Return of the Excluded

Middle that is.

Why should there be only two choices? asks philosopher Barry C Smith in an article/interview at the Institute of Art and Ideas, and I agree that the binary nature of logic may be as much a reflection of how our minds work than an absolute aspect of reality. But his examples are odd.

There are certainly mathematical propositions that are neither provably true nor provably false (from standard axioms by first order logic), but just being unproved (like his Goldbach Conjecture example) does not guarantee being one of those. Also on the other side he identifies acid vs alkaline, and being Carbon or not, as true binary situations in the real world. But pH varies continuously from one extreme to the other and some large molecules may behave as acid in some configurations or contexts and alkaline in others, and it seems quite conceivable (in principle) that in a suitable (but possibly impracticable) intersection of electron and neutrino beams a C14 nucleus might undergo a series of stimulated beta emissions and absorptions with oscillations fast enough that due to quantum uncertainty one could not say whether at any one instant it was either Carbon or Nitrogen. Indeed it is likely that the identity of any particle, no matter how stable cannot really be said to be absolutely this or that if we take account of all possible real and virtual quantum phenomena.

The limits of reason are also explored at AIA in a video (which might be interesting but tl;dw), and Michael Potter discusses the origins and limits of modern logic, including both the "linguistic turn" which seems to be about the attempt to define a perfectly rigorous formal language and the philosophy of ordinary language that is sometimes called "linguistic analysis" (which strikes me as pretty much the opposite so it's no wonder simpletons outside the field get confused - perhaps that's why they do it!).

But when it comes to the "origins" bit he describes Frege's "polyadic quantification logic" as "enormously more powerful than anything ...since Aristotle" - which I would say undervalues Boole (and others of his ilk).

Interestingly Boole's wife helped in his work and considered it to have been influenced (via her uncle George Everest) by ideas about logic from Hindu philosophy. She also wrote this and was an early proponent of both cooperative learning and what illiterate educators now call "manipulatives" (even though what they really are is manipulable).

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