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




Re: Matheology � 278
Posted:
Jun 3, 2013 2:28 PM


"WM" <mueckenh@rz.fhaugsburg.de> wrote in message news:1336c8c699be41f0a012a84c52c1c2ad@m18g2000vbo.googlegroups.com...
Matheology § 278
If, for example, our set theory includes sufficient large cardinals, we might count BanachTarski 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 asyetunimagined 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 settheoretic universe we attend to, as they did in the case of Diracs 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
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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."



