Fractals and Finance
Question: How can we understand financial market fluctuations in fractal terms?
Benoit Mandelbrot: Well, what I discovered quite a while ago in fact, that was my first major piece of work is that a model of price variation which everybody was adopting was very far from being applicable. It’s a very curious story.
In 1900, a Frenchman named Bachelier, who was a student of mathematics, wrote a thesis on the theory of speculation. It was not at all an acceptable topic in pure mathematics and he had a very miserable life. But his thesis was extraordinary. Extraordinary in a very strange way. It applied very well to Brownian motion, which is in physics. So, Bachelier was a pioneer of a very marvelous essential theory in physics. But to economics, it didn’t apply at all, it was very ingenious, but Bachelier had no data, in fact no data was available at that time in 1900, so he imagined an artificial market in which certain rules may apply. Unfortunately, the theory which was developed by economists when computers came up was Bachelier’s theory. It does not account for any of the major effects in economics. For example, it assumes prices are continuous when everybody knows the prices are not continuous. Some people say, well, all right, there are discontinuities but they are a different kind of economics that we are doing, not because certain discontinuities become too complicated and only will the **** look more or less continuous. But it turns out that discontinuities are as important, or more important than the rest.
Bachelier assumed that each price changes in compared of the preceding price change. It’s a very beautiful assumption, but it’s completely incorrect because we know very well, especially today that for a long time prices may vary moderately and then suddenly they begin to vary a great deal. So, even we’re saying that the theory changes or you say the theory which exists is not appropriate. What I found that Bachelier’s theory was defective on both grounds. That was in 1961, 1962, I forgot the exact dates and when the development of Bachelier became very, very rapid. Since nobody wanted to listen to me, I did other things. Many other things, but I was waiting because it was quite clear that my time would have to come. And unfortunately, it has come, that is, the fluctuation of the economy, the stock market, and commodity markets today are about as they were in historical times. There was no change which made the stock market different today than it was long ago. And the lessons which are drawn from **** peers do represent today’s events very accurately. But the situation is much more complicated than Bachelier had assumed. Bachelier, again, was a genius, Bachelier had an excellent idea which happened to be very useful in physics, but economics, he just lacked data. He did not have awareness of discontinuity which is essential in this context. Not having an awareness of dependence, which is also essential in this context. So, his theory is very, very different from what you observe in reality.
Recorded on February 17, 2010
Interviewed by Austin \r\nAllen
The mathematician has long believed the traditional understanding of market fluctuations would need to be replaced with his fractal model. Unfortunately, he says, "my time…has come."
Nazi supporters held huge rallies and summer camps for kids throughout the United States in the 1930s.
- During the 1930s, thousands of Americans sympathized with the Nazis, holding huge rallies.
- The rallies were organized by the American German Bund, which wanted to spread Nazi ideology.
- Nazi supporters also organized summer camps for kids to teach them their values.
A Bund parade in New York, October 30, 1939.
Credit: Library of Congress
1930s AMERICAN FASCIST BUND CAMP HOME MOVIE BERGWALD NEW JERSEY<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="69d54b175b0d317cf9bfd688e4fa04f3"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/gOPeDaDcw3w?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span>
Tea and coffee have known health benefits, but now we know they can work together.
Credit: NIKOLAY OSMACHKO from Pexels
- A new study finds drinking large amounts of coffee and tea lowers the risk of death in some adults by nearly two thirds.
- This is the first study to suggest the known benefits of these drinks are additive.
- The findings are great, but only directly apply to certain people.
Maybe you should enjoy this article with a cup of coffee or tea.<p> The <a href="https://drc.bmj.com/content/8/1/e001252?T=AU" target="_blank" rel="noopener noreferrer">study</a> involved 4,923 type 2 diabetics living in Japan. The average participant was 66 years old. All of the participants were taken from the rolls of the Fukuoka Diabetes Registry, a study geared at learning about the effects of new treatments and lifestyle changes on the health of diabetics. <br> <br> The participants filled out questionnaires concerning their health, diet, habits, and other factors. Among the questions were two focused on determining how much green tea or coffee, if any, the participants consumed over the course of a week. The health of the participants was recorded for five years. During this time, 309 of the test subjects died from a variety of causes. <br> <br> Subjects who drank more than one cup of tea or coffee per day demonstrated lower odds of dying than those who had none. Those who consumed the most tea and coffee, more than four and two cups a day, respectively, enjoyed the most significant reductions in their risk of death. This level of consumption was associated with a 40 percent lower risk of <a href="https://www.sciencedaily.com/releases/2020/10/201020190129.htm" target="_blank" rel="noopener noreferrer">death</a>. </p><p>Most interestingly, the effects of drinking tea and coffee appear to combine to reduce risk even further. Those who reported drinking two or three cups of tea a day and two or more cups of coffee were 51 percent less likely to die during the study, while those who drank a whopping four or more cups of tea and two or more cups of coffee had a 63 percent lower risk of <a href="https://www.medicalnewstoday.com/articles/diabetes-coffee-and-green-tea-might-reduce-death-risk" target="_blank" rel="noopener noreferrer">death</a>. </p>
So, should I start swimming in a vat of coffee and green tea?<iframe width="730" height="430" src="https://www.youtube.com/embed/LY0E-JQxeoY" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> Not quite. </p><p> The primary takeaway from this study is that Japanese adults with type 2 diabetes who drink a lot of green tea and/or coffee die less often than similar people who do not. If this effect is caused by something in the drink, lifestyle choices people who drink that much tea all make, or something else remains unknown. The finding must be considered an association at this point. <br> <br> The eye-popping reductions in mortality rates are compared to the risk of death of others in the study. The people who died reported drinking less tea and coffee than those who lived. Unless you have several demographic and conditional similarities to the subjects of this study, you probably won't suddenly be at a two-thirds lower risk of death than your peers because you drink green tea. </p><p> Like all studies that depend on self-reporting, it is also possible that people misstated how much they consumed any one item. The study also did not look into other factors like socioeconomic status or education level, also known to impact death rates and potentially linked to coffee and tea consumption. </p><p> However, it is yet another study in the pile that suggests that <a href="https://www.healthline.com/nutrition/top-13-evidence-based-health-benefits-of-coffee" target="_blank" rel="noopener noreferrer">coffee</a> and <a href="https://www.healthline.com/nutrition/top-10-evidence-based-health-benefits-of-green-tea" target="_blank" rel="noopener noreferrer">green tea</a> are good for you. That much is increasingly <a href="https://www.health.harvard.edu/press_releases/health-benefits-linked-to-drinking-tea" target="_blank" rel="noopener noreferrer">agreed</a><a href="https://www.rush.edu/health-wellness/discover-health/health-benefits-coffee" target="_blank" rel="noopener noreferrer"> upon</a>. This study also suggests the benefits are additive, which is a new development.</p><p><br> So, while it isn't time to start the IV drip of green tea, a cup or two probably won't <a href="https://www.webmd.com/diabetes/news/20201022/coffee-green-tea-might-extend-life-for-folks-with-type-2-diabetes" target="_blank" rel="noopener noreferrer">hurt</a>. </p>
Logic puzzles can teach reasoning in a fun way that doesn't feel like work.
- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.
The Three Gods Problem<iframe width="730" height="430" src="https://www.youtube.com/embed/UyOGZk7WbIk" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> One of the more popular wordings of the problem, which MIT logic professor George Boolos <a href="https://www.readersdigest.ca/culture/hardest-logic-puzzle-ever/" target="_blank">said</a> was the hardest ever, is:<br> <br> "Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for <em>yes</em> and <em>no</em> are <em>da</em> and <em>ja</em>, in some order. You do not know which word means which."<br> <br> Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja." <br> <br> Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay. <strong><br> <br></strong>George Boolos' <a href="https://www.pdcnet.org/8525737F00588A37/file/31B21D0580E8B125852577CA0060ABC9/$FILE/harvardreview_1996_0006_0001_0060_0063.pdf" target="_blank" rel="noopener noreferrer">solution</a> focuses on finding either True or False through complex questions. </p><p> In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement. </p><p> So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).</p><p> In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ. </p><p> Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains: <br> </p><p> "If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."<br> <br> No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B. </p><p> From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question. </p><p> After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination. </p><p> If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently. </p><p> Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this <a href="https://www.nytimes.com/interactive/2017/02/11/obituaries/smullyan-logic-puzzles.html" target="_blank" rel="noopener noreferrer">interactive</a>. </p>
But most city dwellers weren't seeing the science — they were seeing something out of Blade Runner.
On Sept. 9, many West Coast residents looked out their windows and witnessed a post-apocalyptic landscape: silhouetted cars, buildings and people bathed in an overpowering orange light that looked like a jacked-up sunset.