FOM: Alice, Bob and Carol
wiman lucas raymond
lrwiman at ilstu.edu
Tue Apr 9 00:01:04 EDT 2002
> 7. So, isn't that all there is to say about number?
In some senses, yes, that is all there is to say about a number. A
number
is an intuitively obvious idea, something that even a chimpanzee can
grasp.
Chimpanzees certainly don't understand Fregean ideas! The point is that
those who worked on foundations in 19th century abstracted the
intuitively
obvious ideas of "counting", "set", and "identity" to a rigorous
mathematical
system. These systems don't always agree with reality, but are
interesting
mathematically.
Take the idea of continuity. What is a continuous function? Well,
that's
obvious: it's a function whose graph can be drawn without breaks in it.
That's fine, of course, until you want to start proving things about
continuous
functions. One needs to start working with continuity at a point, and
things of
that nature. One eventually comes up with the standard "epsilon-delta"
notion
of continuity. But this notion, while it seems intuitive, has
counterintuitive
consequences. For example, there are functions which are discontinuous
at every
rational value, but continuous at every irrational value. That's very
counter-
inuitive, but it's an *abstraction* of our real experience of drawing
graphs.
It's not reality.
> If just saying a bunch of objects are all different from each other is
> enough to say they have a number, why not stop right there?
As I said, the idea of "number" is not a difficult one to grasp. The
problem
is how can mathematicians work with them? If we just assume at the
outset that
numbers are intuitively obvious, then what is a proof? We need to state
our
assumptions about them at the outset, and then prove things based upon
those
assumptions. The Peano axioms aren't numbers--they abstract numbers
into a
workable system.
> To sum up a very simple argument, we can communicate information about
how
> many things there are without communicating any information whatsoever
of
> the sort that Fregean or neo-Fregean number says is essential to our
ideas
> about number, or about number "itself".
Yes, because ordinary language is much less precise than mathematical
language.
Our standards for indentifying "three-ness" in everyday language are
much weaker
than Peano's or Frege's. This is fine, but mathematicians need to be
able to
formally study these objects.
> Moreover this information, if correct, reflects the way things are, so
> nothing else is relevant to the *fact* that there are a certain number
of
> things.
>From which I conclude all the Fregean stuff is absolutely irrelevant
either
>to our ideas about number, or to facts about it.
Whoa, that's quite a leap. I find it very dubious to say that Peano
arithmetic
has nothing whatever to do with number theory (the facts about numbers),
and
number theory definitely relates to the real world.
Note that I'm not saying that mathematicians necessarily start out with
Peano
axioms, and work from there. Actually, I doubt that any number
theorists do.
However, if one is to undertake a logical study of number theory, one
needs to
start with an axiom system. Without the Peano arithmetic axioms, it
would not
be possible to prove Goedel's incompleteness theorem. This certainly
says
something signicant. Perhaps not about the immediate notion of
"three-ness",
but about the limits of human reasoning.
-Lucas Wiman
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