A reader emailed me a discussion of and links to several people who have been working on the role of probability and decision making under uncertainty in the General Theory. I don't have a lot of thoughts on it now, although I definitely agree that decision making under uncertainty is crucial for the General Theory, so with his permission I'm just going to reproduce the message:

"Ever heard of the British mathematician Henry Wilbraham? What about the American logician Theodore Hailperin? Or Professor Emeritus David W. Miller of Columbia University's Graduate School of Business? What do these three have in common? They have all dealt with George Boole's Last Challenge Problem, found in "An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities", published in 1854.

Today's "Boolean algebra" is not the original algebraic logic used by George Boole. William Stanley Jevons and Charles Sanders Pierce, adapted Boole's algebraic logic and turned it into something different. That something different eventually evolved into coding for computer programming.

In turn, our mutual intellectual influence, John Maynard Keynes, dealt with George Boole (and other mathematicians) in "A Treatise on Probability", published in 1921. In turn, the urn problem described by John Maynard Keynes was (almost) independently verified by military risk analyst (and later anti-war campaigner) Daniel Ellsberg. I don't think Ellsberg covered Boole, however.

Nevertheless, if you want a full understanding of "A Treatise on Probability", and how the "General Theory" is arguably an application of Keynes's logical-relationist theory of probability, you will need to examine the connection between George Boole and John Maynard Keynes. Unfortunately, of those who understand the Challenge Problem, they belong to a very small minority of specialists in mathematics.

That's why I'm considering delving deeper into maths. Keynes, after all, was hailed by big names in mathematics: Bertrand Russell, Emile Borel, and Alfred North Whitehead. I believe they spoke highly of Keynes's aptitude. F.P.. Ramsey famously criticised Keynes's "TP", but did Keynes really capitulate to Ramsey's conception of Subjective Expected Utility?

I came to this position thanks to a fellow by the name of Michael Emmett Brady - yes, that guy on Amazon.com. Yes, he does seem somewhat eccentric, and highly repetitive. He seems to be stating obvious things. But I think he's onto something that has been sitting underneath the noses of a lot of us. Something that we have a hazy understanding of, but have yet to properly harness and grasp. This Michael Emmett Brady character is certaintly not an idiot, and he's well-read.

Lord Keynes may have taken Thomas Robert Malthus and Silvio Gesell into consideration, but Brady's been making a good argument that one of the GT's most important pillars is in probability - which in turn, is related to decision-making, and has not been sufficiently addressed. It's even more important than the mathematical equations found in Book V of the General Theory (ofn which I can supply a few links).

Keynes's approach to probability and decision-making is non-linear and non-additive - something that the "Platonists" (as Nassim Nicholas Taleb would call the neoclassical orthodoxy) don't grasp, and are still stuck in S.E.U. Most decisions, according to Brady, are actually more like an interval.

0 ≤ w ≤ 1

"w" stands for the weight of evidence. Decisions made don't have to add up to unity. They're often made somewhere in between: the lower-bound being closer to zero and not so likely, and the upper-bound being closer to one. You can ask him more for a better explanation, as I don't fully understand it quite yet.

But the papers on Boole, Keynes, and decision-making are written by Michael Emmett Brady and his Brazilian colleague, Rogerio Arthmar.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1546726

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1632318

M.E. Brady has also written on the mathematics found in Book V of the General Theory. A couple of them are found at Duke's HoPE database, and he's also published in HETSA's publication. Brady's published one of possibly the best modellings of the General Theory I've seen in my entire relationship with economics.

http://hope.dukejournals.org/cgi/pdf_extract/26/4/697

http://hope.dukejournals.org/cgi/pdf_extract/22/1/167

http://www.hetsa.org.au/pdf-back/25-A-13.pdf

Finally, you might want to check out these works by Claudia Heller, who appears to be supporting Brady's position regarding the mathematics of the GT. Note that the second one is in a foreign language, but it contains diagrammatic information.

http://mpra.ub.uni-muenchen.de/12837/1/MPRA_paper_12837.pdf

http://mpra.ub.uni-muenchen.de/5076/1/MPRA_paper_5076.pdf"

For the Weekend...

9 hours ago

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