Re: irrational number continuum

On Jun 23, 2:05 am, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Jun 22, 11:47 am, slartibartfast <tomokane2...@xxxxxxxx> wrote:



an "INFINITENUMBER" of 3's after the decimal point, is no more a
numberthen an ""INFINITENUMBER" of 3's before the decimal point;
both are perpetually undefined.

Not exactly.

First of all, the things you're talking about are representations
of numbers; this is not the same thing as numbers themselves.
You are apparently unhappy that the decimal expansion of
1/3 requires an infinitenumberof digits, but expressing the
exact samenumberas a ratio of integers does not. Why
should we object to 1/3?

If we talk about that samenumberin base 3, now it no
longer has an infinite decimal expansion. Should we
use  a system that requires us to object to numbers
depending on what base we use?

you're missing the point. I'm not objecting to the number 1/3; but
firstly to the possibility to represent it as a finite decimal AND
ALSO secondly to the notion of a completed infinity (of 3's after the
decimal point).

the fact that the differs less and less from 1/3 the more digits we
use is NOT (in combination with the observation that the number 1/3
actually exists) any justification whatsoever for concluding that
"therefore a completed infinity must be notionally possible even if
not actually representable"

That is like arguing that little pixies built the eiffel tower, then
pointing to the existence of the eiffel tower as evidence for the
existence of little pixies!



only the sum's LIMIT is definite. (being "the smallestnumberwhich
the sum shall never reach")

Thenumber2 exists and is definite, but on what basis do we accept
that anumber, r, who's square is 2 exists?

On the basis that to not accept it makes a whole lot of
simple things impossible.

we don't accept the existence of the square root of -1, yet we use it
usefully.
we don't have to accept the actual existence of such a number as the
square root od 2 in order to do useful calculations with that "notion"
either. I've never once had to write it out in full!


If you have a rectangle with sides length 3 and 4, how
long is the diagonal? How long is the diagonal of the
rectangle with sides 1, 1?

it would be exactly as long as it is. But that particular length might
not be representable as a number. you can nearly represent that length
in numbers though, as nearly as you like, but never exactly, OR you
can refer to its length by using the symbol Square-root(2), but what
is at issue here is whether that symbol represents an actual number.

Do you want to make it
so that you can have the first rectangle but not the
second? Or do you want to make it so that you can't
draw the diagonal of the second? What about the
square whose diagonal is 1; how long are its sides?

there's a difference between spatiality and number. You can draw all
the rectangles, but you cannot always give a number for the length of
the hypotenuse regardless of the base.



if a^2 < 2 and b^2>2, is that sufficient justification for accepting
that there must be an r; such that a<r<b, where r^2=2?

why do we say r exists and isirrationalrather than saying:

"2 doesn't have a square root, but there's an infinity of numbers
who's squares will differ from 2 by as little as you need for any
practical purpose"

We'd be a lot better off to say "2 doesn't have a decimal
expansion that can be expressed in a finitenumberof
digits" because that's a lot simpler than trying to
exclude certain numbers.

that already presupposes that there even ARE "such (exact) numbers"
even though they "cannot be represented".

here we come to it.
if something "doesn't have a decimal expansion that can be expressed
in a finite number of digits" (either in the case of dec. rep. of 1/3
or worse the square root of 2)

there is no justification for assuming that there is ANY definite
entity that that (partial) decimal representation partially
represents.

in the case of 1/3, 1/3 definitely does exist and the quibble is only
with the use of an "infinity" of 3's to "represent" it in the decimal
system. (since all of its digits are known)

in the case of the square root of 2 however, we don''t even know what
those digits would be, and there still "an infinity" of them.

in all computations however, we only EVER use "approximations" to
them, and these "approxmations" are always rational.

why do we assume a continuum between 0 and 1?

Because it's simpler if we do.

no it isn't; its completely unnessary! just because space seems a
continuum, doesn't mean number need be.
since when we finish manipulating the symbols for such "irrational
numbers" and progress to actually doing the computations with values,
we only ever use "rational number" approximations" ever.


.

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