Re: Bruijn babbles bull***

Horand.Gassmann@xxxxxxxxxxxxxx wrote:

On Feb 5, 1:55 pm, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

Horand.Gassm...@xxxxxxxxxxxxxx wrote:

On Feb 5, 11:37 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

An error can be zero with integer (discrete) values, but not with reals.
Since an error can be arbitrary small, there is no "arbiter" necessary.

Not so fast. You said 0.333...3 "can be equal to 1/3, depending on the
error allowed in equality". Who is it that allows the error, and if
you
and I set different allowances for the error, who has the final say?

Correction! I should say: since an error MUST be arbitrary small.

Is 0 equal to 1/3? (1)
Is 0.3 equal to 1/3? (2)
Is 0.33 equal to 1/3? (3)
Is 0.333 equal to 1/3? (4)
...
Is 0.33333333333333333333333333333333333333333333333333333 = 1/3? (5)

No. All of them exceed an error of e.g. 1.E-70 .

Uh-uh. This game I will not let you get away with without protest.
Here you clearly say that

0.33333333333333333333333333333333333333333333333333333 is _NOT_ equal
to 1/3.

However, just yesterday you said to the similar question

Would you agree that the finite approximation 0.3333...3 (n threes) is
_not_ equal to 1/3, no matter how large n?

No. Any such finite approximation can be equal to 1/3, depending on the
error allowed in equality. (Absolute rigour is a phantom)

So which is it? I understand that you have trouble accepting "tertium
non datur",
which is fine by me, but here you are simply contradicting yourself.
If I take you by your word (and I intend to, since I am very literal
minded), yesterday's claim says that _any_ (and _all_) of the
questions I numbered (1) through (5) above can
be answered "YES" in some circumstances. You seem to have had a change
of heart since then.

Can I also ask you to elaborate a little further on this remark:

Correction! I should say: since an error MUST be arbitrary small.

What exactly do you mean by this?

I've just changed your "for _all_ epsilon" into my "for _any_ epsilon",

What exactly do you mean by "for _any_ epsilon"? Is that for _every_
epsilon,
or is it that _there_exists_ one, or is something else entirely?

It is: given any error epsilon, I can diminish that error (or rather:
replace it by a smaller error, in case of 1/3 by just calculating more
decimals).

I notice that you go out of your way to say "in case of 1/3". The same
would be true for Champernowne's number or for pi, wouldn't it? (If
not, please explain.)

And how exactly does your "for _any_ epsilon" differ from my "for
_every_ epsilon",
which, by the way, I take as synonymous to "for _all_ epsilon"?

thereby avoiding the _absolute_ rigour, but not the rigour.

Oouugh! Those quick and dirty responses have sucked me even deeper into
the B.b.bs morass and it's becoming increasingly difficult to get out.
But nevertheless, I've decided to give you my very best shot.

Let's define, (but only) for real numbers a,b and delta:

(a like b) iff ( | a - b | < delta where delta > 0 )

It's easy to see that (a like b) is an equivalence relation, because:
(a like a) ; (a like b) => (b like a) ;
((a like b) and (b like c)) => (a like c)
Proof of the latter: let |a - b| < d1 and |b - c| < d2 then
|a - c| <= |a - b| + |b - c| < d1 + d2 .
So transitivity is at cost of an increasing error.

Definition: if (x like a) is defined by |x - a| < da
and (x like b) is defined by |x - b| < db
and db < da
then (x like b) is called _better_ than (x like a)
Examples:
0.3 like 1/3
0.33 like 1/3 : is better
0.33333333333333333333 like 1/3 : is even better
Question:
Is there a BEST 'like' and can it be identified with common equality ?

I like "like" more than "equal" because likeness can always be decided
in a finite number of steps. But what's in a name; we could also have
said (a equals b) / (a = b) approximately. For the sake of clarity, we
should replace "equal" by "like" in my previous responses and you will
see there's not really a contradiction. It's more like replacing the
common black and white for equality by kind of a grey (gray ?) scale.

Another way of accomplishing the latter is with distribution functions.
My favorite one being the Gaussian:
replace (a = b) by exp(-((a-b)/sigma)^2/2) ; note that the value 1
stands for 'equal' and the value 0 for 'not equal'. Anything in between
is also possible.

Simple numerical experiment:

program B_b_bs;

procedure doe(a,b : double);
const
sigma : double = 0.01;
begin
Writeln(exp(-sqr((a-b)/sigma)/2));
end;

begin
doe(0.3,1/3);
doe(0.33,1/3);
doe(0.333333333333,1/3);
end.

Output:

3.86592013947282E-0003
9.45959468906767E-0001
1.00000000000000E+0000

Gaussian functions are also significant when geometric objects, such as
circles, are to be visualized as images:

http://hdebruijn.soo.dto.tudelft.nl/jaar2007/cirkel.jpg

The secret behind it being the function (with sigma = 20 pixels):

f := exp(-abs(sqr(x-a)+sqr(y-b)-sqr(R))/sqr(sigma)/2);

So far so good. Question:
Does HdB's setup lead to different results than common mathematics ?

The answer is yes. A most remarkable consequence is that we are forced
to agree with Brouwer's Continuity Theorem (from intuitionism). This is
readily seen from the common definition of continuity:
f is continuous in (a) iff for all epsilon there is a delta such that
if |x - a| < delta then |f(x) - f(a)| < epsilon
Now read the last part of this sentence as follows:
if (x like a) then ( f(x) like f(a) ) or rather
(x = a) => ( f(x) = f(a) ) : "approximately"
With other words: the fact alone that f is a real-valued function just
_means_ that it is continuous. This may be called pink intuitionism :-)

Another little consequence is my old friend the SINC function, commonly
defined by:

SINC(x) = sin(x)/x for x <> 0
= 1 for x = 0

It is suggested in common mathematics that other definitions would be
possible, for example:

SINC(x) = sin(x)/x for x <> 0
= 2 for x = 0

HdB's theory says that NO such alternative definitions are possible.
The reason is that the latter SINC is _not_ a function, according to
Brouwer's Continuity Theorem (and pink intuitionism, and physics).
The fact alone that SINC is a function implies that it is 1 at zero.

Another more serious consequence is the removal of _singularities_ from
physics; a problem commonly recognized in Quantum ElectroDynamics/QED.
But in my (not so) humble opinion it can be found all over the place.
It's caused by the same phenomenon as with the SINC function: solutions
are allowed with common mathematics, which should be "made continuous"
in the first place, before being accepted in a scientific environment.
It turns out that any singularity of the form 1/r^d is renormalized in
a space of dimension > d . Thus bringing renormalization into the realm
of mathematics; it's not a problem of physics, at all:

http://hdebruijn.soo.dto.tudelft.nl/QED/singular.pdf (typo's in here)

I've been working on the above ideas for quite some time and many, many
other applications can be found on my web site.

Among these is the "Inside / Outside Problem" with our "jewel formula"
I've never seen before and which could only be derived by employing the
idea that equality is a limiting case of something fuzzy:

http://hdebruijn.soo.dto.tudelft.nl/www/grondig/crossing.htm

And yes, only the "best like = equals" has been programmed in the end.

The "law of excluded middle" is seen in action again at page 35 of the
following paper, where numbers at a "dangerous interval" cannot really
be distinguished into harmful (: lead to division by zero) or harmless.

http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf

Well, I guess it's more than enough for today ..

Han de Bruijn

.

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