Sunday, February 13, 2011

Gems from Ruzsa and McGilchrist

Making some progress with Additive Combinatorics. It's fascinating stuff, though I'm not getting in really deep and attempting the exercises. However I came across this gem. They can prove that if A and B are additive sets:
|2B - 2B| <= 16 (|A + B|4 |A - A|)/|A|4
(in Additive Sets A+B is the set of all possible sums of an element of A and an element of B etc.. and 2B is B + B, so 2B - 2B will only be 0 if B has only one element)

However they can get rid of the factor of 16, by taking M to be a large integer, considering the M-fold cartesian product of A and B in the group ZM, showing that:
|2B - 2B|M <= 16 (|A + B|4M |A - A|)/|A|4M
and then taking Mth roots and letting M tend to infinity! Very elegant idea due to Ruzsa.

Also getting back to The Master and his Emissary. McGilchrist refers to an unpublished study he made of degree subjects undertaken by university students who then went on to be admitted to the Bethlem Royal and Maudsley Hospital druring a psychotic episode. He found that subsequent diagnosis of schizophrenia was most closely associated with having studied Engineering, followed by Philosophy.

Also listen to this:
Neuropsychology is inextricable bound up with philosophy...this has been increasingly recognised, mre by philosophers than neuroscientists...[but]...what science is actualll doing when it delivers its revelations goes unexamined: the scientific process and the meaning of its findings is generally taken for granted. The model of the body, and therefore the brain, as a mechanism is exempted from the process of philosophical scepticism...As a result, in a spectacular hijack, instead of a mutually shaping process...the naive world view of science has tended by default to shape and direct what has been called 'neurophilosophy' (p135
and this:
The word 'true' suggests a realtionship between things: being true to someone or something, truth as loyalty, or something that fits....It is related to trust and is fundamentally a matter of what one believes to be the case. The Latin word verum (true) is congnate with a Sanskrit word [{ वॄ } {vRR} c/f वेद veda] meaning to choose or believe: the option one chooses, the situation in which one places one's trust. Such a situation is not absolute - it tells us not only about the chosen thing but about the chooser. It cannot be certain: it involves an act of faith, and it involves being faithful to one's intuitions. (p151)

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