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Topic: Matheology § 176
Replies: 10   Last Post: Dec 11, 2012 12:33 AM

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Virgil

Posts: 6,993
Registered: 1/6/11
Re: Matheology � 176
Posted: Dec 11, 2012 12:32 AM
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In article
<225e3678-2256-4cf2-858a-5a9ccdc70001@g7g2000pbi.googlegroups.com>,
"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:

> On Dec 10, 1:11 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <c293798d-5b7e-4eba-a4df-6332461dc...@f19g2000vbv.googlegroups.com>,
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > Matheology § 176
> >
> > > Here's a paradox of infinity noticed by Galileo in 1638. It seems that
> > > the even numbers are as numerous as the evens and the odds put
> > > together. Why? Because they can be put into one-to-one correspondence.
> > > The evens and odds put together are called the natural numbers. The
> > > first even number and the first natural number can be paired; the
> > > second even and the second natural can be paired, and so on. When two
> > > finite sets can be put into one-to-one correspondence in this way,
> > > they always have the same number of members.

> >
> > > Supporting this conclusion from another direction is our intuition
> > > that "infinity is infinity", or that all infinite sets are the same
> > > size. If we can speak of infinite sets as having some number of
> > > members, then this intuition tells us that all infinite sets have the
> > > same number of members.

> >
> > > Galileo's paradox is paradoxical because this intuitive view that the
> > > two sets are the same size violates another intuition which is just as
> > > strong {{and as justified! If it is possible to put two sets A and B
> > > in bijection but also to put A in bijection with a proper subset of B
> > > and to put B in bijection with a proper subset of A, then it is insane
> > > to judge the first bijection as more valid than the others and to talk
> > > about equinumerousity of A and B.}}

> >
> > > [Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2
> > > (1998) 1-59]
> > >http://www.earlham.edu/~peters/writing/infinity.htm#galileo

> >
> > > Regards, WM
> >
> > Note that the part in  {{ }} above is WM's addition, which runs totally
> > counter to the Peter Suber's own conclusion which reads:
> >
> > "Conclusion
> > Properly understood, the idea of a completed infinity is no longer a
> > problem in mathematics or philosophy. It is perfectly intelligible and
> > coherent. Perhaps it cannot be imagined but it can be conceived; it is
> > not reserved for infinite omniscience, but knowable by finite humanity;
> > it may contradict intuition, but it does not contradict itself. To
> > conceive it adequately we need not enumerate or visualize infinitely
> > many objects, but merely understand self-nesting. We have an actual,
> > positive idea of it, or at least with training we can have one; we are
> > not limited to the idea of finitude and its negation. In fact, it is at
> > least as plausible to think that we understand finitude as the negation
> > of infinitude as the other way around. The world of the infinite is not
> > barred to exploration by the equivalent of sea monsters and tempests; it
> > is barred by the equivalent of motion sickness. The world of the
> > infinite is already open for exploration, but to embark we must unlearn
> > our finitistic intuitions which instill fear and confusion by making
> > some consistent and demonstrable results about the infinite literally
> > counter-intuitive. Exploration itself will create an alternative set of
> > intuitions which make us more susceptible to the feeling which Kant
> > called the sublime. Longer acquaintance will confirm Spinoza's
> > conclusion that the secret of joy is to love something infinite."
> >
> > http://www.earlham.edu/~peters/writing/infinity.htm#galileo
> >
> > --

>
>
> Baruch Spinoza, a 17th century enlightenment thinker, sees the natural
> integers as a continuum, of individua (thanks Eco, for the word).
>
> http://en.wikipedia.org/wiki/Baruch_Spinoza
>
> That's quite a fine quote, Suber's. Yet, we might not need the
> infinite regress when reason gives us already the firmament, of the
> continuum, as the individua, what it is.
>
> And it's turtles: all the way down.
>

If that's a quote, put it in quote marks, since otherwise it marks you
as the wiseass little old lady who originated the phrase.
--





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