamental; it looked like a brick out of which all mathematics could
> > be constructed. Even arithmetic could be downgraded (or upgraded)
> > from primary to secondary rank, for the natural numbers could be
> > constructed, as we have just seen, from nothing ie., the empty set by
> > operations of set theory.
> > Any comments or opinions on whether this theory is the basis for the
> > natural numbers and their relations as is described in the quote
> > above?
> > Thanks
> Hi Pzomby,
> Nice post, but I need to point out that that von Neumann's
> construction depends on the ability to bracket the singleton an
> arbitrary number of times to generate the pairs, triplets, etc. which
> implies that more exists than just the singleton. What is the source of
> the bracketing? I have long considered that this bracketing is a
> primitive form of 'making distinctions' which is one of the necessary
> (but not sufficient) properties of consciousness.
> Stephen- Hide quoted text -
The full three paragraphs are from the book. The sentence ‘Once I
have the empty sets, I can use a trick of Von Neumann as an
alternative way to construct the number 1.’ is Hersh’s words.
I was looking for opinions, as you have given, on Hersh’s
conclusions. Your comment on ‘making distinctions’ is the direction I
was heading in understanding the role of primitive mathematics (sets,
numbers) underlying human consciousness.
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