Re: Bruijn babbles bull***

Horand.Gassmann@xxxxxxxxxxxxxx wrote:

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

Horand.Gassm...@xxxxxxxxxxxxxx wrote:

On Feb 5, 9:25 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

Horand.Gassm...@xxxxxxxxxxxxxx wrote:

Do you accept the infinite expansion 0.3333... as an equivalent to the
ratio 1/3?

Yes. (Because the infinite expansion can be understood as a limit)

Very good. We are making progress. Any countably infinite process that
produces a well-defined limit is pukkah. Would you agree that the
error of this infinite expansion is 0?

No. But due to the definition of a limit, the error is arbitrary small,
which is not the same as zero.

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)

Alright... Who gets to choose the precision, and who is to be the
arbiter if there are discrepancies between, say, you and me? Also, is
zero error theoretically allowable in your system?

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?
Is 0.3 equal to 1/3?
Is 0.33 equal to 1/3?
Is 0.333 equal to 1/3?
...
Is 0.33333333333333333333333333333333333333333333333333333 = 1/3?

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

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).

thereby avoiding the _absolute_ rigour, but not the rigour.

I'll leave the rest for later.

Han de Bruijn

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