The hope of the Critique of the Diagonal Method was that math teachers would realize the many flaws of the method and do a better job of presenting it. After reading Everything and More, I decided that maybe I should move away from subtle hints that teachers improve their method to a stronger refutation.
In the article on the diagonal method on finite binary strings, I decided to emphasize that the diagonal method is simply a new variety of the liar's paradox. The liar's paradox is any statement that includes a self negation, like "This sentence is false."
The goal of the article dialectics is to show that the method was one of several foundational theories based on oppositional logic that were popular during the German Idealist movement. Freud, Hegel and Marx have been widely refuted. The dichotomy created by the diagonal method is really one of the last of these massive theories of everything left standing.
The next article will explore the Galileo paradox. That piece is really necessary to show the true origins of Cantor's dichotomy.
Thursday, November 27, 2003
Wednesday, November 19, 2003
I finished reading David Foster Wallace's everything and more [See Notes]. The story provides an extremely one sided view of history with Aristotle (and his bastard child logic) as the greatest enemies of civilization. Zeno, Plato, Hegel, Cantor and that crowd are the greatest champions of truth (I should say truth after it is properly redefined) Justice (again, justice after it is properly sanitized) and the academic way.
Anyway, I now need to take a couple days and write a rebuttal in which the people who have defended logic, liberty and the individuals are the heros. The problem is that Cantor has given us such a twisted view of the infinite that it is extremely difficult to unravel. Discussions about the transfinite simply turn into shouting matches with one group saying "I can see the completed infinity." the other group saying "no, I cannot." I guess I should add a third group. The Wittgenstein's who know the whole thing was just a game.
Anyway, I now need to take a couple days and write a rebuttal in which the people who have defended logic, liberty and the individuals are the heros. The problem is that Cantor has given us such a twisted view of the infinite that it is extremely difficult to unravel. Discussions about the transfinite simply turn into shouting matches with one group saying "I can see the completed infinity." the other group saying "no, I cannot." I guess I should add a third group. The Wittgenstein's who know the whole thing was just a game.
Thursday, November 13, 2003
This quote is from a blurb that appeared in yahoo about an Alabama Judge who defiantly had a large statue of the 10 Commandments at the courthouse.
This quote is about a dispute between atheists and christians. When I first read the quote, something about it appeared wrong. I had to read the thing twice, then it jumped out at me. The Ten Commandments is not "a reminder of basic values". A true believer should hold first and foremost that the Ten Commandments are a direct covenant with God. These are not simple values, these are a law that transcend any written human law.
If the judge does not understand that his having this statue carved in stone at the courthouse tells the world that he holds his religious beliefs to be higher than the laws of the land, then the Judge Roy Moore is showing that he either does not understand the premise of the rule of law, or that he does not understand the nature of the ten commandments.
Commandments are not basic values. In the Judeo-Christian tradition, the Ten Commandments are a law that was written in stone by the hand of God. A true Christian who understood the theology would know that spending taxpayer's dollars to put up an expensive statue with the Ten Commandments carved in stone directly tells the public that the court holds The Ten Commandments to be a higher law than the Constitution, and to be a higher law than any law passed by the legislator.
A true Christian would know that the ten commandments is a set of laws that they themselves must obey. However, the rule of law does not permit us all to run around with our own laws. A Judge who swears an oath to uphold the laws of the land is derelict if he holds another set of laws in supremacy to the laws of the land. If there was an Islamic Judge that claimed Shari`ah was a higher law than the Constitution, he would be drawn and quartered in Alabama.
Although I think the Atheist's movement to prevent the Ten Commandments from being discussed in schools is horrible, I agree 100% that Judge Roy Moore needs to be tossed on his keister for this absurd little symbolic battle of his.
Personally, I think Christians should be more up in arms about these people who are claiming that the Ten Commandments are just a moral principle than about the statue issue. A true Christian who understand both the nature of the covenent with God, and the nature of the rule of law would understand the conflict and why a Courthouse should not have a statue of the ten commandments on their steps.
The Alabama case has fueled a long-running U.S. debate pitting defenders of the First Amendment against Christians who believe the Ten Commandments should be viewed in schools or courthouses as a reminder of basic values.
This quote is about a dispute between atheists and christians. When I first read the quote, something about it appeared wrong. I had to read the thing twice, then it jumped out at me. The Ten Commandments is not "a reminder of basic values". A true believer should hold first and foremost that the Ten Commandments are a direct covenant with God. These are not simple values, these are a law that transcend any written human law.
If the judge does not understand that his having this statue carved in stone at the courthouse tells the world that he holds his religious beliefs to be higher than the laws of the land, then the Judge Roy Moore is showing that he either does not understand the premise of the rule of law, or that he does not understand the nature of the ten commandments.
Commandments are not basic values. In the Judeo-Christian tradition, the Ten Commandments are a law that was written in stone by the hand of God. A true Christian who understood the theology would know that spending taxpayer's dollars to put up an expensive statue with the Ten Commandments carved in stone directly tells the public that the court holds The Ten Commandments to be a higher law than the Constitution, and to be a higher law than any law passed by the legislator.
A true Christian would know that the ten commandments is a set of laws that they themselves must obey. However, the rule of law does not permit us all to run around with our own laws. A Judge who swears an oath to uphold the laws of the land is derelict if he holds another set of laws in supremacy to the laws of the land. If there was an Islamic Judge that claimed Shari`ah was a higher law than the Constitution, he would be drawn and quartered in Alabama.
Although I think the Atheist's movement to prevent the Ten Commandments from being discussed in schools is horrible, I agree 100% that Judge Roy Moore needs to be tossed on his keister for this absurd little symbolic battle of his.
Personally, I think Christians should be more up in arms about these people who are claiming that the Ten Commandments are just a moral principle than about the statue issue. A true Christian who understand both the nature of the covenent with God, and the nature of the rule of law would understand the conflict and why a Courthouse should not have a statue of the ten commandments on their steps.
Wednesday, November 12, 2003
First off, I believe that Rational and Irrational numbers are different things. For the reasons I express in Rich Theory, I believe that there are more irrational numbers than there are rational numbers. Essentially, the crossectional mapping creates a set of rational numbers that is less than n^2. Building the power set implies that the irrational numbers are better described with exponential functions...2^n. 2^n > n^2.
Anyway, Set Theorists determine that the size of the set of irrational numbers is great than the size of the rational numbers through a reverse reasoning. They start with the assumption that the Real numbers are at a higher level of infinity than the rationals. The term irrational number refers to a real number that is not rational. As the reals are the union of rational and irrational numbers, set theorists conclude that the irrational numbers must be part of this higher level of existence shared with the real numbers. In fact, the theory is essentially saying that the irrational numbers are so dense that they give the real line dimension.
I need to repeat that. The rational numbers do not have any dimension. As you know, a point has no dimension. If you took all of the rational numbers together, you would not have any dimension. The supposition is that the irrational numbers are so dense. The irrational numbers are at a higher level of existence. There are so many of them that they actually create dimension. If you draw a line from point A to Point B, you do not see any rational numbers. If you look at the line going from point A to B below. You don't see any rational numbers...you only see irrational numbers:
A _________ B
Let's jump out of dialectical metaphysics, and get back on track.
Premise A: I understand that set theorist hold to the belief that all irrational numbers can be expressed with a unique infinite string of digits. (NOTE, this is a supposition that may or may not be true).
Anyway. If you hold to premise A, then you come across a very interesting point. There is a rational number between any two irrational numbers. This falls instantly from premise A. If two irrational numbers have a unique digital representation, then there must be some digit in their representation that they differ. You can create a rational number simply by finding this point where the two numbers differ and truncate the larger of the two digits to that precision. Voila. You have a rational number that is between your two irrational numbers.
This is easier to discuss in binary than in decimal mode. Binary numbers are expressed as strings of 0s and 1s. You can use binary digits to express numbers less than 1. For example 0.1 is 1/2, 0.01 is 1/4. The nth digit in the string is 1/2^n. The number .1011 is 1/2 + 1/8 + 1/16. This happens to equal 11/16.
The rules for binary digits follows are the same as the rules for decimal numbers. Rational numbers will
either end in an infinite string of zeros, or in a repeating pattern. .01010101... Where 01 repeats forever is equal to 1/3.
Okay, so we have two irrational numbers a and b where a < b. By definition neither a nor b end in an infinite string of zeros. If I find the first digit where a and b differ and simply truncate b to that digit, then I have a rational number that is between a and b. This is a demo:
Note, the number ...11101101010101011 is between a and b. Not only that. I can append an infinite combinitions of repeating strings (rational numbers) to the string and show that there is an infinite number of rational numbers between each irrational number. (BTW, it is possible to show that there is an infinite number of irrational numbers between each pair of rationals using the same technique.)
Back to Premise A.
If I hold to the position that irrationals can be represented by a unique infinite string--the keyword being unique. Then I must conclude that for irrational number a, there is some digit at which a differs from all other irrational numbers. Truncating at that digit, I get a rational numbers. With a little bit of finagling, I can create a theoretic 1-1 mapping between rationals and irrationals. It is a theoretic mapping because we are dealing with strings so large that they are transcomputational, but I know of no limits that says I can't imagine a set of irrational numbers larger than the number of quantum states on the planet earth.
As dialecticians are granted the divine right to change definitions at a whim. The route around this dilemma is probably just to say that two numbers might have the exact same digital representation, but still be unique.
Lesser folk, like myself, simply stare and wonder we've pushed this convoluted dialectical system into the foundations of modern mathematics.
The integers, rational numbers and real numbers are all different types of numbers. This transfinite theory that tries to claim that the rational numbers are just a different type of integer is really one of the most bogus theories ever to be forced on man.
Anyway, Set Theorists determine that the size of the set of irrational numbers is great than the size of the rational numbers through a reverse reasoning. They start with the assumption that the Real numbers are at a higher level of infinity than the rationals. The term irrational number refers to a real number that is not rational. As the reals are the union of rational and irrational numbers, set theorists conclude that the irrational numbers must be part of this higher level of existence shared with the real numbers. In fact, the theory is essentially saying that the irrational numbers are so dense that they give the real line dimension.
I need to repeat that. The rational numbers do not have any dimension. As you know, a point has no dimension. If you took all of the rational numbers together, you would not have any dimension. The supposition is that the irrational numbers are so dense. The irrational numbers are at a higher level of existence. There are so many of them that they actually create dimension. If you draw a line from point A to Point B, you do not see any rational numbers. If you look at the line going from point A to B below. You don't see any rational numbers...you only see irrational numbers:
A _________ B
Let's jump out of dialectical metaphysics, and get back on track.
Premise A: I understand that set theorist hold to the belief that all irrational numbers can be expressed with a unique infinite string of digits. (NOTE, this is a supposition that may or may not be true).
Anyway. If you hold to premise A, then you come across a very interesting point. There is a rational number between any two irrational numbers. This falls instantly from premise A. If two irrational numbers have a unique digital representation, then there must be some digit in their representation that they differ. You can create a rational number simply by finding this point where the two numbers differ and truncate the larger of the two digits to that precision. Voila. You have a rational number that is between your two irrational numbers.
This is easier to discuss in binary than in decimal mode. Binary numbers are expressed as strings of 0s and 1s. You can use binary digits to express numbers less than 1. For example 0.1 is 1/2, 0.01 is 1/4. The nth digit in the string is 1/2^n. The number .1011 is 1/2 + 1/8 + 1/16. This happens to equal 11/16.
The rules for binary digits follows are the same as the rules for decimal numbers. Rational numbers will
either end in an infinite string of zeros, or in a repeating pattern. .01010101... Where 01 repeats forever is equal to 1/3.
Okay, so we have two irrational numbers a and b where a < b. By definition neither a nor b end in an infinite string of zeros. If I find the first digit where a and b differ and simply truncate b to that digit, then I have a rational number that is between a and b. This is a demo:
a ...111011010101010100010...
b ...111011010101010110110...
^
Note, the number ...11101101010101011 is between a and b. Not only that. I can append an infinite combinitions of repeating strings (rational numbers) to the string and show that there is an infinite number of rational numbers between each irrational number. (BTW, it is possible to show that there is an infinite number of irrational numbers between each pair of rationals using the same technique.)
Back to Premise A.
If I hold to the position that irrationals can be represented by a unique infinite string--the keyword being unique. Then I must conclude that for irrational number a, there is some digit at which a differs from all other irrational numbers. Truncating at that digit, I get a rational numbers. With a little bit of finagling, I can create a theoretic 1-1 mapping between rationals and irrationals. It is a theoretic mapping because we are dealing with strings so large that they are transcomputational, but I know of no limits that says I can't imagine a set of irrational numbers larger than the number of quantum states on the planet earth.
As dialecticians are granted the divine right to change definitions at a whim. The route around this dilemma is probably just to say that two numbers might have the exact same digital representation, but still be unique.
Lesser folk, like myself, simply stare and wonder we've pushed this convoluted dialectical system into the foundations of modern mathematics.
The integers, rational numbers and real numbers are all different types of numbers. This transfinite theory that tries to claim that the rational numbers are just a different type of integer is really one of the most bogus theories ever to be forced on man.
Monday, November 10, 2003
My presentation on transfinite theory did not go quite as well I would have hoped. Of course, this was the first time I tried my hand at public speaking since high school. Not only that, I had given myself some extremely difficult obstacles to surmount. The first was that half the audience were not familiar with the theory. I wanted to give these people an unbiased view that would show both the positive and negative side of the debate.
I also forgot to mention transfinite theory was essentially based upon the same dialectical method developed by Kant and Hegel that were used by Freud, Marx, Jung and others. The basic formula was to inject, at a metaphysical layer, a thesis, antithesis and catharsis.
The truth is that Cauchy had developed an adequately rigorous foundation for the Calculus. What Cantor did was to inject a the dialectics at a metaphysical level. Mathematics is simply waiting for enough of the vanguard that is propounding the theory to die off. When they are dead and gone, the disease will eventually cure itself, just as economics and psychology have gradually been curing from the diseases caused by Freud and Marx.
I also forgot to mention transfinite theory was essentially based upon the same dialectical method developed by Kant and Hegel that were used by Freud, Marx, Jung and others. The basic formula was to inject, at a metaphysical layer, a thesis, antithesis and catharsis.
The truth is that Cauchy had developed an adequately rigorous foundation for the Calculus. What Cantor did was to inject a the dialectics at a metaphysical level. Mathematics is simply waiting for enough of the vanguard that is propounding the theory to die off. When they are dead and gone, the disease will eventually cure itself, just as economics and psychology have gradually been curing from the diseases caused by Freud and Marx.
The Bureau of Labor Statistics says that companies started hiring again. There was positive growth in the number of employed and a drop in the unemployment rate.
Sadly, the most interesting job listed in the Computer section of the trib was:
I think combining web development and warehouse work is a good combo. As warehouse workers tend to get paid more than $6.00 my guess is that the company values web development at less than minimum wage, but can pull the web developer salary up to $6.00 if they do some real work.
Sadly, the most interesting job listed in the Computer section of the trib was:
Web Developer good pay.
light warehouse work
$6/hr.
light warehouse work
$6/hr.
I think combining web development and warehouse work is a good combo. As warehouse workers tend to get paid more than $6.00 my guess is that the company values web development at less than minimum wage, but can pull the web developer salary up to $6.00 if they do some real work.
Friday, November 07, 2003
The Night of a Thousand Dinners had a good showing. There was about 150 people. The night featured middle east dancing, and everyone had a good time. I am anxious to find out how much the dinner raised.
With the land mine ban accepted by most nations, the program is making great strides in cleaning up mines around the world. Hopefully, there won't be a need for such fund raisers in the near future.
The US is one of the few nations that has not signed the ban. The main reason is the minefield between North and South Korea. Considering the delicacy of that situation, I have to agree that it is best not to address the problem.
Anyone interested in hosting a dinner can go to www.1000Dinners.com.
With the land mine ban accepted by most nations, the program is making great strides in cleaning up mines around the world. Hopefully, there won't be a need for such fund raisers in the near future.
The US is one of the few nations that has not signed the ban. The main reason is the minefield between North and South Korea. Considering the delicacy of that situation, I have to agree that it is best not to address the problem.
Anyone interested in hosting a dinner can go to www.1000Dinners.com.
Tuesday, November 04, 2003
I wasted a day and a half reading the new Harry Potter and I now have a Harry Potter head ache.
Winter's hit Salt Lake with a vengeance. We got a foot of snow in the first snow of the year.
I will be giving a presentation on the diagonal method at the next iies meeting. It would be an interesting topic to discuss, but I suspect there will be zero attendees. The last meeting had 6 people.
Winter's hit Salt Lake with a vengeance. We got a foot of snow in the first snow of the year.
I will be giving a presentation on the diagonal method at the next iies meeting. It would be an interesting topic to discuss, but I suspect there will be zero attendees. The last meeting had 6 people.
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