Kurt Gödel was a logician and set theorist. Usually set theory is not an exciting subject. Gödel brought this subject some excitement.
You see the whole thing started when people like Gerog Cantor and others started discussing when sets are bigger or smaller than other sets. See a set with three elements has more members than a set with two elements. The interesting things happen when sets have a infinite number of elements. For example you can add an element to an infinite set and its "size" (called cardinality) does not change. People then began to wonder if these infinite sets were in some kind of order. That is, given an infinite set can we construct the next bigger set, without missing one.
Well, that's where Gödel comes in with his "Incompleteness Theorem." The incompleteness theorem basically states that the axioms of set theory do not imply an answer. He proved that the axioms of set theory are incomplete. The reason Gödel is so praised for this result is that this is not an answer that people search for. He showed that "I don't know, and you don't either."
It also seemed that Gödel was paranoid. He often didn't eat because he thought someone was trying to poison him. He had a companion who made sure he ate. It's been reported that after the death of his companion died, he died from starvation afraid that he might get poisoned.
Oh the irony, starving yourself to death because you're afraid that food might bring your death.
Wednesday, January 9, 2008
So you say you hate math?
Do you hate math? Do you really? All things being equal, math is just a subject. Math won't punch you, it won't call you names, it won't even give you a bad grade. A teacher may give a bad grade, in math even, but it's not math that is harming you.
So let's consider what you might actually hate. Maybe you hate math homework. Maybe you just rather be spending time with people you love instead considering mathematical topics. It is perfectly understandable that you would rather do a slew of things in the place of thinking about math.
The secret behind why people love thinking about math, believe me there are people who love thinking about math, is that they see beauty behind the logical structure of the subject. For some people a well proved theorem is like a well thought out work of art. To others, mathematics is a self existing field of study. The answer is out there, it just needs to be found. Oddly enough, sometimes the answer is "we don't know" as Gödel found.
I don't think that there is anything I can write that will change your opinion of math right away. The fact that you are reading this means that you care enough to alter your opinion. Check out my Inspiration in algebra for adults if you wish to learn more.
So let's consider what you might actually hate. Maybe you hate math homework. Maybe you just rather be spending time with people you love instead considering mathematical topics. It is perfectly understandable that you would rather do a slew of things in the place of thinking about math.
The secret behind why people love thinking about math, believe me there are people who love thinking about math, is that they see beauty behind the logical structure of the subject. For some people a well proved theorem is like a well thought out work of art. To others, mathematics is a self existing field of study. The answer is out there, it just needs to be found. Oddly enough, sometimes the answer is "we don't know" as Gödel found.
I don't think that there is anything I can write that will change your opinion of math right away. The fact that you are reading this means that you care enough to alter your opinion. Check out my Inspiration in algebra for adults if you wish to learn more.
Inspiration in algebra for adults
To understand algebra you must first understand numbers. First understand the integers (positive and negative whole numbers), then rational numbers, then the real numbers. The integers should be simple: There are five sheep, I have two pens. The negative numbers can be understood thinking of money. I have $1000 in the bank and a $500 debt, so my worth is $500. That debt plays the role of a negative number.
After the integers are understood, rational numbers are the next step. You should be able to add and multiply them together without a calculator or much effort (for simple problems). After that the really silly stuff begins. The rational numbers kind of have gaps in them. They don't explain every magnitude that one might consider. For example there is no rational number whose square is 2. The real numbers fill in these gaps. Once real numbers are understood, with all their operations then algebra can be a target of study.
Our discussion of algebra will involve grandmother's house. Say your grandmother moved to a house 120 miles away. You get in your car and drive at 60 miles an hour. How long will it take you to get to grandmother's house? The great thing about algebra is that it will actually answer this question. It'll take two hours. The point is that you can actually find out stuff you didn't know using the stuff you do.
This example is rather simplistic, but that's the catch 22 of math. More interesting things take more interesting (harder) math. The more interesting math is not accessible until the simpler math is understood.
Another hurdle of mathematics is the language aspect of math. Math really creates its own language. There are ideas that are deemed important, and words are created to help express the ideas. Also, there's the aspect that letters are used to represent quantities. For our grandmother's house example could have been stated as: Let d be the distance from your house to grandmother's house, r be the speed you travel, and let t be the time it takes to travel to grandmothers house. Since we know that these quantities are related by the relation d=rt we have 120=60*t for our case. Solving this equation yields that t=2.
The reason one would use such verbose language is because it is more precise. The line of reasoning is easier to follow once this type of language has been adjusted to. Also the use of the equation "d=rt" is crucial in explaining how to solve similar problems.
Don't sell yourself out. Don't go into the fight thinking you will lose. You can do it, the fact that you're reading this suggests that you are the type that can.
After the integers are understood, rational numbers are the next step. You should be able to add and multiply them together without a calculator or much effort (for simple problems). After that the really silly stuff begins. The rational numbers kind of have gaps in them. They don't explain every magnitude that one might consider. For example there is no rational number whose square is 2. The real numbers fill in these gaps. Once real numbers are understood, with all their operations then algebra can be a target of study.
Our discussion of algebra will involve grandmother's house. Say your grandmother moved to a house 120 miles away. You get in your car and drive at 60 miles an hour. How long will it take you to get to grandmother's house? The great thing about algebra is that it will actually answer this question. It'll take two hours. The point is that you can actually find out stuff you didn't know using the stuff you do.
This example is rather simplistic, but that's the catch 22 of math. More interesting things take more interesting (harder) math. The more interesting math is not accessible until the simpler math is understood.
Another hurdle of mathematics is the language aspect of math. Math really creates its own language. There are ideas that are deemed important, and words are created to help express the ideas. Also, there's the aspect that letters are used to represent quantities. For our grandmother's house example could have been stated as: Let d be the distance from your house to grandmother's house, r be the speed you travel, and let t be the time it takes to travel to grandmothers house. Since we know that these quantities are related by the relation d=rt we have 120=60*t for our case. Solving this equation yields that t=2.
The reason one would use such verbose language is because it is more precise. The line of reasoning is easier to follow once this type of language has been adjusted to. Also the use of the equation "d=rt" is crucial in explaining how to solve similar problems.
Don't sell yourself out. Don't go into the fight thinking you will lose. You can do it, the fact that you're reading this suggests that you are the type that can.
Sunday, January 6, 2008
Differential equations
Differential equations rule the world! Well, it's not that simple. You see, differential equations explain the world around us. Every physical process seems to have a differential equation behind it. For example Maxwell's equations explain the classical behavior of electrically charged particles. Even Newton's famous "F=ma" is a differential equation. Notice that a stands for acceleration which is nothing but a second derivative. If Newton is too simple for you, then you should study Lagrangian mechanics. Lagrangian mechanics is a more general way of solving classical problems.
Oh, if you're not happy with only a classical study of nature, you may want to study quantum mechanics. Guess what, there's a differential equation behind that to. Quantum mechanics does influence the real world. The so-called "Electron leak problem" is explained by quantum mechanics. This "problem" means that there is a limit on how small traditional computers can get. Once a computer gets too small it's darn near impossible to keep an electron where you want it. Noticed I used the term "traditional computers" that's because maybe with a better understanding of quantum mechanics one could make a computer we only dream of today. Quantum mechanics also helps explain the properties of matter.
There are other phenomena which are not classical in nature. These are explained by special and general relativity. Special relativity doesn't have any differential equations behind it, but it alters Newton's differential equations and makes them harder to solve. General relativity, well it has differential equations behind it, a whole new kind of differential equation.
These are reasons people who study the natural sciences want to understand differential equations. Differential equations also are used in the study of engineering, economics, biological science, epidemiology, and probably a host of subjects that I'm leaving out.
Differential equations really do rule the world. Good thing they don't have an agenda.
Oh, if you're not happy with only a classical study of nature, you may want to study quantum mechanics. Guess what, there's a differential equation behind that to. Quantum mechanics does influence the real world. The so-called "Electron leak problem" is explained by quantum mechanics. This "problem" means that there is a limit on how small traditional computers can get. Once a computer gets too small it's darn near impossible to keep an electron where you want it. Noticed I used the term "traditional computers" that's because maybe with a better understanding of quantum mechanics one could make a computer we only dream of today. Quantum mechanics also helps explain the properties of matter.
There are other phenomena which are not classical in nature. These are explained by special and general relativity. Special relativity doesn't have any differential equations behind it, but it alters Newton's differential equations and makes them harder to solve. General relativity, well it has differential equations behind it, a whole new kind of differential equation.
These are reasons people who study the natural sciences want to understand differential equations. Differential equations also are used in the study of engineering, economics, biological science, epidemiology, and probably a host of subjects that I'm leaving out.
Differential equations really do rule the world. Good thing they don't have an agenda.
Schrödinger's equation
Schrödinger's equation is the equation of quantum mechanics. The Ψ symbol is the function that carries all the information about a particle, or particles that one can know. This equation describes how that function will change with time. The ∇2 is a differential operator when replaced by a second space derivative the resulting equation is the one dimensional version. The symbol V represents potential. This is different for different situations and as a result this equation may be very hard to solve. The equation shown is a partial differential equation, or the multi-variable version of ordinary differential equations.
Notice that this equation has an i in it. This i is a root of -1. Numbers that involve i are called complex numbers. Complex numbers were developed for mathematical reasons and it's surprising that they have a explicit role in physical equations.
Maxwell's Equations
Maxwell's equations classically explain all that is electro-magnetic. They are the four equations shown above. This is a short hand notation that makes them really compact. In fact E and B are vector quantities, the electric and magnetic fields. The triangle thing (&nabla) represents a differential operation being applied to E or B. The symbol &rho is the charge distribution, t is time, &epsilon0 and &mu0 are constants, j is any electrical current, c is the speed of light . The wave equation can be derived (when &rho =0, j=0) from these four equations. This wave is the light we see, as well as the radio waves and X-rays we use. These differential equations are actually partial differential equations, or the multi-variable version of ordinary differential equations.
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