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  • feedwordpress 09:01:22 on 2019/01/06 Permalink
    Tags: , Georg Cantor, , history of mathematics, , , Peter Carruthers, , set theory,   

    “Control of consciousness determines the quality of life”*… 


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    2D02EB62-9945-4BB8-A5E9507B396FEF67_source

    Peter Carruthers, Distinguished University Professor of Philosophy at the University of Maryland, College Park, is an expert on the philosophy of mind who draws heavily on empirical psychology and cognitive neuroscience. He outlined many of his ideas on conscious thinking in his 2015 book The Centered Mind: What the Science of Working Memory Shows Us about the Nature of Human Thought. More recently, in 2017, he published a paper with the astonishing title of “The Illusion of Conscious Thought.”…

    Philosopher Peter Carruthers insists that conscious thought, judgment and volition are illusions. They arise from processes of which we are forever unaware.  He explains to Steve Ayan the reasons for his provocative proposal: “There Is No Such Thing as Conscious Thought.”

    See also: “An Anthropologist Investigates How We Think About How We Think.”

    * Mihaly Csikszentmihalyi, Flow: The Psychology of Optimal Experience

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    As we think about thought, we might spare one for Georg Ferdinand Ludwig Philipp Cantor; he died on this date in 1918.  Cantor was the mathematician who created set theory, now fundamental to math,  His proof that the real numbers are more numerous than the natural numbers implies the existence of an “infinity of infinities”… a result that generated a great deal of resistance, both mathematical (from the likes of Henri Poincaré) and philosophical (most notably from Wittgenstein).  Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor, a devout Lutheran, vigorously rejected.

    These harsh criticisms fueled Cantor’s bouts of depression (retrospectively judged by some to have been bipolar disorder); he died in a mental institution.

    220px-Georg_Cantor2 source

     

     
  • feedwordpress 08:01:51 on 2018/08/29 Permalink
    Tags: Hermann Hankel, , , history of mathematics, , , , supertasks, , Zeno   

    “I am incapable of conceiving infinity, and yet I do not accept finity”*… 


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    Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?

    Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.

    The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…

    More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)

    * Simone de Beauvoir, La Vieillesse

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    As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873.  A mathematician who worked with Möbius, Riemann, Weierstrass,  and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.

    220px-Hankel source

     

     
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