Re: Why does everyone do it?

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Virgil wrote:

In article <a43d8$48bf926f$82a1e228$11516@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David R Tribble wrote:

david petry wrote:

My claim is this: set theory includes a mythology about things beyond
what we can observe (it implies the existence of objects that cannot
be identified).

David R Tribble wrote:

When was the last time you observed, say, a seven?
What did it look like?

David R Tribble wrote:

I'd still like to know if you or Petry have ever observed an
actual number (not just a representation of a number).

Han de Bruijn wrote:

Any decent programmer can convert any representation of any number into
any other representation of the number. Isn't a number just "the set" of
all its representations? How else would we be able to communicate about
numbers?

Or rather ..

A digital computer is the implementation of mathematical _ideas_,
an OCR program is the implementation of mathematical _ideas_.
One must first have an _idea_. And those ideas come essentially

from _nowhere_. Is that what you're trying to tell us?

Except the part about ideas coming from "nowhere", yes,
that's pretty close to my answer to Petry.

He was rambling about a "mythology of things beyond what we
can observe". But of course, you can't observe or measure
an abstract idea, can you?

How do we communicate an abstract idea? How can you be sure that _your_
abstract idea of a straight line or the number 7 is the same as mine ?

We can't. But we can, by setting up a suitable axiomatic system, get
close enough for mathematical purposes, i.e., that the two ideas are
effectively isomorphic in all relevant respects.

For example, given the group axioms, it really doesn't matter whether
your idea of a group is the same as mine as long as they both satisfy
all those axioms.

What are the axioms of the number seven ?

Seven occurs in the context of a mathematical structure, say N. It is
defined in that context.

Thus, given a structure satisfying the axioms of PA, 7 is the element
otherwise written

s(s(s(s(s(s(s(0))))))).

--
"Now, once [James's research] is accepted, number theory is the wild,
wild, west of the intellectual field and the hottest field on the
planet in terms of potential for new entries. [...] The future in
number theory belongs to the kids." -- James S. Harris corrupts youth
.

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