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From: David Bernier on 18 Jan 2010 20:55
Archimedes Plutonium wrote:
>
> David Bernier wrote:
>> A wrote:
> (snipped)
>>> I find this extremely hard to believe. Can you provide some evidence
>>> that this actually happened?
>> Seconded. Without standard quoting conventions or the equivalent,
>> we don't know if Archimedes Plutonium is paraphrasing, re-wording,
>> re-phrasing or interpreting & "re-wording". It's pretty hard
>> to figure out what the original source material actually was ...
>>
>> David
>
> A private survey is a private survey. But it is easy to check and
> verify in
> a roundabout method. We look at the entire math literature for anyone
> who has tried to define finite-number with any precision. Even "A"
> says
> the literature is empty, dearth, zip, zero of a definition of "finite-
> number."
>
> Now you ask anyone in mathematics or you consult the math literature
> for what kind of numbers that the Peano Natural Numbers are, and the
> literature
> is overflowing with the announcements that the Natural Numbers are
> each and every
> one of them a "finite-number".
>
> I find that horribly odd and unsettling that mathematics, the science
> of precision
> is replete of the fact that the Peano Natural Numbers are all "finite
> numbers" yet
> noone, ever bothered to define "finite-number". That is worse than
> going to buy
> a brand new pickup truck and find out that the company forgot to
> install a engine.
>
> Look up MathWorld for definition of "finite number". Look up
> Wikipedia for definition of "finite number". Type in "finite-number"
> in Google
> to see if anyone defines it with precision.
>
> What you end up with is zero, zip. Because everyone who has been
> trained
> in mathematics puts the definition of "finite-number" as one of those
> assumed items, a fuzzy assumed piece of hardware in mathematics, so
> long as noone asks poignant questions about your assumed "finite
> number definition"
> everyone goes their merry inconsistent ways.
>
> In fact, the only place in all of math literature that defines "finite-
> number" is not even
> in math but the Webster's New World 4th ed, 2002, dictionary:
>
> Finite: (Math) (a) capable of being reached, completed, or surpassed
> by counting (said
> of numbers or sets) (b) neither infinite nor infinetesimal (said of a
> magnitude)
>
> So that definition of Finite by Webster's falls apart because
> 999....9999 is reached by
> 1 + 1 + ....+ 1, and completed. But it is not surpassed by counting,
> but that 9999....998
> is surpassed by counting so it would be finite according to Websters.
>
> So that definition fails, but few mathematicians expect a dictionary
> to be at the cutting
> edge of math definitions.
>
> Now there is another aspect to the previous post where I lambasted "A"
> about his
> set definition of finite-set. I forgotten who it was who defined
> "infinite set" as a set that
> can be placed in a 1-1 correspondence with a proper subset, (I think
> it was Riemann but
> my memory is not as good as when I was younger, and it seems as though
> Riemann was
> far brighter than to be making such a goofy definition of infinite
> sets, so I am hoping it was
> not Riemann) and that a finite-set would
> thus be unable to perform that feat. And as I wrote about a year or
> two ago, the flaw of
> that definition of infinite-set and finite-set. The question becomes
> which is more important
> of a concept, a finite-number or finite-set? I believe numbers come
> first or are more
> primal than is set theory. Regardless, here is a counterexample that
> tells us that finite-set
> and infinite-set based on the notion of a 1-1 correspondence with a
> proper subset is fakery.
>
> The infinite set of singlet numbers {0, 1, 2, 3, . . . . 9999....998,
> 9999....999}
>
> Now the reason Cantor and others got away with the fake definition of
> infinite-set as a
> 1-1 correspondence with a proper subset, is because they never defined
> finite versus
> infinite for numbers and always just looked at the Peano Natural
> Numbers as this:
>
> { 0 , 1, 2, 3, . . . .} and those four dot ellipsis just swept the
> fake definitions away.
>
> So try putting the even Numbers into a 1-1 correspondence with all the
> numbers in the
> set
>
> {0, 1, 2, 3, . . . . 9999....9999}
>
> You see, your 1-1 correspondence with a proper subset fizzles away
> into the trash dump.
>
> The reason the junky definition of infinite-set worked for Cantor et
> al and for "A", is because
> they kept looking at
>
> {0 , 1, 2, 3, . . . . } at that four-dot ellipsis to rescue them with
> a 1-1 correspondence. But when
> you realize that the Successor Axiom goes to 999....9999 then your
> convoluted infinite-set
> definition is destroyed.
>
> So, David Bernier, you have been in math for a good long time, and
> what is your understanding
> of defining a Finite-Number versus an Infinite-Number. What is your
> definition? What is your
> definition when you first learned the Peano Axioms of the Natural
> Numbers and what is your
> definition of finite-number since it is key to the Peano Natural
> Numbers.
[...]

I'd say small finite numbers are ideas. Just like colors are qualities of
objects around us, abstracted from everything else about those
objects, small finite numbers are qualities of small collections of
objects. It doesn't seem reasonable to me that I would have
had in my life more than 10^(10^100) ideas. So it seems there
is also the idea of a generic finite number, i.e. the idea
corresponding to "some finite number". With small finite
numbers, there's the idea of adding them and obtaining
their sum. For two generic finite numbers, there's also
the idea that they can be added to obtain their sum.

In the course of growing-up, at some point there dawns the idea
that any two generic finite numbers can be added to arrive at
their sum, which is also a finite number. From there many are
likely to conclude that there is no largest finite number,
and that there is no end in principle to counting-up from
one by adding one each time.

Some philosophers such as Plato believed in a world of ideas
that exists independently of humans. This is possible, but
I can't imagine how one could argue very convincingly for
it, or prove it. On the other hand, people mention small
finite numbers every day, so it seems very reasonable that
small finite numbers exist as ideas for them. One can ask if
two different mathematicians have equivalent conceptions
of finite numbers. This is generally believed by mathematicians
but it seems that it would take forever to prove that this is so.

I think many mathematicians are more interested in solving
problems than in philosophy. Thus one comes upon
very large numbers mentioned in proofs, for example
the famous first and second Skewes' number(s).

I doubt that one can give a definition of "finite number"
that will be satisfactory to most or all philosophers
with an interest in mathematics.

David Bernier
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