Occupy Math and his former student, Colin Lee, have published a textbook that introduces the art of mathematical proof, a core discipline of pure mathematics. Learning to do formal proofs is a new way of thinking, farther from reality than most ways of thinking, but if you learn it you gain a clarity of thought that may help quite a bit. The book is entitled An Introduction to Proofs with Set Theory. If you want a print copy, it is available from Morgan and Claypool Publishers. It is also available for no added charge to students at Occupy Math’s university and any other university that subscribes to Morgan and Claypool’s synthesis series. Click through to browse; check your institution’s library for availability. The rest of this post skims through what the books is about and talks about the interesting (no cost to students!) publishing model employed by Occupy Math’s publisher, one answer to outrageous textbook prices.
This is a new example of the warm color Newton’s method fractal.
Chess is an interesting game because the game, overall, is fairly hard, but the rules can be explained in a few minutes. Problems in the real world are even more like this — even determining whether you have the best answer to a particular problem may require a supercomputer or, in some cases, can be shown to be unknowable. This goes off into the really deep end of mathematics and is beyond the scope of an Occupy Math post. Today we are going to go over a fraction-teaching game that has a best answer, but it’s pretty difficult to know if you have found that best answer.
The core of this Occupy Math post is a new activity for the fraction-teaching game FRAX. Look at the 3×3 array of FRAX cards at the top of the post. The goal is to lay the cards out so that as many rows and columns as possible add to a whole number. Scoring this layout of cards, the first row sums to 1 and the third row sums to 2. The score is the sum of the whole numbers, meaning that the score for this configuration is 3
This week, we have another warm color palette Newton’s method fractal.
“If I have to go back to school, why can’t I visit my friends?” This question is the starting point for trying to explain a mathematical notion that answers the question and explains something that a whole lot of people have exactly wrong about risk management during the pandemic. Things are not “safe” or “unsafe”! Rather, the more contacts individual people have, the lower safety is for everyone. This is a good starting point for figuring out if you want to go back to in-person school or attend virtual school from home. Another problem is that some people do not have the hardware to attend virtual school. Return to school is a Gordian knot of disease and safety, money and psychology.
A new color scheme for Newton’s method fractals.
We now know that there are thousands of planets beyond the eight or nine in our own solar system. Most of the ones we have found are a slice of hell, because planets very close to their own sun are easier to find. This means that the poignant question “are we alone?” is not much closer to being answered, though the fact planets are common does improve the odds. Occupy Math would love to meet aliens, or even detect an alien ecosystem at long range, and right now we are at the dawn of an era where that might happen. This is both exciting and frustrating and this post is a background briefing on what we know now.
Planet Bis a name that is supposed to evoke the notion of Plan B and mean a new, back-up Earth. The transiting exoplanet survey satellite (TESS from now on) recently located something that might be Planet B, except that it is much, much too far away. When Occupy Math was reading the comments from the announcement, he developed an acute sense that fans of extra-solar planets do not have the scale of the problem correct and are in fact pretty math-deficient. That makes it harder to understand the scale of interstellar space and the Drake equation which attempts to estimate the number of communicating alien civilizations in our galaxy. The Drake equation is much closer to speculative fiction than mathematics, but it is a nice way to frame the discussion; this view was shared by the originator of the equation.
A ribbon bridge from the middle conjugate 232-Mandelbrot set.
Occupy Math uses math to understand how cooperation arises — cooperation between animals, cooperation between humans, and even cooperation between pieces of software. An incredible example of cooperation is that of the community of mathematics. Over a period of thousands of years, people who share neither culture nor language have collaborated, across time and past their own deaths, to solve a large collection of mysteries and to reveal even more mysteries. Mathematics is a truly never-ending story and mathematicians are the people telling its eternal first part.
The MacTutor archive at St. Andrews University has the biographies of nearly 3,000 mathematicians spanning the ancient to the modern world. It is a wonderful resource for those interested in the history of mathematics. One of the questions that Occupy Math thinks about is that of the universality of mathematics. If an alien species discovered math, as their civilization formed, how much overlap would there be with the various human versions of mathematics? In the modern world math has fused into a single, diverse discipline but it arose in China, India, Africa, South America, and Europe in different flavors and to different degrees. The MacTutor archive is earth’s record of the formation of math in our civilizations. If we ever get to look at another species’ mathematics, this will be the basis for comparison. This post will give you a sampler of the people in the MacTutor archive.
A 2-3 Cojugate-2 Mandelbrot view, fists of dark.