*Jaime Alfonso Navas, Krzysztof Zawisza*

**Marilyn and the Monty Hall Paradox**

At the beginning of the 1990’s – American journalist Marilyn vos Savant proved the superiority of innate intelligence over learned knowledge. Since then some scholars who only learned their profession at universities hate her, are afraid of her and try to discredit her in various ways. The more so that she later discredited the allegedly greatest achievement of XX-century mathematics – the supposed Andrew Wiles’ proof of Fermat’s Last Theorem.

Well, some people – along with their natural intelligence – get a title to practice science from Nature or God; others – not having this gift – must receive it, on the official and contractual path, not in that natural way but from other people.

Marilyn vos Savant does not have formal higher education. She studied philosophy for two years. However, she is very interested in logical – i.e. correct – thinking. In the second half of the 1980s she was included in the Guinness Book of Records as the most intelligent person on Earth. She’s known for some witty quotes such as:

“Be able to defend your arguments in a rational way. Otherwise, all you have is an opinion”^{[i]},

“to acquire knowledge, one must study; but to acquire wisdom, one must observe”^{[ii]}.

Her intelligence was reflected in the early 90’s when she solved the well-known Monty Hall paradox. Mathematicians usually “tactically” forget about this history today. However, it has been brilliantly described by famous physicist and populariser of physics Leonard Mlodinov, in *The Drunkard’s Walk*:

“As newspaper columns go, Parade magazine’s “Ask Marilyn” has to be considered a smashing success. Distributed in 350 newspapers and boasting a combined circulation of nearly 36 million, the question-and-answer column originated in 1986 and is still going strong. One might ask, especially if one knows a little something about mathematics and science, ‘Who is this guru Marilyn?’. Well, Marilyn is Marilyn vos Savant, famous for being listed for years in the Guinness World Records Hall of Fame as the person with the world’s highest recorded IQ (228). She is also famous for being married to Robert Jarvik, inventor of the Jarvik artificial heart. Marilyn, is most famous for her response to the following question, which appeared in her column one Sunday in September 1990 (I have altered the wording slightly): Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, ‘Do you want to switch to the other unopened door?’ Is it to the contestant’s advantage to make the switch?’ This issue has immortalized both Marilyn and *Let’s Make a Deal* because of the vehemence with which Marilyn vos Savant’s readers responded to the column. After all, it appears to be a pretty silly question. Two doors are available—open one and you win; open the other and you lose—so it seems self-evident that whether you change your choice or not, your chances of winning are 50/50. What could be simpler? The thing is, Marilyn said in her column that it is better to switch. Despite the public’s much-heralded lethargy when it comes to mathematical issues, Marilyn’s readers reacted as if she’d advocated ceding California back to Mexico. Her denial of the obvious brought her an avalanche of mail, 10,000 letters by her estimate. If you ask the American people whether they agree that plants create the oxygen in the air, light travels faster than sound, or you cannot make radioactive milk safe by boiling it, you will get double-digit disagreement in each case (13 percent, 24 percent, and 35 percent, respectively). But on this issue, Americans were united: 92 percent agreed Marilyn was wrong. Many readers seemed to feel let down. How could a person they trusted on such a broad range of issues be confused by such a simple question? Was her mistake a symbol of the woeful ignorance of the American people?

**Almost 1,000 PhDs wrote in, many of them math professors, who seemed to be especially irate.**

‘You blew it’, wrote a mathematician from George Mason University: ‘Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice — neither of which has any reason to be more likely — to 1⁄2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful’. From Dickinson State University came this: ‘I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake’. From Georgetown: ‘How many irate mathematicians are needed to change your mind?’ And someone from the U.S. Army Research Institute remarked, ‘If all those PhDs are wrong the country would be in serious trouble’.

[…] **The army PhD who wrote in may have been correct that if all those PhDs were wrong, it would be a sign of trouble. But Marilyn was correct.**

When told of this, Paul Erdös, one of the leading mathematicians of the twentieth century, said, ‘That’s impossible’. Then, when presented with a formal mathematical proof of the correct answer, he still didn’t believe it and grew angry. Only after a colleague arranged for a computer simulation in which Erdös watched hundreds of trials that came out 2 to 1 in favor of switching did Erdös concede he was wrong. How can something that seems so obvious be wrong? In the words of a Harvard professor who specializes in probability and statistics, ‘Our brains are just not wired to do probability problems very well’. But the Monty Hall problem is one of those that can be solved without any specialized mathematical knowledge. You don’t need calculus, geometry, algebra, or even amphetamines, which Erdös was reportedly fond of taking”^{[iii]}.

According to Mlodinow, the matter in favor to Marilyn vos Savant was finally resolved by computer simulations carried out at MIT, and above all by numerous simulation games that teachers began to conduct at American schools at Marilyn’s request. These games gave results provided by a journalist. However, these were empirical arguments only. Arguments based on logical reasoning, as one can see, almost to the end could have not be understood or properly evaluated by mathematicians^{[iv]}.

There were, however, exceptions. For instance, James Simmons, a mathematician at the University of Virginia wrote:

“If I am the contestant and my strategy is always to stick with my initial choice, then the probability that I *win* is 1/3. If my strategy is always to switch, then the probability that I *lose* is 1/3, since I lose only if the car is behind the door I first chose […]”[v].

However, as Marilyn herself stated:

“Overall nine out of ten readers completely disagree with my reply”[vi].

For my part, I’d like to note that it can be very easily shown that the simplest correct solution of the Monty Hall problem results directly from the logical principle of *Identity of Indiscernibles*, which was applied by Ms vos Savant, albeit unconsciously (since she has no formal education in the field of logic)[vii]. The point is that in the problem under consideration there are no essential features that distinguish the remaining two doors, of which the host opens one. Since these doors do not differ significantly in any given situation, it does not matter which of them was opened by the host. In this case, the opening of the door by the host does not provide any relevant new information and must therefore not alter the original probability of winning assigned to the door initially selected by the contestant.

The fact that no one of the mathematicians was able to solve the problem by referring to the logical *principium* *identitatis indiscernibilium* proves lack of knowledge of the rules of logic and lack of the ability to apply these rules in practice among modern mathematicians.

The whole discussion around it can be traced on the Marilyn’s web page^{[viii]}. The reading of this web page is instructive, and somewhere – shocking. As recently one of the readers of the Parade magazine – Ellen B. from Spokane in Washington – wrote:

**“**My college math professor assigned the game show problem to us as an exercise. Her instruction included a review of the problem and letters published in *Parade* magazine. I was surprised at how many negative (and judgmental, sexist, or offensive) responses there were even–and maybe especially- among highly educated readers. Your comments in return, however, were polite, thorough, and unruffled. My question to you is how did you accomplish such tact under fire? It seems the comments flooding in at the time were sweeping, and in the face of so many qualified readers telling you that you were wrong, you responded with surety and grace. What strategies do you use for holding onto such absolute certainty in the face of strong opposition? Thank you for setting such a strong example of how we can treat one another even when we disagree**”**^{ }^{[ix]}**.**

Marilyn responded:

“I was confident that I was correct. It’s easy to be cool when you know you’re right. It’s the people who aren’t really sure of themselves who get all hot and bothered when challenged”^{[x]}.

It is worth to mention that even one of the greatest professional mathematicians of the 20th century – Paul Erdös^{[xi]} – could not assess the theoretical proof presented to him. He was finally convinced by the results of computer simulation^{[xii]}. As Tibor Musing writes:

“It was Andrew Vazsonyi who tried to convince Paul Erdös about the game show problem solution. His personal account of the events is available in short format in Zentralblatt^{[xiii]} and in longer format in a Decision Line paper^{[xiv]} and in an even longer format in Vazsonyi’s book ‘Which Door has the Cadillac: Adventures of a Real-Life Mathematician’. The book contains more side details and other stylistic changes, but the heart of the story and the quotes about Erdös’s reactions are the same as in the above Decision Line paper. Vazsonyi’s account seems to suggest that Erdös was genuinely puzzled about the problem”^{[xv]}.

The discussion on Monty Hall paradox was continued by Marilyn in the next book, refuting in her subsequent – sometimes very sophisticated – allegations^{[xvi]}.

The moral of this story is that intelligence and logical thinking are inevitably needed to solve new math problems, while the average – and even the most outstanding – mathematicians do not have enough of the first one, and – because of that – they are unable to use the latter^{[xvii]}. As one of the Harvard probabilistic specialists scholars wrote: “Our brains are just not wired to do probability problems very well”^{[xviii]}. However, as one can see, maybe the brains of Harvard probabilists are not adapted to solve probabilistic problems, but Ms. vos Savant’s brain is doing quite well^{[xix]}. That’s nothing unusual when average intelligence of today’s American professor (tenure) is 15SDIQ **135**. It might be a bit more in the area of mathematics at Harvard, let say ca. 140. The measured intelligence quotient of the probably most honored mathematician today – sir Andrew Wiles – is ca. 170. But the known Marilyn vos Savant’s Intelligence Quotient is situated at 186 SD15. This means that the American journalist is already located beyond the 15-point *window of comprehension* of the greatest contemporary (professional) mathematician.

If mathematicians would think, the Monty Hall Paradox story could give them something to think about. But because they do not think, they have not learned anything from the whole thing^{[xx]}.

**Marilyn, Fermat’s Last Theorem and its Lost Proof**

Around two years after the Monty Hall paradox case, Marilyn vos Savant wrote a critical book about the proof of *Fermat’s Last Theorem* presented by Adrew Wiles^{[xxi]}. Predictably, we could observe again a clash of high intelligence and logical thinking with the mediocrity behind the commonly accepted mental schemes, which are replacing logical thinking in nowadays math.

The main thought of this book was – again to dismay of professional mathematicians – as follows:

“[…] squaring the circle is “known to be impossible, and so any purported solutions can be rejected out of hand.” So has Fermat’s last theorem been proved, or has it not? That is, if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat’s last theorem!”^{[xxii]}.

Then the American high IQ journalist develops her thought this way:

“This is not a matter of merely changing the rules (for example, using a ruler instead of a straight edge.) It’s much more significant than that. Instead, it’s a matter of changing whole definitions, including the definition of what constitutes a contradiction. And regardless, it is logically inconsistent to reject a hyperbolic method of squaring the circle and accept a hyperbolic method of proving F.L.T!”^{[xxiii]}.

She then ends her argumentation ironically:

“The next thing you know, someone will use non-Euclidean geometry to prove Euclid’s parallel postulate! (And then what a fix Einstein will be in.)”^{[xxiv]}.

In turn at the end of her book, Ms. vos Savant turns “To All the Unknown Genius in the World” and she appeals:

“[…] you might want to consider performing the following mathematical feats:

[1 ] Wiles proved F.L.T with concepts from hyperbolic (non-Euclidean) geometry in the way that Bolyai squared the circle in hyperbolic geometry, a construction that was proven impossible in abstract algebra. Use Euclidean geometry or abstract algebra to prove that F. L.T is impossible to prove.

[ 2 ] Better yet, use Euclidean geometry to prove that EL.T is true, which will be more of an achievement than a non-Euclidean proof. Or use Euclidean geometry to prove that F. L.T is true with a direct proof. (This is guaranteed to thrill every formalist on the planet and can hardly be surpassed.)

[ 3 ] And for the most ambitious, use hyperbolic geometry to prove that F. L.T is impossible to prove. A contradiction of this magnitude (and fame) could cause the entire field to collapse. (But don’t expect any of the experts in hyperbolic geometry to want to verify your proof.)

[ 4 ] To demolish Einstein’s theories of relativity (in elliptic geometry), you could go back and prove the parallel postulate, bringing down not only Einstein, but all of the non-Euclidean geometries, as well. (Or you could go after Einstein selectively by the route of proving a contradiction in elliptic geometry.)”^{[xxv]}.

As a result, the American high IQ journalist – *although* she is not a modern mathematician (or precisely *because* she is not) – using irony usually belonging to the self-conscious minds[xxvi] – started to see and create mathematics as a self-conscious, open science; as a dialogical and dialectical, reasonable cognition.[xxvii]

What was the answer from *professional* mathematics? The reviewers of the Marilyn vos Savant’s book (i.e. professional mathematicians) wrote:

“You probably know of Marilyn vos Savant. She became famous because the Guinness Book of World Records presented her as the person with the highest IQ ever recorded, as a child and as an adult. (As we shall discuss, her intelligence shines through in many places in the book, making it that much sadder that she could have written such drivel.) In her “Ask Marilyn” ‘problem-analysis column’ each Sunday in Parade magazine, she popularises mathematical puzzles and conundrums, reaching out to a potential audience of seventy million people. And now she has challenged the orthodoxy of the mathematics world by refuting Wiles’s purported proof of Fermat’s Last Theorem, by claiming that it is wrong because it is illogical, relying, she believes, on ridiculous inconsistencies accepted by mathematicians. For example, the concept of a non-Euclidean geometry. For another example, proofs using induction. Would you expect that such a controversial opinion, from the person with the world’s highest IQ, is based on a careful analysis of the ideas involved in these concepts, by studying the appropriate literature at hand, and by justifying her substantial findings with an irrefutable argument? Well don’t expect such an argument in this book, because you’ll be disappointed. In fact, she even boasts that she wrote the book in just three weeks! Not that she began as an expert; no, she just dived in, read a few popular mathematics books, and then proceeded to her startling conclusions […].

This book was written, start-to-finish, in three weeks. Judging from the lengthy bibliography, vos Savant did a lot of valuable reading in that time and culled some interesting ideas. It seems, however, that there were a number of things which she came across that did not make much sense to her. Rather than allow an expert to explain these things to her, she just decided that they were wrong. After all, she seems to reason, if these things are not self-evident then how could they possibly be correct? From Mazur’s article she seems to have gleaned the fact that the Taniyama – Shimura conjecture may be re-phrased in part in terms of hyperbolic geometry. Thus she concluded that Wiles gave a ‘hyperbolic method of proving FLT’ In fact, her central theme is that non-Euclidean geometry, and indeed any mathematics related to non-Euclidean geometry, is nonsense. Her thesis seems to be that, since it was proved in 1882 that ‘squaring the circle’ is impossible in a Euclidean setting, and since Bolyai managed to ‘square the circle’ in an appropriate non-Euclidean geometry, thus non-Euclidean geometry is inconsistent with Euclidean geometry. However since Fermat’s Last Theorem is a statement consistent with regular geometry, it cannot be proved by arguments that involve any non-Euclidean geometry. After all, ‘one of the founders of hyperbolic geometry [J . Bolyai ] managed to square the circle?! Then why is it known as such a famous impossibility?’ Therefore, she concludes, ‘if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat’s Last Theorem’. This is typical of the inane reasoning (and hyperbole) that pervades this book. For those readers unfamiliar with the peaceful co-existence possible between non Euclidean and Euclidean geometry, let us now give a simple (not unrelated) example of such coexistence between concepts that seem inconsistent to the very naive: In decimal arithmetic, 1 + 1 = 2 . In binary arithmetic, 1 + 1 = 10 . This, by vos Savant’s way of thinking, is not logical. Binary presents a different formal system of logic from the normal one, so we cannot accept any mathematics done on computers (since they work in binary). So let’s unplug all computers! Likewise, clock arithmetic tells us 7 + 6 = 1 , so clocks are out too. Different systems of logic can peacefully co-exist. One might visualise two giant computers hooked up, one called ‘Euclidean geometry’ , the other “non-Euclidean geometry”. There are some things one can do accurately on one computer that would produce error messages galore with the other. However they can still work precisely together, to solve problems, by translating information between them”^{[xxviii]}.

As you can see, the authors of the review (“professional”, i.e. “official” mathematicians) *ex officio* disagree with the thesis Ms. vos Savant has presented. However, they do not try to enter into a rational discussion [officials do not discuss things; officials make decisions based on their official, formal criteria]. However, to give some sort of justification for their decision, they begin (like in previous discussion) with a logically and rhetorically illegal *ad personam* argument (“As we shall discuss, her intelligence shines through in many places in the book, making it that much sadder that she could have written such drivel”). This “argument” used already in the beginning was obviously to “set” the narrative so that it would be harder to notice the subsequent lack of logical arguments on the side of the reviewers. Meanwhile, her argument is logically simple: Wiles used an inadequate method, using non-Euclidean geometry structures for proving theorems from Euclidean geometry. It was therefore the erroneous method called *ignoratio elenchi* (“ignorance of the thesis proved”). You can’t prove something with the help of methods designed to prove a logically different case. The reviewers didn’t show their thesis that Ms vos Savant is allegedly wrong and that this generally inadequate method could be particularly used in this case.

The reviewers (“professional”, i.e. “official” mathematicians) have shown neither the adequacy of the inadequate method nor anything close to it [officials, i.e. bureaucrats do not discuss, they just tend to make a decision based on *ipse dixit* argumentation]. Instead, to give *pro forma* reasons for their decision, they used logically flawed arguments like “false analogy”. They claim that since there are different computer languages that can be translated into each other (and different numerical notations of the same number) it allegedly follows that also theorems from the area of Euclidean geometry can be proved using non-Euclidean geometry methods. However, the fact that there are different but mutually translatable computer languages, or the fact that there are different notations of the same number in different number systems does not imply (*non sequitur*) that one can prove a given theorem in the area of Euclidean geometry using methods which are correct in the area of non-Euclidean geometry only. Different geometries are not a record of the same mathematical reality in different languages. They are records of different realities in the same (mathematical) language. While a given number can be written simultaneously in different numerical notations, different geometries cannot be realized at the same time to the same subject. The above observation proves that different number systems are numerically **heterogeneous**, while different geometries are geometrically **contradictory** to each other. While different numerical entries may have the same content (they can be a notation of the same numerical value), different geometries (by definition) cannot describe the same geometrical reality. For example, if the universe has negative curvature, then it should be described by hyperbolic geometry, but cannot be correctly described by Euclidean geometry

The fact that the use of an incorrect method can sometimes lead to correct results does not change the fact that the incorrect method has no probative value, and if something is formally correct only, it merely means that it looks like to be correct, but it does not have to be correct in its essence. Thus, until a logically correct proof of Fermat’s Last Theorem is found, this theorem remains unproven. It remains unproven despite the conviction of today’s mathematicians and the convention they have adopted. For even the strongest conviction and the commonly recognized convention also don’t have in thyself any probative value.

In other words, Marilyn vos Savant pointed out that while mathematics may be considered science, it has to be reasonable knowledge, that is, it has to follow the principles of logic in its proof procedures. But then, in addition to proving the validity of a theorem by means of consensually accepted proof, the mathematics also needs a logically valid proof of the logical validity of this consent proof. For assuming the correctness of a mathematical proof only on the basis of the consensus of prominent mathematicians is a typical textbook fallacy of *consensus gentium*. In the case of the Monty Hall paradox, there was a consensus among mathematicians that Marilyn had provided an incorrect solution. As it turned out, however, Marilyn’s solution was correct, and the consensus of even the most successful scholars does not decide what is and what is not true.

Wiles proof is very sophisticated, while Marilyn’s argument is quite simple. But simplicity is the ultimate sophistication[xxix].

I think that if professional mathematicians would be able to think, they would manage to think it over and to jump to the same conclusion as Marilyn did. But bureaucrats don’t think [otherwise they would not be bureaucrats: otherwise they would be just persons and they’d take the truth personally, i.e. responsibly].

In summary, Marilyn vos Savant accused Wiles that his method was inadequate to the subject [the *ignoratio elenchi* fallacy]. In response, the mathematical reviewers of her book used arguments inadequate to the subject they criticize; i.e. they also made the same fallacy of *ignoratio elenchi*.

Just hit the table and the shears will call you.

**Marilyn and the Further Discussions**

Later, the discussion of the Marilyn’s book moved to the Internet, inspiring to ask questions on the *Quora* educational portal. Mathematicians just clung out to their opinions there. For lack of thoughtful arguments, they tried to deprecate the high IQ journalist’s way of thinking by means of a thoughtless semantic gibberish.

An example of such gibberish is the answer to the question asked on the portal Quora “What do math professors think of Marilyn Vos Savant’s book”, The World’s Most Famous Math Problem: The Proof of Fermat’s Last Theorem “? from 2016. The answer was given by Shai Simonson, a mathematics teacher with over 30 years of experience and a graduate of Columbia University. His answer gained “upvots” from “serious” mathematicians with PhDs. Here it is:

“Unfortunately, Marilyn’s columns, in general, do not make mathematics more clear or interesting. They serve to make her look smart by intimidation and obfuscation. It is entirely in character for her to write a book filled with nonsense and believe that she had some point.

Here is an example of her work.

**Starting at 3 o’clock, at what time will the minute hand catch up with the hour hand? —Bill Beachey, Berne, Ind.**

**At 16.3636 minutes after 3. That’s the short answer. Not very satisfying, is it?! So—as readers often ask how I solve problems—here’s the long answer:**

**Say that X is the distance the hour hand travels in an hour and Y is the distance the minute hand moves in the same time. The minute hand moves 12 times as fast, so Y equals 12X. Now say that P is a fraction of the distance travelled by the hands in an hour. We’re looking for the point between 3 o’clock and 4 o’clock at which PX and PY coincide. That would be the spot where 3X plus PX equals PY. As Y equals 12X, this means 3X plus PX equals P times 12X.**

**We’re in the home stretch! Solving the equation 3X + PX = P (12X) gives us a fraction of 3/11. Which means the minute hand will catch up with the hour hand at 3:16.3636. Oh, well. I’ll bet this at least will tamp down the tendency to ask how I solve problems.**

Don’t groan. I too have a hard time following her reasoning. The only thing I am convinced of after reading Marilyn’s explanation is that *she* knows how to solve the problem, but I am not at all convinced that she knows why her idea is correct and she certainly has no clue how to explain the idea. Don’t let her article intimidate you. It is hard to read math, and it is hard to write it as well.

Marilyn’s final answer of 3:16.3636 is close but technically incorrect. The correct answer is 3 and 3/11 hours, or 3:16:21 and 9/11 seconds. If the .3636 repeated forever, then she has the correct answer.

Now why is the answer 3 3/11 hours? One can add up the following geometric series:

3(1+1/12+1/122+1/123+…)3(1+1/12+1/122+1/123+…) representing the motion of the minute hand in stages as it catches up to where the hour hand had been at the beginning of the previous stage. After 3:00, the minute hand moves 3 clock units, during which time the hour hand (which moves 12 times more slowly than the minute hand) travels 3/123/12 units further. To catch up, the minute hand moves 3/123/12 units, during which time the hour hand moves forward 3/1223/122 units, etc.

I used this puzzle to illustrate infinite geometric series in “real life,” and how to sum them. Nonetheless, a clever 6th grade student of mine, who is today a PhD student in robotics at Carnegie Melon, had a much more clever and simple idea. Every 12 hours, the minute hand passes over the hour hand exactly 11 times. Get an old clock with hands and try this if you need to see why. Thus the time between “passovers” is 12/11 hours. Three of these gets us the correct answer.

Marilyn – the voice of popular mathematics – so sad….”^{[xxx]}.

At this point, you can write (using the same kind of argumentation): “Shai – the voice of ‘professional’ mathematics – so sad …”. As you can see, the person answering the question [“a mathematics teacher with over 30 years of experience”] is unable to answer the question asked, but tries to deprecate Marilyn’s competence through the classic nit-picking. The crowning “argument” against the solution proposed by Ms. vos Savant is that she has found her own logical, simple and brilliant method, apparently incomprehensible to a mathematics teacher with over 30 years of experience. According to that teacher, she should apply (unknown to lay people for whom the answer was intended) the method of adding elements of an infinite series. This method is less effective and less elegant than the Marilyn method, but it has the “advantage” over the latter that it was memorized by a mathematics teacher in college.

[Let me in this case to refrain from any further comments].

All the above leads to conclusion that today’s professional mathematicians are not qualified for professional discussions.

The idea that people with lower intelligence are able to evaluate the thought products of people much smarter than them can only come from people with low intelligence, who, due to their low intelligence, do not understand the importance of higher intelligence. Instead, these people are susceptible to the Dunning-Kruger effect^{[xxxi]}.

That is that simple.

**Conclusions concerning (not only) Marilyn**

Ms. vos Savant’s discussion with mathematicians illustrates one more point: way the high IQ people discuss with scientists (who are nowadays not very high IQ people). High IQ people at Marilyn’s cognition level are by nature reasonable individuals and strive to understand. Therefore, they do not understand why other people do not understand what is self-understandable. High IQ persons, by their respect for logic and reason, think that if something is not understood by someone in a reasonable way, it should be explained with an even more logical and more reasonable method. As a result of this belief, they add even more logical explanation to the logical explanation gave before. However, this method of explaining issues is counterproductive in the case of low and average IQ people. Low and average IQ people cannot understand reason in reasonable way, and thus – they usually react emotionally to the rules of pure logic and thinking. Their usual “thinking rule” is to replace thinking with associative algorithms learned by them in their colleges or at universities.

In other words, for people with reduced intelligence, a significant amount of logical arguments used in the discussion overloads their intellect. Then, from a certain point of the discussion, we are dealing with the phenomenon that the more logical the arguments are and the greater their number is, the more emotionally the other side of the dispute reacts. It could be seen when the Monty Hall paradox was discussed, it was only the change of the explanatory method from logical to empirical (experimental) that allowed the problem to be closed successfully for todays’ mathematicians.

The theses contained vos Savant’s book on the Wiles’ “proof” of Fermat’s Last Theorem, were never defend by her later and she did not intend to defend them anyway. She’s surely intelligent enough to knows by this time that intelligence has no chance of intelligent arguing with non-intelligent entities. Another thing is that, however, Marilyn wrote in her book about ideas (concepts) that were absolutely logically correct and valid, she haven’t created theories based on that ideas. More specifically – and referring to Buckle’s model – Ms. vos Savant, in her book on Fermat’s Last Theorem, presented mathematical facts (i.e. things) and her own ideas, but did not bring things and ideas together in a one single whole. In order to replace the old theories with new, one should discern new ideas in the known facts and derive new facts from these new ideas, creating in such way a new theory. Ms. vos Savant unfortunately does not create new theories, deliberately leaving this job to other high IQ persons. Perhaps the prospect of a hopeless fight against ignorance terrified her.

The question remains, how much the intelligence of Ms. vos Savant could achieve in science (including mathematics) if, instead of dealing with ignorance, she obtained the support of another intelligence.

And how much could contemporary mathematics and nowadays’ mathematicians gain from it?

[i] Read more at: https://www.brainyquote.com/quotes/marilyn_vos_savant_365179.

[ii] https://www.goodreads.com/author/quotes/44295.Marilyn_Vos_Savant.

[iii] Mlodinow, L. *The Drunkard’s Walk: How Randomness Rules Our Lives*, Pantheon Books, 2008.

[iv] Cf. Mazen Alrahili, “Simulation of the Monty Hall Problem”, *International Journal of Computer Applications* (0975 – 8887), Volume 152 – No.6, October 2016, https://pdfs.semanticscholar.org/344f/ 0c38c9391a02ad26d82f20263b646ba895f1.pdf.

[v] Vos Savant, M. *The Power of Logical Thinking*, St. Martin’s Press, New York 1996.

[vi] Ibid.

[vii] From the beginning of the game it was known that the host would open one of the two doors not chosen by the player. These doors are indistinguishable from each other in terms of the game universe – therefore, whether the host chose the first or the second one has no logical significance in this situation and does not change the probability that the player made the right choice the first time. The probability is still 1/3.

[viii] http://marilynvossavant.com/game-show-problem/. Archived at Internet Archive on 29 April 2012: https://web.archive.org/web/20120429013941/http://marilynvossavant.com/game-show-problem/.

[ix] Vos Savant, M. *The Game Show Problem*, https://parade.com/506669/marilynvossavant/the-game-show-problem/.

[x] Ibid.

[xi] https://www.britannica.com/biography/Paul-Erdos.

[xii] Por. Snider, I. *Monty Hall, Paul Erd**ö**s**, and Monte Carlo*, https://www.mwsug.org/proceedings/ 2010/stats/MWSUG-2010-87.pdf.

[xiii] Andrew Vazsonyi, Ph.D., *Erdos, Cars and Goats, and Bayes’ Theorem*, http://www.emis.de/classics/Erdos/textpdf/vazsonyi/bayes.pdf.

[xiv] ANDREW VAZSONYI (Feature Editor, McLaren School of Business, University of San Francisco), *Which Door Has the Cadillac?*, https://web.archive.org/web/20140413131827/http://www.decisionsciences.org/ DecisionLine/Vol30/30_1/vazs30_1.pdf.

[xv] http://tiborsimko.org/erdos-game-show-mistake.html.

[xvi] Vos Savant, M. *The Power of Logical Thinking*, op. cit.

[xvii] Cf. Waller, S. “6 Differences Between Book Smarts And True Intelligence”, *Spirit Science*, June 27, 2017, http://thespiritscience.net/2017/06/27/6-differences-between-book-smarts-and-true-intelligence/.

[xviii] Mlodinow, L., op. cit.

[xix] Cf. Papakyriakopoulou, K. “10 Differences Between Book Smart People And Genuine Intelligence”, https://www.stevenaitchison.co.uk/10-differences-book-smart-people-genuine-intelligence/.

[xx] Cf. Sinclair, G. “12 Key Differences Between Book Smarts And Genuine Intelligence”, *Awareness Act*, July 23, 2018, https://awarenessact.com/12-key-differences-between-book-smarts-and-genuine-intelligence/.

[xxi] vos Savant, M. *The World’s Most Famous Math Problem**: The Proof of Fermat’s Last Theorem and Other Mathematical Mysteries*, St. Martin’s Press, New York, 1993.

[xxii] Ibidem.

[xxiii] Ibidem.

[xxiv] Ibidem.

[xxv] Ibidem as well.

[xxvi] Cf. Kierkegaard, S. “On the Concept of Irony with Continual Reference to Socrates”, in: Kierkagaard, S., Hong, H.V., Hong, E.H. *Kierkegaard**‘s Writings, II, Volume 2: The Concept of Irony, with Continual Reference to Socrates/Notes of Schelling’s Berlin Lectures*, Princeton University Press, 1989.

[xxvii] It is worth noting that mathematics as a form of dialogue is to some extent attempted by one of the most intelligent living mathematicians – Terence Chi-Shen Tao. For years he has been discussing mathematical concepts on scientific blogs he runs. As we learn from the comments posted there, “Terrence Tao is an excellent, accessible writer. Thanks for sharing this link. I did not know about his (frequently updated!) google buzz […]. And he replies to people’s comments” [Tao, T. “A modern refinement of classical deduction is that of Bayesian probability”https://www.reddit.com/r/math/comments/bnvty/terry_tao_a_modern_refinement_of_classical/]

[xxviii] Boston, N. Granville, A. “Reviewed Work: The World’s Most Famous Math Problem (The Proof of Fermat’s Last Theorem and Other Mathematical Mysteries) by Marilyn vos Savant”, *The American Mathematical Monthly*, Vol. 102, No. 5 (May, 1995), pp. 470-473, https://dms.umontreal.ca/~andrew/PDF/VS.pdf.

[xxix] The last quote is commonly attributed to Leonardo da Vinci. Cf. “Stuffed Shirts by Clare Boothe Brokaw (Clare Boothe Luce)”, Chapter 17: “Snobs, New Style”, Published by Horace Liveright, New York 1931: “I have resolved to grow old, naturally and gracefully, content in the knowledge that the greatest intellects are the homeliest ones, and that the height of sophistication is simplicity”.

[xxx] https://qr.ae/TWNXkz.

[xxxi] Krueger, J. Mueller, R.A. “Unskilled, unaware, or both? The better-than-average heuristic and statistical regression predict errors in estimates of own performance”, *Journal of Personality and Social Psychology*, 82 (2), 2002, p. 180–188.