Re: Cantorian pseudomathematics

Han.deBruijn@xxxxxxxxxxxxxx wrote: 
Han.deBru...@xxxxxxxxxxxxxx wrote:


Our goal is to establish, though, that "materialization" may be _more_
than just a hollow phrase and it's covering more than some well known
habits of our scientists and engineers.



Here is our basic picture again. Post- instead of pre-processing:

                    materialization
        mathematics ===============> real world

Let's start with the real numbers as an appetizer. The obvious question
is: what is the materialization of the real numbers, in the real world?
When looking for an answer, you are _not_ free to grab something out of
the big blue sky. You should look instead to that real world itself and
ask yourself what is the most significant & outstanding materialization
of the real numbers. As i.e. David Petry has said on many occasions, it
is beyond doubt that the answer is found in: our computing devices. I'm
not going to call this an axiom, because it would suggest something we
are not going to do here: abstract mathematics. Instead, I would like
to call it something that is "essential". And no, it doesn't mean that
there are no other possibilities, like continued fractions or some
such.
It only means that mainstream materialization is adopted as significant
enough for our purpose.

Essential 1
-----------
The (best / most common) materialization of the real numbers is the
type
called floating point / double precision in a modern digital computer.

We will take an arbitrary but quite important piece of mathematics:
just
common calculus. Our reference will be the book by James Stewart,
called
"Calculus, early transcendentals", fifth edition, Thomson (2003).
(Happy
to feel firm ground under my feet again! And an _excellent_ book,
IMNHO)

              x^2 - 1        (x + 1)(x - 1)
Example: lim  ------- = lim  -------------- = lim  (x + 1) = 2
         x->1  x - 1    x->1     (x - 1)      x->1

But suppose that the function f is defined mathematically as follows:

f(x) = (x^2 - 1) / (x - 1)  for x <> 1 ;  f(x) = 1  for x = 1 .

Now suppose we have a real number close to 1, but which is "actually"
like this, namely in the ideal world (heaven) of abstract mathematics:

1.000000000000000000000000000000123597059137504570...

Suppose that x and f(x) are both implemented on a digital computer with
the following material limitation, as far as the ideal number of digits
is concerned. (Note that "material" is a proper subset of the "ideal")

1.000000000000000000000000000000123597059137504570...
|----------------------------||---------------------> oo
        material                       ideal

Then, inside the computer at hand, _this_ value of x cannot possibly be
distinguished from x = 1. And the machine calculates f(x) = 1. But now
suppose we buy a somewhat better computer, with some extended precision
such that:

1.000000000000000000000000000000123597059137504570...
|-------------------------------------||------------> oo
                 material                  ideal

Then suddenly the value which was supposedly equal to one becomes just
close to 1 instead. And our new machine calculates the value f(x) = 2 .

An unacceptable result. Therefore we formulate:

Essential 2
-----------
Conclusions drawn with materialization should be independent of machine
details. The only essential property being that a machine is _finite_:
David Petry's microscope has a _limited_ aperture.

It has been demonstrated that the value f(x = 1) = 1 leads to a problem
with materialization of the results. No such problem would have arised
if instead the following definition had been adopted:

f(x) = (x^2 - 1) / (x - 1)  for x <> 1 ; f(x) = 2  for x = 1 .

Meaning that the limit for x->1 is also adopted as the value for x = 1.
This can be generalized to the following: the values at isolated points
of a continuous function must be equal to the limiting values at those
points (if they exist). Otherwise, these values cannot be materialized,
in an unambiguous manner, with different "microscopes".


Then all functions should be continuous ?

Most mathematicians in this group agree about the idea that mathematics
is _not_ a science. Right?


No, but it's only a detail in the discussion.

So it shouldn't be a slap in your face when
somebody tells you that materialization is essentially the process that
makes _science_ out of your mathematics. Right?

Still no. *Your* materialization is strange: floating point numbers are very far from ideal, mainstream reals. And you dismiss all mathematics history before computers, and those many interpretations and/or constructions of real numbers. Difficult to accept, unless you show
a very good reason for that. What is it ?

Considering again the above example, this means that the isolated point
x=1 in f(x)=(x^2-1)/(x-1) can be defined in _mathematics_ as f(1) = 1,
but it can _not_ be defined in _science_ as f(1) = 1 . In science, the
outcome can only be f(1) = 2 , being the same as the limit value there.

Excercises:
----------
- How about the distinction between rational and irrational numbers?
  Is it scientific or purely mathematical?

Guess. Hint: draw a square of side 1. Is it true that no rationnal
number is the length of the diagonal. Is it mathematical or scientific ?
Maybe you mean that in practice, no square is perfectly square, and
no length is perfectly measured. Right, but then you don't do mathematics anymore. Worse, I doubt you'll find something interesting
to study.

- How about the scientific meaning of a function that assigns value 0
  to any rational number and value 1 to any irrational number in [0,1]?

This one is more tricky ;-) Although it's not conceptually very difficult to accept, I admit it's not often useful in practical computations - though the theories behind *are* useful.

Remember: Een vos verliest wel zijn haren, maar niet zijn streken :-)

Can you translate, please ? I tried automatic translation from dutch on a website, it was rather surrealistic :-)

Han de Bruijn 

.

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