Nancy and Gregg, re inner dialogue and external talk (longer than I planned, I hope you can stick with it) -- so apparent in psychotherapy, therapist aware of his inner talk before his outer talk, observing other's facial expressions and body movements, and slight sounds --intuitions arising -- then others talk, like a light just went on-- all relevant to observables. I've been texting to Gregg about my mental process as I've worked on a math problem -- and resulting thoughts about mental process, inner steps I had taken and then my putting it on paper, and as I was doing that, more inner thoughts about the meaning of the steps. I awoke this am -- stimulated by Gregg's &  Nancy's blogs -- reviewing this from past times, and developing alternative ways to work (often using simple arithmetical steps). This am -- to illustrate :  to expand the number A with exponent n, example, A=9 n=3,  9^3=729, 9x9=81, 81x9=729. An alternative I was using, was to work with fractions, i.e., A<1, so .9x.9=.81, then translated to the simple arithmetic,  .9=1-.1, to arrive at how much is deducted from .9, .[.9x .1= .09] , [.9 - .09 = .81], then how much deducted from .81, [.81x.1=.081], .81-.081=.729. The new step came from looking at this .81 and .729, and the question came to mind, how can I go from .729 to .81? Can I reverse? I believe that this question arising in my mind illustrates a mental process  -- the brain is structured such that looking at the .729 and .81, it automatically sees a question there and we consciously experience the question (akin to Gestalt old hypothesis how the brain works, automatically creating wholes, completing closure to unclosed circles.) So,  .81-.081=.729, then .081=.729/.81, a step we learned in grammar school. So what?  I was working on the equation B^n = 1-(A^n). B^3= 1-.729, B^3=.271.  And the question of the problem is, does .271 have a root that is rational? If you take your computer and seek the cube root, the answer is a number that is non-ending -- there is no rational number that can be cubed to give .271. So here is where my simple is innovative; I use the ratio for A^n and apply it to B^n. X=.271/.81, a fraction, which is ratio, which also has to be a rational number. By reversing the steps a B that expands to .271 can be found, though not according to hoyle. It only introduces a strange anomaly. I am using it here to how the brain works to innovate. Once this step is taken it is obvious -  mathematicians call it trivial -- which is a way to ignore some possibility.
[ignore the rest if you have had enough of this long text, or return to later]
        Nancy, to give just little background: The problem I worked on is A^n + B^n = C^n. where n is larger than 2 and A,B,C are whole numbers. After puzzling on this for a long time, my mind made a leap to  "divide both sides by C^n", and work with decimals, A^n+B^n=1 (C^n/C^n=1), A rational and <1. Then the A^n and B^n are also rational numbers <1. This intuitive step which simplified the work enabled my brain to automatically see alternatives. The bottom line of this monologue
is to suggest how the brain may work to find abstractions -- perhaps how the pigeons brain found the abstract math principle (re the Scarf research I quoted yesterday.) Martin Johnson

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