Higher Variations of the Monty Hall Problem (3.0, 4.0) and Empirical Definition of the Phenomenon of Mathematics, in Boole’s Footsteps, as Something the Brain Does

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References

[1] L. Depuydt, “The Monty Hall Problem and Beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,” Advances in Pure Mathematics, Vol. 1, No. 4, 2011, pp. 136-154. doi:10.4236/apm.2011.14027

[2]
Cf. J. Rosenhouse, “The Monty Hall Problem,” Oxford University Press, Oxford and New York, 2009 (history of the problem and its context).

[3]
Cf. R. Deaves, “The Monty Hall Problem: Beyond Closed Doors,” 2006 (additional evidence of the interest in the problem). www.lulu.com

[4]
Cf. M. vos Savant, “Q(uestion) & A(nswer) (involving the Monty Hall Problem),” Parade, 9 September 1990 (article serving as principal catalyst of the interest in the problem).

[5]
H. H. Goldstine, “The Computer from Pascal to von Neumann,” Princeton University Press, Princeton, 1972, p. 37.

[6]
L. Depuydt, “The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,” Advances in Pure Mathematics, Vol. 1, No. 4, 2011, pp. 135-154, at p. 148.

[7]
A. W. F. Edwards, “Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea,” Johns Hopkins University Press, Baltimore, 2002, p. xiii.

[8]
L. Depuydt, “The Monty Hall Problem and Beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,” Advances in Pure Mathematics, Vol. 1, No. 4, 2011, pp. 135-154, at p. 145.

[9]
L. Euler, “Elements of Algebra,” Springer Verlag, New York, Berlin, Heidelberg, and Tokyo, 1984, pp. 110-120.

[10]
S. F. Lacroix, “Traité élémentaire du calcul des probabilités,” Fourth Edition, Mallet-Bachelier, Paris, 1864, p. 30.

[11]
L. Depuydt, “The Monty Hall Problem and Beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,” Advances in Pure Mathematics, Vol. 1, No. 4, 2011, p. 150.

[12]
D. W. Miller, “The Last Challenge Problem: George Boole’s Theory of Probability.”
http://zeteticgleanings.com/boole.html

[13]
G. Boole, “The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning,” Macmillan, Barclay, & Macmillan, Cambridge and George Bell, London, 1847.

[14]
G. Boole, “Studies in Logic and Probability,” Dover Publications, Mineola, New York, 2004, pp. 45-124 (reprint of [13] whose pagination is used in what follows).

[15]
G. Boole, “Studies in Logic and Probability,” Watts & Co., London, 1952 (original edition reprinted in [14]).

[16]
It should be noted that Boole’s “0” is electrical engineering’s “1” and vice versa, Boole’s “0” (AND) is electrical engineering’s “H” (AND) and vice versa, and Boole’s “+” (OR) is electrical engineering’s “H” facts that I have failed to appreciate in the introduction to my “The Other Mathematics: Language and Logic in Egyptian and in General,” Gorgias Press, Piscataway, New Jersey, 2008, even if this oversight does not affect the arguments presented in this work. It is difficult to find any published obs

[17]
G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854.

[18]
I have used the reprint of 1958 by Dover Publications, New York.

[19]
J. Venn, “Symbolic Logic,” Second Edition, Macmillan and Co., London and New York, 1894.

[20]
A. N. Whitehead, “A Treatise on Universal Algebra with Applications,” Cambridge, 1897, p. 11.

[21]
N. I. Styazhkin, “History of Mathematical Logic from Leibniz to Peano,” Cambridge, Mass., 1969, p. 214.

[22]
G. Boole, “Studies in Logic and Probability,” Dover Publications, Mineola, New York, p. 53.

[23]
Th. Hailperin, “Boole’s Logic and Probability,” Second Edition, North-Holland Publishing Company, Amsterdam, New York, Oxford, and Tokyo, 1986.

[24]
B. Russell, “Recent Work on the Principles of Mathematics,” International Monthly, Vol. 4, 1901, pp. 83-101, at p. 366 of the reprint in [25]. I owe the reference to [26].

[25]
G. H. Moore (ed.), “The Collected Papers of Bertrand Russell, Vol. 3,” Routledge, London, 1993, pp. 366-379.

[26]
G. Bornet, “Frege’s psychologism criticism (of Boole),” In: I. Grattan-Guinness and G. Bornet, Eds., George Boole: Selected Manuscripts on Logic and Its Philosophy, Birkh?user Verlag, Basel, Boston, and Berlin, 1997, pp. xlviii-l.

[27]
G. Boole, “The Mathematical Analysis of Logic,” Dover Publications, Mineola, New York, p. 47.

[28]
G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854, p. 1.

[29]
G. Bornet, “George Boole: Selected Manuscripts on Logic and Its Philosophy,” In: I. Grattan-Guinness and G. Bornet, Eds., Science Networks Historical Studies, Birkh?user Verlag, Basel, Boston, and Berlin, 1997, p. lxiv.

[30]
G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854, p. 11.

[31]
G. Boole, “Studies in Logic and Probability,” Dover Publications, Mineola, New York, p. 52.

[32]
G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854, p. 11.