[FOM] What numbers can happily be (a reply to Slater)

[Arnon Avron]

I would not have taken part in the discussion (re)started in Slater's
posting from September 18 (concerning what numbers cannot be) had my
name not mentioned in Slater's posting from  September 24 as someone
who accepts his point that numbers are not sets. So I should clarify
my position, which not quite sympathtic with Slater's views.

The only thing I agreed with Slater's at the posting he mentioned is that
a rational number cannot be taken to be a set of rational numbers 
because of obvious circularity. I pointed out that a formal solution 
to this problem (used e.g. in Fikhtengolt's classic
"the fundamentals of mathematical analysis")
is to officially identify only the irrational numbers 
with sets of rationals, to take then R to be the union of the set of 
rationals and the set of irrationals, and let the operations
and relations on R be induced by the corresponding operations
and relations on the set of all Dedekind's cuts.

  On the other hand I find as extremely odd Slater's argument 
that identifying the natural numbers with the finite von Neumann ordinals is 
a grammatical mistake. On the same ground he could have
rejected modern chemistry by stating that saying that an atom
consists of a nucleus and electrons is a grammatical mistake
(since atoms cannot be further analised). Much worse: the whole 
modern mathematical conception of what is a function is one big grammatical
mistake according to this very unmathematical way of thought . 
I assure Slater (from my experience as a teacher) that students
do not find "0 belongs to 1" as queerer than "The pair <0,1>
belongs to the exponential function" or "the exponential function
is a subset of the upper plane". They see that these
statements are true for the *graph* of the exponential function,
but would happily agree with anyone who will write a paper 
explaining "what functions cannot be" on the ground that by 
type distinctions, functions can have no elements and are not
subsets of anything! in fact, my experience shows me that the 
speed in which a student internalize the identification of a function
with its graph is a good indication of her/his mathematical ability.
Bad students never really get used to the idea that functions
are *sets" of pairs.

  To sum up: in science in general, and in mathematical in particular,
concepts frequently change their original meanings and definitions
in various productive ways,
and what is a "grammatical mistake" in one period might become a 
trivial truth later - *without* changing the eternal truths of Mathematics
(another example: the current official definition of basic
"trigonometric functions" is VERY remote from their original definition)!

 As for the question what are exactly the natural numbers, I admit
that (like most fomers who come from mathematics) I dont find
this question as particularly important. For me the main thing is
that we know how to represent natural numbers in a finitary,
effective way(s),and  how to effectively compute with them or
to compare them (all the things we *cannot* do with 
the real numbers!). Under these circumatances the question
what are "really" the natural numbers has almost the same
interest for mathematicians as the question what are "really"
the chess-pieces is for chess players. 

  Having say this, I would like to add that I like very much the 
identification of the natural numbers with the finite 
von Neumann ordinals, I  happily accept it, and I find it
much better than any other suggestion (including those ofFrege's and Holmes').
The reason are as follows: 

1) The natural numbers are FINITE objects, and so it seems to me
   very wrong to identify them with infinite sets.

2) Both Slater and Holmes concentrate on the role of the natural numbers
   as cardinal numbers, ignoring their role as ordinal numbers. However,
   it is in the latter role that everyone of us has realized (as a child)
   that the collection of natural numbers is infinite, because there is
   no "biggest" natural number: we can always add 1 (i.e. there is a successor)
   to every natural number). When a child gets it (and each of them does
   at some point or another) the question whether there are infinite
   number of objects in the world so that each natural number is indeed also
   the cardinal number of some set simply does not seem relevant (although 
   it causes great difficulties to Russel!). Moreover: although I never
   did an experiment, but I bet that most people, if pressed about this point, 
   will say at the end that a number n is always the cardinality of the
   set of numbers less than n (including 0):

   n=Card({k| k<n})
 
   Is there any more natural or basic example/characterization of n as 
   a cardinal number (for n not too small, at least)?

   Now this characterization refer to the misterious "Card", of which we 
   have no definition. Given this, what can be more natural, and in the
   spirit of mathematics then to change the above equation into a definition
   which does not contain the unnecessary "Card":

   n=({k| k<n})

   which is exactly von Neumann's definition.
  
   Moreover: in my opinion, the essence of the series of natural numbers,
   in their role as ordinal numbers, is given by the inductive clause:
   every natural number has a successor, which is different from that
   number and all the natural numbers that precede it. Denoting 
   temporarily {k| k<n} by S_n, this essence is capture by the inductive
   characterization:

   S_(n+1)=S_n\cup {n}

   Now again, what can be mathematically more natural and pleasing then
   abolishing the unneeded distinction between n and S_n (like that between
   a function and its graph) and turning this into:

   n+1=n\cup {n}

   Starting with 0=\emptyset, this inductive clause again gives us von-Neumann
   natural numbers. 

To sum up: both as ordinal numbers and cardinal numbers, and given that
the notion of a finite set is anyway unavoidable, I cant think of something
that can be so natural, so appealing, and so mathematically correct (meaning:
in the true spirit of mathematics) as von-Neumann's definition of 
the natural numbers.

Arnon Avron