Sunday, September 5, 2010
I've got a real treat in store today. I've taken some time to build more pages into this blog. I'm going to be writing and publishing tutorials that I've written using Mathematica. I'm trying to take a more analytical approach to these topics. Rather than list a bunch of theorems or definitions like a typical text book, I try to work through the topic as if I am discovering it on my own. The reason I'm leaving out the intensive rigor is because that has been done time and time again via textbooks. I highly recommend learning the rigor because that gives and added level of depth you just can't get otherwise. However, I think the tutorials, as they stand, will be helpful for many to garner some insight.
I don't want people to misunderstand my approach, so I've included a "Recommended Reading" tab, so that you can go and find where I am getting a lot of my influences. I've read multiple text books on the same topics so I'm only recommending the ones I think are the best. I'll be including links to Amazon.com where you can usually find textbooks at more reasonable prices. I realize over time this could get outdated with new editions, but the books' quality still stands so it may not matter what edition you decide to get, should you decide to purchase a book.
I am also going to try and list some "fun" books on the topics I've come across. I'm going to try and build these recommendations around people that might not have any math background, so the reading should not be overwhelming. My intention here is that this will hopefully get people interested in the subjects and perhaps get the confidence to delve into the topic even though the readers confidence may have been squashed by years of rote learning in classroom settings that didn't help people learn anything.
In the future I am going to include executable webMathematica programs where users can launch a page that you can actual practice topics discussed in the tutorials. All of this right from the web. I am also considering including video tutorials of me actually giving lessons on a subject. That might be much further down the road though. Like a year or more away.
Anyway, I hope people enjoy and appreciate the work I've put into building these things.
Sunday, August 1, 2010
I read a rather insightful book about how the education system treats children. It’s written by a PhD Mathematician, so it carries some weight. However, I am not sure how widely read it is outside of the university community. The book is called “A Mathematician’s Lament” by Paul Lockhart and I definitely recommend reading it if you get the chance. It’s not a high level math book by any means, it is just a good read to help get insight into the world. I have read reviews on Amazon insisting that education theory has been applied better than when this book was written. However, I should mention my mother is a high school math teacher and her curriculum has only been changed by standardized tests. So, whatever new “theory” these people are professing… people are still losing out on math in literal droves. As Paul Lockhart put it “In school you learn that math is not something you do, it’s something that is done TO you.” Lockhart’s essay is post year 2000, however, I find it very interesting to read similar commentary from Heaviside in the 1890’s, over a hundred years earlier.
“‘Mathematics is gibberish.’ Little need be said about this statement. It is only worthy of the utterly illiterate.
‘What is the use of it? It is all a waste of time. Better be doing something useful. Why, you might be inventing a new dynamo in the time you waste over all that stuff.’ Now, similar remarks to these I have often heard from fairly intelligent and educated people. They don’t see the use of it, that is plain. That is nothing; what is to the point is that they conclude that it is of no use. For it may be easily observed that the parrot-cry ‘What’s the use of it?’ does not emanate in a humble spirit of inquiry, but on the contrary, quite the reverse. You can see the nose turn up.
“But what is the use of it, then? Well, it is quite certain that if a person has no mathematical talent whatever he had really better be doing something ‘useful,’ that is to say, something else than mathematics, (inventing a dynamo, for instance,) and not be wasting his time in (so to speak) trying to force a crop of wheat on the sands of the sea-shore. This is quite a personal question. Every mind should receive fair development (in good directions) for what it is capable of doing fairly well. People who do not cultivate their minds have no conception of what they lose. They become mere eating and drinking and money-grabbing machines. And yet they seem happy! There is some merciful dispensation at work, no doubt.
“‘Mathematics is a mere machine. You can’t get anything out of it that you don’t put in first. You put it in, and then just grind it out again. You can’t discover anything by mathematics, or invent anything. You can’t get more than a pint out of a pint pot.’ And so forth.
“It is scarcely credible to the initiated that such statements could be made by any person who could be said to have an intellect. But I have heard similar remarks from really talented men, who might have fair mathematical aptitude themselves, though quite undeveloped. The fact is, the statements contain at once a profound truth, and a mischievous fallacy. That the fallacy is not self-evident affords an excuse for its not being perceived even by those who may (perhaps imperfectly) recognize the element of truth. But as regards the truth mentioned, I doubt whether the caviler has generally any distinct idea of it either, or he would not express it so contemptuously along with the fallacy.” (Section 9, Electromagnetic Theory, 1893)
I think the first part of this is great and really illustrates the divide some people have. Mathematics is, literally, its own language. The array of symbols used in mathematics indeed looks like gibberish to the uninitiated. But take someone that only speaks English and send them to Russia and you have the same problem. People are incessantly lazy, and learning new languages later in life is actually quite difficult (well for most people). Not to mention a language that is based on pure logic where certain “slang” is not allowed, but it is not different than any language. Such as in English “I can’t not do that”, grammatically makes no sense. Just as 2 = 1, mathematically makes no sense. If you want to start incorporating “slang” in math, you need to prove it makes sense or define it in such a specific way. Take the concept of a gradient for example? I won’t bore you (yet), but it’s sort of similar.
Like any language it has evolved from many rudimentary levels, but, like any language, it has great descriptive power for what is happening. If you ever attempt to read old school math proofs from something like Euclid’s “Thirteen Elements” you won’t see anything like you see in modern math. Instead you see long droning paragraphs setting up a situation that would essentially say something like “sin (x + y) = sin(x)cos(y) + sin(y)cos(x)”. Isn’t that way easier to look at than a long paragraph explaining the same thing? When you look into math, you have to remember it really is short hand for a giant paragraph discussing a specific situation. While it may seem intimidating with all the crazy symbols, trust me its better. Read Newton’s “The Principia” to find out…
In this treatment I am not sure Heaviside is completely clear towards the end. However, he is attempting to illustrate the power of Mathematical Theory versus Application. This reminds me of an interesting thing concerning Tesla and Edison. In America we tend to venerate Edison as the “great American inventor”. However, very little appears to be known as to how inefficient his methods were (never mind how horrible of a person he actually was, but that’s another topic). Anyway, the Edison method can probably best be described as the “brute force” method, as one of my math teachers would have said. Basically, if you want to make a light bulb then you keep trying every kind of material for a filament until something finally starts to glow. It’s pretty messy and not a very nice way to go about doing things. Tesla on the other hand would do the math; he’d study the physics, the heat transfer of materials and so forth. He’d come to solutions much faster than his contemporaries and on top of that they are elegant solutions. This is why most people now work in this fashion and why the concept of Applied Mathematics is so powerful. I think people sorely miss out on that kind of power.
I don’t want to lead anyone astray; this stuff can get real hard. In fact if you are doing mathematical research sometimes you might delve into a particular aspect of the field that has five other people that MAY know what you’re talking about. But this is not a problem exclusive to mathematics; you can find it in about anything, even something like Economics. I even encountered Economics text books that are using Quantum Mechanics to help explain what is happening for Risk Pricing in the stock market. If that’s not a mind bowing application of Quantum Mechanics… then I’m sure there are many others, but I thought that was an interesting one.
Thursday, July 22, 2010
I’ve been quite inspired in reading the introduction to Oliver Heaviside’s “Electromagnetic Theory: Volume 1”. It is surprising to me how little is known or revered by this amazing Applied Mathematician. Then again I am still surprised over the lack of fame Tesla has today. While instead the “Great American Inventor” title is reserved for people like Edison. Anyway, I simply will not rob Heaviside’s words, thus I quote him directly and it appears he had a similar problem that is still rather pervasive in our modern times.
“There are men of a certain type of mind who are never wearied with gibing at mathematics, at mathematicians, and at mathematical methods of inquiry. It goes almost without saying that these men have themselves little mathematical bent. I believe this to be a general fact; but, as a fact, it does not explain very well their attitude towards mathematicians. The reason seems to lie deeper. How does it come about, for instance, that whilst they are themselves so transparently ignorant of the real nature, meaning, and effects of mathematical investigation, they yet lay down the law in the most confident and self-satisfied manner, telling the mathematician what the nature of his work is (or rather is not), and of its erroneousness and inutility, and so forth? It is quite as if they knew all about it.
“It reminds one of the professional paradoxers, the men who want to make you believe that the ratio of the circumference to the diameter of a circle is 3, or 3.125, or some other nice easy number (any but the right one); or that the earth is flat, or that the sun is a lump of ice; or that the distance of the moon is exactly 6 miles 500 yards, or that the speed of the current varies as the square of the length of the line. They, too, write as if they knew all about it! Plainly, then, the anti-mathematician must belong to the same class as the paradoxer, whose characteristic is to be wise in his ignorance, whereas the really wise man is ignorant in his wisdom. But this matter may be left for students of mind to settle. What is of greater importance is that the anti-mathematicians sometimes do a deal of mischief. For there are many of a neutral frame of mind, little acquainted themselves with mathematical methods, who are sufficiently impressible to be easily taken in by the gibers and to be prejudiced thereby; and, should they possess some mathematical bent, they may be hindered by their prejudice from giving it fair development. We cannot all be Newton’s or Laplace’s, but that there is an immense amount of moderate mathematical talent lying latent in the average man I regard as a fact; and even the moderate development implied in a working knowledge of simple algebraically equations can, with common-sense to assist, be not only the means of valuable mental discipline, but even be of commercial importance (which goes a long way with some people), should one’s occupation be a branch of engineering for example.” (Section 8, Electromagnetic Theory, 1893)
Please take the time to note the date in which this writing was published. Even in the late 1800’s men like Heaviside were aware of the spreading of the ignorant, by generating ridiculous concepts like the erroneous representation of Pi. Today, I don’t see many meddle in the realms of mathematics, perhaps the discipline has grown so far beyond the grasp of the charlatans that nothing they say will be believed? For if you look at the time frame the charlatans were arguing mathematics from the 1500’s or so, whereas today it has become such a solidified and widely taught subject that people do not question the mathematical conclusions anymore. If they do question something as mundane as Pi, then it seems they are not studying mathematics very clearly.
What we do experience today is the “paradoxer”, as Heaviside put it, trying to undermine science, especially Biology. Little is argued against Physics these days, perhaps it is so far beyond people (just like math) that no one can argue it. Although, I see all kinds of nonsense coming out of the Quantum Quackery camp, perhaps that is the plight of Physics today. However, as Bob Stanek of CERN, commented on the documentary “Through the Wormhole”: “Most people know about the gravitational force, some people know about the Electromagnetic forces, no one knows about the weak nuclear force, and no one knows about the strong nuclear force.” (Possibly re-ordered by me.) This tells me that the divide between what people know and where Physicists are is very great.
However, people do not act like this when it comes to Biology. With Biology the concept of “natural selection” is even put into question! When I discuss this topic with people they are arguing 19th century Biology. These are the paradoxers arguing old world and already solved problems, yet somehow thinking they are relevant to modern science. For example in a recent discussion someone tried to discuss how Hoeckel’s recapitulation theory was untenable, while my girlfriend would quickly retort: “of course it is, that’s why Evo-Devo exists today”. This is clearly a case of someone NOT keeping up with the field they wish to criticize. If you really want to have something to say about science, especially a critical something, then you must know what the modern science says about the topic. It is, generally, not hard to find out.
You see this happen quite often today and I just ran into it yesterday in a discussion with a tenacious twelve year old named Brianna. She has shown an interest in things like Physics, but instead of finding information that is relevant, she stumbled upon a site filled with all kinds of nonsense. She obviously wouldn’t know any better; luckily for her someone was able to help her out. I looked over this site briefly and on the main page they had a video of someone who invented an “incredible machine” that only runs on magnets. The basic gist is that it was a rotary motor that rotates within a magnetic field. They had a group of people around the table looking on in awe, and they were all amazed at how close we were to having “powerless energy.” I think everyone there missed out on one major fact… Tesla invented that in the late 1800’s. This is how out of touch people are with concepts and inventions related to something as common as Electromagnetism. This simply blows my mind. Instead of progression we have regression, these people could be reasonably smart people, but if they spent less time trying to combine science with spirituality and spent their time working ACTUAL modern science, maybe our species could accomplish even more than we have already!
In sum, I am both awed and dismayed at what I stumble upon in my daily readings and research. While Oliver Heaviside did impressive work with Electromagnetism and really helped further the field, while Tesla gave the world the power of Alternating Current, they still faced dissenters that could not get over the already solved problems. As it is with history… little has changed.
Here’s to progress. Tomorrow I may discuss more on Mathematics and Electromagnetism in terms of Theory and Application.
Wednesday, June 16, 2010
Currently Watching: The Ninth Gate
Currently Reading: Electromagnetic Theory by Oliver Heaviside
Mathematical Fallacies and Paradoxes by Bryan Bunch
In my travels I have talked with many people and throughout my years I never quite grasped the massive gap between the lay person and science. Like many others out there I would revel in the science section at the local Barnes & Noble or Borders, sifting through the latest books on science. I would read things like “The Universe in a Nutshell” by Stephen Hawking or “Hyperspace” by Michio Kaku. Now I didn’t find all these books perfectly accessible, but I have always been fascinated by science.
When I got out of high school I decided to attempt majoring in Computer Science, but I did not do well in my computer classes and I outright failed at the mathematics end. After a single attempt I gave up and switched my major to business. After working just four years in that industry I couldn’t take it anymore and my fascination for science started to loom over me. However, I knew the only key that would fit into the lock of science, the only thing that would open the gateway, was mathematics. So I simply taught myself and went back to school. In this newfound expedition of my life I’ve noticed quite a stark difference between actually doing the science and reading about it in the general science books.
Even though I am still at the beginning of my journey, I can see, quite clearly, that there is a major difference between actually doing and just observing. Reading books on the concepts is great and interesting, but actually working through the equations and drawing the same conclusions as others is simply powerful. Having done this when I read these books designed for the “lay” person, I get a great deal more insight. It is so much clearer to understand and follow what the author is discussing that it is simply impossible to describe.
If anyone reading this is at least partially interested in science and you read the types of books I am referencing then I highly recommend you try doing some of the work. It will provide a great deal of insight into the “general” books being read. (No, I don’t recommend starting with this like Lie Algebra or Particle Physics, but look into what you need to start out with, not just at the top.)
Alas, this brings me to a gripe I have recently noticed. How do you go about bridging this gap to an incredibly uninformed populace? I was travelling with my girlfriend in Ohio recently and we stopped at a Borders close to the airport, which was pretty large and kept an excellent stock of books. Naturally I went over to the science and math section to peruse it. In the Physics section of the store it was filled with books on Quantum Mechanics. The sheer volume of books on this topic was staggering. I can’t even imagine how the lay person can even delve into this kind of material without having previous exposure to other fundamental fields of physics. Granted Quantum Mechanics is essential to physics research today, but to have so many books on it… well it is simply outrageous. Going through the High School system in the U.S., they barely covered topics on Gravitation, never mind something as intense as Einstein’s Relativity… and never mind Special Relativity. How can you expect to glean understanding from a book about Quantum Physics without understanding things like Electromagnetic Theory, Conservation of Energy, Impulse, Momentum, Atomic Theory etc… all these great fundamental things. But alas I am left wondering where people would get this information outside of college level Physics.
In our travels there was another Borders near her hometown where the science section is something to be lamented. They barely had any books on advanced sciences. The best that could be said about mathematics were books to help with things like “SAT’s”, this is pathetic. These will not inspire the imagination in any kind of a field. Other than that it was geology and field guide’s to nature. This is truly a pathetic state. While the section on Metaphysics and Religion dwarfed the science section. There is truly something wrong today when people think “Metaphysics” has anything to say about the real world which is more powerful than the actual facts we face with science.
In either case the end results is quite detrimental. We wind up with a populace that is so scientifically illiterate there is barely anything we can do to stop the downward spiral. On one hand we have a huge selection of books that are impossible for people to truly grasp, which lend itself to all kinds of nonsense in the form of Quantum Quackery. On the other hand is such a deficient exposure to science that Metaphysics actually has a legitimate voice to help people answer questions of “why” and “how”. No wonder nonsense like “Intelligent Design” is passing for “science” in the minds of the general public, or at least a lot of people are unable to tell the difference.
I make no mistake to offer a solution at this time, I am still at the beginning of my journey, but I thought it would be wise to document my observations. I realize the need to fire the minds of the imaginations of the public, or else there would be little hope of getting scientific funding. But the question I am coming into is: What is more important? Inspiring the minds of the public with fancy bells and whistles so they will fund us for more bells and whistles OR generating a knowledgeable populace that actually understands the work we are doing and the merit of it? For me, the latter goal is the preferred.