Global Math Week: The Universal Language

by Oscar Fernandez

FernandezFill in the blank: Some people speak English, some speak French, and some speak ____. I doubt you said “math.” Yet, as I will argue, the thought should have crossed your mind. And moreover, the fact that mathematics being a language likely never has, speaks volumes about how we think of math, and why we should start thinking of it—and teaching it—as a language.

To make my point, consider the following fundamental characteristics shared by most languages:

  •  A set of words or symbols (the language’s vocabulary)
  •  A set of rules for how to use these words or symbols (the language’s rules of grammar)
  •  A set of rules for combining these words or symbols to make statements (the language’s syntax)

Now think back to the math classes you have taken. I bet you will soon remember each of these characteristics present throughout your courses. (For instance, when you learned that 𝑎2 means 𝑎 × 𝑎, you were learning how to combine some of the symbols used in mathematics to make a statement—that the square of a number is the number multiplied by itself.) Indeed, viewed this way, every mathematics lesson can be thought of as a language lesson: new vocabulary, rules of grammar, or syntax is introduced; everyone then practices the new content; and the cycle repeats. By extension, every mathematics course can be thought of as a language course.

Now that I have you thinking of mathematics as a language, let me point out the many benefits of this new viewpoint. For one, this viewpoint helps dispel many myths about the subject. For instance, travel to any country and you will find a diverse set of people speaking that country’s language. Some are smarter than others; some are men and some women; perhaps some are Latino and some Asian. Group them as you wish, they will all share the capacity to speak the same language. The same is true of mathematics. It is not a subject accessible only to people of certain intelligence, sex, or races; we all have the capacity to speak mathematics. And once we start thinking of the subject as a language, we will recognize that learning mathematics is like learning any other language: all you need are good teachers, and lots of practice. And while mastering a language is often the endpoint of the learning process, mastering the language that is mathematics will yield much larger dividends, including the ability to express yourself precisely, and the capacity to understand the Universe. As Alfred Adler put it: “

Mathematics is pure language – the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms.” Galileo—widely regarded the father of modern science—once wrote that Nature is a great book “written in the language of mathematics” (The Assayer, 1623). Centuries later, Einstein, after having discovered the equation for gravity using mathematics, echoed Galileo’s sentiment, writing: “pure mathematics is, in its way, the poetry of logical ideas” (Obituary for Emmy Noether, 1935). Most of us today wouldn’t use words like “language” and “poetry” to describe mathematics. Yet, as I will argue, we should. And moreover, we should start thinking of—and teaching—math as a language.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.

Global Math Week: Counting on Math

by Tim Chartier

The Global Math Project has a goal of sharing the joys of mathematics to 1 million students around the world from October 10th through the 17th. As we watch the ever-increasing number of lives that will share in math’s wonders, let’s talk about counting, which is fundamental to reaching this goal.

Let’s count. Suppose we have five objects, like the plus signs below. We easily enough count five of them.

 

 

 

You could put them in a hat and mix them up.

 

 

 

 

 

 

 

If you take them out, they might be jumbled but you’d still have five.

 

 

 

 

 

 

 

 

 

 

Easy enough! Jumbling can induce subtle complexities, even to something as basic as counting.

Counting to 14 isn’t much more complicated than counting to five. Be careful as it depends what you are counting and how you jumble things! Verify there are 14 of Empire State Buildings in the picture below.

 

 

 

 

 

 

 

If you cut out the image along the straight black lines, you will have three pieces to a puzzle. If you interchange the left and right pieces on the top row, then you get the configuration below. How many buildings do you count now? Look at the puzzle carefully and see if you can determine how your count changed.

 

 

 

 

 

 

 

Can you spot any changes in the buildings in the first versus the second pictures? How we pick up an additional image is more easily seen if we reorder the buildings. So, let’s take the 14 buildings and reorder them as seen below.

 

 

 

 

 

 

 

Swapping the pieces on the top row of the original puzzle has the same effect as shifting the top piece in the picture above. Such a shift creates the picture below. Notice how we pick up that additional building. Further, each image loses 1/14th of its total height.

 

 

 

 

 

 

 

Let’s look at the original puzzle before and after the swap.

 

 

 

 

 

 

 

 

 

 

 

 

 

This type of puzzle is called a Dissection Puzzle. Our eyes can play tricks on us. We know 14 doesn’t equal 15 so something else must be happening when a puzzle indicates that 14 = 15. Mathematics allows us to push through assumptions that can lead to illogical conclusions. Math can also take something that seems quite magical and turn it into something very logical — even something as fundamental as counting to 14.

Want to look at counting through another mathematical lens? A main topic of the Global Math Project will be exploding dots. Use a search engine to find videos of James Tanton introducing exploding dots. James is a main force behind the Global Math Project and quite simply oozes joy of mathematics. You’ll also find resources at the Global Math Project web page. Take the time to look through the Global Math Project resources and watch James explain exploding dots, as the topic can be suitable from elementary to high school levels. You’ll enjoy your time with James. You can count on it!

ChartierTim Chartier is associate professor of mathematics at Davidson College. He is the coauthor of Numerical Methods and the author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing.

Global Math Week: Around the World from Unsolved to Solved

by Craig Bauer

BauerWhat hope do we have of solving ciphers that go back decades, centuries, or even all the way back to the ancient world? Well, we have a lot more hope than we did in the days before the Internet. Today’s mathematicians form a global community that poses a much greater threat to unsolved problems, of every imaginable sort, than they have every faced before.

In my Princeton University Press book, Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies, I collected scores of the most intriguing unsolved ciphers. It’s a big book, in proper proportion to its title, and I believe many of the ciphers in it will fall to the onslaught the book welcomes from the world’s codebreakers, both professionals and amateurs. Why am I making this prediction with such confidence? Well, I gave a few lectures based on material from the book, while I was still writing it, and the results bode well for the ciphers falling.

Here’s what happened.

Early in the writing process, I was invited to give a lecture on unsolved ciphers at the United States Naval Academy. I was surprised, when I got there, by the presence of a video camera. I was asked if I was okay with the lecture being filmed and placed on YouTube. I said yes, but inside I was cursing myself for not having gotten a much needed haircut before the talk. Oh well. Despite my rough appearance, the lecture went well.[1] I surveyed some of the unsolved ciphers that I was aware of at the time, including one that had been put forth by a German colleague and friend of mine, Klaus Schmeh. It was a double transposition cipher that he had created himself to show how difficult it is to solve such ciphers. He had placed it in a book he had written on unsolved ciphers, a book which is unfortunately only available in German.[2] But to make the cipher as accessible as possible, he assured everyone that that particular bit of writing was in English.

 

VESINTNVONMWSFEWNOEALWRNRNCFITEEICRHCODEEA

HEACAEOHMYTONTDFIFMDANGTDRVAONRRTORMTDHE

OUALTHNFHHWHLESLIIAOETOUTOSCDNRITYEELSOANGP

VSHLRMUGTNUITASETNENASNNANRTTRHGUODAAARAO

EGHEESAODWIDEHUNNTFMUSISCDLEDTRNARTMOOIREEY

EIMINFELORWETDANEUTHEEEENENTHEOOEAUEAEAHUHI

CNCGDTUROUTNAEYLOEINRDHEENMEIAHREEDOLNNIRAR

PNVEAHEOAATGEFITWMYSOTHTHAANIUPTADLRSRSDNOT

GEOSRLAAAURPEETARMFEHIREAQEEOILSEHERAHAOTNT

RDEDRSDOOEGAEFPUOBENADRNLEIAFRHSASHSNAMRLT

UNNTPHIOERNESRHAMHIGTAETOHSENGFTRUANIPARTAOR

SIHOOAEUTRMERETIDALSDIRUAIEFHRHADRESEDNDOION

ITDRSTIEIRHARARRSETOIHOKETHRSRUAODTSCTTAFSTHCA

HTSYAOLONDNDWORIWHLENTHHMHTLCVROSTXVDRESDR

Figure 1. Klaus Schmeh’s double transposition cipher challenge.

When the YouTube video went online, it was seen by an Israeli computer scientist, George Lasry, who became obsessed with it. He was not employed at the time, so he was able to devote a massive amount of time to seeking the solution to this cipher. As is natural for George, he attacked it with computer programs of his own design. He eventually found himself doing almost nothing other than working on the cipher. His persistence paid off and he found himself reading the solution.

I ended up being among the very first to see George’s solution, not because I’m the one who introduced him to the challenge via the YouTube video, but because I’m the editor-in-chief of the international journal (it’s owned by the British company Taylor and Francis) Cryptologia. This journal covers everything having to do with codes and ciphers, from cutting edge cryptosystems and attacks on them, to history, pedagogy, and more. Most of the papers that appear in it are written by men and women who live somewhere other than America and it was to this journal that George submitted a paper describing how he obtained his solution to Klaus’s challenge.

George’s solution looked great to me, but I sent it to Klaus to review, just to be sure. As expected, he was impressed by the paper and I queued it up to see print. The solution generated some media attention for George, which led to him being noticed by people at Google (an American company, of course). They approached him and, after he cleared the interviewing hurdles, offered him a position, which he accepted. I was very happy that George found the solution, but of course that left me with one less unsolved cipher to write about in my forthcoming book. Not a problem. As it turns out there are far more intriguing unsolved ciphers than can be fit in a single volume. One less won’t make any difference.

Later on, but still before the book saw print, I delivered a similar lecture at the Charlotte International Cryptologic Symposium held in Charlotte, North Carolina. This time, unlike at the Naval Academy, Klaus Schmeh was in the audience.

One of the ciphers that I shared was fairly new to me. I had not spoken about it publicly prior to this event. It appeared on a tombstone in Ohio and seemed to be a Masonic cipher. It didn’t look to be sophisticated, but it was very short and shorter ciphers are harder to break. Brent Morris, a 33rd degree Mason with whom I had discussed it, thought that it might be a listing of initials of offices, such as PM, PHP, PIM (Past Master, Past High Priest, Past Illustrious Master), that the deceased had held. This cipher was new to Klaus and he made note of it and later blogged about it. Some of his followers collaborated in an attempt to solve it and succeeded. Because I hadn’t even devoted a full page to this cipher in my book, I left it in as a challenge for readers, but also added a link to the solution for those who want to see the solution right away.

Bauer

Figure 2. A once mysterious tombstone just south of Metamora, Ohio.

So, what was my role in all of this? Getting the ball rolling, that’s all. The work was done by Germans and an Israeli, but America and England benefited as well, as Google gained yet another highly intelligent and creative employee and a British owned journal received another great paper.

I look forward to hearing from other people from around the globe, as they dive into the challenges I’ve brought forth. The puzzles of the past don’t stand a chance against the globally networked geniuses of today!

Craig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

 

[1] It was split into two parts for the YouTube channel. You can see them at https://www.youtube.com/watch?v=qe0JhEajfj8 (Part 1) and https://www.youtube.com/watch?v=5L12gjgMOMw (Part 2). A few years later, I got cleaned up and delivered an updated version of the talk at the International Spy Museum. That talk, aimed at a wider audience, may be seen at https://www.youtube.com/watch?v=rsdUDdkjdQg.

[2] Schmeh, Klaus, Nicht zu Knacken, Carl Hanser Verlag, Munich, 2012.

Vickie Kearn kicks off Global Math Week

October 10 – 17 marks the first ever Global Math Week. This is exciting for many reasons and if you go to the official website, you’ll find that there are already 736,546—and counting— reasons there. One more: PUP will be celebrating with a series of posts from some of our most fascinating math authors, so check this space tomorrow for the first, on ciphers, by Craig Bauer. Global Math Week provides a purposeful opportunity to have a global math conversation with your friends, colleagues, students, and family.

Mathematics is for everyone, as evidenced in the launch of Exploding Dots, which James Tanton brilliantly demonstrates at the link above. It is a mathematical story that looks at math in a new way, from grade school arithmetic, all the way to infinite sums and on to unsolved problems that are still stumping our brightest mathematicians. Best of all, you can ace this and no longer say “math is hard”, “math is boring”, or “I hate math”.

Vickie Kearn visits the Great Wall during her trip to our new office in Beijing

I personally started celebrating early as I traveled to Beijing in August to attend the Beijing International Book Fair. I met with the mathematics editors at a dozen different publishers to discuss Chinese editions of our math books. Although we did not speak the same language, we had no trouble communicating. We all knew what a differential equation is and a picture in a book of a driverless car caused lots of hand clapping. I was thrilled to be presented with the first Chinese editions of two books written by Elias Stein (Real Analysis and Complex Analysis) from the editor at China Machine Press. Although I love getting announcements from our rights department that one of our math books is being translated into Chinese, Japanese, German, French, etc., there is nothing like the thrill I had of meeting the people who love math as much as I do and who actually make our books come to life for people all over the world.

Because Princeton University Press now has offices in Oxford and Beijing, in addition to Princeton, and because I go to many conferences each year, I am fortunate to travel internationally and experience global math firsthand. No matter where you live, it is possible to share experiences through doing math. I urge you to visit the Global Math Project website and learn how to do math(s) in a global way.

Check back tomorrow for the start of our PUP blog series on what doing math globally means to our authors. Find someone who says they don’t like math and tell them your global math story.