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?
Posts Tagged: Math
7
Jan 10
Alice, Wonderland, and Math
29
Jun 09
The Summit of Math Education: Statistics, not Calculus
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.
23
Jun 09
Flickr: Mathematics Photos
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
29
May 09
The Consequences of Trusting Computers
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.
- Verizon employees are used to hearing customers complain about how they have been mischarged, and generally speaking the customer is wrong.
- These Verizon employees do not understand the math being explained to them by the customer.
- 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.
15
Jan 09
Carl Friedrich Gauss Facts
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!
