The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit-and as much as possible by means of the algebraic conventions-of George Boole's Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn's Symbolic Logic (second edition, 1894), which is squarely based on Boole's Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical Equations are developed and presented and illustrated by means of examples.

136-154, the mathematical structure of the much discussed problem of probability known as the Monty Hall problem was mapped in detail. It is styled 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...

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The design of this paper is to consolidate a comprehensive model pertaining to the evolution of Egyptian calendars over three millennia of Pharaonic history, as an extension of this writer's earlier work on calendars. This model is a variation on the model advanced by Ludwig Borchardt and Richard Parker. While hardly immune from criticism, the Borchardt-Parker model has been prevalent in the second half of the twentieth century. According to this model, there were three calendars in ancient Egypt, two lunar and one non-lunar called civil. According to the variant model, there are only two calendars at any one time, the dominant civil calendar and a marginal lunar calendar of religious purport and of incomplete articulation. After the creation of the two calendars in prehistory and early history, only one truly significant event took place in all of Egyptian calendar history, around the fourteenth century B.C.E. Before the event, the lunar year began around the rising of Sirius in July. After the event, it began around the first new moon following civil New Year's Day. Owing to the backward wandering of the civil year, civil new year came to coincide with the rising of Sirius in the later fourteenth century B.C.E. The lunar calendar was unhooked from the rising, as it were, and attached and subordinated to the civil calendar. A double calendar, spiraling forward in time like a double helix, was the result. If the earlier and later beginnings of the lunar year are counted as two different calendars, there were three calendars, one civil and two lunar. However, it seems preferable to count just one lunar calendar, one that changed in regard to just one feature, its year's beginning.