Wednesday, September 26, 2018

Different forms of set theory

Continuing this post, yet Lakoff said this about set theory, which is built at least in part on the container schema:

"The same is true of set theory. There are lots and lots of set theories, each defined by different axioms. You can construct a set theory in which the Continuum hypothesis is true and a set theory in which it is false. You can construct a set theory in which sets cannot be members of themselves and a set theory in which sets can be members of themselves. It is just a matter of which axioms you choose, and each collection of axioms defines a different subject matter. Yet each such subject matter is itself a viable and self-consistent form of mathematics. [...] There is no one true set theory." (WMCF, 355).

 He also explains why the above is not postmodern relativism: 

"In recognizing all the ways that mathematics makes use of cognitive universals and universal aspects of experience, the theory of embodied mathematics explicitly rejects any possible claim that mathematics is arbitrarily shaped by history and culture alone. Indeed, the embodiment of mathematics accounts for real properties of mathematics that a radical cultural relativism would deny or ignore: conceptual stability, stability of inference, precision, consistency, generalizability, discoverability, calculability, and real utility in describing the world" (362).

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