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

Affiliation(s)

Department of Egyptology and Ancient Western Asian Studies, Brown University, Providence, USA.

Mathematisch Instituut, Universiteit Leiden, Leiden, The Netherlands.

Department of Egyptology and Ancient Western Asian Studies, Brown University, Providence, USA.

Mathematisch Instituut, Universiteit Leiden, Leiden, The Netherlands.

ABSTRACT

In*Advances in Pure Mathematics* (www.scirp.org/journal/apm), Vol. 1, No. 4 (July 2011), pp. 136-154, the mathematical structure of the much discussed problem of probability known as the Monty Hall problem was mapped in detail. It is d here as Monty Hall 1.0. The proposed analysis was then generalized to related cases involving any number of doors (*d*), cars (*c*), and opened doors (*o*) (Monty Hall 2.0) and 1 specific case involving more than 1 picked door (p) (Monty Hall 3.0). In cognitive terms, this analysis was interpreted in function of the presumed digital nature of rational thought and language. In the present paper, Monty Hall 1.0 and 2.0 are briefly reviewed (§§2-3). Additional generalizations of the problem are then presented in §§4-7. They concern expansions of the problem to the following items: (1) to any number of picked doors, with *p* denoting the number of doors initially picked and *q * the number of doors picked when switching doors after doors have been opened to reveal goats (Monty Hall 3.0; see §4); (3) to the precise conditions under which one’s chances increase or decrease in instances of Monty Hall 3.0 (Monty Hall 3.2; see §6); and (4) to any number of switches of doors (s) (Monty Hall 4.0; see §7). The afore-mentioned article in APM, Vol. 1, No. 4 may serve as a useful introduction to the analysis of the higher variations of the Monty Hall problem offered in the present article. The body of the article is by Leo Depuydt. An appendix by Richard D. Gill (see §8) provides additional context by building a bridge to modern probability theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of computation now available on the Internet. The cognitive component of the earlier investigation is extended in §9 by reflections on the foundations of mathematics. It will be proposed, in the footsteps of George Boole, that the phenomenon of mathematics needs to be defined in empirical terms as something that happens to the brain or something that the brain does. It is generally assumed that mathematics is a property of nature or reality or whatever one may call it. There is not the slightest intention in this paper to falsify this assumption because it cannot be falsified, just as it cannot be empirically or positively proven. But there is no way that this assumption can be a factual observation. It can be no more than an altogether reasonable, yet fully secondary, inference derived mainly from the fact that mathematics appears to work, even if some may deem the fact of this match to constitute proof. On the deepest empirical level, mathematics can only be directly observed and therefore directly analyzed as an activity of the brain. The study of mathematics therefore becomes an essential part of the study of cognition and human intelligence. The reflections on mathematics as a phenomenon offered in the present article will serve as a prelude to planned articles on how to redefine the foundations of probability as one type of mathematics in cognitive fashion and on how exactly Boole’s theory of probability subsumes, supersedes, and completes classical probability theory. §§2-7 combined, on the one hand, and §9, on the other hand, are both self-sufficient units and can be read independently from one another. The ultimate design of the larger project of which this paper is part remains the increase of digitalization of the analysis of rational thought and language, that is, of (rational, not emotional) human intelligence. To reach out to other disciplines, an effort is made to describe the mathematics more explicitly than is usual.

In

Cite this paper

L. Depuydt and R. Gill, "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,"*Advances in Pure Mathematics*, Vol. 2 No. 4, 2012, pp. 243-273. doi: 10.4236/apm.2012.24034.

L. Depuydt and R. Gill, "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,"

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.

[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.