Re: The modern mathematical concept of infinity is indefensible
Aatu Koskensilta
> Han de Bruijn wrote:
>
>> Why do people become so _emotional_ then, as soon as some "crank"
>> tries to declare certain axioms of set theory (ZFC in particular)
>> as futile ?
>
> Because "futile" has no mathematical meaning. It's like accusing
> Beethoven's 7th symphony of being too green, or too expensive.
"Futile", "fruitful", "interesting", etc. indeed have no mathematical
definition. This doesn't mean that to say of a mathematical theory or
a field of study that it is or is not futile, fruitful, interesting
etc. is as non-sensical as noting of Beethoven's 7th symphony that it
is not sufficiently green, only that such things do not turn on purely
mathematical matters.
> No, "actual infinity" is a purely crank notion.
No, it's not.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
george
> Brian Chandler <imaginator...@despammed.com> writes:
> > No, "actual infinity" is a purely crank notion.
On Jan 28, 12:01 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> No, it's not.
One ONLY needs to BOTHER saying "actual infinity" if there
is, conceptually in the intellectual realm or room in which the
discussion is being conducted, "potential" infinity with which
to contrast it. It is (arguably; I am trying to argue it) the
DISTINCTION
that either is or isn't a wrong notion.
To most of us, "actual" infinity is simply the ONLY kind and therefore
is
referred to simply as "infinity". It is POTENTIAL infinity whose
"crankhood"
might NEED to be debated, and even there, we don't so much dismiss
that concept as "crankish" as simply DEFINE it in terms of infinite
sets with finite members.
herbzet
george wrote:
>
> > Brian Chandler <imaginator...@despammed.com> writes:
> > > No, "actual infinity" is a purely crank notion.
>
> On Jan 28, 12:01 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > No, it's not.
>
> One ONLY needs to BOTHER saying "actual infinity" if there
> is
Some non-actual infinity from which it needs to be distinguished.
Yeah.
>, conceptually in the intellectual realm or room in which the
> discussion is being conducted, "potential" infinity with which
> to contrast it. It is (arguably; I am trying to argue it) the
> DISTINCTION
> that either is or isn't a wrong notion.
>
> To most of us, "actual" infinity is simply the ONLY kind and therefore
> is
> referred to simply as "infinity". It is POTENTIAL infinity whose
> "crankhood"
> might NEED to be debated, and even there, we don't so much dismiss
> that concept as "crankish" as simply DEFINE it in terms of infinite
> sets with finite members.
I think "potentially infinite set" is a swiz. It just means "finite
set" as far as I can see.
--
hz
george
> I think "potentially infinite set" is a swiz. It just means "finite
> set" as far as I can see.
Not exactly. It doesn't mean "set" AT ALL as far as we're concerned:
it is about a set VARIABLE that can take DIFFERENT values.
The mathematical entities that WE talk about are, by contrast,
CONSTANTS.
WE DON'T DO *time*. THEY (the "potential infinity" people) are
talking about
ONE AND THE SAME SET having MORE members at a LATER time,
i.e., "finite" in the sense that at every given time, it has a finite
number
of members, but (potentially) infinite in the sense that there are an
infinite
number of things that CAN POTENTIALLY EVENTUALLY BECOME members of it.
Obviously, for us, the memebrship of a set has to stay constant in
order for
the set itself to be existing and well-defined.
"They" DON'T EVEN UNDERSTAND THE DIFFERENCE between
a constant and a variable.
herbzet
george wrote:
> herbzet wrote:
> > I think "potentially infinite set" is a swiz. It just means "finite
> > set" as far as I can see.
>
> Not exactly. It doesn't mean "set" AT ALL as far as we're concerned:
That's being nice. In actual usage, it seems to mean nothing other
than "finite set".
I have also given some thought that by "potentially infinite
set" is meant something that could equally be termed
an "essentially extensible set", meaning that there's
always <spooky voice> "room for one more", i.e., if
A is "essentially extensible" then we can form A' by
adding an element, and A' will also be "essentially extensible".
> it is about a set VARIABLE that can take DIFFERENT values.
> The mathematical entities that WE talk about are, by contrast,
> CONSTANTS.
> WE DON'T DO *time*. THEY (the "potential infinity" people) are
> talking about
> ONE AND THE SAME SET having MORE members at a LATER time,
> i.e., "finite" in the sense that at every given time, it has a finite
> number
> of members, but (potentially) infinite in the sense that there are an
> infinite
> number of things that CAN POTENTIALLY EVENTUALLY BECOME members of it.
Ha, that's an angle I hadn't thought of -- there's an implicit suggestion
that there's an ACTUAL infinity of potential elements lying around.
Of course the idea of the "potentially infinite" has always been
associated with the rejection of a static, finished universe of
pre-existing objects -- the idea has always been that these new
elements of the potentially infinite sets are conjured into
being by our thinking of them.
It's funny, come to think of it, how from ancient Greece we get both
the idea of a static Platonic universe of perfect forms, and a certain
fundamental horror of an actually infinite alogos, beyond our power
to rationally construct.
> Obviously, for us, the memebrship of a set has to stay constant in
> order for
> the set itself to be existing and well-defined.
> "They" DON'T EVEN UNDERSTAND THE DIFFERENCE between
> a constant and a variable.
--
hz
Herbert Newman
>
> It's funny, come to think of it, how from ancient Greece we get both
> the idea of a static Platonic universe of perfect forms, and a certain
> fundamental horror of an actually infinite alogos, beyond our power
> to rationally construct.
>
Right: Plato vs. Aristotle (his pupil).
Herb
P.S.
Concerning Aristotle we KNOW for sure that he _erred_ at least in (the
domain of) physics ...
herbzet
Herbert Newman wrote:
> herbzet wrote:
>
> > It's funny, come to think of it, how from ancient Greece we get both
> > the idea of a static Platonic universe of perfect forms, and a certain
> > fundamental horror of an actually infinite alogos, beyond our power
> > to rationally construct.
> >
> Right: Plato vs. Aristotle (his pupil).
Following your hint, I did a lot of reading at Wikipedia on
Aristotle's metaphysics and philosophy of mathematics.
Alas, none of what I read survived in my head for 24 hours. :-(
> P.S.
> Concerning Aristotle we KNOW for sure that he _erred_ at least in (the
> domain of) physics ...
Of course, since Aristotle invented the syllogism, he's still a god.
--
hz