Certainly, we all have had moments where we explain something to someone. For a given Why...? or How come...? in a conversation, there is a natural reaction to answer that question (assuming one knows the answer), and this involves a teaching activity in a minor or mayor scale. Let's stop here for a moment. We have two main processes involved here: knowing and teaching. It might be obvious for some people, but believe me, not for everyone. If you are going to teach something to someone, make sure you first know about the subject you are going to explain. Otherwise, you are about to propagate a wrong message/concept/idea to the people, and wrong ideas will be the perfect seed to wrong assumptions and further actions. How to make sure we know about a certain topic? Contrast what we receive from different channels with original sources of information. A permanent contrast of information should be a mandatory activity in our life. Put that new information in practice, understanding the assumption used to build that explanations and what you can (and cannot) explain with that. (At this point, I'll keep in mind to write something about the important and holy act of understanding something in another post). Then, and once again, assuming you understand a certain topic, it will come the moment of teaching it, maybe in a random conversation with friends, or at work, or to your grandma or son. Teaching is a fantastic activity because you share knowledge you have to one or multiple persons. It's a gift!, actually one of the wonderful gifts you can ever give/receive to/from another person. One doesn't need the typical elements a teacher has (blackboard, chalk, etc.) to teach something to someone, but I guess that's the classical picture one has in mind. Please keep in mind the following elements when teaching: Teach something just to yourself, aloud, and note the time it takes to make your point. The previous step will give you the idea of the following: - Pick the proper amount of phrases to explain something. Very few words are as bad as many words. - Use several resources: your voice, pen and paper, body movements, examples (for example, graphs, pictures, clips, etc.). - Verify with your audience that the message was understood. Otherwise, and this is so important, make sure you have another way (another example, other words, other resources) to explain the same topic. This, by the way, will make you to understand even more the topic you know. Richard Feynman, apart from a great physicist of the modern era, he was considered an extraordinary teacher who transmitted complex ideas in an extraordinary and clear way. His lecture books are very famous, and one can find recorded lectures and interviews in internet:
https://www.feynmanlectures.caltech.edu/ Richard has some remarkable phrases related to the act of teaching that I want to share with you: "If you want to master something, teach it" "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong." "The first principle is that you must not fool yourself and you are the easiest person to fool". "You do not know anything until you have practiced it". See you around, Jesús I remember the first time I saw the reductio ad absurdum in a class of mathematics. Also known as argument to absurdity, that elegant way that tries to establish something by showing the opposite situation, after some development it eventually ends up to absurdity or contradiction. This method has been applied in mathematics, the classical example relates to the discovery by Pythagoras – disclosed to the chagrin of his associates by Hippasus of Metapontum in the fifth century BC – of the incommensurability of the diagonal of a square with its sides. The reasoning is as follows: Let d be the length of the diagonal of a square and s the length of its sides. Then by the Pythagorean theorem we have it that d² = 2s². Now suppose (by way of a reductio assumption) that d and s were commensurable in terms of a common unit u, so that d = n x u and s = m x u, where m and n are whole numbers (integers) that have no common divisor. (If there were a common divisor, we could simply shift it into u.) Now we know that (n x u)² = 2(m x u)² We then have it that n² = 2m². This means that n must be even, since only even integers have even squares. So n = 2k. But now n² = (2k)² = 4k² = 2m², so that 2k² = m². But this means that m must be even (by the same reasoning as before). And this means that m and n, both being even, will have common divisors (namely 2), contrary to the hypothesis that they do not. Accordingly, since that initial commensurability assumption engendered a contradiction, we have no alternative but to reject it. The incommensurability thesis is accordingly established. What about philosophy? I personally consider it as a fascinating way to take your thoughts in a different path and see the types of results you can get, it is like watching how the solution is emerging in front of you. Defining the absurdity as the conflict the humankind has between the tendency to find meaning or the value of life, and the inability to find it within any certain, a fight between curiosity to know and the incapacity to obtain it. Albert Camus, an absurdist philosopher stated that individuals should embrace the absurd condition of human existence. Looking at it in more general terms, the absurdity has so many expressions, so many shapes, reaching areas like art: theater, music, literature, painting... But going back to what we see everyday, how is something categorized as absurd?, what or who declares that something is indeed absurd. The critic?, the expert? I have no idea... In art my first thought is the Fountain by Marcel Duchamp (see the picture on the right). Is that Absurd?, art?, both?, none? Is time to end this writing now?, I think so... Let's talk more about all this in another time. See you around, Jesús One of the topics in mathematics that I always find fascinating is logics. I can find myself reading and practicing few problems in logics, sometimes getting nice and interesting results, sometimes ending with a headache. Some days ago, reading the (by the way, great!) book Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem, I found an aspect in the history of logics that takes me time to reflect every time I read about it: the Russell's Paradox. Everything started with an incoherent thought. Bertrand Russell (British philosopher, logician, mathematician, historian, writer, social critic, political activist, Nobel laureate in Litereature-1950) was the person who got that incoherent thought. It is known as the Russell's paradox and can be explained with the tale of the librarian: One day, a librarian discovers a collection of catalogs. There are different catalogs, for novels, essays, tales, etc. The librarian finds that some catalogs include themselves and some others not. In order to simplify the system, the librarian elaborates two new catalogs, one catalog who includes catalogs that includes themselves, and another catalog who doesn't. After completing this task, the librarian finds out a problem: the catalog that includes the list of the catalogs that don't include themselves, should it catalog itself? If it does, by definition it shouldn't be included, whereas if it doesn't, it should includes itself by definition. The librarian is in impossible situation! This incoherency was a consequence of working with mathematical axioms, which until that time they were considered as evident and good enough to define all the mathematics. A way to tackle the problem was to create an additional axiom that prohibits to any group be a member of itself. This would reduce the Russell's paradox and converts it as superfluous the question about including in the catalog those who are not included themselves. Russell expend the next decade examining the axioms. In 1910, together with Alfred Whitehead (English mathematician and philosopher), published the first of three volumes of Principia Mathematica, trying the problem that came up from his own paradox. Apparently, the building of a coherent logics, strong enough to answer any question, was on a good track. Then, an unknow 25 years old mathematician, came to say something new, something that would stop that thought of an ideal world created with a pure and perfect mathematical structure, this man was Kurt Gödel. But Gödel deserves a separate post.
See you around, Jesús "His name is Greek, his nationality is French and his history is curious. He is one of the most influential mathematicians of the 20th century. His works are read and cited widely throughout the world. He has fervent supporters and staunch detractors in any group of mathematicians to meet. The strangest fact about him, however, is that he does not exist." From: P. R. Halmos, Nicolas Bourbaki. Scientific American, (1957), May, 88–99. https://www.scientificamerican.com/article/nicolas-bourbaki/ In mathematics, once you proof a theorem, it is valid forever and ever. This is crucial because the work of every new generation of mathematicians is based on legacy work It was in Paris 1934 where the name Nicolas Bourbaki first appeared in a place rocked by turmoil at a volatile time in history. World War I had wiped out a generation of French intellectuals. As a result, the standard university-level calculus textbook, written more than two and half decades ago, was out of date. Newly minted professors André Weil and Henri Cartan wanted a rigorous method to teach Stokes’ theorem, a key result of calculus. After realizing that others had similar concerns, Weil organized a meeting. This meeting took place on December 10, 1934 at a Parisian café called Capoulade. The nine mathematicians in attendance agreed to write a textbook ‘to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis,’ which they hoped to complete in just six months. As a joke, they named themselves after an old French general who had been duped in the Franco-Prussian war. Bourbaki's mode of exposition, consisting of the arrangement of all the text under: Definition, Theorem, Preposition, Lemma, Proof, Example, Remark, etc., is without doubt welcome. Personally, I like his "tournant dangereux" (a "danger bend" sign), warning the reader not to make a wrong conclusion. Soon, Bourbaki realized that in addition to writing a text as mentioned above, it was also necessary to have discussions on contemporary topics that the text itself cannot cover in a reasonable time frame. To address this problem, Bourbaki met twice or thrice a year and arranged lectures on areas of the then current and topical interest. These “exposés” were also published in the widely popular series Bourbaki Seminar. Sometimes the meetings took place in the countryside to not get distracted from their main objective: rewrite the mathematics from top to bottom so to restructure the field on solid foundations.
Notations such as the symbol ∅ for the empty set, and terms like injective, surjective and bijective owe their widespread use to their adoption in the Éléments de mathématique. The group believed in the inherit unity of mathematics. They wanted to present the set of mathematics in its more pure and simple way. The results were abstracts and ambitious. At that time, people started to think that the attempt of a new possible war about to begin. Weil escaped to Finland, hoping that he can keep Bourbaki alive from there, but ended up in jail as suspected of being a Soviet spy. According to his memories, he escaped from execution thanks to Rolf Nevanlinna (Finnish mathematician, who made significant contributions to complex analysis). Back in France, Weil was charged with failure to report for duty, and was imprisoned in Le Havre and then Rouen (France). During his time in prison, Weil started to develop a completely new approach of mathematics, based on the Bourbaki's work. An approach that allows him to transform the algebra of equations to something more tangible... Taking René Descartes as reference, who showed us how to convert shapes into numbers using coordinates: would it be possible to do something similar but in the opposite direction? Could it be possible to use intuitive ideas of geometry and shapes so to solve problems related to numbers and equations? Weil showed how to think in equations as shapes!, complicated even for the mathematicians of that time. And this was the beginning of a new concept: algebraic geometry. And since then, that concept has allowed us to solve equations that for centuries seemed impossible. The most famous case was that it paved the way for the solution of Fermat's notoriously difficult Last Theorem, proved by British mathematician Andrew Wiles in 1995. Weil was released from prison in May 1940 and, as many European mathematicians during that time, he moved to America. The Institute for Advanced Study at Princeton opened its doors in 1933 and at the end of the war there was a well-funded research center with countless emigrated scientists and many superstars, like Albert Einstein. With Europe ravaged by war and Göttingen destroyed by Hitler, Princeton quickly became the place to study mathematics. The success of Bourbaki is linked to the scientific quality of his members. All of them have been extraordinary mathematicians. Five of them have obtained the Fields medal, most important international recognition of excellence in Mathematics, awarded at the International Congresses of Mathematicians that are held every four years: Laurent Schwartz (1950), Jean Pierre Serre (1954), Alexander Grothendieck (en 1966), Alain Connes (1982) and Jean-Christophe Yoccoz (1994). Best, Jesús If you visit the Historiska Museet in Stockholm (Sweden), you will have the chance to see a fantastic collection of Medieval art and history of Sweden, impressive collection of gold and silver artifacts from Viking period. One one of the museum's rooms, I found a short description about Eva Ekeblad (1724 - 1786). Later on, reading about her was fascinating. Eva, was a Swedish countess, agronomist and scientist. The Ekeblad's house (Stockholm) hosted a cultural salon and was described by the wife of the Spanish Ambassador de marquis de Puentefuerte as "one of few aristocratic ladies whose honor was considered untainted". The first concert performances of the mass music of Johan Helmich Roman were performed in her salon at the Ekeblad's house. A look at Johan's music can be listened here. The potatoes are relevant in this discussion (and you might have an idea after seeing the picture above) because Eva Ekeblad is known for discovering a method in 1746 to make alcohol and flour from potatoes, which allowed using scarce grains for food production, and contributed to reduce Sweden's period of famine. In 1748, Eva Ekeblad was the first woman elected as member of the Royal Swedish Academy of Sciences. In November 5 of that year it was announced that "Countess Ekeblad had made several attempts to use potatoes for starch and powder". She sent sample and description of how to proceed. The sample could also be mixed with oats and bread or to burn potatoes brandy. Here, the relevant fact of burning brandy on potatoes was that you didn't have to use the small stock of oats and barley that you had for brandy production. Moreover, the residual product could be used for cattle feed. Funny fact: the residual powder she made was good for powdering the wigs! Although she was now a member of the academy, she personally never attended its meetings. She nevertheless submitted new research, including a soap for cotton linen bleacher (December 1751) and a bleaching of cotton yarn (April 1752) and these contributions were also printed in the documents. It would be 209 years before the next woman was included in the Royal Swedish Academy of Sciences. It happened in 1951, Lise Meitner (Austrian-Swedish physicist) who contributed in discovering the element protactinium and to the nuclear fission. See you around, Jesús In the past, you watched a movie and the final scene usually came with the classical words "The End", clear indication that the movie is over. Nowadays, these words might shows up, but in both cases there is something is common: the final song of the movie. I cannot image how difficult it is for a director/editor/compositor to decide the proper song for each part of a movie, including the final part.
I highlight this topic because in my case, when I remember a particular movie, those images come with the soundtrack, like an unavoidable attachment. Even more, in some cases the soundtrack is much better than the movie itself, so I tend to remember the songs than the movie in those cases. Same thing happens with series: I associate series with the intro song they had. Then, thanks to this great choice of songs for movies, I tend to discover that there are fantastic bands and musicians out there I was completely ignoring so far! This is a nice surprise for me, to have the chance to explore new territories of bands, rhythms, folklore,etc. Discussing about movies with some friends, we came out with some interesting are unforgettable songs. The list is so huge, but here I would like to share at least few soundtracks I find remarkable: - "Urami Bushi" by Meiko Kaji. Movie: Kill Bill 2. https://www.youtube.com/watch?v=yT9zJ2V9lfw - "Between the bars" by Elliot Smith. Movie: Good Will Hunting. https://www.youtube.com/watch?v=8aomt1gQ6So&list=RD8aomt1gQ6So&index=1 - "Yumeji's theme" by Shigeru Umebayashi. Movie: In the Mood for Love. https://www.youtube.com/watch?v=vBWCphAu8ik - "Bel Ami" by Rchel Portman. Movie: Bel Ami. https://www.youtube.com/watch?v=C-_OIffF6D8 - "Toop Toop" by Cassius. Movie: Il Divo. https://www.youtube.com/watch?v=wBZM6RSqW-Y - "I Think I Like It" by Fake Blood. Movie: We Are Your Friends. https://www.youtube.com/watch?v=3xKUiva2WSQ - "Won't Fooled Again" by The Who. Series: CSI Miami. https://www.youtube.com/watch?v=qEQpsbTPADs - "When I Am Through With You" by VLA. Series: Damages. https://www.youtube.com/watch?v=KNU2nOo1EwM For me, apart from a good story behind, good dialog, photography, for a movie/series to be considered great it needs to have a proper selection of songs, mainly for intro/ending. A good story can be destroyed or will not have the same impact of there is a bad selection of the songs selected for them. See you around, Jesús "Only in the state of delirium one can compose the most elevated poetry". These words are from Democritus (c. 460 – c. 370 BC, Greek philosopher and mathematician), and this might be a good starting point for this post. Delirium, even if it might sound as an out-of-control state, I always relate it with a certain sublime state of thinking, where a certain group of dissociated ideas are placed together harmonically, in a certain order, apparently chaotic from outside but incredibly ordered from inside. From the literature, below you will see a plot given by Google about the frequency of the word "delirium" during the last 550 years, exactly from year 1550 to 2018. It is interesting to see two regions with maximum values, around year 1778 and 1880. I cannot say exactly what was the reason for these two maximums, but probably important contributions in the narrative or in novels happened in those periods. According to the Oxford dictionary, "delirium" has two main definitions: - An acutely disturbed state of mind characterized by restlessness, illusions, and incoherence, occurring in intoxication, fever, and other disorders. - Wild excitement or ecstasy. Delirium has the following synonyms: derangement, dementia, dementedness, temporary madness/insanity, incoherence, raving, irrationality, hysteria, wildness, feverishness, frenzy, hallucination, rare calenture, Perhaps this was the state that Jackson Pollock (American painter, 1912-1956) felt every time he was creating his marvelous paintings. Pollock, who started to show his paintings and becoming a legend in the abstract expressionism movement, at the time when Pablo Picasso (Spanish artist, 1881-1973) and Henri Matisse (French artist, 1869-1954) were the most famous painters, Picasso with the cubism and surrealism movement, and Matisse with the modernism and post-impressionism movement. See below a picture of Pollock in action and one of his paintings: Going back in time, I must mention what Hilma af Klint (Swedish artist, 1862-1944) proposed. Klint is considered one of the first artists in the abstract art movement. From her notes, she described that her paintings were coming from spiritual thoughts, which she tried to represent in a visual way. This was mainly motivated by the early death of her sister. The spiritism was in vogue at that time, and together with different thinking theories could have given an important influence to Klint's paintings. One will never discover if Klint was immerse in deliriums, but the idea might not be far from the real explanation. See below a picture of Klint and one of her paintings. At this point of my thoughts about delirium, it always comes to my mind a particular guy, Georg Cantor (German mathematician, 1845-1918). Before Cantor, nobody had understood well the meaning of infinity. This was a tricky subject that did not have a clear definition, but Cantor demonstrated that the notion of infinity was perfectly understandable. In fact, there was not only one but many infinities, some of them larger (e.g. decimals) than others (e.g. natural numbers and fraction numbers). Some mathematicians, when reading Cantor's works, get a bit nervous because as log as one goes into the reading, one starts to ask: what's going on here?, where does he want to go?. Besides his enormous talent and effort in this field, Cantor had also an important difficulty, he was a manic depressive, dealing with the illness at the same time he was immersed in his work, and from time to time he ended up in the University's sanatorium. There was not medical treatment at that time, however the sanatorium was a good place to be comfortable and quiet, where Cantor found strength to continue his studies about infinity. The delirium also brings me the sounds of Rachmaninoff (Russian pianist, composer and conductor, 1873-1943), specially of one of his famous concerts: Piano concert N.3, where you can listen here together with the score, with a special sublime moment at minute 8. It is not surprising that Rachmaninoff is in the list, together with Chopin, and Liszt, as composer with very demanding solo piano pieces. Listen some examples here.
This topic about delirium gives the chance for more discussion, but I think is better to stop now, for the moment, before the delirium comes and I cannot finish this post. See you around! Jesús All of us, at least once in our life, have been in a particular social activity called dancing party. It means that, given a promising good evening, you made a decision of having fun in a certain place where people decided to follow the rhythm of the music in the best possible way. This decision came from multiple reasons: cause you have some friends who told you to go and have fun, or cause you wanted to go to meet your friends, or you are new in town and want to explore new places, or you are taking dancing lessons and want to practice somewhere all the things you just learnt. Once you are up for some nice dancing time, which means that you have selected the proper place with a good dance floor, there are some other crucial factors to take into account in order to have a good time. Each of these factors intrinsically has a certain probability to happen, let list them in detail and in order of occurrence:
- The rhythm of the song: The DJ (or band) typically plays one or few types of (somehow) similar rhythms in a party. For example: (i) tango, tango vals and milonga; (ii) salsa, bachata and kizomba; etc. In a typical situation, let assume that there is a DJ playing songs randomly from a playlist, and the list has M songs. Assume you want to dance with a stranger, which is the most critical situation (since you don't know the partner's preferences). Assume also that the partner always has 50% chance (Yes or No) to accept dancing with you (related to different factors such as the impression you give to the partner, how tired the partner is, how late the evening is,...) and this chance is present in every song. And assume that you know how to dance all the rhythms. You browse for a possible partner to dance with, and I guess you want to calculate the probability to dance, denoted P, the first song you ask for. Then you might have different scenarios: Scenario 1A: The partner knows how to dance all the rhythms. The P value is simply the 50% chance to get accepted by the partner: P=0.5. Scenario 2A: The partner only knows how to dance n songs from the DJ list (where n <= M). In this case P can be calculated using the specific multiplication rule, which is valid in this case since the two events (type of song the partner is able to dance, and the partner answer) are independent, therefore: P = (n/M)*(1/2)= n/(2M) Note that P in scenario 2 is lower or equal to P in scenario 1. Is equal when n=M. Scenario 1B: You succeed to enjoy scenario 1A and want to dance the next song with the partner. The P value is calculated by the multiplication of chances given by the decision of the partner. Then: P = 0.5*0.5 = 0.25. Note that for more songs, the P value will decrease towards zero (since the partner is getting tired, is getting late,... Scenario 2B: You succeed to enjoy scenario 1B and want to dance the next song with the partner. We still can use the multiplication rule probability, but in this case we have: P(a ∩ b) = P(a) P(b|a), where P(a) is the probability that event a happens, and P(b|a) is the probability that event b happens given a. In our case, P(a) has already been calculated in scenario 2A (i.e. n/2M), now (assuming that a song is played only once), the second probability (without including the final decision of the partner, which is 50%) is then (n-1)/(M-1), all together gives: P = (n/M)*(1/2)*((n-1)/(M-1))*(1/2) = (n*(n-1))/(M*(M-1)*4) Note that P is decreasing faster towards zero than in scenario 2A, meaning that in this situation you will rapidly have no chance to dance a certain song with your current partner. More about the multiplication rule probability can be found here, and about conditional probability here. After these numbers, in the middle of the party, I guess you still has the courage to go and invite somebody to dance. Don't think twice and go! See you around! Jesús Hi there! It is more than a year now that I haven't post anything new. So many things have happened in the last months: interesting topics to research, new colleagues to work with, teaching activities, new students with amazing way of thinking,....many things indeed! Speaking about "many", about "quantities", sometimes I forget the amount of people that is living in this planet. We are many in this world. The amount of people in this planet has been increasing. Actually, this number has increased exponentially in the last century, as you can check here!!! Among all this amount of people, sometimes I wonder about certain facts that might be happening right now, in some unknown places, done by unknown people. An infinite number of good and bad facts that are constantly occurring. Regarding the bad facts, they would be affecting the environment, the people, the near future of a person, family, society, population or nation. We can enumerate several types of bad actions cause we see them daily, or read or hear about them in the news, and I prefer not to think about them when writing these lines.
Instead, I prefer to think about those actions that could improve or help to every person. Actions you probably don't read in the newspaper or see in the street, but they are for sure happening now. And the examples could be so diverse. Right now, one or more of the following things are occurring: - There is a guy somewhere in South America, trying to paint as good as possible something that he has in mind, finding the proper colors and lines, discovering a new technique. This person is not a professional, has never sold a painting, but this detail is not important now. - There is a girl in an apartment in Asia, trying to play the entire music score of a concerto for violin and orchestra. The rehearsal is taking many hours a day, she knows is tough, but she also knows that the time to get a nice execution is very close. - In a certain village in Europe, an unknown rock band is playing in a bar, the accords from the guitars are simply great and the drums push you to enjoy the gig. They are right now the best rock band in the world. - After some years raising her children, a mother in Australia is ready to retake her tango lessons. She has been listening tango for years, by chance while she was walking close to a music shop, and got fascinated. For her, every tango lesson is like an open window where she takes fresh air. Each of these actions is just a tiny step away from the glory, from a personal glory, certain kind of plenitude. I prefer to think that everyday somebody is creating some fantastic thing after a long time rehearsing, practicing, improving. See you around! Jesus |
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