Monday, May 18, 2009

Topoi 2: strange logics/hypertopias

The panoply of possibilities (truths, many worlds, the field of argument) > a reality (Truth, the world, an argument).

= the collapse of the wave-function in observation—or in writing, the assumption that you must reduce your argument to a singular position.

Cf. Kochen & Specker (1967): every statement about a quantum system must
a. rely on a host of assumptions, and/or
b. refuse to obey the standard rules of logic (which are distributive and use the Boolean operators "and," "or," and "not").

Standard logic and mathematics fit the topos of set theory. (They are straight.)

Quantum logics do not. (They are fuzzy and strange.)

has identified the topos of quantum theory, the logic of which serves for the quantum and real worlds, i.e. it is distributive (C. J. Isham and J. Butterfield, "Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity," Found. Phys. 30.10 (2000): 1707-35).

But, rather than finding a topos that fits (inductively [yields a law]), we can also opt for a certain topos (abductively [yields a hypothesis or possibility]). (See
Robert Matthews, "Impossible Things for Breakfast, at the Logic Café," New Scientist [14 April 2007], online at <>.)

That would allow us to assume a partial truth to be true—or a hypertopia to exist as a genuine possibility.

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