[FOM] Finite Set Theory

[Harvey Friedman]

On 2/21/06 5:00 PM, "Dean Buckner" <d3uckner at btinternet.com> wrote:

I wrote:

>>  What is a grammatical mirage? Can you explain this as well as I have
>> explained the emptyset and singletons?
> 
> Mirages are difficult to explain, especially to those convinced of the
> reality of what they appear to represent.

I really appreciate this frank answer of no. I.e., "not as well as I have
explained the emptyset and singletons".

> I'll try.  Some words like 'shoe' apply to just one thing at a time.
> But collective nouns like 'pair' apply to more than one thing at a time.
> The word 'one' in 'one pair' tells you that 'pair' applies once, but to
> two things.  Thus if there is just one pair of things, there are only
> two things, no third thing.

This is incorrect. There is a third thing here, in the pair written <x,y>.
There is x, there is y, and there is the ordered pair <x,y>. (Or we could do
this with the unordered pair).

This is the way students are taught analytic geometry starting in high
school.

The mirage that there is only x and y, and not a third thing <x,y> can be
argued to have held up the development of mathematics - or at least certain
branches of mathematics - for very long periods of time.

E.g., we now say "let x be an element of R^n" or "let x be an n-tuple of
reals". This a bit more advanced, since here there is also a variable n
representing a natural number, which may be far bigger than 2.
 
> One who imagined that it applied in fact to one thing, a pair-thing, is
> thus in the grip of the mirage, and a powerful one.

Now I am getting the idea that mirages are very good things that drive the
progress of science.

>  But 'pair', like
> 'set' is just a way of denoting many things at once.  There is not one
> thing denoted, but many.

This banning suggestion is quite bad for the development of mathematics and
science.

> Thus: no singleton set, no empty set.  A single of shoes is one shoe.  A
> nothing of shoes is nothing.

What about the set that has nothing in it? And what about the set that has
my computer in it and nothing else? And what about the set that has my
computer in it and my house in it and nothing else? Not only are all of
these crystal clear as single objects, but clarity about these obvious
constructions is essential for the progress of mathematics and science.

I am sure that you are not proposing a ban on such constructions just
because you personally dislike them. But I am at a loss to know just why you
want to ban them? Are people who claim to understand them perfectly well and
use them in mathematics and science doing something bad or dangerous? Or do
they understand something that you don't?

Harvey Friedman