Date: Dec 10, 2012 11:52 PM Author: ross.finlayson@gmail.com Subject: Re: Matheology § 176 On Dec 10, 1:11 pm, Virgil <vir...@ligriv.com> wrote:

> In article

> <c293798d-5b7e-4eba-a4df-6332461dc...@f19g2000vbv.googlegroups.com>,

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> 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.

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> > 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.

Regards,

Ross Finlayson