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Topic: Matheology § 278
Replies: 4   Last Post: Jun 3, 2013 2:59 PM

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Scott Berg

Posts: 2,111
Registered: 12/12/04
Re: Matheology � 278
Posted: Jun 3, 2013 2:28 PM
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"WM" <> wrote in message

Matheology § 278

If, for example, our set theory includes sufficient large cardinals,
we might count Banach–Tarski as a good reason to model physical space
[...] From this I think it is clear that considerations from
applications are quite unlikely to prompt mathematicians to restrict
the range of abstract structures they admit. It is just possible that
as-yet-unimagined pressures from science will lead to profound
expansions of the ontology of mathematics, as with Newton and Euler,
but this seems considerably less likely than in the past, given that
contemporary set theory is explicitly designed to be as inclusive as
possible. More likely, pressures from applications will continue to
influence which parts of the set-theoretic universe we attend to, as
they did in the case of Dirac’s delta function; in contemporary
science, for example, the needs of quantum field theory and string
theory have both led to the study of new provinces of the set-
theoretic universe {{with negative result. There is no meaningful
application of a meaningless theory possible}}.
[Penelope Maddy: "How applied mathematics became pure", Reviev
Symbolic Logic 1 (2008) 16 - 41]

Regards, WM


continue with gernic poop ;

Penelope Maddy was interested in what we can know and cannot know about
infinite numbers. In math, there isn't just one "infinity," Maddy notes.
There are many infinite numbers of different sizes. To begin with, there's
the size of the set of natural numbers (1, 2, 3, 4…). However, the set of
real numbers (those corresponding to all the points on a line, including
between those numbers), which is also infinite, is bigger than the set of
natural numbers. All the different infinities can be lined up -- the
smallest, then the next biggest, and so forth -- and many of the familiar
operations, like multiplication or raising numbers to an exponent, can be
defined on these infinite numbers.
These different infinite numbers also present some perplexing problems: For
instance, what happens if you take the number 2 and raise it to the smallest
infinite number? "The answer will have to be infinite, but which infinite
number is it?" she asks. The smallest, the next smallest…? Something called
"The continuum hypothesis" (CH), proposed by Georg Cantor in the 1870s, says
that the answer is the second infinite number, but whether the CH is true or
false cannot be proved via the normal methods, Maddy says. You can't show
whether it's true "without adding some new fundamental axiom"—that is, a
basic assumption that can't be founded on anything more basic. "And nobody's
yet found a satisfactory way of doing that."

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