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.
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Ars Technica has an interesting article today about a paper about to be released in the Proceedings of the National Academy of Sciences (PNAS) about variations to an experiment in Game Theory called the Ultamatum Game. As the Ars article explains,
The basic rules of the Ultimatum Game are simple. One person is given a stack of cash, and told to divide it between themselves and a second party. That second party is then given the chance to accept or reject the offer; if it’s rejected, neither of them get any money. Clearly, any of this free money should be better than nothing, so under assumptions of strictly rational behavior, you might expect all offers to be accepted.
It turns out that tweaking some of the premises of this game leads to some interesting results in terms of human rationality, economic systems, and guilt. I highly recommend reading the article in its entirety. Also, if you’re interested in reading the abstract of actual paper itself, it’s available on the PNAS website here. It also looks like the full paper is available in PDF format if you’d like to read it. Not sure how long it will be available, but it’s there now. Toshio Yamagishi, the lead author, has a website here.
(Photo by cljo)
In 1997, chess champion Gary Kasparov was beaten in a 6-point match against a computer. It was the first time this had ever happened. The computer, named Deep Blue, was developed by IBM after some Carnegie Mellon University graduates joined the company. Here’s what Wikipedia has to say about the hardware computing power of Deep Blue:
The system derived its playing strength mainly out of brute force computing power. It was a massively parallel, RS/6000 SP Thin P2SC-based system with 30-nodes, with each node containing a 120 MHz P2SC microprocessor for a total of 30, enhanced with 480 special purpose VLSI chess chips. Its chess playing program was written in C and ran under the AIX operating system. It was capable of evaluating 200 million positions per second…In June 1997, Deep Blue was the 259th most powerful supercomputer according to the TOP500 list, achieving 11.38 GFLOPS on the High-Performance LINPACK benchmark.
Brute force. That’s how the computer got the job done. Of course, it’s never that simple. But there is one thing that can be said for certain: If you lose a game of chess, it is because you were outplayed. Plain and simple. And I think it’s for this reason that chess became an apt metaphor for modernist notions of intelligence. Stereotypically speaking, if you ask a person the question of what game smart people play, I would guess that chess would be the most common answer in the western world (perhaps Go in the eastern world). The fate of this game is in the hand of the players entirely. There is no chance involved, with the one exception of which player plays first.
As a child, I had a hard time enjoying games that involved a substantial amount of probability. “What’s the point,” I thought, “of playing a game skillfully if it’s possible for me to lose at the last possible moment due to a bad roll of the dice or a badly dealt card?” But as I’ve grown older, I’ve come to enjoy games like this MORE on average than straightforward skill games like chess. Enter backgammon.
For those of you who don’t know backgammon, I suggest checking out the Wikipedia page here. Backgammon has been played for 5,000 years, and has evolved substantially over that time. For example, of the additions to the game, the doubling cube, drastically changed play and was introduced less than 100 years ago. Backgammon is not like chess. In a single game of backgammon, it’s quite possible for a novice to beat a master due to elements of chance. Said another way, it’s possible to play the best possible game of backgammon you can based on your dice rolls and still lose. And this is the aspect of the game that makes it an apt metaphor for the 21st century. While the 20th century dealt with certainty, the 21st will deal with probability.
And this is not to say that games like backgammon are somehow more subjective than games like chess. There are some amazing machine learning techniques used to study the game (e.g. TD-Gammon), and there are quite a few computer programs, such as GNU Backgammon, that use these techniques to outplay human opponents. Poker games like Texas Hold’em also involve an element of probability, and have grown wildly popular over the last many years. And those of you who know poker know that there are rules that govern “right” playing. Though the cards dictate play, there are strategies that maximize gain and minimize risk. The same is true of backgammon. And with the game popping up in popular culture a bit more, like in the television show Lost, I can only see backgammon growing in popularity.
(Photo by Jeephead)
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
I’ve spent some time over the last week looking at resumes. I’ve had about 100 or so cross my email inbox from a variety of job posting sites, and I was reminded of a few quirks that people tend to fall into that are not at all helpful for getting yourself a job. I’ve been on both sides of the hiring manager divide, and I thought I would relate some resume writing tips. There are a few examples given below that are IT-centric, so feel free to fill in your own examples as you’re reading.
- Do not include an objective section: Objective lines are always generic; they say nothing that differentiates you from others who are applying. I would much rather that a person has an overview section that includes your career highlights or technical capabilities. Telling me that you’re “interested in using your skills in an innovative and challenging environment” says nothing and wastes precious space. Rather, tell me that you “have 5 years experience in data warehousing technologies, including the deployment of 3 large scale data cubes.” The former is a statement that tells me nothing specific. The latter gives me a much better idea of who you are professionally and what you’re capable of accomplishing.
- Proofread: Your resume is the first piece of work you’ve created that I see. Do you really expect me to trust your level of conscientiousness if you’re not capable of adequately proofreading your public facing professional document? The answer should be no. This includes not only spelling and grammar, but consistent formatting.
- Do not use a generic resume template: Again, when a hiring manager is looking at stacks of resumes, differentiation makes a difference. If your resume blends in with 50 others, it’s a safe bet that I’m not going to remember yours. It’s worth your while to spend some time planning out the formatting of your resume for uniqueness.
- Tell me what you’ve done; tell me what it accomplished: Most people do the former, but few do the latter. I oftentimes read resume blurbs like “program effectively in C#.” A sentence like this relates to me your skill set, but it doesn’t tell me what you’ve done with this skill. A blurb like “programmed a replacement CRM system in C#, increasing application performance and saving the company $50K over the previously licensed CRM system” not only relays your skills, but it tells me what your skills have accomplished.
- Use white space: White space is capable of focusing the attention of the reader on particular pieces of the resume. More often than not, I receive what I refer to as “machine gun” resumes. These are resumes that use 8 point font, have 0.15in margins, and have full lines of text on every line. The thinking here seems to be that if you’re able to throw every possible thing you’ve ever done or read about (or whatever) at the manager, they’ll be impressed. Actually, it’s quite the opposite. If it’s difficult for me to focus on your resume, and there’s no indication of what pieces of the document you’d like for me to focus on, sensory overload takes over and it’s nearly impossible for me to remember anything about your resume.
- Do not refer to yourself in the third person: It sounds ridiculous, but I’ve seen this in a lot of resumes. You cannot vouch for yourself. By referring to yourself in the third person you sound silly. Do not do this.
- No images: Please don’t include any images. I’m sure some people will disagree with this one, but I don’t think that it’s appropriate. For tech resumes, I understand that people are sometimes interested in including graphics that represent certain received certifications. While these certifications should certainly be listed in the resume, please exclude the graphics. There have been several times where people have included graphics for certifications that have NOTHING to do with the position they’re applying to. And this tells me that they’re simply blanketing job sites with resumes rather than to tailor their search to specific positions.
There are a lot of other recommendations I could give, but others have already done a great job of this. Please check out these other resume tip sites:
Resume Tips from Taos
44 Resume Writing Tips from Daily Writing Tips
12 Important Resume Tips (YouTube)
Photo by woodleywonderworks
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.
I’ve seen Craig Damrauer’s New Math pictures across a bunch of different places on the web, and they always make me take pause. A lot of them are downright hilarious, while others are more thoughtful. Check this one out:
I actually laughed out loud the first time I saw this. And then I thought about pirates in recent news history. And then I thought about Pirates of the Caribbean. And then I realized that in my brain there was a giant chasm between these two impressions.
It’s not very often that I blog about my hometown of Pittsburgh, but today is one of those days. Congratulations to Pittsburgh, which will be hosting the G-20 summit this coming September. Good stuff. You can read the AP version of the story here. Of course, when the White House made the announcement today the media’s response was along the lines of “Whaaaaaat?!” But as a person whose lived in Pittsburgh for a long time, I’m very excited for this positive attention. We have a lot to offer in this city, and I’m glad that to show that to the world.
In addition, I must say that I’m ridiculously excited to see the Pittsburgh Penguins back in the Stanley Cup finals for the second year in a row. Sports, of course, are one of the major reasons people know about Pittsburgh, and between the Steelers and the Penguins, the city has had a good run as of late. The Pens didn’t pull it out last year, but I’m feeling good for this year.
LET’S GO PENS!
Happy Pi Day, everyone. It seems that the U.S. Congress has actually declared it this year. That’s funny. It’s also funny to surf around to various websites to see how much merchandise is available to commemorate this day. I guess everybody needs a holiday, huh? Well, if you celebrate Pi Day in any way, I suggest not getting caught up in all the glitz, but instead think back on the various ways that Pi has influenced your life; e.g. your trig class when you were 15. Check out the first link in this post to see the official Pi Day website, including some fun ideas of how to celebrate the day!
And if you’ve never seen the following visual representation of Pi, enjoy!
My post from a few days ago that mentioned the public domain status of Bertrand Russell’s “The Problems of Philosophy” reminded me to check whether LibriVox had any audio recordings of Russell’s works that are now in the public domain. If you’ve never heard of LibriVox, the site explains that:
LibriVox volunteers record chapters of books in the public domain and release the audio files back onto the net. Our goal is to make all public domain books available as free audio books.
The site has some amazing content, including a complete audio copy of “The Problems of Philosophy” (as well as a few other Russell titles). So for those of you who prefer to listen to audio books, there are some great philosophy, logic, and mathematics related books are available. Here are links for a few:
- The Problems of Philosophy, by Bertrand Russell
- Flatland: A Romance of Many Dimensions, by Edwin Abbott Abbott
- The Apology of Socrates, by Plato
- On the Origin of Species by Means of Natural Selection, by Charles Darwin
(Photo by Phil Moore)
While browsing Apple’s website for various openly available math related downloads, I came across JAME, the Java Real-Time Multi-Threaded Fractal Platform. It’s awesome. The JAME website provides the following description:
JAME is a Java real-time, multi-threaded fractal graphics platform which supports images and animations. The core of JAME is the graphics engine which supports layers, filters, effects and alpha composition. JAME creates Mandelbrot and Julia fractals and supports zoom, rotation and colours shifting.
The software is entirely free (under GPL3) and is available for Windows, Mac OS X, and Linux as long as you have met some minimum memory requirements and have an up-to-date version of the Java Runtime Environment installed. The website for the project contains some wonderful tutorials to help get you started, and your creations are exportable to various photo and movie extensions. I created the following clip entirely with JAME:
Now, the rendering of this clip took a while on my Macbook Pro, so if you start playing around keep in mind that the rendering of various movie clips could take a significant amount of time, especially if you increase the frame rate, etc. But the clip above took about 5 minutes to create minus the rendering time. That’s pretty awesome for out of the box capability, let alone all of the customizations that are possible once you learn more about the environment.
There’s a gallery of photos on the site to give you an idea of what’s possible to create more a more advanced user, and some awesome photos have been bundled together into a book that can be purchased here. I’ve seen various movies online that show fractal exploration, but it’s a whole other thing entirely to be in control of the exploration. If you have any interest at all in fractals or mathematical art, I highly suggest checking out the software.
It has always driven me totally crazy on my Mac that drop down lists were skipped when tabbing through web forms. This is especially infuriating for at least two reasons: 1) I’ve never heard of this issue in Windows, and 2) Almost every web form that you fill out includes a drop down list for either state or country, which means you have to spend the extra time to use the mouse to move back into the list and find the correct value. Just so that you know, it’s easy to change this behavior. Open System Preferences and go to the Keyboard & Mouse section. Toward the bottom of the screen, you should see the option for “In windows and dialogs, press Tab to move the keyboard focus between:“. Change the value from Text boxes and lists only to All controls, and you’re ready to roll. The screen shot provided below will show you exactly what you’re looking for. Hopefully this saves at least one of you out there some frustration!
In 2007 BBC Four released a documentary named Dangerous Knowledge, which is summarized on the official page as follows:
In this one-off documentary, David Malone looks at four brilliant mathematicians – Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing – whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.
Sounds uplifting, huh? Well, David Malone, the British filmmaker, does a really great job tracing how mathematical ideas tie together the four individuals through history. While some of the films claims are a bit…over the top and dramatic, overall the content is solid and quite informative. There’s a lot of information given not only about mathematics, but about how these four individuals dealt with various difficult aspects of their personal lives. These videos may still be up on YouTube, but I wouldn’t hold my breath for them being there for long given the copy protection issues involved. Having said that, I won’t even attempt to embed one of the videos here. But you CAN watch a clip directly from the BBC 4 official page by clicking here. If provided with the opportunity, you should take advantage of checking it out the program in its entirety.
I’ve decided this year to read or reread as much of Bertrand Russell’s work as possible. I’ve been reading The History of Western Philosophy and The Problems of Philosophy (the latter of which is now public domain), and have been trying to digest On Denoting by reading it every few weeks. With this in mind, I was excited to recently find a set of several articles written by Douglas Anele for Vanguard Online on Russell. They serve as a good biographical introduction, and also provide a bit more information than average on Russell’s view of sexual ethics and religion. Perhaps most interesting is that the essays are written from a Nigerian perspective. They’re certainly worth reading.
Part 1: The Power of Reason: A Celebration of Bertrand Russell
Part 2: The Power of Reason: A Celebration of Bertrand Russell
Part 3: The Power of Reason: A Celebration of Bertrand Russell
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!
So I certainly won’t be covering any new ground by bringing up the debate of whether mathematics is created or discovered. There are plenty of resources online covering this topic. Some of the better ones I’ve read are here and here. I’ve been thinking about this question over the past day given the Radiolab program I mention in my last post. Suffice is it to say that Plato certainly believed that mathematics was discovered, as evidenced through his theory of forms. While thinking over the issue, I remembered an interview that I had read earlier this year in Discover Magazine with Max Tegmark, an Associate Professor of Physics at MIT. Though most of his works centers around conventional cosmology, he has some interesting theories about the universe, and how it is physically composed of mathematics. As Max puts it in the interview, “Mathematical things actually exist, and they are actually physical reality”. In this sense, he doesn’t align himself with Plato’s theory of forms given that mathematics IS reality, rather than mathematical forms existing in some ideal way outside of reality. The interview is informative from a cosmological perspective as well, particuarly in regard to the four levels of multiverse he describes.
The latest edition of WNYC’s Radiolab is named “Yellow Fluff and Other Curious Encounters“, and brings up a few interesting questions. Namely, it asks whether scientific knowledge is discovered, or whether it is created. With its typical casually educational tone, this issue brings us stories from individuals such as Steve Strogatz, who sets the theme of the hour long program with a story about parabolas. Steve is currently a professor in Applied Mathematics with Cornell, and in the opening of the program he talks about being a young student, and how a simple experiment in school led him to a eureka moment. Though Radiolab is an audio program (which can be downloaded or listened to on their site), they’ve bundled this episode with a video interpretation of Steve’s story, which can be watched below.
Kurt Gödel died 31 years ago today. From the little I’ve read of his life, and from the even smaller amount that I truly grasp from his work, I believe that only in reality could such a fantastic and somewhat lamentable figure come to be. He was included in the infamous Vienna Circle, but was himself a Platonist. He was shy, reclusive, and prone to illnesses both physical and mental. He was a friend to Albert Einstein. And he shook the world of mathematics in a way that destroyed the Hilbert program. In simple terms, he showed that the mechanization of mathematics could not be fully automated, or that mathematics was not something that could be neatly placed in a box and tied up with a bow.
John W Dawson Jr. explains the first of Gödel’s Incompleteness Theorems by saying, “In his 1931 paper Gödel showed that, no matter how you formulate the axioms for number theory, there will always be some statement that is true of the natural numbers, but that can’t be proved. (That is, objects that obey the axioms of number theory but fail to behave like the natural numbers in some other respects do exist.)”
John Von Neumann, certainly one of the greatest mathematicians of the 20th century, had the following to say in a letter shortly after the publication of the Incompleteness theorems:
Thus today I am of the opinion that 1. Gödel has shown the unrealizability of Hilbert’s program. 2. There is no more reason to reject intuitionism (if one disregards the aesthetic issue, which in practice will also for me be the decisive factor). Therefore I consider the state of the foundational discussion in Königsberg to be outdated, for Gödel’s fundamental discoveries have brought the question to a completely different level.
Another way of summing this up is to say, “this work has changed the way we must view mathematics.” I have to imagine that the fame of the majority of famous people peaks in the prime of life, only to wane with time and death. Only the smallest number of people see their influence grow with time, as reflection shows their achievements to be truly monumental. Gödel, I believe, sits comfortably in the latter group.
Obviously, I have a bit of a crush.
Dan Gilbert has a few talks posted on the TED website, and the one that’s embedded here has to do with the idea of expected value, and how this mathematical idea applies to practical life. The idea of expected value was formalized by Bernoulli, and Dan explains it as the product of “the odds that an action will allow us to gain something, and the value of the gain to us”. Quite simply, an expected values is a way of quantifying whether or not a decision is a good one. If you watch the first 5 minutes of the video below, there are some simple examples given, but I’ll throw one out as well.
Imagine that I walk up to you, and I say the following: “I’ll tell you what. Right now, I’m thinking about a color that’s in a rainbow. If you can guess what color I’m thinking about, I’ll give you $14. If you guess incorrectly, you give me $1.” Do you take the bet?
Well, assuming that I’m not cheating in any way, and there are 7 colors in a rainbow (ROYGBIV), you have a 1/7 chance of guessing correctly. And if you win, you get $14. Given the definition from above, the expected value of this bet is equal to (1/7) * ($14) = $2. In other words, given this betting situation you stand to make $2. And our arrangement dictates that if I win, you give me $1. Comparing your possible dollar gains or losses, this is certainly a good bet. You should take it.
Where this idea gets extremely interesting is when it’s applied to human psychology and everyday decision making. Basically, we’re very bad at computing expected values in our daily lives. Obviously, I don’t expect that any of you generally has someone coming up to you offering bets of the aforementioned type, but decisions bombard us daily.
The reason I enjoy this talk so much is that it takes a mathematical idea and tells us why it’s PRACTICAL. Gasp! There are a lot of interesting examples given in this talk, and Dan is an engaging speaker. I certainly recommend this video as a worthwhile way to spend half an hour!
I’ve come across Roy Tanck’s weblog several different times, and in several different ways. And I must say, it’s just about the most stylish use of Wordpress I’ve ever seen. Amongst other skills, Roy has developed several plugins, including WP-Cumulus, which is shown below. I’m thinking about making this a permanent part of my sidebar, but at the very least I wanted to include it into this post just to show you one of the many amazing things that’s easily possible to include within a Wordpress blog. This plugin, which is visually fantastic, took me about 5 minutes to install at max. Its also been ported to various other blogging platforms as well. There’s really good stuff to find over at Roy’s blog. Check it out.
WP Cumulus Flash tag cloud by Roy Tanck requires Flash Player 9 or better.
I really enjoyed reading this post about the physicality of a calculator over at Social Mathematics. The idea of having a simple calculator available to you in your workplace or home office setting is intriguing to me. I must admit that I do enjoy having a basic little calculator at my disposal. I know that I can open up a spreadsheet on my computer to take care of some maths, or that I can simply use my brain, but there’s something about using a calculator for basic arithmetic that is appealing to me. I tend to use the calculator as a sort of extension to my short term memory.
Of course, it’s quite possible to take this idea to the extreme. Many moons ago at a previous job, there was an individual who worked there that was not the sharpest crayon in the box. We’ll call this individual “Slowbe”, and we will assume for this story that Slowbe was a man.
Slowbe was well known for his occasional mental lapses, and I remember hearing a story about him sitting in a meeting with several others, going over various numbers in a spreadsheet. This was a spreadsheet that Slowbe had created himself. As the group was looking at the numbers, Slowbe decided that he wanted to add up several numbers in the spreadsheet. So he left the conference room, walked around 300 meters round trip to his desk, and returned with a pocket calculator.
Now, I’m not sure if you, the reader, are familar with Excel. But when you have a spreadsheet open, and there are several numbers within this spreadsheet which you’d like to add up, doing this requires about 0.4 seconds worth of work. In essence, Excel can act quite easily as a really stinkin’ powerful calculator. Everyone in the meeting (most of whom were IT people), sort of looked at one another with “What just happened?” kind of faces.
This, my friends, is overkill.
Picture by draggin of a Little Professor Calculator created by Texas Instruments in 1976.
For those of you interested in the iPhone, a developer named alpheccar has released an application for the device named Mandelbrot, which is a fractal generation tool of the famous Mandelbrot set. It’s a fantastic little free application that allows the user to zoom in on various pieces of the fractal using the typical touch screen commands of the iPhone. Users are also able to discover Julia sets within the fractal, and to switch back and forth between the Mandelbrot and Julia sets.
Many color palettes are included, and the most recent version (1.3.3) allows for palette animation, which creates a great visual effect. The animation is relatively fast, and can be either automatically or manually set. By selecting manual animation, users have the option of setting a slide bar that controls both the speed and accuracy of animation. It’s also possible to save animations to the photo book on the phone.
It’s a great for education and visual entertainment. I really appreciate that this developer took the time to develop such an application and then to distribute it freely. That’s a great gesture. For those of you with an iPhone, a direct link to the application can be found here.
FYI, the photo from this post was taken within the Mandelbrot application and later exported to my computer. Good stuff. Good job, alpheccar!
I mean, how can you argue with this?
UPDATE: Thanks to ll for finding attribution for this image. It comes from GraphJam, and the pie chart is here. The author is Jamie Schimley. I’m looking forward to browsing through more hilarity at this link.
ORIGINAL: Thanks to AdArena.net for this one. You should check them out. They’ve got some great stuff posted. I’m not sure where this picture originates, but it certainly made me laugh out loud. If anyone has further attribution on this photo, let me know. It’s truly hilarious.
There are a few math stories that I tell rather often, and this is my favorite of the bunch. Other people seem to enjoy it as well. Given that I’ve never committed the story to writing, I thought that it was time to do so.
During my undergraduate experience, we had an optional two week winter session that allowed for individuals to take simple introductory courses in order to meet various graduation requirements. One particular winter, I signed up to take a psychology 101 course, and the second day of the class we learned about behaviorism and conditioning. I started thinking about various ways that I had conditioned myself, and I realized that there was a single recurring thought that oftentimes pass through my mind with no discernible pattern or regularity. The thought started when I was around 17 years of age, and at the time of this story I think I was 22. It didn’t matter if I was brushing my teeth, out with friends, driving my car, etc. It didn’t matter if I was daydreaming or having a serious conversation. The thought was this:
“Nineteen squared is three hundred sixty one.”
Now, I have no recollection of learning this fact. I never memorized it, and I cannot think of any practical reason that this thought would stick so strongly in my brain somewhere (I learned later that a full size Go board was 19×19, alas). But the thought came to me nonetheless.
During my senior year of undergraduate work, I had a number theory class, where the professor wouldn’t allow us to use calculators for arithmetic problems we would work through. One day, in class, we were working on a rather extensive arithmetic calculation when the professor suddenly hesitated at the blackboard. He turned to us, and asked, “Does anyone know what nineteen squared is?”
Well, I instantaneously answered, “Three hundred sixty one.” Not even one second later. Everyone in the class sort of turned and looked at me with expressions that said something like, “Who in the world keeps track of the square of nineteen?” or “How did he calculate that so fast?”
It was as if the whole of my life had led up to that one moment. If was as if the math gods had prepared me for this solitary moment, where the haunting thought that appeared for no reason whatsoever would ACTUALLY be useful.
Needless to say, when class ended, I was slightly afraid to leave the room. Given that I had obviously fulfilled my purpose in life, I felt that an anvil would imminently fall from the sky and put an end to my misery. But to my great benefit, no anvil fell. And ever since that day, the recurring thought has left my mind. Well, except when I tell this story.
(Photo by rexhammock)
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, Ξ_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 came across this article today on Science Daily that talks about the Pirahã, which, according to Wikipedia, are “an indigenous hunter-gatherer tribe of Amazon natives, who mainly live on the banks of the Maici River in Brazil”. The Science Daily article introduced me to the fact that this tribe has no concept of precise numbers. While they do use indefinite numerical terms such as “some” and “more”, this group does not seem to have any representation for concepts such as “one” or “two”. As MIT professor Edward Gibson states, “here is a group that does not count. They could learn, but it’s not useful in their culture, so they’ve never picked it up.” Absolutely fascinating. You should certainly check out the two links above, especially the portion in the Science Daily article that describes some of the experiments carried out by Gibson and his MIT team that have further illuminated this portion of the Pirahã culture.
This article intrigued me so much that I dug a bit deeper, and found that Daniel L. Everett, the Chair of Languages, Literatures, and Cultures from Illinois State University, has spent a good portion of his career working with the Pirahã people. He has collaborated in the past with Gibson on various projects in the past. Some info can be found here. There’s a great New Yorker story that was published in April 2007 on Dan here that’s certainly worth a look. Here’s a teaser from this article:
The Pirahã, Everett wrote, have no numbers, no fixed color terms, no perfect tense, no deep memory, no tradition of art or drawing, and no words for “all,” “each,” “every,” “most,” or “few”—terms of quantification believed by some linguists to be among the common building blocks of human cognition.
It’s a very long article, but it paints a beautiful picture of linguistics, cognition, faith, and personal relationships. It’s packed full of great questions. There’s a LOT that’s in these writings I’ve linked to that I haven’t even brought up (including the idea of recursion in linguistics), so I urge you all to read more! There are also some great links for further reading in the Wikipedia article linked to above, including several scholarly papers.
A friend let me know quite a while ago about this story presented on NPR’s site entitled “Mathematicians Explain Tape’s Tendency to Tear”. It’s an explanation of a recent Pedro Reis article in the journal Nature Materials describing the annoying tendancy of tape to narrow while unpeeling it from the roll. As the article explains, Reis’ work “could help engineers test thin films for strength and reliability” The audio of the story is also available on the NPR site.
I love this story because I can imagine Pedro first thinking about this problem while unpeeling a roll of tape. I don’t know if the inspiration actually came this way, but its a great mental image that conveys the idea that some of the most interesting problems to solve are right under our noses.
Here’s the abstract of the paper from Dr. Reis’s website:
Thin adhesive films have become increasingly important in applications involving packaging, coating or for advertising. Once a film is adhered to a substrate, flaps can be detached by tearing and peeling, but they narrow and collapse in pointy shapes. Similar geometries are observed when peeling ultrathin films grown or deposited on a solid substrate, or skinning the natural protective cover of a ripe fruit. In this work, we have shown that the detached flaps have perfect triangular shapes with a well-defined vertex angle; this is a signature of the conversion of bending energy into surface energy of fracture and adhesion. In particular, this triangular shape of the tear encodes the mechanical parameters related to these three forms of energy and could form the basis of a quantitative assay for the mechanical characterization of thin adhesive films, nanofilms deposited on substrates or fruit skin.
I spent some time today reinstalling the LaTex Render plugin for Wordpress. I’ll now be able to put some wonderful looking 
graphics into the blog. I’d like to thank Steve for spending time both writing and debugging the code. So for the time being here is your mathematical moment of zen, for which I also need to thank Steve:
![\pi = \sum _{k=0} ^{\infty} \dfrac{1}{16^k} \left[\dfrac{4}{8k+1}-\dfrac{2}{8k+4}-\dfrac{1}{8k+5}-\dfrac{1}{8k+6}\right]](/latexrender/pictures/39f6d1160a5dc4b74aeb55fa063bdf58.gif "\pi = \sum _{k=0} ^{\infty} \dfrac{1}{16^k} \left[\dfrac{4}{8k+1}-\dfrac{2}{8k+4}-\dfrac{1}{8k+5}-\dfrac{1}{8k+6}\right]")
Feel free to also check out the LaTeX in Wordpress post that I wrote a while ago. It should still be mostly up to date. It’s also possible to leave comments using LaTeX syntax as long as they are wrapped in [ tex ] [ / tex ] tags.
If you haven’t yet been introduced to the Stuff White People Like blog, you’re in for a treat. There is a new article on statistics that you can read here. Basically, the blog is comprised of witty, yet strangely accurate descriptions of…well…things that white people like. It’s truly hilarious. For instance, take the first line of this article, “White people hate math. If you want to befriend white people, mention “that weird Asian calculus teacher who drew perfect circles” and how much you hated his class…” Awesome.
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’m sorry, but this XKCD comic was so wonderful that I simply had to post it. If you don’t read this comic religiously, shame on you. I had to shrink the photo down a bit to make it fit, so feel free to click on the comic to head over to the XKCD page.
I must confess that I’ve never learned to use an abacus (or a slide rule, for that matter). I came across the following video, and thought that it would act as another great view into the wonderful world of mental math. It’s quite tremendous what the human brain is capable of. Check it out:
I’ve always hated when professors have midnight due dates for assignments. This is because there doesn’t seem to be a grand understanding of what a midnight due date actually means. For instance, consider that I’m taking two courses, and that each of them has an assignment due at midnight on April 1. For Professor X, midnight on April 1 actually corresponds to 00:00:01 on April 1, i.e. the very first moment after midnight on April 1. For Professor Z, midnight on April 1 corresponds to 23:59:59 on April 1, i.e. the very last moment before midnight on April 2. This has bothered me so much that I’ve done a small amount of research, and have come to realize that Professor X is actually correct.
From the official Greenwich Mean Time website, “Every day starts precisely at midnight and A.M. starts immediately after that point in time e.g. 00:00:01 A.M.” This may seem to be obvious, but there’s a remarkable amount of confusion over the issue. Professors who understand this call for due dates of 11:59:59 PM on a particular day. The airline industry always rounds similarly so as to not confuse customers. So if you ever have an assignment due at midnight, it may be in your best interest to check with your professor about what she or he really means!
Here are a few links other than the official one above that will give you more information about noon and midnight:
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.
This is perhaps one of the most amazing videos I’ve every watched on the Internet. I was literally left speechless. I can accomplish certain small feats of mental math, but this is absolutely unbelievable. Arthur Benjamin shows us some inspiring abilities of the human mind. I’m sure he has spent a fair amount of time learning his methods, and that to him his abilities are perfectly normal (in some sense!), but it’s great to watch someone with this talent. I highly recommend watching the video in its entirety.
Slashdot is linking today to an article that considers the implications of the 10 year anniversary of the defeat of chess grandmaster Gary Kasparov by IBM’s Deep Blue computer. The article (here), written by philosopher Daniel Dennett, considers the possible differences, or lack of differences, between humans and machines. I’ve linked to other pieces considered by Daniel Dennett on this blog, and I consider him to be an articulate and fair judge over matters of this type. It is highly worth your time to read this piece and to think it over a bit.
A few days ago on Slashdot there was an article about a statistical model that claims to be able to accurately predict the result of a war nearly 4 out of 5 times. Here’s a snippet from the University of Georgia’s press release on Dr. Patricia L. Sullivan’s study: “‘If you know some key variables – like the major objective, the nature of the target, whether there’s going to be another strong state that will intervene on the side of the target and whether you’ll have an ally – you can get a sense of your probability of victory,’ said Sullivan, whose study appears in the June issue of the Journal of Conflict Resolution.” Very interesting. Statistics is a beautiful, and very misunderstood, field. When I hear about claims like this my ears definitely perk up. In general, studies like this propose that particular variables (such as a poor military strategy) are predictive of other events (such as a military victory). There’s obviously a cause/effect chain reflected in this type of idea. And believe it or not, there is a LOT of study in the area of cause/effect relationships. People like Peter Spirtes at Carnegie Mellon University spend a lot of time studying these causal relationships.
So while that claim that a statistical model can predict the outcome of wars should be taken with a grain of salt, everyone should consider the fantastic amount of research (and quality science) that is going into these types of causal models.
I came across this wonderful introduction to several famous paradoxes quite a while ago, but haven’t taken the time to inform you all about it. Daniel Haggard presents a non-technical explanation of five age-old paradoxes that have both delighted and confused humanity. It’s a very accessible read and I recommend it for everyone interested in the strange logical conundrums that surround us. I particularly enjoy his section on Newcomb’s Paradox, which boggles my brain every time I think about it. Honestly, I’m glad that paradoxes exist. I mean, nothing I say is true, right?
There’s a wonderful article over at ars technica about the mathematics of sudoku. Agnes M. Herzberg and M. Ram Murty recently wrote an article in which they explain some of the mathematical underpinnings of the popular puzzles. They explore several fundamental questions such as “does a given sudoku have a unique solution?” The link above provides a great introduction to the formal paper, and also includes a link to the PDF version of the paper. Check it out. One of the great tidbits from the article includes the stat that there are over 5.5 BILLION unique sudoku puzzles. Yikes. That’s keep even the most committed addicts busy for a while!
I can’t remember exactly how I came across this hilarious article, but I highly suggest checking it out (warning: it contains what some might deem slightly offensive language). It’s all about a guy named Jason who keeps being mistaken for a robot in his instant messaging conversations. It’s highly amusing. For those of you who don’t know, Alan Turing proposed a test in 1950 to gauge whether or not a computer can think. Here’s briefly (and incompletely) how the test goes:
- Recruit two humans, one to participate in the test and the other to judge it.
- Recruit a computer whose inventor claims it can think.
- Put the human judge in a room that contains only a device capable of receiving and sending text messages.
- Have the human judge type questions into this device which she would like to ask the human participant and “thinking” computer.
- One question at a time, the “thinking” computer and human participant answer these questions by writing out text answers and transmitting them back to the device in the room with the judge.
- If the judge cannot determine through the answers to these questions who is the human participant and who is the “thinking” computer, the computer wins and passes the test. In other words, since the computer tricked the judge, it can be said to think.
This is a simplistic version of the test, but it’s definitely the gist of it. In Turing’s paper he guessed that by the year 2000 a computer would have been built that was able to pass the test. He was wrong. Even now, in 2007, a computer hasn’t been built that has consistently passed this test. Interesting, huh? The full text of Turing’s paper which details the Turing test can be read here. The article is called “Computing Machinery and Intelligence”. It’s an accessible paper for anyone with these types of interests. And for more information regarding the Turing test, check out its Wikipedia article here.
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.
Wolfram, the makers of the software Mathematica, are offering a $25,000 prize to the first person who can prove whether the above 2, 3 Turing machine is universal. From the website (here):
“A universal Turing machine is powerful enough to emulate any standard computer.
The question is: how simple can the rules for a universal Turing machine be?
Since the 1960s it has been known that there is a universal 7,4 machine. In A New Kind of Science, Stephen Wolfram found a universal 2,5 machine, and suggested that the particular 2,3 machine that is the subject of this prize might be universal.
The prize is for determining whether or not the 2,3 machine is in fact universal.”
What a great idea! I’m really curious to see how long it take for someone to claim the prize. If you’re interested in understanding more about what a Turing machine is, please check out the above links.
I don’t know how many of you are familiar with the TED conference, but if you don’t know anything about it I suggest checking out their website here. The yearly conference focuses on technology, entertainment, and design, and it defines its mission as “spreading ideas”. You can watch most of the conference talks on this website. There are some serious gems that you should watch. I suggest the following:
- Hans Rosling: Debunking third-world myths with the best stats you’ve ever seen
- Dan Dennet: Can we know our own minds?
- Richard Dawkins: The universe is queerer than we can suppose
- Majora Carter: Greening the ghetto
Also, I just watched the following presentation made by the Reverend Tom Honey. As the brief synopsis of the talk indicates, “It’s a classic problem in theology: How can the existence of evil be reconciled with a God who is supposed to be all-loving, all-knowing and all-powerful? Many Christian thinkers have attempted answers to this question. In the days following the thousands of personal tragedies recorded during the South Asian tsunami of 2004, Tom Honey pondered those answers and found them wanting. Instead, he penned his own, personal, and sometimes dramatic response to the tsunami. This is a courageous talk for a Church of England vicar to have given. It concludes that certain traditional concepts of God just won’t do … and calls for believers and nonbelievers alike to dig deeper in their quest for truth.”
I was extremely impressed with his thoughts. None of them were particularly new to me, but this talk does provide a good theological redux of the issues involved in the current debate over the problem of evil. It’s an extremely personal and honest appraisal of thought, and I enjoyed listening to it.
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.
I’m writing this post with the hope that it will be helpful to people who face the same computer predicament that I did a few days ago. Here’s a little bit of background information: Last Tuesday I met John Chol Daau, who is from Sudan. He grew up as one of the Lost Boys of Sudan, forced to leave his home and wander hundreds of miles through Africa to survive. If you don’t know much about this particular humanitarian issue, I suggest spending a small amount of time reading up on it. Anyway, John told me that his PC was experiencing a debilitating virus, and asked if I would look at it. I said that I would. After spending quite a bit of time reading through various website forums, here’s a short description of the problem and its solution:
Problem: The PC (which runs Windows XP with SP2) starts normally. The Windows splash screen appears correctly and then the login prompt correctly loads. You can then enter your user name and password like normal, but as soon as you try to login you are IMMEDIATELY logged back out again. The desktop doesn’t even load. It moves immediately back to the login window where you can then enter your user name and password again. No matter how many times you try to login you always experience this immediate logout. Even if you try to login to the computer in safe mode you still experience the same problem. This problem is documented on Microsoft’s website here.
Solution: I’m sure this behavior can be caused by many different problems, but the most common cause is a virus. If you’re familiar with the Windows registry, this virus changes a few registry key values that makes it impossible to login to your computer. If you’re not familiar with the registry, don’t panic. I’ll post links to a few articles that very clearly explain how to fix this problem. Basically, the virus makes two very simple changes to your computer that render it useless. In order to fix the problem, you have to change these two things back to the way they were while your computer was working.
Easy Fix: The “easy” solution to this problem can be found here. In order to use this fix you have to have your Windows XP install CD. This is the CD that contains the files necessary to install the operating system on your computer. You probably have this disk stashed in a drawer somewhere. You should note that there’s a difference between the Windows XP install CD and the recovery CD that may have shipped with your computer. It’s actually possible that when you bought your computer that it didn’t actually come with a Windows XP install CD. Sometimes computer manufacturers will only ship you a recovery disk, which is altogether different. You need your Windows XP install CD so that you can run an application called the Recovery Console. The link above should provide documentation on how to use the Recovery Console. Unfortunately, this fix didn’t work for John’s computer, but it may work for yours.
Slightly Harder Fix: This fix is the one that ended up working to fix John’s computer. A detailed explanation of this fix can be found here. It requires you to have access to another Windows PC with a CD burner (even if it’s a friend’s computer). You have to download a program called BartPE, which is one of the greatest recovery tools that exists. For this particular problem, BartPE will enable you to quickly change the two settings that the virus messed up. You may need a Windows XP install CD for this method as well. But it may be possible for the program to find what it needs from your friend’s computer without having to have access to this disk.
If you have any questions, please feel free to contact me. The above links should give you the tutorials you need to fix the problem. And if you use a PC you should use a virus protection program! If you don’t, you’re asking for trouble! Good luck!
Here’s yet another reason why you should make sure to learn basic math. I suppose that this is one way to lose your job…
Is it possible to teach a horse how to do math? Around the year 1900 it looked like the answer was a resounding yes. My friend Trevor suggested that I write something up about Clever Hans, who was a horse capable of performing mathematical feats on par with a young human teenager. Hans could add, subtract, multiply, divide, work with fractions, differentiate musical tones, and understand the German language. At least, Hans could apparently do all of these things. Several other pages have described the Clever Hans case in great detail, so I’ll refer you to them. Please check out one or more of the following links to learn about an interesting phenomenon that has less to do with math and more to do with psychology:
There was an announcement yesterday that a collaboration of mathematicians from the United States and Europe have mapped the structure of E8, which is a 248-dimensional Lie group. It’s actually even more rich than that, but I think the concept of a Lie group is intense enough for one post. What interests me most about this particular problem is that there was some SERIOUS computer horsepower that went into the solution. As the Yahoo! news story (link) indicates, “While the human genome, which contains all the genetic information of a cell, is less than a gigabyte in size, the result of the E8 calculation, which contains all the information about E8, is 60 gigabytes in size.” Yikes. Amongst other practical applications this result will provide some good information for physicists who study string theory. The reason for this is that structure of E8 is both symmetrical and extremely complex. Please check out the American Institute of Mathematics page on the E8 project here for more information. There’s a lot of great information on their site. So what does the structure of E8 look like? Here’s the picture:
As if we didn’t already have enough to thank Guinness for, I learned something very interesting today about a connection between this fine beer and statistics. I’m reading the book “Randomness” by Deborah Bennett (Amazon link), which is an introductory text concerning the basics of probability theory and statistics. It’s an accessible read, and contains several nuggets of interesting historical information. I suggest checking it out if you’re interested in this sort of thing. Here’s what Dr. Bennett has to say concerning the Guinness connection to statistics:
“The best-known early demonstration of a random sampling experiment was performed by William Sealy Gosset, a research chemist working for the Arthur Guinness Son and Company Ltd. in Dublin. Gosset was studying the relationship between the quality of Guinness beer and various factors in the beer’s production. The brewery was continually experimenting with soil conditions and grain variety that might produce improvements in crop yield, and Gosset was intent on bringing all the benefits of statistics to the brewery’s agricultural experiments.”
In short, Gosset discovered the t-distribution while working on this problem for Guinness. If you’ve ever taken an introductory course in statistics you’ll probably remember working with t-tests. They’re a way of correctly analyzing small sample sizes, where “small” usually means samples less than size 30. There are some other fun facets of Gosset’s work on this problem, including the fact that he published his findings in papers under the pseudonym of Student. Read Gosset’s Wikipedia page here for other general pieces of information.
I’ve decided to try to do a weekly feature called “Proof of the Week,” where I’ll explain a mathematical proof that I find particularly illuminating or intriguing. Part of the reason that I write so many math posts on this blog is that I feel that much of the beauty of math is an acquired taste. So my desire is to help serve as a “waiter” who introduces people to some of the fascinating tidbits of the subject. I know a lot of people who run (or roll their eyes) when they hear the word “math.” It brings back terrifying memories of grade school multiplication tests and what not. I don’t blame you. My fourth grade math teacher used to slam a book shut at the end of every minute long mad-dash times test. It scared the bejeus out of me every time. Even so, I still love math.
Most of the proofs I’ll be talking about from week to week won’t be overly intense. I’m sure that many of them will require some general knowledge background, but nothing too academic. My hope is that by explaining some interesting results that you too might see a little bit more of the grandeur contained in this subject. I remember when I took my first proof-based math class during my sophomore year of college. I knew that a lot of rigorous math had to do with proofs, but it wasn’t until my 20th year of life on this planet that I learned what they were really all about. And here’s one of the many revelations I came to rather quickly:
Math is nowhere near as objective as I thought it was growing up. In other words, I always thought that there was a unique answer to every problem. Because of this, I think that many people regard math as some sort of rigid 60 year old person wearing starched clothing who eats the exact same three meals a day and whose house is painted a single shade of grey. To use another image, many people view math problems as some sort of assembly line. You insert a problem at the beginning of the line, perform a bunch of robotic methods, and the answer plops out at the end of the line. If this is your view of math, no wonder you think it’s boring! There’s no art in these images. There’s no movement or color in these pictures.
Math is nowhere near as simple as an assembly line. At least not at its heart. But since most of us grow up learning rote methods to solve problems many of us find the subject to be too tedious or mundane. And I don’t blame you for thinking that. What I WOULD like for you to consider is that you’ve been misled. Like any other academic discipline, math is a growing organism. Hopefully in these “Proofs of the Week” I’ll be able to illuminate some of the beauty that is contained in math. The first of the series will be up in a day or two. Stay tuned!
Since today is March 14 (3.14), I’d like to wish everyone a happy pi day. Read more about the wonderful number of pi here and here. I’m sure that some of you have seen the following visual representation of pi before, but this animated gif should give everyone a refresher on exactly what pi is. This informative animation was created by John Reid. [Note: I have this animation set so that it will only loop 6 times in total. If you'd like to see it again please refresh or restart your browser.]
As a lifetime resident of Pittsburgh, Pennsylvania, I’d like to take this moment to express my happiness at learning that the Pittsburgh Penguins hockey organization will be staying in the city. Thank you to Mario Lemieux and the countless others who put forth both effort and patience in making this deal a reality. If you have no idea what I’m talking about, read a little bit about the outcome of this long ordeal here. I grew up in a family that followed hockey closely, and I loved watching Lemieux and company win two Stanley Cup championships for the city of Pittsburgh in the early 90s. I’m glad that I’ll be able to see more hockey in this city! Let’s go Pens!
Fractals are beautiful things. If you don’t know what a fractal is, you should read this for a general overview. The most famous fractal (and one of the most mathematically simple) is the Mandelbrot Set, which is named after its discoverer Benoît Mandelbrot. For awhile I’ve wanted to include some sort of video of the Mandelbrot Set “in action”. The following video shows what happens when you “zoom in” on a portion of this fractal. It’s quite interesting. Suffice is it to say that if I ever fall into a bottomless pit, I hope that bottomless pit is like falling into the Mandelbrot Set. At least that way there would be good stuff to look at. There are several other videos out there on the web that show other perspectives of zooming into this particular fractal, so if you like what you see here head over to YouTube or what not and search for some more! The math rock song in the video was written by Jonathan Coulton. If you listen to the lyrics they actually explain a little bit about how to graph this particular fractal. Check out his website here. [Warning: For those of you with sensitive ears, the song that accompanies the video has a few curse words scattered throughout!]
Voila! Turing machine muffins! What a delicious idea. If I had used this method while learning about these universal machines I probably would’ve been much happier. What’s a Turing Machine, you ask? Read about them here and here. Check out other pictures of muffin madness here and here. Thanks to Boing Boing for this info!
Yesterday I listened to a fantastic podcast from the NPR program Intelligence Squared U.S.. From the website, “Intelligence Squared U.S. brings Oxford-style debating to America – one motion, one moderator, three panelists for the motion and against.” The specific program I listened to examines the question, “Is America Too Damn Religious?”, which is a particularly fascinating question to me. The panelists present a scope of different opinions on this issue, and most of the comments are well thought out. Everyone is generally respectful, which is a trait I find important in this type of programming. The reflections are at times theological, political, and practical, which was an interesting mixture to listen to. The entire program can be found here. It’s possible to download a free MP3 version of the program on this page. For those of you with iTunes, you can also find an abbreviated version of the debate by looking for the Intelligence Squared U.S. podcast (which is also free). The entire debate runs about 1.5 hours (the abbreviated podcast is about 1 hour), but I think it’s worth it. For those familiar with the issues surrounding this debate I wouldn’t expect to find too much new information, but what I thought was interesting was the particular representation of viewpoints associated with the panelists.
I just spent about 20 minutes trying to figure out how to take screen shots in Windows XP on my MacBook Pro (using Boot Camp). I thought I would write up the simplest answer of how to do this, in the sense that you don’t have to install anything or remap your keyboard. Here’s all you have to do:
Under the Start menu, select All Programs, then Accessories, then Accessibility, and then On-Screen Keyboard. When you do this the Windows XP on-screen keyboard will appear on your screen. This on-screen utility includes the PrintScreen button (labelled psc in the same position as where the F13 key should be located. That’s it.
If you want to take a screen shot of only the active window (using the Alt+psc command), make sure that the on-screen keyboard isn’t in the area of the active window, otherwise it will be included in the screen shot.
Anyway, I searched around for far too long to find that little piece of information, so hopefully someone in the same predicament will easily be able to find this post (and the answer)! Since the Apple external keyboards actually include the F13 key (while the MacBook and MacBook Pro built-in keyboards do not), you can use PrintScreen slightly easier with the plugged-in keyboard.
UPDATE: As pointed out by Stuart in the comments below, if you’re using Boot Camp v. 1.3, Fn + F11 now maps to print screen. You can print only the active window with Option + Fn + F11. Thanks for the tip!
Are you looking for the best web site to buy printer ink? We have the best brands of printer ink cartridges including Epson ink cartridges on the Internet. We also have toner cartridges at Printerinks.com.
Yes, it seems that Edwin A. Abbott’s wonderfully original novel about the travels of the square named A. Square through one, two, and three dimensional space will soon be brought to video. The website for Flatland: The Movie can be found here. The trailer for the movie is available on the website or on YouTube here. Here’s the synopsis of the movie:
Flatland: The Movie is an animated film inspired by Edwin A. Abbott’s classic novel, Flatland. Set in a world of only two dimensions inhabited by sentient geometrical shapes, the story follows Arthur Square and his ever-curious granddaughter Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms with the truth of the third dimension, risking dire consequences from the evil Circles that have ruled Flatland for a thousand years.
Well, it sounds like there has definitely been some license given to modify the orginal plot of the novel. But I have to say that the plot modifications were immediately forgiven once I found out that Martin Sheen was going to do the voice of A. Square. Who can argue with that? Also, Tony Hale, of the late TV show Arrested Development, will be playing the King of Pointland. At any rate, as the website explains, “The movie will be part of an educational DVD, which will include the original text from Abbott’s book.” Also, it looks like it will be coming out in spring 2007, which isn’t too far away! If you’re dying to get a copy you can sign up on the website for priority access to the DVD. While you’re anxiously awaiting its release, I suggest reading Abbott’s original work. It’s a really quick read and is imaginative and original.
Though I’m a little bit late on this, Science Magazine recently published a great article on the scientific breakthroughs of 2006. Topping the list was the proof of the Poincare Conjecture, which I’ve posted about several times on this blog. You can read their synopsis of the breakthrough proof here. It turns out that from the media’s perspective the drama behind the proof is almost greater than the mathematical result. Basically there was a lot of name calling among some members of the mathematical community concerning who made certain contributions toward the eventual proof. Sad. Apart from the soap opera, the author explains the Poincaré Conjecture in a very accessible way, which should be understandable by anyone who’s interested in reading it. This proof will be a huge deal for mathematics over the coming decades, and should help mathematicians better understand topics such as the “Navier-Stokes equation [of fluid dynamics] and the Einstein equation [of general relativity].”
Perhaps the most interesting thing to note is that the article focuses not only on the result of the problem (the proof itself), but also the methods used to solve the problem. This is an hugely understated part of the mathematical process. I’m of the opinion that when the general populace thinks about math that they are fixated on two things: the problem and the answer. What people tend to overlook is the process of problem solving. In math, there are not always clear-cut methods that explain how to get from point A to point B. A lot of thought is sometimes necessary to figure out how to traverse the path. The Poincaré Conjecture is a monumental achievement not only because of the end result, but also because of the original steps the solvers of the problem (especially Grigori Perelman) took to get there. These steps will be used in other problems; they are not exclusively tied to this one specific problem. Once again, congratulations to Perelman and the other mathematicians who had a hand in making this historic achievement!
If you haven’t read this article written by Jeff Tietz for Rolling Stone magazine, I highly suggest reading it. The piece presents the pork industry through the business of Smithfield Foods, which is the largest pork producer in America. The article suggests that one of every four pigs in America is slaughtered by this company. There’s a lot of familiar ground covered for those who know about industrial food production, such as the living conditions of the pigs, unsanitary excrement levels, and animal antibiotic consumption. The article paints a bleak picture, but manages to do so while giving the reader a fair amount of readable statistics. There’s also a historical portion that tells of the story of Joseph Luter III, the chairman of Smithfield Foods. It’s interesting to read the economic implications of the growth of such a large company. The keyword of the entire article is pollution. This specific company (and others like it) exposes nearby people and land to a copious amount of pollution. One statistic relayed is that Smithfield’s largest processing plant “dumps more toxic waste into the nation’s water each year than all but three other industrial facilities in America.” Yikes.
Part of what I like about this writing is its focus on the impact on humanity. Oftentimes these types of stories are sad stories only from the perspective of the animal (which is still true), but doesn’t measure the ways that businesses like Smithfield Foods are adversely influencing human lives. Tietz focuses on the ways that pollution generated on the farms sickens people and keeps them from leading normal lives. Fish who used to live in the areas waterways are now dead, ending the employment of countless local fishermen. Workers in the hog plants die while becoming overwhelmed with the toxic fumes they breathe. People living in the area have contaminated drinking and bath water. The list goes on.
I’ve never been the type of vegetarian who gets angry with omnivores. But articles like this one give me reason to keep up my chosen eating habits. Our current methods of food production are not sustainable. When millions of gallons of pig shit per year are dumped into our rivers, that is NOT sustainable. When pigs are pumped full of drugs that breed antibiotic resistant germs that make people sick, that is NOT sustainable. And on and on. Okay, my rant is over. Read the article, it’s full of information that people should know.
I am the proud owner of a 15″ Macbook Pro. It has 2GB of RAM. My iBook only had 256MB of RAM. Think about it. I must say that I’m extremely pleased so far with the purchase. Even Dreamweaver and MS Office are running really well under Rosetta. I really can’t complain at all about the performance. I’ve also installed Boot Camp, so I’m also running Windows XP. I’ve decided not to worry about either Vista or Office 2007 until the summer. By that point a lot of people and businesses will have moved over. So far I have to say that Windows is running quite well. Once I do some heavier duty stuff I’ll report back on what I’ve found. I thought about using Parallels but decided against it for the time being. I’ll wait until Leopard to see if Apple will enable users to switch between operating systems without having to reboot. If that’s the case I have no reason to buy Parallels (other than to easily play with Linux). Otherwise I’ll have to reconsider. As far as my other software needs are concerned, TeXShop was already available as a universal binary, and TeX installed really easily on the machine. Time will tell. If anyone has any particular questions about what I think about the computer let me know. The self-portrait in this post was taken with the built-in iSight camera.
I love when people intentionally mix together mathematics and art, and one of the best examples of this merger that I’ve seen for awhile can be found here. As the site itself says, “this experiment attempts to convert the first 10,000 digits of pi into a musical sequence.” You have the ability to choose several preset music scales, or can choose 10 notes either manually or randomly. It takes a few minutes to play through the sequence, and the sounds are quite transfixing. Even though this meshing of pi and music is somewhat artificial, the result is wonderful. It’s worth checking out.
My friends Tim and Megan over at The Franktuary recently sent me a math puzzle and asked me why it worked. I thought it would be a fun little exercise to explain on the site. So here’s the puzzle:
1. First pick the number of times a week that you would like to go out to eat (more than once but less than 10).
2. Multiply this number by two.
3. Add five.
4. Multiply by fifty.
5. If you have already had your birthday this year add 1757. If you haven’t, add 1756.
6. Subtract the four digit year you were born.
You should have a three digit number.
The first digit of this was your original number (i.e., how many times you want to go out to eat in a week).
The next two numbers are your age.
It’s said that this is the only year (2007) that this will work.
Here’s why this works:
First, you should notice that this puzzle has nothing to do with the number of times that you’d like to eat out every week. You could ask a person to randomly pick a number between 1 and 9 (including 1 and 9) and the puzzle would work out the same way.
If we go through step by step here’s why this works:
1. Let’s call the number that you initially pick x. It’s important to the puzzle that this number be between 1 and 9, including 1 and 9. You cannot pick 10. After we go through the explanation of the problem you should quiz yourself and ask why you cannot pick 10.
2. Multiply your number by 2. Now we have 2x.
3. Add 5 to this number. Now we have 2x + 5.
4. Multiply this number by 50. Now we have (50)(2x + 5). If we multiply this out we have 100x + 250.
5. Let’s assume that you’ve already had your birthday this year. According to the puzzle we should next add 1757 to our number. So we have (100x + 250) + 1757. Again, if we simplify we get 100x + 2007. Notice that the second half of this equation (after the plus sign) is the four digit year. Once we’re done with the puzzle you should ask yourself why we only add 1756 if we haven’t had our birthday yet this year.
6. At this point we’re supposed to subtract our four digit year of birth from our number. Since I was born in 1981 I’ll subtract this number. So our number is now (100x + 2007) – 1981. Simplifying, we get 100x + 26. This final equation is split into two parts. The second part (after the plus sign) will be your age. I am 26 years old. This is correct. The first part of the equation will be a multiple of 100, and will always be a three digit number. In other words, if you consider the three individual digits of this number, it will always be digit x followed by two digits of 0. So the number represented by the first part of this equation will basically be the number x00, if that makes sense. If you add 26 (or whatever your age is) to x00, you’ll always get x26.
The first digit will always be the number of times you’d like to eat out every week and the last two digits will be your age. And there’s the answer!
Here’s another two quiz questions for you:
1. Will this puzzle work for people of ALL ages, or just for specific ages?
2. How could you modify this puzzle so that it works in 2008 (or any other year for that matter)?
Well, hopefully that explains the puzzle. Let me know if you have any questions!
If you grew up in Pittsburgh you always knew that it was officially the holiday season when you saw this commercial. Seriously, I know it’s dumb, but this 30 second TV spot always makes me feel a bit sentimental. I thought I would post it here since I know quite a few Pittsburgh folks who read this site. Enjoy!
Yet another reason why I’m a total dork. If you don’t understand what’s going on just laugh when the people in the video laugh. Trust me, it’s funny. Also, this is actually what you do in graduate school. 
Watch the music video here.
I don’t know what the hell this article seems to be talking about. I think that the article would like me to believe that several professors at a school in the UK have somehow “cracked” the lottery system. Well, let’s examine their method. They “bagged the big prize by using two boxes, 49 pieces of paper and a large amount of brain power”. Check. It took the group 4 years and about $9000, but they actually hit the six digit number to the tune of $13 million.
Ahem…
Luck. These people hit the lottery. These people didn’t “crack” anything. This so-called system of “brain power”: luck. Though I don’t know all of the particulars regarding the lottery game they won, I can be sure that their method didn’t vastly improve their odds. (If anyone knows exactly how the odds were changed using the method described in the article please let me know.)
It’s PROBABILITY, people! The state wouldn’t run a lottery if they didn’t make money off of it. The lottery is designed to make the state money. And I assure you that they make a LOT of money off of us playing.
So please, don’t let articles like this fool you. There is no system for winning the lottery. At least not one that’s profitable. Or legal. So go invest your money in a house or a mutual fund. Consider this my public service announcement of the day.
This evening I went to the theater to see the movie The Prestige, which was a fantastic. I highly recommend it. The acting was superb and the subject matter was intriguing. And hell, David Bowie was in it! Magic, and ultimately its relationship to science, were key themes in the story. Though there were a few minor holes in the plot it was definitely the type of movie that spurred some discussion after the viewing.
I don’t want to spoil anything with this post, but I will say that Nikola Tesla plays an important role in the story. When I got home I immediately turned to Tesla’s Wikipedia entry where I learned that three pieces of information played up in the movie were at least partially true:
1. There was some bad blood between Edison and Tesla, not least of which because Edison seems to have screwed Tesla out of a large chunk of change for some brilliant work he did for Edison.
2. Tesla did move his base of operations for a time to Colorado Springs, Colorado, where he did some work on “wireless telegraphy”.
3. Especially later in life Tesla held some fantastic theories regarding subjects such as UFOs and the like. It seems to be the case that some of these beliefs came out of an untreated psychiatric disorder (probably OCD).
Interesting. Feel free to read more about Tesla at the link above. And I’d highly recommend seeing the movie. It’s one of the best I’ve seen this year.
How could I NOT include this picture of a pumpkin carved into the likeness of Alan Turing? It just wouldn’t be right to overlook this one! Read more about Turing here. In case you don’t know much about Turing, let’s just say he laid a substantial foundation for computer science.
Also, here’s a post from Boing Boing describing some mystery explosions that seem to be occurring around the world. More Halloween goodness!
I thought I should let everyone know that my friends over at Hot Dogma, which happens to be the best hot dog shop in the city of Pittsburgh, have recently had a syndicated news story come across from the AP. You can read that article here. Also, there’s another local news article here and some local news video of the story here (link may not be Mac friendly). I don’t understand the situation fully, but Hot Dogma is being legally forced to relinquish its name due to a certain type of copyright infringement with another restaurant located in Miami, FL. The so-called Dogma Grill believes that it’s necessary to squash the name of a business located hundreds and hundreds of miles away. Okay. Good for you. At any rate, Hot Dogma won’t be closing its doors, but will be born again under a different name. If you’re curious about what that new name may be, click on the link above!
Well good grief, how can I possibly improve on this headline? In the spirit of Halloween, check out this article over at Live Science, which attempts to deny the existence of vampires through some simple exponential reasoning.
I have always enjoyed Halloween. It conjures up images of mischievousness in my mind, rather than images of gore or fear. Then again, I’m a sucker for the mysterious, so Halloween fits that category quite nicely. Also, we’re coming up on All-Saints day as well, which you can read about here. At any rate, Emily and I will be passing out candy at our house this Halloween, so stop by! Also, thanks to my mom for this article, who recently celebrated her birthday on one of the most mysterious days of the year: Friday, October 13!
I saw this photo over on Boing Boing this morning and just had to put it up here. I’m not well versed in the specifics of windmill driven energy, but damn these things are good looking. It’s a great merger of technology and nature in my opinion.
I’ve been meaning to write out my thoughts on the Engage Pittsburgh event that Emily and I attended a few weeks back. This event, sponsored by The Sprout Fund, was meant to serve as an idea-roundup for ways that individuals wanted to change the city of Pittsburgh. There were ideas fitting many categories, including transportation, economic development, housing, arts & culture, etc. Based upon the ideas that came out of the event The Sprout Fund will be issuing around $100,000 to help with the implementation of some of these initiatives. If you’d like to see one of the ideas I helped to conceive you can link to it here. The idea centers around changing the Pittsburgh busways so that bikers could use them as well. Though there would definitely be some infrastructure issues I think an idea along these lines would be extremely beneficial for the city.
The whole event, though noble in intent, was lacking in several ways. First, Emily and I were immediately split up. Though we came to the event to experience and brainstorm ideas together the staff separated parties that came together. Though I could understand the intent of this procedure there was no advance warning that such a split would occur. This was surprise one of the day.
Though I was really hoping to engage personally with individuals there wasn’t much time for such interactions. The whole event was extremely structured, and we were meant to keep a stringent schedule. In that sense the event felt extremely inorganic. While there was a large contingent of people who knew one another (the “in” crowd), I was not one of these people. It seemed that both the staff and “in” individuals weren’t interested in getting to know me. I was saddened by this reality. But toward the end of the day I did get to interact with a few people a bit more fully, so I wouldn’t call the day a relational waste.
I commend The Sprout Fund for their initiative on this event, but I thought that the details could’ve been carried out in a friendlier and more efficient manner. As a silly example, though there was a lunch provided only about 25% of them were vegetarian. But this was an event for the GRANOLA crowd of Pittsburgh. I’m being a bit facetious but you get my point. By the time Emily and I got there the vegetarian option had vanished. And I know that a lot of other folks experienced the same problem.
I’m hoping that the event actually advances change in the city. I’m weary that there was a lot energy generated that will vanish into the ether. Hopefully I’ll be proved wrong on that point.
I was meandering through the Slashdot archives this evening and came across this discussion about philosophy’s role in computer science. I think the conversation is illuminating on several levels. There are blatant IT professionals coming from one angle versus hard-core philosophers coming from another. While there’s a lot of overlap in perspective each person tends to accentuate a particular part of the (dis?)connection. I think the threads are worth reading both from a professional and an academic standpoint.
There are obvious links between the disciplines, notably the fact that concepts such as computability were born from the work of folks like A.M. Turing, but I often wonder if computer scientists think about this with any sort of regularity.
Five years ago I heard the news while sitting in Harbison Chapel at Grove City College. I was recently watching a bunch of Daily Show clips on YouTube and hadn’t seen the following segment for a long time. I thought it would be good to share today. It’s Jon Stewart’s monolugue from the first show on air after 9/11/2001. Watch it here.
There’s an interesting article over at ABC News explaining that the number of Americans buying into evolutionary theory has been in decline. The story attributes this surge of opinion to several sources, including “religious fundamentalism, inadequate science education, and partisan political maneuvering”. While I’m not currently interested in discussing the relevancy of these possibilities, the article also talks about the misuse of probability theory by creationists. Here’s a snippet of the argument:
A bit more specifically, the standard argument goes roughly as follows. A very long sequence of individually improbable mutations must occur in order for a species or a biological process to evolve. If we assume these are independent events, then the probability of all of them occurring and occurring in the right order is the product of their respective probabilities, which is always an extremely tiny number. Thus, for example, the probability of getting a 3, 2, 6, 2, and 5 when rolling a single die five times is 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/7,776 — one chance in 7,776.
Check out the above link to read the rest of the story.
Yesterday I read a paper by Tom Mitchell, the chair of the Machine Learning Department at Carnegie Mellon University, about the field of Machine Learning. This very readable introductory paper (a 7 page PDF document) can be found here. As stated by Dr. Mitchell, “Machine Learning seeks to answer the question ‘How can we build computer systems that automatically improve with experience, and what are the fundamental laws that govern all learning processes.’” The article surveys areas such as the current application successes of Machine Learning, active research questions, and ethical considerations in the discipline. If you’re at all interested in interdicsiplinary computer goodness, I highly suggest checking it out.
Part of what I love about the idea of Machine Learning is that it unavoidably collides computer science, statistics, philosophy, psychology, neuroscience, and related learning fields. Researchers in this area are actively exploring the overlap (which is sometimes messy, no doubt) of various disciplines, following the trail of questions. I plan on writing more about this later. Good stuff. Check it out.
I’m sure most of you have heard by now, but on Tuesday Grigori Perelman refused to accept the Fields Medal for his work on the Poincaré Conjecture. I’m sure there are people with all sorts of opinions about this, but the First Post has an interesting take on the situation. Also, if you’re looking for another explanation of why any of this matters anyway, I suggest reading Jordan Ellenberg’s Slate article here.
If you have no idea at all what I’m talking about, then you should read my first and second posts on the subject, which should supply a bit of mathematical background information and redirect you to more extensive resources.
Since I’ve been getting more and more traffic as of late I’ve decided to change over the site to its own dedicated domain name. Logic Nest was the first phrase that came to mind, so I went with it. Please take a moment to update your RSS feeds for this site if you use them, and if you’re here from StumbleUpon I would appreciate a thumbs up! Thanks everyone.
Also, I’ve added a Contact Me page so that you’re able to get in touch with me easily if interested. If you have any news/questions that you’d like for me to address on this site please shoot me a message.
I started thinking about the RSA Factoring challenge the other day when I received my RSA SecureID® fob to log into the UPMC network offsite. According to the RSA website, “The RSA Factoring challenge is an effort, sponsored by RSA Laboratories, to learn about the actual difficulty of factoring large numbers of the type used in RSA keys. A set of eight challenge numbers, ranging in size from 576 bits to 2048 bits is posted here. Each number is the product of two large primes, similar to the modulus of an RSA key pair.” So if you feel like trying to make yourself an easy $200,000, try to factor the following number as the product of two primes:
25195908475657893494027183240048398571429282126204
03202777713783604366202070759555626401852588078440
69182906412495150821892985591491761845028084891200
72844992687392807287776735971418347270261896375014
97182469116507761337985909570009733045974880842840
17974291006424586918171951187461215151726546322822
16869987549182422433637259085141865462043576798423
38718477444792073993423658482382428119816381501067
48104516603773060562016196762561338441436038339044
14952634432190114657544454178424020924616515723350
77870774981712577246796292638635637328991215483143
81678998850404453640235273819513786365643912120103
97122822120720357
If you’re interested in learning about the history of RSA, which is an algorithm for public key encryption that helps to make internet security tick, you should read this Wikipedia article or check out the RSA Laboratories website.
It looks like the New York Times has a story today about the proof of the Poincaré Conjecture. You can read the article here. I just love how the media always plays up the “insatiably crazed introvert” image of mathematicians, in this case overly dramatizing Grigory Perelman, the Russian mathematician responsible for much of the proof methodology. I blogged about this a while back, but this article definitely gives a much more in-depth explanation of the history of the conjecture. Happy reading everyone!
I’ve noticed a trend…
My high school calculus teacher used a calculator that looked like it had been built in 1985. My undergraduate algebra professor used a calculator that looked like it had been built in 1990 (perhaps earlier).
And though I hate to admit it in some strange sense, my calculator of choice is my TI-85, which was introduced to the world in 1992.
Maybe it’s nostalgia that I always reach for my TI-85 when sitting down to do personal finances. Maybe it’s the fact that I must’ve spent YEARS of my life playing Tetris on that sucker. But it’s probably because I’ve learned how to use it with incredible efficiency over the years. For some reason the calculator seems to violate my desire for extreme technological innovation. I constantly desire the cutting edge in consumer electronics. So what is it exactly that keeps me from updating to a TI-89? I may never know.
So does anyone out there have a specific piece of technology (calculator or otherwise) that has followed you through the years? Leave a comment if you have something. And if you’re looking to have some Texas Instruments nostalgia of your own you should check out ticalc.org. They’ve got some good stuff over there.
Thanks to Boing Boing I found Randall Munroe’s website this morning, which describes itself as “a webcomic of romance, sarcasm, math, and language”. Some of the comics are tremendously funny, especially if you’re a dork, which I am. Here’s one of my favorites:
Here are a few of my other favorites from the site (warning: there’s a bit of swearing in some of these if you’re offended by that kind of thing):
Computational Linguistics
Centrifugal Force
dPain Over dt
Science
Su Doku
Fourier
Happy laughing everyone!
First off, I want to welcome everyone from StumbleUpon. Hopefully you’ll enjoy your stay while you’re here. Also, I’m going to be shifting the focus of this blog away from the theological for the indefinite future. I’ve had my fill of adding to the vast landscape which is the emergent conversation. I’ve reached saturation point, so to speak.
So into the future I’m going to try to post more on the topics of math, logic, and the philosophical underpinnings thereof. Check out the Logic/Math category on this blog to see what I’ve written so far on these topics. I’m also going to install a LaTeX plugin in the near future so I’ll be able to post some more symbol intensive stuff. Also, if you have any recommendations of topics you’d like to read about or related articles/blogs that would be good for me to link to, please let me know by leaving a comment. Take it easy everyone.
Check out this article over at Science News Online that explores some of the wonderful mathematics references that have found their place in episodes of The Simpsons. For instance, in one episode “Kwik-E-Mart proprietor Apu brags that he can recite pi to 40,000 decimal places. “The last digit is 1,” he announces. To get that detail right, the Simpsons writing team faxed a query to NASA, where mathematician David Bailey obliged with the digit in question.”
It’s good to know that great comedy doesn’t always have to be brainless.
There’s a great article over at Seed Magazine about the seventh annual “Gathering for Gardner” conference, which in short celebrates the literal magic of mathematics. As the write-up explains, the Gathering for Gardner conference is “a bi-annual pilgrimage honoring Martin Gardner, who, from 1957 to 1981, enraptured mathematicians and scientists, hobbyists and professionals, magicians and puzzlists and skeptics alike with his “Mathematical Games” column in Scientific American.”
I highly recommend reading their account of the conference, which is highly entertaining. Ah, mystery and magic!
264 years ago today Christian Goldbach wrote a letter to fellow mathematician Leonhard Euler in which he conjectured a very simple idea.
This conjecture, though hauntingly simple, has resisted proof for over two and a half centuries. Though I have no evidence to support it, I would say that the so-called Goldbach conjecture has led to the insanity of more than one mathematician.
Simply stated in its modern form, here is the Goldbach Conjecture: Every even number greater than four can be expressed as the sum of two prime numbers. The number 2 is excluded since it is itself prime. Sounds simple, doesn’t it? Well, one can only wish that every easily stated problem had an easily stated answer.
In fact, sometimes the simplest sounding problems are the most resilient. This conjecture would be a prime example. Although millions of even numbers have been computer tested in confirmation to the conjecture, there is always that faint possibility that there’s an untested even number out there that doesn’t fit in (does anyone remember the Fermat Primes?). So while the Goldbach Conjecture seems experimentally obvious, its proof is still hidden.
I love mystery. I suppose that’s at least partially why I love math and logic. There are century old mysteries that exist in these disciplines. These mysteries provide a portion of the shared history of mathematicians and logicians. Also, I believe these mysteries provide hope. They give hope that even though math and logic have historically come a long way that there are still beautiful discoveries to be made. In other words, the mathematical landscape is lush.
It’s a shame to me that the popular perception of math views the discipline as total “objective” fact. When seen in this way the actual organic nature of the subject is skewed or forgotten. Math is not as “objective” as you’d think. At least not in the way you were taught as a child. To borrow a famous example, when first learning about the structure of an atom, a physicist may be asked to imagine a model similar to our solar system, with the planets revolving gracefully around the sun. It’s only later when the true nature of the atom is revealed as a much more complex reality governed by probability. A similar type of metaphor may be given for the mathematical sciences.
Aim to think deeper. Aim to revel in mystery. Thank you Christian Goldbach.
The Xinhua News Agency website is reporting that two Chinese mathematicians, Zhu Xiping and Cao Huaidong, have proved the Poincaré Conjecture. Or rather, I should say, that the proof is currently being scrutinized by a panel from the Asian Journal of Mathematics, where the proposed proof will be published upon consensus.
According to its MathWorld entry, the Poincaré Conjecture states “that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes”, where “a three-sphere is simply a generalization of the usual sphere to one dimension higher”.
The Clay Mathematics Institute also had listed the conjecture as one of its Millenium Problems, each of which carried a one million dollar reward for a proof. Their explanation of the problem puts it this way:
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.
As the article makes clear, the movement toward this proof spanned both decades and nationality. American, Russian, and Chinese mathematicians were instrumental in progression of ideas which finally cracked the conjecture. I hope that this example illustrates the global and community effort of current mathematicians in their work.
Congratulations to Zhu Xiping and Cao Huaidong!
While driving home from Bethel Park today I saw two somewhat “interesting”, if not troubling, sites:
1. On a church marquee I saw the following: “Roadmap to heaven: turn right and go straight.” There are so many ways to disagree with this assertion that I don’t even know where to start. So I won’t. I relate it merely for your possible amusement.
2. I saw a PA license plate with the specialized message “GCC GRAD”, which I’m assuming stood for “Grove City College Graduate”. Grove City College, my undergraduate alma mater, is a fairly right-wing Christian school. The car which the license plate was attached to was a large SUV. Inconsistent? Sheesh…
Last week Emily and I had the fortune to visit St. Louis, Missouri for several days during our Midwestern tour vacation. We asked several locals what sites they recommended seeing while there, and each person without exception told us that we needed to see the Glass in the Garden exhibition at the Missouri Botanical Gardens. The installation was designed by Dale Chihuly, who amongst other achievements holds both a Masters of Science in Glass Blowing and a Masters of Fine Arts in Sculpture.
I must admit that art museums are often tiresome to me. Large galleries are a sensory overload. Walking through one is an inundation of passive information in a generally sterile environment. In other words, it’s hard for me to actually experience the art. With exceptions, I am merely an observer rather than a participant. Part of what I enjoyed so much about the Chihuly pieces was their experiential nature. I felt that I was part of the art, and that my being present actually mattered.
If I had to summarize my impression of the installation, I would use the phrase “entwinement of the inorganic with the organic”. While walking through the nearly uncountable species of plants in the gardens there are pieces of ornamental glass decorating the landscape. Sometimes it actually appears as if a plant is growing glass branches. In one of my favorite pieces there are tubes of iridescent blue glass standing amongst an army of cool green bamboo. It almost looks as if the two could easily coexist together in nature. Around every corner was a merger of glass and plant. I almost felt like a child on an Easter egg hunt, just waiting for the next treasure to appear unexpectedly. If you’d like to see more examples of what I mean you can see Emily and my pictures here.
While I wouldn’t count Chihuly as my favorite artist (this “honor” is held by James Turrell), I have to thank him for his creative methods. Both Turrell and Chihuly design art which is experiential rather than merely observable. Both use light and color with beautiful effect. Both blur the lines between organic and inorganic. These traits, to me, embody art at its fullest.
According to its Wikipedia entry, a numbers station is “a shortwave radio station of uncertain origins.” According to both direct and indirect evidence a numbers station is a radio station used by a government to communicate secretive information to its spies. A stereotypical numbers station broadcast is basically comprised of a seemingly nonsensical sequence of numbers or letters read by a human voice. The idea is that the spy listens to the broadcast and then is able to decipher the sequence using a previously received decoder. In theory, if the spy has kept the decoder secretive, the code is almost literally unbreakable. In other words, this is a fairly secure way to pass sensitive information.
You can listen to a good example of a numbers station broadcast here. I suggest not listening to this alone at night, as it sounds a bit creepy. This recording is one of many included in the Conet Project, which according to Archive.org is “the first comprehensive collection of Numbers Stations recordings released to the public”.
Given the news of various NSA activities in America over the last several months I thought it would be interesting to write something about “spy” activity. There’s a lot of great in-depth information (including many audio examples of numbers station broadcasts) located in both the Wikipedia article and on Archive.org (both previously linked), so I suggest checking these out if you’re interested at all. Who knows, the next time you turn on a shortwave radio maybe all you’ll hear is numbers.
So this isn’t new at all, but when I saw it about a month ago for the first time I thought it was hilarious. Yeah yeah I know it should be “as x approaches 8 from above”, but come on now, let’s just let ourselves laugh a bit. Let me know if you have any funny math jokes/quotes!
For those who don’t know, April happens to be Mathematics Awareness Month (MAM). According to the MAM website, “Mathematics Awareness Month is held each year in April. Its goal is to increase public understanding of and appreciation for mathematics.” The organization which sponsors this month is the Joint Policy Board for Mathematics, which “is a collaborative effort of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.”
Each year has an appropriately labeled theme, and this year is no different. 2006 is the year of Mathematics and Internet Security. There are a few links to articles related to math and internet security on the website, including topics such as internet voting, public key cryptography, internet password security, computer viruses, and secure data storage. So if you fall into the category of people who doesn’t understand what math does in the real world, read some of this stuff. There is no escape from math, MWAHAHAHAHA!
The website for MAM can be found here.
Happy Mathematics Awareness Month Everyone!
Aldo Coffee Company is probably responsible for brewing the best coffee in Pittsburgh. It’s located in Mt. Lebanon, and is less than a five minute walk from the Mt. Lebanon trolley stop. They take coffee quite seriously in this cozy place, and they also happen to serve some scrumptous Panini sandwiches as well. The owners are friendly, the decor is inviting, and the drinks are just plain good (my favorite so far is the Cappuccino Aldo).
Also, it just so happens that the establishment boasts as a barista Ms. Belle Battista, who will be traveling to Charlotte this weekend to compete in the United States Barista Championship. In other words, she makes some dang good coffee. Actually, the Pittsburgh Post-Gazette newspaper wrote an article about her yesterday. You can read that article here. So if you’re looking for some good coffee, I highly suggest checking this place out.
Good luck and happy brewing this weekend Belle!
It happens to be the case that there are two different verses in the Old Testament which provide for an approximation of π. In the NIV translation both I Kings 7:23 and II Chronicles 4:2 give the following measurements for a tank which would be enclosed in the “First” Temple:
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.
Taking these measurements along with the ratio for π given as circumference over diameter we have a coarse approximation for π of 3. While not astounding in accuracy, I’m always excited to see how subtle tidbits of mathematics invade even religious scripture. In other words, sometimes math is beautifully inescapable.
In you’re interested in learning more about these two verses I suggest reading an article called “On the Rabbinical Exegesis of an Enhanced Biblical Value of π” written by Shlomo Edward G. Belaga. An online version of the article can be found here. The article surmises that the Biblical narrative lends itself to a much more accurate approximation of π. It’s worth a look if you’re intrigued by such ideas.
While on a walk in the nearby forest I came across a lamb grazing in an open pasture. The lamb was home in this place. The sun beat down on the late autumn day as the animal trod graciously along the earth, frequently stopping to taste the fragrant grass beneath it. Wanting to learn more, in the way of my training, I took a step forward. Cautiously at first, I approached the lamb.
As I was equipped with a scale and a ruler, I took the lamb’s measurements. I observed its feeding methods and took note how it traveled along the yielding ground. I examined its wool, wondering the temperature the lamb underneath would be kept in the wintertime. I scrutinized its sound against other tones cataloged in my mind.
Having expended my options I took the lamb away with me to a lab, in order to continue experimentation. Wanting to capture its true nature in order to extend my findings to all who would study lambs after me, I painlessly took its life. Dissecting the lamb I noticed its organs and how its digestive system worked. I noted the distinction of its physiology to that of other animals. I cross-sectioned and test tubed and electrified and poked and prodded along in my studies.
Until in the end, as I gazed around me, I realized that the lamb was gone. Having taken it away with me, in the way of my training, I had taken the lamb to a place it did not want to travel. If I could relive that autumn walk, equipped with a scale and a ruler, I simply would have stopped to watch the lamb, for in the end I would have known it in a greater fullness.
I wanted to take a moment to remind everyone that there are well over 6 BILLION people living on the planet this very day. In fact, we’re now closer to 7 billion than we are to 6 billion. Soak in the thought…
6 billion. The most people I’ve probably ever seen in person at one time is about 94,000 (at a home Penn State football game in Beaver Stadium). In the metropolitan Pittsburgh area there is a little over 1 million. Utterly ridiculous.
6 billion people. As a “math person” I’m inclined to think about large things. Some of them are countably large and some of them are uncountably large. When I think about abstract hugeness, like the amount of real numbers (an uncountably large thing), it doesn’t faze me. But 6 billion people?! Surely you must be joking…
Also, over half of the world’s population lives on less than $2 a day. Think for a moment, on an average month, about how much you live on a day when factoring in rent/mortgage, utilities, internet, phone, pets, food, entertainment, healthcare, etc. It’s staggering.
Too be honest, I’m at a loss. I’ll let the thought end here, though my brain is swimming.
While Emily and I are in the process of totally remaking our downstairs space we’ve moved all of our belongings to other parts of the house. This includes our television, which has made its way upstairs to our “guest room”, which is now serving as our dorm living headquarters. It’s basically the only living space in the entire house right now, so it’s grand central station for the time we’re spending in the house and not sanding, stripping paint, or whatever. Since the TV is upstairs we currently don’t have it hooked up to cable. On Saturday evening, while Em was out with some old friends, I was bored and tried to check out what could be found on the television. Amidst the static I found three channels which offered actual content. The content was distressing. Rather, the content was a reminder of some of the worst parts of the American condition. Here’s what I found on the stations:
1. A generic home shopping network program, i.e. the worst possible caricature of American consumerism, where the watcher is asked to consume mostly worthless cultural artifacts. I believe when I was flipping through that the fare of the moment was some sort of terribly ugly fake jewelry.
2. An infomercial, sponsored by GM, for their OnStar system, i.e. the most blatant and strange type of marketing message that exists. While the show tried to pawn itself off as a source of safety information in reality it was just a long commercial for GM vehicles.
3. Christian religious programming of the worst kind. I made myself watch it for about a quarter hour. I was sickened. I’m so sorry to all of you who believe that this is what Christianity either is, or what it has been reduced to. The message of the show was that (for the price of a lofty sum of money) God would give you success, whether it be in occupation, finances, or whatever. Sigh…
Needless to say I turned off the television quite soon after turning it on. But I still can’t quite shake the message that the three programs sent to me. It’s disturbing. But what shall I say, is this not America? At least in some of its worst forms surely it is.
On this St. Patrick’s Day I’d like to take a moment to let everyone out there know about the wonderful sport of Irish hurling. It’s the national sport of Ireland, and if you haven’t heard of it then you’re in for a treat. It’s often described as the fastest field game in the world. If you’d like a terse explanation of the game think field hockey on a rugby field. It’d actually be much easier for you to watch some video clips of the game in action, so if you’re interested check out some clips here and here. These snippets are from the All Ireland Hurling Finals of the years 2005 and 2004, respectively. Cork, the county in which I once lived for a summer, now stands as the All-Ireland hurling champions. Happy St. Patrick’s Day everyone! Anyone caught today drinking light beers will be punished most severely!
I’m reading a book on the history of zero which I’m sure I’ll talk more about later, but for the time being I want to reiterate a proof as to why division by zero is a mathematical no no. This isn’t a complete answer at all but it’s a quick little proof by contradiction that is easily grasped. Assume for the time being that division by zero is okay.
Take, as an assumption, that 3 does not equal 11.
We know that both 3 * 0 = 0 and 11 * 0 = 0.
So 3 * 0 = 11 * 0.
Then (3 * 0) / 0 = (11 * 0) / 0.
On both the left and right sides of the equal sign cancel the zeroes in the fraction.
So 3 = 11.
But we know from our initial assumption that 3 does not equal 11.
Contradiction.
The only other assumption we made was that division by zero was possible. Since this assumption led to a contradiction it is a false assumption. So division by zero is a questionable practice. I used 3 and 11 as example numbers, but the same argument holds for arbitrary numbers x and y, where x does not equal y.
For those of you who have at any point imagined that I was in some way intelligent- you may question that appraisal after reading this post. This past week I made one of the harder decisions I’ve had to make over the past two years, and I’d like to share a bit about the experience.
If you’ve either read the “About Me” section of this website or know me personally you’re aware of the different jobs/environments that fill a typical day for me. Like so many others I juggle multiple roles simultaneously, such as husband, son, brother, employee, graduate student, pilgrim, friend, homeowner, etc. Most of the time I manage to travel through life holding the (apparent) ability to manage the varying demands of these various roles. It’s no surprise for me to say that sometimes certain roles are elevated over others, whether because of the amount of time spent in one environment, the amount of importance placed upon a role in a given time, or whatever. It’s also no surprise to say that when you find yourself focusing exclusively on only a few life roles that the others lose a position of prominence. In other words, you forget about them or ignore them.
This is simple fact. I suppose this is just another way of saying that we can’t do everything. But I tell you reader, I wish that I could do everything. I wish that I could be an exemplary example of each of my life roles. I wish I could fit everything in. I put so much pressure on myself to do everything well. Some would call this perfectionism. I would call it not only that, but an intense desire to live a balanced life. Historically I carry too heavy of a burden, but hopefully the conclusion of this story will convince you of my desire for change.
I know so many people who are doing so much more than they can handle. At least on paper. And while I do believe in a God that sometimes calls us into apparently impossible situations, I also believe that sometimes humans place themselves in unrealistic situations. Some of us are just doing too much. And even though I think we each know it, we keep pushing into the future, believing that the unwelcome intensity will one day end. Lies, I say, lies. Let’s pick up the story from here…
So last fall I took one graduate level logic class at Carnegie Mellon. The class work took up much of the free time that I didn’t have anyways. There were many days (especially Wednesdays and Thursdays) where I either didn’t really sleep or couldn’t really focus on other life roles because the graduate student role took on an urgent quality. Notice that I said urgent, not necessarily important. And while I do believe that the class was important as a whole, there was so much happening outside of class that I was missing. I wasn’t able to spend nearly enough time with certain friends. I didn’t put much emphasis on spending time with my family, especially my brother and sister whom I’d like to see more. I had no time to read or really relax. I just kept on doing. And you know what, I made it through. I finished the class while earning a fantastic grade. And yet…
During the holidays, after the semester had ended, I actually had some unbudgeted time. I had time to think again. Time to read and relax and to see a few people. And I really started to realize that I was missing something in my intense scheduling of life. In my constant doing I lost sight of both being and becoming. I could no longer ignore the ever-present voice within me whispering, “Too much.”
So although I again started another graduate class at Carnegie Mellon this spring, I won’t be finishing it. I’ve decided to wait until the fall to start full-time, when I will no longer be working a 40 hour work week along with the demands of life’s maintenance and the general joys of my existence.
I dropped the class a few days after the official grace period for taking a leave of absence ended. The financial price of my error for Emily and I: $1200, give or take. Let me state this plainly: I am paying money not to be a student this semester. So today I can conclusively state that I’ve put a price tag on the sanity of my life. Insane, isn’t it? Or maybe not. Sometimes something must break. My time for such a break is now.
Also, on this February 14, I would like to say that I have the most unbelievable wife there has ever been. She understands so much of what is truly important, and has been so supportive of me in these past several weeks where I’ve had so much happening in my brain.
Finally, I would like to publicly thank the philosophy department of Carnegie Mellon for being so supportive of me in my decision. In my experience they truly care for their students. So if anyone is checking out CMU, here’s another positive note for you to think about.
Those of you who have ever ready anything by Kurt Vonnegut know that his style is full of the type of sarcasm that reads as apparent fact. The following excerpt from his book Cat’s Cradle is no exception when it comes to what the author is actually saying about technology. This fiction traces the effects of pure knowledge manifested through technology, as well as the role of religion and belief in a world of tech. Cat’s Cradle is a quick read, and is well worth the time. The book maintains of level of uncomfortable humor throughout, which made me ask myself more than once why I was laughing.
“[Research] isn’t looking for a better cigarette filter or a softer face tissue or a longer-lasting house paint, God help us. Everybody talks about research and practically nobody in this country’s doing it. We’re one of the few companies that actually hires men to do pure research. When most other companies brag about their research, they’re talking about industrial hack technicians who wear white coats, work out of cookbooks, and dream up an improved windshield wiper for next year’s Oldsmobile.”
“But here…?”
“Here, and shockingly few other places in this country, men are paid to increase knowledge, to work toward no end but that.”
“That’s very generous of General Forge and Foundry Company.”
“Nothing generous about it. New knowledge is the most valuable commodity on earth. The more truth we have to work with, the richer we become.”
I’ve read two books recently that have traced the same type of theme, though from different angles. The first is the aforementioned Cat’s Cradle. The second is the late Neil Postman’s Technopoly. This is my first reading of anything by Postman, though he has been on my “to read” list for several years now. His legacy is one of cultural criticism, especially concerning the subjects of education, entertainment, and technology.
The book Technopoly expounds upon the current trend of wholesale faith in all things technical. In other words, in our common belief that anything created through an application of pure science will lead to a better life both in the here and now and in the future. This belief holds in medicine, education, entertainment, food consumption, as well as other areas.
As an example, Postman recounts an experiment he would perform on fellow colleagues. He would walk up to a fairly education person and say something like “Did you hear that there was an article in the New York Times today about a Harvard scientific study which found that individuals who drank coffee every morning are 35% smarter than those who don’t?” He found, more often than not, that though the subject of this study would have initial reservations about the study, the common response was something like “Wow, that’s strange”, or “Where did you say that study was from again?”. In other words, the common response was one of belief rather than unbelief.
The modern mindset states still that science is king. If it can be wrought by science, not only is it true, but it is most likely good as well. His term Technopoly applies to cultures who express an unreserved belief in the process of technological innovation. We eat it up without fail, oftentimes without a thought to the consequences. In this thread Postman traces some of what has been lost in education due to this reliance. One is the lack of focus in using memory. Since all information is immediately available it has become even more unnecessary to use memory as a discipline (in contrast to the era of print, in which reliance on memory was greatly reduced, but not totally eliminated). Also, since our culture has generally adopted the mindset that “what is newer is better”, education has shifted its focus away from the “classic founts” of education, such as logic, grammar, early philosophy, and the like.
I don’t totally agree with Postman’s conclusions. He seems to a certain extent to pine for a bygone age. But I do think his voice sounds a necessary warning. At the very least we should as a culture stand back and think of the consequences of our faith in science expressed through technology. Of course, I say this as a technological “early adopter”. I very often love looking into the latest and greatest in regard to technology, especially when it comes to personal computing and “gadgets”. I need to do a better job of critically engaging our culture on this front.
I do think that our culture has responded to some areas of technology in a critical way. Looking at the organic food movement of the past several years I would have to conclude that there has been at the very least a noticeable minority of people who have become interested in eating food the “natural” way. Sales of organic products in places like Whole Foods, local farms, co-ops, and markets has been stunning to me. People are interested in hormone-free milk, cage-free eggs, grass-fed beef, and pesticide-free produce. As far as the food market is concerned, the consumer doesn’t seem to be doing a terrible job letting business know what it wants. I can only hope that this particular trend will expand.
Consider this as an introduction to a topic that I’d like to continue writing about. Just because it’s new doesn’t mean it’s better. And just because we can do it doesn’t mean that we ought to do it, either because it is ethically questionable from a philosophical perspective, or because the consumer isn’t calling for it from an economic perspective.
I wanted to take a moment to let everyone out there know about the summer school in logic and formal epistemology that will be offered through Carnegie Mellon Philosophy this summer. Here’s a short excerpt from the webpage:
There is a long tradition of fruitful interaction between philosophy and the sciences. Logic and statistics emerged, historically, from combined philosophical and scientific inquiry into the nature of mathematical and scientific inference; and the modern conceptions of psychology, linguistics, and computer science are the results of sustained reflection on the nature of mind, language, and computation. In today’s climate of disciplinary specialization, however, foundational reflection is becoming increasingly rare. As a result, developments in the sciences are often conceptually ill-founded, and philosophical debates lack scientific substance.
In 2006, the Department of Philosophy at Carnegie Mellon University will launch a three-week summer school in logic and formal epistemology for promising undergraduates in philosophy, mathematics, computer science, linguistics, and other sciences. The goals are to:
* introduce promising students to cross-disciplinary fields of research at an early stage in their career; and
* forge lasting interdisciplinary links between the various disciplines.
Other points to note are:
1. The program is open to all undergraduates or those who will have just received their undergraduate degrees.
2. The costs of housing and tuition are $0. In other words, this opportunity won’t break the bank. Some costs will apply, such as food, but all in all the price is quite inexpensive.
3. The dates of the program run from Monday, June 12 to Friday, June 30, 2006
For more information please reference the web page for the summer school here. There is also a PDF flyer for the program located here.
I have about 3 posts that are about 75% complete, but that aren’t quite there. Since I don’t feel like dealing with any of them at the given moment I’ll just say a few words about one of the books I’ve recently read.
First I should say that Emily and I have been vegetarians for about 6 months now. We had talked about making the change for a long while before we did it, but once we bought our house we thought it would be a natural time to institute a new diet. So far we’ve had a few days of “cheating” (such as Thanksgiving), but for the most part we’ve been extremely consistent. Generally speaking I’ve felt healthier, both physically and spiritually. I can’t say that its changed my life dramatically, but I can say that I believe it to be a step in a positive direction.
Having said this I just read Fast Food Nation by Eric Schlosser, which is a book that surveys the history of the fast food industry, and how it has changed American (and worldwide) views on vegetable farming, cattle farming, marketing, meatpacking, health, cultural identity, and more. I first heard of this book in one of the special features of the great documentary Supersize Me, which chronicles a man’s month long eating binge exclusively of McDonald’s food. All in all the book is filled with relevant facts, and though the content openly opposes the Bush administration I would say that underlying political nature of the book is fair and well researched. It’s a good read, and not only if you’re a vegetarian. The book is a snapshot of current culture. It helps to shed light on some of the reasons we all now stand in the culture that surrounds us. Having read the book mostly on this cultural angle I must say that I truly appreciate Schlosser’s work. In summary, here’s why:
I realized after reading this book how disconnected I am from my food supply. I take it for granted (much like my kitchen sink). I grew up in the suburbs of Pittsburgh, so though I grew up near to farms, I never had much exposure to the following:
1. Growing vegetables or grains.
2. Raising animals for milk, eggs, or meat.
3. Slaughtering animals for meat.
Like most of us I only ever see the end products of the many layered process of food production while shopping at my local grocery store. In fact, if I couldn’t gather food from a grocery store, I really don’t know what I would do. Mostly, I would be in huge trouble. But since I only ever see the end products of processed food, I never have to think about the methods of production.
I am not the type of vegetarian who damns all meat eaters to hell. In fact, if I could always have local meat produced with what I deemed responsible methods I would take up meat eating again (granted that the food is also affordable). But when I think about the lengths that McDonald’s goes through to stock their restaurants, I am mostly appalled. As the largest buyer of beef in the world, the second largest of chicken, and as an extremely profitable industry, McDonald’s (and other fast food companies) has employed many harmful methods to rake in the dough. If you’re curious in further details I recommend reading the book.
What I’d like to get at here is how many decisions in life that I passively make. I don’t often think of the consequences of my simple everyday decisions. But a question like “What will I eat today?” has far reaching consequences not only for myself, but also for both a myriad of other people as well as the environment. All of humanity is in this together. And each of our decisions influences someone else. When I try to wrap my mind around this I get dizzy. Whether it’s where I put my spare money or what I clean my house with or what I wear or what I buy, or whatever, my decision is not just for me. My decision reverberates through countless others. What does it mean to love my neighbor? I believe that it’s more than buying food for the homeless guy down the street or watching my friend’s kids when they need a babysitter. I believe that it means being intentional and responsible with simple everyday decisions. And that’s my point for today.
If you have a stray hour here or there I highly suggest checking out my multimedia link for “The Elegant Universe”. It’s a 3 hour (broken into smaller segments) PBS adaptation of Brian Greene’s book of the same name. It surveys 20th century physics into the present, showing how the developments and subsequent mismatches of relativity and quantum mechanics are potentially bridged by string theory. The program is extremely accessible and informative. Even if you don’t consider yourself scientific I would recommend this one. I’ve said for a long time that if you wanted to follow the zeitgeist of the 20th century that all you had to do was follow the theories of the physicists (my hypothesis for the 21st is that we’ll be following the theories of the biologists). In other words, watching the program will bring you more than just a physics tutorial. As least I hope it does.
In this vein…
Once upon a time a man was branded as a heretic by the church for claiming that the earth was not the center of the universe. You may have heard of this man at some point in your life travels. It turns out he was right. Well, he was least slightly more right than the presently accepted truth. He posited that the sun was the center of the universe. Now we know that this is in fact not the case. Not only is the sun moving along with every other star in our galaxy, but our galaxy is also moving. Every galaxy is moving. To and fro the heavens move. What I’m getting at is that we’re never finished with discovering the beautiful and complex facts about existence. We shouldn’t be surprised when the popular understanding of reality is shaken beyond recognition. Nor should we be worried.
Even as I write this post scholars at the forefront of physics question whether our universe is unique or whether it is just one of an infinite number of universes. Does that possibility bother you? Think about it. An infinite number of universes. Well, I don’t think it should bother you. In fact, my belief is that it should comfort you. As we discover the deeper complexities of our universe I see more and more only the hands of God.
Similarly, I feel the same way about the so-called emerging worldviews. I embrace postmodernity. I embrace relativism. Not without question, but I do embrace them. New perspectives bring both greater clarity and greater confusion. It’s the job of those who care to help work out which is which. Will postmodernity or relativism destroy Christian or any other religious belief? Of course not. And if you believe that it will then you are naïve.
Look at relativity and quantum mechanics. They are two branches of physics that smart people have been trying to unite for the past half century. Alone, each tells us unique facts. But when these facts try to mesh together in the current spheres of thought there are inconsistencies. Does this mean that the two oppose one another? I don’t think it does. On one levels there is definite disagreement. But I believe with no doubt that there is a larger canvas through which the two fit in harmony. It is beyond right and wrong.
Why can’t we get past right and wrong? Why can’t we let go of trying to control everything we do? It is so utterly binding. There is always, I am convinced, one step farther down the line that explains the currently unanswerable. And at the end of this line I do believe that God stands with a smile.
Until today, I did not know what a sink tail pipe was. If I didn’t own a house I would probably never know what a sink tail pipe is. A sink tail pipe is the piece of pipe that fits directly to the base of a sink. It is different from a normal piece of pipe in that it has a lip on it. The reason it has the lip is so that when it is screwed into place that it doesn’t slip away from the sink, causing the water from your faucet to end up all over the floor. I learned what a sink tail pipe is the hard way.
I’m not sure what to make of this. Spending part of my Sunday fixing my kitchen sink wasn’t something that I thought would ever be on my agenda. In fact, I take things like kitchen sinks for granted. I think that they will always work, and that I will always be able to use them. In reality this is not the case.
Owning a house has been both rewarding and frustrating. It isn’t just about the sink spilling water all over the place. It’s about having to constantly deal with fixing things up and making things better and replacing broken stuff. Generally I don’t put a ton of value on taking care of this type of stuff. Then again I’ve never owned a house before.
I wonder if I will ever put more value on doing this type of thing. I suppose that growing up in a house that was brand new didn’t lend itself to dealing with the issues that one would face by owning a 100 year old home. I just don’t know what to expect. Hopefully I’ll have a better attitude as time goes on. I have to admit though that I sometimes believe that I’ll always find these type of home improvement tasks less full of meaning than a myriad of other things.
Here’s to either “being an adult” or furthering the process of figuring out what parts of “being an adult” are worth embracing.
Enter 2006. I’m always a bit weary of even years. There’s something unsettling about it. At least I’ll turn 25 (a fantastic odd number, though not prime, and yet still a perfect square) in July. Generally speaking I don’t usually consider January 1 as the New Year. Having been raised in a country where the school year begins in the late August or early September I’ve always considered the advent of fall as the New Year.
As far as 2006 goes, I must say that I have high expectations. I’ve dubbed this year “The Year of Deeper Newness”. I’m not even sure what I mean by this exactly. The feelings I have when I hear the phrase are akin to the feelings one gets while listening to a song which embodies a fantastic amount of meaning.
The past several years of my life have been like setting up a chess board. It seems that all the pieces are now in place. In fact they may have been set for some time now. But I’ve been afraid to move a piece. Moving a piece means commitment. Moving a piece means embracing the adulthood (read “summer of life”) that my existence now embodies. Moving a piece means moving forward, not back. And sometimes, just sometimes, I’m afraid how the game will play out.
I think that both Emily and I will have made some big decisions come the end of next December. I hate to speculate on the possibilities, but I know that when it all plays out it will be both totally different and utterly fuller than we could both imagine.
A toast to 2006, prime factorization 2*17*59.
What follows is a list of the most influentual songs to my life in 2005 with a short commentary of each song. Some of them are newer, some are older. Some are well known and some are not. Hopefully there will be something good in here.
“This is the Picture (Excellent Birds)” – Peter Gabriel
This is definitely the strangest song on the list. If I were to make a movie I think this would be my closing song. I love Peter Gabriel. Ever time I hear this song I inexplicably start bobbing my head.
“Suitcase” – Over the Rhine
Karin Bergquist. That’s all I have to say. Her voice is a weapon. I remember when I saw them play live. When she wasn’t singing she just leaned up against the pillar on the stage, cool as anything, like she could just kick your ass at a moments notice.
“Somebody’s Gotta Do It” – The Roots
“The Tipping Point” from The Roots is probably my personal album of the year. The beats on the album are so above and beyond most other hip-hop that it’s unfair. The hook on this song pretty much sums of what I’ve concluded in 2005. Check this one out.
“Holiday”-“Boulevard of Broken Dreams”-“Are We the Waiting” – Green Day
The first time I really listened to Green Day’s “American Idiot” album was on a long car ride to New York for a wedding. It was night, I was driving, and the rest of the car was sleeping. There aren’t many times when I remember truly hearing a song for the first time, but when I heard the guitar at the beginning of “Boulevard of Broken Dreams” bleeding in from “Holiday” I was just in awe. And then “Are We the Waiting” to end the triad. Amazing. An incredible record that was in no way over-hyped.
“Lovers in a Dangerous Time” – Bruce Cockburn
I always really loved the Barenaked Lady’s cover of this song. For a long time in fact I thought they wrote it. This song contains one of the greatest lines I can remember hearing in a song for a long time. “Nothing worth having comes without some kind of fight, got to kick at the darkness til it bleeds daylight”. This line was also sung by U2 on the song “God Part II” on Rattle and Hum, which is why many people know about it. If you haven’t heard this one I highly recommend it. The production on the recording is terrible, but the beauty of the song shines through.
“City of Blinding Lights” – U2
One of those U2 songs that gives you hope. I wish I could’ve experienced this song live. It was meant to be played live. I remember the first time I heard Bono sing the climactic line in this song, “Time won’t leave me as I am, but time won’t take the boy out of this man”. Shivers went up my spine.
“Brand New Colony” – The Postal Service
I discovered this one earlier in the year and loved it. It’s a unique album with all the best elements of electronic music. The tempo changes in this song are stellar, and the words are like good milk chocolate. Just strong enough and smooth.
Last night, mostly on a whim, Emily and I decided to try to see a midnight showing of the new Harry Potter flick. When we arrived at the theatre we found that our only unsold-out option was the 12:50am showing. So we went for it, and I’m glad we did. My body is slowly shutting down due to having only 8 hours of sleep in the past two nights (dang I’m getting old!), but I must say that I’m in fairly good spirits.
As far as content on this blog is concerned I’m working on several posts right now that I want to mesh through a bit more before I post them. I haven’t forgotten my promise to post a paper on the problem of esse in Christianity, and I’m working on something about induction and another something on the cardinal numbers. Next week there should be some sizeable content to enjoy here.
In the meantime, here is my response to being “tagged” by k flanagan (in other words, 5 random facts about myself):
1. At the age of 9 it was discovered that I had a third misshapen front tooth that needed to be removed via oral surgery before my two normal front teeth could grow in correctly. I think my mom still has it somewhere…
2. I attended Penn State University for the fall semester of my freshman year of undergraduate study before transferring to Grove City College. At PSU I was a math/philosophy major before changing to math/theology at Grove City.
3. For the most part I believe that paper books should be replaced by digital books.
4. One day I would like to open up a philosophical/theological think tank with my wife, Emily. We would tend the house and grounds while helping to shape the conversation that occurred on the premises.
5. I consider one of my intellectual flaws to be my memory.
It’s a slower day at work today so I’m trying to reorganize some small stuff around the office. Since I spend a lot of the day on my computer(s) here, the reorganization process usually involves changing my desktop pictures and deleting\reorganizing files and what not.
I was looking through my Firefox bookmarks and started to think about the sources that I use daily on the web to hear about “the news”. As far as technical news goes I have several really great resources, but I currently don’t have many fantastic sources outside of the mainstream ones for finding national or world news (ABC News, Fox News, Drudge report (shudder…), etc.)
So my question is this: Where do you go online to find news that isn’t gossip or consumer based? I hate wasting time on the Internet, and want to try to find some informational news sources. Anyone have any opinions?
So thanks to Boing Boing I came across a beautiful page of art created by a person named Justin Mullins (Mathematical Photography). His art focuses exclusively on mathematical equations. I must say that I enjoy this site immensely, both for its vision and information. What adds to the beauty of this site is the brief poetic explanation the artist provides for most of his pieces. I especially suggest that you read the entries for the pieces entitled “Beauty” (Euler’s Relation) , “Ugliness” (the Four Color Theorem), “Mystery” (Godel’s Incompleteness Theorem), and “Power” (Aleph One).
I have such a hard time when someone (usually a speaker who knows nothing or relatively nothing about mathematics) uses an analogy that paints the field as the dull black and white objectified reality. First, this is not math. This is some strange twisted view of math that either you learned in second grade or that you have been fed by our culture. This view is pervasive and ugly. Math has so much more to do with elegance and beauty and the greater search of making sense of possibility.
I’d like to see two changes in the popular perception of mathematics. First, I’d like to see a greater effort by the mathematics community to make their material accessible to the popular masses. I believe this can be done by pursuing some of the artistic or philosophical implications and questions brought on by a particular technical result or problem. There are a few people who have done this, and off the top of my head I’d name Ian Stewart as an example. For instance, if one really looks at Euler’s Relation, and digests it like one would a Monet, I believe there’s just as much beauty to behold.
Second, I’d like to see postmodern theologians taking a more mature view of mathematics. Rather than painting it as the personification of the Modern era (and thus mostly an unhelpful and inorganic subject), I’d like each and every one of them to gain a familiarity of the subject and what it means for the twenty-first century. With this aim, I suggest that these readers refer to Godel’s Incompleteness Theorem, with some of the philosophical questions behind it.
Check out the site. And if you have a moment try to engage one of the pieces. Thank you Justin Mullins for taking such a simple idea and running with it.
Emily and I bought a nearly 100 year old arts and crafts (verging on modern) house several months ago in Edgewood (the East side of Pittsburgh). It’s a beautiful house that was designed by Frederick Scheibler, who was a fairly well-known Pittsburgh architect around the turn into the 20th century. He has many spaces in Pittsburgh, and they’re all gorgeous. If you’ve been around the area you’ve probably seen some. For instance, one of his buildings is the Shadyside Variety Store building in Shadyside, which has gorgeous apartments on the higher floors.
Somewhere between the time that our house was built and the present there have been some substantial changes to the space, and not all of them have been for the better. For instance, several built-in bookcases and window seats have been removed and sold off. It sucks, but I suppose that’s the economics of a 100 year old home.
One of the changes to the space was that a coat closet was created on the lefthand side of the entryway. It is ugly. So ugly, in fact, that Em and I decided that it needed to go. So on Monday night I decided that I was going to begin to rip this sucker down. So I did. I started at the bottom and worked toward the ceiling, removing wood and nails and spraying dust around into the air.
There was a piece os wood paneling covering the space between the top of the closet and the ceiling, which a started to knock out with a hammer. But after several swipes I immediately noticed that there was something behind the covering. What I found was books. OLD books. There were about 30 of them and many of them were over 100 years old. Several of them still have the inscriptions inside the covers, indicating where they came from whose they were. Cool. Very cool.
Neither Em or I has any idea of how or why they got there. It’s extremely random. It’s helping us to uncover a bit more history about this house, and we might even be able to donate or sell some of them. So here’s a reason to live in an old house: it may be full of wonderful and inexplicable old stuff!
I’m not sure what to do with this website ultimately. I feel tripolar about it. First, I have all these random “state-of-the-union in-Ian’s-life” stuff, comprised of dreams and musings and incoherent nonsense. Some of the posts in this category are probably better off somewhere else, but I know that there’s probably someone out there that likes to read them.
Next I have the logic and math stuff, which I’m currently pursuing educationally. I love this stuff, but when I post about it I realize that just about no one is going to read it or understand it (unless they expressly came here to check out what I’m doing at Carnegie Mellon, which is a possibility). I do recommend that even if you don’t have any desire to learn anything about math/logic that you try to read one or two papers/posts anyway. Some of the ideas behind the papers are easily grasped and philosophically extensive. It’s good stuff, and wherever I end up in life I’ll always respect these disciplines.
Lastly, I have all this theology stuff everywhere, which is something I’m passionate about. But the people who are coming here to read about logic/math are probably not at all interested in reading about Christian concepts of absolute truth in the postmodern age. Alas, for now they’ll just have to sift through the muck, although from a philosophical and cultural point of view some of the theology writings and website links are extremely informative even for thos who hold no faith.
So there it is. My tripolar nature. And I don’t apologize about it. It’s just not externally consistent. I hesitate to split this up into three separate blogs, but I may have to do it. Any suggestions from anyone out there?
So I just read Brad Wray’s explanation of both what he does and what he wants to do. Ultimately, Brad wants to teach social studies. I think this is a noble and wonderful pursuit, and his dreams have got me thinking. Its got me thinking on something that I’ve had in my head for several years but haven’t articulated much verbally because I’m half-scared to death that it just might happen if I start talking about it.
Here’s the dream: I’d like to start a school. I’m not sure if it would be a classical high school or a liberal arts college. It would be a place where kids learn to think. I think about this a lot because so many of my friends (you are probably one of them if you’re reading this) are so damn talented. Most of them would be fantastic teachers. I wouldn’t require anyone to have a Ph.D. if it were a college, although that would be great; and I wouldn’t even care if people didn’t have their secondary education teaching certificates, cause I think that’s all a bunch of bullshit anyway. Seriously folks, how many people do you know who aren’t “educated” to the maximum are absolutely brilliant in comparison to those who have all sorts of meaningless degrees? I know several.
So I just wanted to put that out there. I have a mental list of people who I’ll ask to teach if this ever goes down, but if you’re interested just let me know and I’ll put you on the list.
The next time someone asks me how I am, I am resolved to not give the following reply:
“Busy.”
I am resolved not to whine about what I’ve chosen for myself. I have chosen to be married, a homeowner, a full-time employee, a part-time graduate student, a dog owner, a member of the vestry for my church, a member of my church, and so forth. And I have also chosen to do all of this as a 24 year old. I have no room to bitch about being busy. If I didn’t like what I was doing, I could change it. No one held a gun to my temple and demanded for me to enter graduate school or to buy a house. I chose it all.
Nor will I answer: “Tired.” Same issue. If I were so flipping tired, I could change my routine.
I am not a victim, and I have lost this kind of patience for myself. So if you ever hear me whining, please smack me.
A mental math installment…
While walking to Carnegie Mellon the other day I started thinking about squaring two digit numbers. So instead of hoarding my findings and short amount of research I thought I would write a tutorial on how to square two digit numbers, and then some.
Method 1: If You Know the Previous Square
This method is only marginally helpful, but will come in handy if you know how to easily square the number previous to the number you’re trying to square. Let’s say you’re trying to find x2. If you know what (x-1)2 is already, all you have to do is add x and (x-1) to (x-1)2 to find x2.
Proof:
(x-1)2 + x + (x-1) = (x2 – 2x + 1) + x + x – 1 =
(x2 – 2x + 1) + 2x – 1 =
x2 + (2x – 2x) + (1 – 1) =
x2
q.e.d.
Example:
Let’s find 312 knowing that 302 = 900.
312 = 302 + 31 + 30 = 900 + 31 + 30 = 961.
This method is mostly helpful for squaring numbers which are one more than a multiple of ten, since humans can square multiples of ten without much thought (more on this later). Also, you’re not restricted to squaring two digit numbers with this method, which is quite fantastic.
Method 2: Multiplying One Up, One Down, and then Adding One
In my opinion this method is a bit more fun, but it definitely requires some mental multiplication. In fact, you’ll almost be doing as much work (or more) using this method as you would be in outrightly squaring a number, but it is quite an amusing trick.
Let’s say you want to square a number x. If you multiply (x-1) and (x+1) together, and then add 1, you’ll find x2.
Proof:
(x-1)(x+1) + 1 =
(x2 + x – x – 1) + 1 =
x2 + (x – x) + (1 – 1) =
x2
q.e.d.
Example:
Let’s find 312 again using this trick.
312 = (30)(32) + 1 = 960 + 1 = 961.
Like I said, if you can’t quickly calculate that (30)(32) is 960, then this method isn’t saving you much mental energy, but this averaging method could be a shortcut for some.
Method 3: Squaring Two Easy Numbers Instead of One Hard Number (Plus a Step)
In my opinion, this method is best practice. This explanation will be a bit lengthier, and also a bit harder to write out in plain English, but it’s the most fruitful by far of the three methods.
Like I said in my previous post on mental math, perhaps the most extensive repository most folks have for math is their times tables. With some exception, people are able by their mid-teens to multiply together two one digit numbers with relative ease, all the way up to (9)(9) = 81. Let’s make use of this fact, plus the fact that folks can easily multiply multiples of ten, to square any two digit number easily.
First, one needs to realize that any two digit number is at most five numbers away from a multiple of ten. For instance, 34 is 4 numbers away from 30, and 65 is 5 numbers away from both 60 and 70. So when we’re thinking of squaring large two digit numbers it’s best to think of it as a multiple of ten (e.g. 10, 20, 30, 40, …) plus or minus a number no greater than five.
So when we go to square a number like 74 mentally (yikes!), it’s better for one to imagine this number not as 74 but as (70 + 4), or to imagine the number 66 not as a single number, but as the difference between two, i.e. (70 – 4). It’s a matter of breaking down a difficult single process into several easier ones.
So let’s find 742. Instead of looking at the single number 74, let’s break it down into (70 + 4). Now let’s square this number.
(70 + 4)2 =
(70 + 4)(70 + 4)
At this point in the game we’re multiplying two binomials together. Remember the FOIL (Front, Outer, Inner, Last) method from way back when? I thought so. Continuing with this method we have
702 + (70)(4) + (4)(70) + 42
4900 + 280 + 280 + 16 (a)
We’ve broken the process down into more or less multiplying single digit numbers together and then adding zeroes at the end. We know 72 = 49, (7)(4) = (4)(7) = 28, and 42 = 16, and it’s then easy to multiply these numbers by powers of ten.
Adding these numbers up we get 5476. We’ve taken a difficult multiplication problem and turned it into an easier addition problem (though some would beg to differ, I’m sure). It works the same for a number like 66, which looks like (70 – 4) when we break it down. The only difference is that we’ve traded our plus sign for a negative one. The only part of the arithmetic that changes is that instead of adding 280 twice to expression (a) above, we subtract it twice. So similarly to 742 we have that
662 =
4900 – 280 – 280 + 16 =
4356
And this method, in my opinion, is by far the easiest way to mentally square two digit numbers. A similar process will work with three digit numbers. Perhaps I’ll write on that later.
There you go. Hopefully the next time you need to square a number quickly you’ll be more equipped for usual. So for now, happy squaring, and let me know if anyone has any suggestions or additions.
Note: For all you nitpickers out there let it be known that when I’ve used the word “number” at any point in this post I’ve actually meant this word to mean “positive integer”.
As many of you know part of my work as a student was and is in mathematics. So you shouldn’t be surprised to learn that I’ll be posting some thoughts in this space every now and again of a mathematical nature. Here’s my introduction on my mathematical posts…
I love the calculator. I love the idea that a machine can perform computations faster than I can. Calculators (or computers if you will) vastly outdo humans in both speed and accuracy when it comes to computational ability, just as humans outdo computers when it comes to other varieties of tasks. In fact, as an aside, much of the aim of artificial intelligence is to teach a computer how to perform actions that humans have mastered, but at which computers are currently mediocre. One common example is the task of face recognition in human beings, which in part is being developed as a security measure and in part is being developed because why the hell not?
And in case you were wondering, the origin of our modern use of the word “computer” comes from our historic human “computers”, or those men and women who were employed to compute various tedious and repetitive calculations by hand. Those were in the days when there were such things as slide rules and logarithmic tables. Those days are now over. This is at least what I’ve been told.
I remember taking a course in college called Linear Algebra, which is concerned mostly with giving its learners near death levels of frustration, but which is also concerned here and again with matrices. I remember spending over an hour performing some stupid computation by hand. It was a homework problem and we had to show all our work. When I checked my answer via computer, I was supposing that it would take the computer program at least a minute or so to answer. Nope. Its answer was instantaneous, and its answer, oddly enough, was correct.
Having said all this, though, I regret to see that human capacity for mental math has decreased in these present times. In fact, the greatest feat of mental math that many of us perform with any sort of frequency is related to our simple times tables. And generally as adults we don’t “perform” any mental math when we use them because we memorized the facts years ago. Cognitively we generally don’t mull over how or why 6 x 5 = 30. We learned the mechanics of how simple multiplication worked in grade school, so we take it for granted that 6 x 5 = 30. Indeed.
It amuses me that many of our cell phones come with a built in “tip calculator”, allowing us to quickly generate what 15% of $23.15 is. What amuses me even more is how often I actually see people using this calculator. And in case you were wondering, the answer rounded to the nearest cent is $3.47. And no, I did not use a calculator to figure that out. 10% of $23.15 is $2.315 so to speak and half of that (5%) is $1.1575. Add those together to get $3.4725, which is 15%. Round to $3.47.
I don’t support mental math because of the math part of it. I support it for the puzzle part of it. Oftentimes tremendous ideas come out of puzzles, and I espouse mental math for this art of thought. More on this idea at a later date.
I realize that I’m in the vast minority of people who walk around this earth actually thinking, for fun, about numbers. And I realize that this fact is a strange and scary fact for many, if not all, of you. What I want to try to do in my mathematical posts is try to give you all a glimpse of something beautiful in math. It’s the beauty that drove me to it in the past, and this same beauty is driving me back to a type of math (applied logic) for graduate school at Carnegie Mellon University. Here and there I’ll try to provide a helpful tip as well.
I remember on my 18th birthday walking into a tobacco shop and buying my first legal pack of cloves. It was a liberating moment. I had also earlier that buy bought a journal. I went to a nearby pastry/coffee shop and smoked and wrote to my heart’s contentment. So today at year marker 24 I’m starting, for serious, this webspace. We’ll leave it at that. I’ll be reposting some writing I’ve done other places, so I’ll most likely just be throwing that stuff in at some point soon. One step at a time. So in the meantime I’ll try to spend at least a few moments of my 24th birthday smoking, and perhaps writing a bit more as well.
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