Having just completed both of Lewis Carol’s books concerning Alice and her adventures in Wonderland, the recent Boing Boing post about Melanie Bayley and her research into the idea that scenes were added into the narrative after the initial draft in order to mock new math of the day, namely symbolic algebra. As one example, Bayley likens the Mad Hatter tea party scene to the concept of the quaternion introduced by William Rowan Hamilton. Without giving away the punchline, Bayley paints an interesting picture of why the three guests at the tea party are stuck at their table, constantly swapping seats. Read the full article at New Scientist here, which gives many more examples of how Carol lampooned the so-called “new math”. Who likes imaginary numbers, anyway?

The following TED video, given by mathemagician and professor Arthur Benjamin (about whom I’ve previously blogged about here), embodies the best idea I’ve heard about math education in a LONG time. Perhaps ever. Just as I recently posted about how games like backgammon embody the 21st century in replacement of games like chess for the 20th, statistics is the central branch of mathematics for the 21st century rather than the calculus centric view of the 20th century. If you’re into math and math education, this will probably be the best 3 minutes you’ll spend today.

As an artistic diversion, I decided to search Flickr for the words “mathematics”, “math”, and “probability” on Creative Commons licensed photographs. The results were wonderful. Some of my favorites are below. Click on the photos to see explanations from the authors or to see more of their work!

Klein bottle (procrastination), by Pragmagraphr

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Love & Mathematics, by Lost Archetype

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Vegetable Meets Mathematics, by anroir

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n(n+1), by Jan Tik
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Torus with pairs of Villarceau circles, by Seb Przd

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Railroad Math, by Adamcha

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The Portrait of Conditional Probability, With A Third Ear Maybe, by DerrickT

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One in a billion, by Micah Sittig

Computers were created in large measure to solve problems. And the programs that run on computers are designed to solve these problems. And those programs generally run to do exactly what we tell them to do. And much of what we tell them to do is straightforward in the sense that the problems they solve follow the law of non-contradiction, i.e. an answer provided by a computer for a specific problem is either true or not true, but never both simultaneously.

I can program a computer to answer for me the question, “What is three factorial?”
The answer provided, hopefully “six”, is either true or not true, but is quite obviously not both.

I’m ignoring some gray areas here, particularly in the places where problems are solved by computers learning, a la genetic algorithms in the case of Roger Alsing’s EvoLisa program or neural nets in the case of GNU Backgammon. But even in these arenas, computers are programmed to perform specific tasks that solve (or approximate) particular problems. For the rest of this post, I’m generally referring to the simpler class of problems, though I will touch on how decisions made within the financial sector over the last several years have in part caused our current global economic situation based on solutions to incomplete mathematical models.

I really started thinking about this issue in relation to the now famous Verizon Math site and associated videos that show just how hapless humans can be when we depend entirely on computers to return the correct answer. What I’m saying here is that we’ve more or less reached the point where we believe that computers will always return the correct answer, and forget that while computer programmers aim to have their programs answer on the “true” side of the law of non-contradiction, sometimes this unfortunately isn’t the case.

If you’d like a poignant example, please watch this video, where several Verizon employees fail to recognize how their computer system has overcharged the customer on the phone. I don’t bring this video up to pick on Verizon specifically, but this is an issue that has gained a lot of attention over the last several months:

Now, here’s the point: Though Verizon is in the wrong, the employees are not willing to recognize the error. And why is this the case? I can think of several reasons.

1. Verizon employees are used to hearing customers complain about how they have been mischarged, and generally speaking the customer is wrong.
2. These Verizon employees do not understand the math being explained to them by the customer.
3. These Verizon employees are trusting what their computer system is telling them without fail.

And I think that all three issues played a part in the lack of understanding of the employees. But the issue that bothers me the most is the third, that the employees infallibly trust their computer system. What bothers me most about this story is that even in the face of blatant mathematical reasoning, the belief of the employees was to side with the answer provided by the computer. And the computer was incorrect. Due to a variety of circumstances, the math provided by the computer program did not match the price quote delivered by Verizon. And rather than viewing the computer as the product of human intellict, they viewed the computer as the objective arbiter.

Using the computer as an objective arbiter is a dangerous business for a variety of reasons, including most notably that the program returning the answer can be incomplete or incorrect. In the case of the recent financial meltodown, at least part of the blame can be placed on mathematical models that viewed sets of risk transactions (e.g. credit default swaps) as indepdent events. As it turns out, these events were NOT independent. Here’s an article about this. But an assumption of the program was to treat them independently. So was the computer wrong? Practically speaking, in retrospect, yes. But I don’t think that’s the right way of looking at it. The computer was answering the question based upon the programmer’s intent. And it was answering the question correctly in that sense.

What’s the moral of the story? Basically, it’s that computers answer problem in EXACTLY the ways they are programmed to do so. No more and no less. Computers are designed to be “right”, but it doesn’t mean that it will always pan out this way. Treating them as flawless objective arbiters is farming out your intellect. And while I’m certainly not saying that computers and their programs can’t be trusted (hell, it’s what I do for a living), I’m also saying that it’s a good idea to treat them as if they’re a product of humanity.

I’ve come across a few sites that list facetious facts about Gauss, similar to the wonderful Chuck Norris Facts that we’ve all come to love. Gauss, if you don’t know it, was one of those hyper-intelligent individuals who may have in fact been a space alien. It’s the only natural explanation, right? It’s hard to tell who originated the facts, but the two people who have listed the most are Matt Heath and Andrew Dolphin. My favorite two facts from these links are:

• Gauss never needs the axiom of choice, and
• Gauss didn’t discover the normal distribution, nature conformed to his will.

I thought I would give it a go as well. So here are 20 original Gauss facts coined by me this evening in a state of tiredness. Please keep in mind that if you understand at least 3 of these, you’re every bit as much of a geek as I am. Fair warning.

• Gauss can trisect an angle with a straightedge and compass.
• Gauss can get to the other side of a Möbius strip.
• “Uncountably Infinite” was a phrase coined to explain the intelligence of Gauss.
• There are no Fermat Primes greater than 65,537 because Gauss saw that Fermat was on to something, and well…he put an end to that.
• For Gauss, arithmetic is consistent AND complete.
• It only takes Gauss 4 minutes to sing “Aleph-Null Bottles of Beer on the Wall”.
• When Gauss tells you that he’s lying, he’s telling the truth.
• Gauss once played himself in a zero-sum game and won $50.
• For Gauss, point nine repeating equals whatever he wants it to equal.
• Gauss did not prove theorems, he simply stared at them until they yielded their solutions.
• Occam’s Razor – The principle stating that the explanation of any phenomenon is equal to the explanation that came out of Gauss’ mouth.
• Gauss drinks his beer from a Klein bottle.
• For Gauss, there are no indefinite integrals.
• Gauss once started falling asleep in his complex analysis class. The result…singularities.
• Imaginary numbers are simply those that Gauss has not deemed worthy of existence.
• The shortest distance between two points is Gauss.
• Once, while playing chess, Gauss solved the Knights Problem in six moves.
• Gauss is neither a Frequentist nor a Bayesian. For Gauss, the probability is always 1.
• Fermat once made Gauss angry. The result…Fermat’s Last Theorem.
• In Gauss’ mind, there is no such branch of mathematics as “Number Theory”. This is because he knows it as “Number Facts”.

Have any more? Leave one in the comments!

Welcome to the 37th edition of the Carnival of Mathematics!

In preparation for this edition, I actually managed to secure an exclusive interview with the number 37, and have included a small portion of our conversation below:

Logic Nest (LN): So 37, what have you been up to lately?

37: Oh, not much. I’ve always had a fairly good life given that I’m not only a prime number, but a lucky, irregular, AND unique prime. It’s summertime where I live, so mostly I hang out by the pool with my good friends 16, 21, and 28. We’re in a band together called the Padovan Sequence.

LN: Wow. That’s very interesting. I’ve heard that some people think you’re unlucky though. What do you say about that?

37: That’s totally a fabrication. Just because I’m the number 666 divided by its digits added together [37=666/(6+6+6)] doesn’t mean a thing.

LN: Understandable. I can see the confusion. I’ve heard that there’s a website out there that’s all about you, is that true?

37: Yes, and I must admit that I’m a bit embarrassed about it. Just because I pop up in all sorts of scientific, cultural, and historical situations doesn’t mean that I should have a fansite. I mean, come on now, people…

And it went on like that for a while…

Speaking of prime numbers, let’s kick the carnival off with this article submitted by Jeffrey Shallit from Recursivity about a Rutgers graduate student named Eric Rowland who has proved a new prime-generating formula that’s quite simple. There are some great comments on this post that include various programming implementations of the formula.

Over at Walking Randomly, Mike Croucher has posted his second Integral of the Week involving an exponential function and the square root of pi. The twist on this problem is that he gives you the evaluation and asks you to prove it. In addition, he’s asking readers to exclude the common evaluation method of converting the integral to polar coordinates. He’s taking solutions via the comments on the site. There are already a few proposed solutions, but take some time to think it over before jumping straight to the comments!

“A” presents an editorial on Being Bad at Math posted at It’s the Thought that Counts. This post is about the popular idea that it’s acceptable to confess a total lack of math ability, even though equivalent statements about difficulty in something like one’s native tongue would be seen as embarrassing. This post explores a cultural brushing off of mathematics, and how this idea should no longer be tolerated in the twenty-first century.

Another great lesson in math and culture comes from Barry Leiba, who points out a personal pet peeve of mine in his article That’s a mean median posted at Staring At Empty Pages, namely that people often incorrectly equate “median” with “average”, even at the New York Times. This one should get the blood of you stats people out there boiling!

Given the impending American presidential election, Barry Wright, III presents an educational post entitled Plurality Winner, Condorcet Loser? at fashionablemathematician – mathematics. The contents of the article explores various ideas that Barry is exploring from Donald Saari’s Basic Geometry of Voting, which is a text he is using “both for research purposes and to prepare to TA a class on the mathematics involved in Democracy, voting systems, and the like”. By definition, “a Condorcet winner is one which is ranked higher than every other alternative in a majority of decisions” while a “plurality winner is an alternative which receives more first-place votes than any other alternatives”. As the title implies, there is an interesting case when one can be both a plurality winner and a Condorcet loser. Good stuff.

In The Universe of Discourse : Period Three and Chaos posted at The Universe of Discourse, Mark Dominus gives us some information about Möbius functions, which are of major importance in complex analysis, where they correspond to certain transformations of the Riemann sphere. In particular, he looks at Möbius functions with real coefficients. In this post he talks about functions with a periodic point of order 3 (where f(f(f(x))) = x for some x) in connection to the Sharkovskii’s theorem. Both of these concepts are explained more fully at the link above.

Denise presents Math History on the Internet posted at Let’s play math!. She presents links for some WONDERFUL historical resources available on the web. As she says, “the story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.” There’s some really great stuff available at this link for anyone interested in picking up some mathematics history!

In his post Playing with Permutations at Reasonable Deviations, Rod Carvalho proposes a 2-player game. The goal is to find out whether a necessary condition is also sufficient. This game blends Combinatorics with Algebra, and even Algebraic Geometry. It’s an interesting game to consider and builds on a few other posts that Rod has written since January 2008.

Ron Cook from The Endeavour gives us an explanation of Random Inequalities in this three part series. Random inequalities are often used in Bayesian clinical trial methods, and should interest all the stats people who are reading. The first part introduces the reader to the concept of random inequalities, the second part shows how they are analytically evaluated, and the third shows how they are numerically evaluated when analytical evaluation is not possible.

Lastly, Heather wants us to think about Burnt Pancakes and Godzilla at her article 4, 6, 8, 10, 12, 14,?.What comes next? posted at 360. As she explains, “the Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another.” It’s great content explained in an entertaining manner. FYI, Godzilla evidently wears a chef’s hat when cooking pancakes.

Here are a few more submissions that have come in since I initially published last night:

Alvaro Fernandez presents Top 10 Brain Training Future Trends posted at Sharp Brains.This article discusses the concept of “brain training”, or how we keep our brains fit. This is particularly interesting given that mathematics is commonly perceived as a game for the young, as evidenced by this XKCD comic. Take care of your brains, people!

Are you aware that there is an Encyclopedia of Triangle Centers? David Eppstein is, and he describes another kind of triangle center, different from the ones at the Encyclopedia, here.

Catsynth asks the question, “What do you get when you mix a cat and a Fourier Transform?” in this post. Education and entertainment ensue! The lesson to be learned is simple: be careful of what mathematical transforms you perform on your pets. Obviously.

That’s all for this edition! If you’d like to post any additional articles to this edition of the Carnival, please contact me. I’ll be taking submissions through Sunday evening. Otherwise, stay tuned for the next edition which will be hosted by CatSynth.

I mean, how can you resist this? Hilarious, cute, and math oriented. I hope this brings a smile to your face today!

more cat pictures

Another caption given for this picture in the comments is “Delta Kitteh knows the difference”. Hehehe.

I hadn’t heard of the “Look Around You” BBC television series before yesterday, but I was nearly on the floor laughing by the end of the following video segment about “Maths”. Read up on the series at their Wikipedia entry. I’m sure the other episodes are equally as hilarious. In my opinion, there are few things better than British humor mixed with math. Enjoy the video!

Let’s be honest…there are certain subjects that a math-ish kind of blog must mention at some point. One of these obligatory topics happens to be the “0.9999999… = 1″ proof. It’s one of those facts that delights the mathematically inclined. It’s sort of like the joke that Grandpa always tells when the family gets together: you know it’s coming, and you know how much pleasure he gets out of relaying the joke, but for goodness sake, this is the 99th time you’ve heard the punchline. At any rate, there is a set of about 15 math facts that people love to talk about simply because they’re all totaling mind-blowing or sound totally nonsensical. I tend to think that the “0.9999999… = 1″ proof belongs in the latter category.

The previous digression leads me to mention the Things of Interest blog, and their absolutely fantastic post on various forms of the “0.9999999… = 1″ proof. You can find that post here. In case one proof doesn’t do it for you, this site offers several, each of which occurs at a various level of mathematical rigor. There will definitely be a proof for you here that you’ll understand.

I’m usually quite interested in the attempts of individuals to apply math to the Bible, and the Church Hopping blog has a fun little article about some people that are actually using interesting mathematical principles on the text of the Bible. Check it out.

UPDATE: Daniel from the Logos Blog has contacted me about a great post over on that site. As he said, “Thought you might be interested in today’s Logos Blog post looking at The Top 50 People in the Bible and using the IBM Many Eyes visualization. Cool thing is that anyone can play around with the charts and data…” Check it out here.

Memory is a strange thing. I was thinking about this the other night in the context of a few simple math facts that have somehow always alluded my memory. For instance, whenever I have to mentally compute either 7+5 or 8+5 I really have to think about it. I’m not quite sure why. My conjecture is that I was absent from primary school on that particular day, and simply never recovered. I had a similar experience with the lower-case cursive letter “k”.

Does anyone else out there have a similar experience? I’m sure that this phenomenon is fairly universal. Science is teaching us some amazing and unexpected stuff about how the mind works, and so I suppose that it’s not surprising that sometimes “easy” facts escape us.