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Re: Matheology § 176
Posted:
Dec 10, 2012 11:52 PM


On Dec 10, 1:11 pm, Virgil <vir...@ligriv.com> wrote: > In article > <c293798d5b7e4ebaa4df6332461dc...@f19g2000vbv.googlegroups.com>, > > > > > > > > > > WM <mueck...@rz.fhaugsburg.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 onetoone 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 onetoone 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) 159] > >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 selfnesting. 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 > counterintuitive. 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



