(no subject)
97.
15.
I found some fabulous quotes online today. It's math related, I know. I'm a math nerd remember? These quotes are so exciting to me!
"It is impossible to establish the logical consistency of any complex deductive system except by assuming principles of reasoning whose own internal consistency is an open question as that of the system itself."
-Kurt Godel's Incompleteness theorm.
"Our conception of substance is only vivid so long as we don't face it. It begins to fade when we analyze it. ...It's amazing how complimentary Godel's [incompleteness] theorm and the [Heisenberg] Uncertainty principle are - they both devistate the idea of a solid physical world filled with 'truths.' There are no objects, no people, no truth. It seems like you can never find ultimate truth peace, and the purest of love because you are trying to get these things under the false assumption that they exist in some real way."
-Ken Korczak, commenting on the mathematically proven fact that reality is simply an illusion.
It's an interesting concept really. This is how I understood it, but as you will see, it is only an illusion, as is everything we know and 'understand.'
The only reason mathematical proofs are so important is because they transcend language. To quote Mean Girls, "It [math] is the same in every language." According to Godel, language is the first method of human illusion. It's the gateway to believing that the world is real when in fact it's not. For example, think about the civil war. We've all read in text books about it, we've all seen movies about it, etc. We all know what happened. But what if some of recorded history isn't true? The only basis for our belief of what happened is language-written or expressed. Expressed language includes artwork and later photograpy and videos. We believe what we do about the past even though we have not seen it with our own eyes through our belief and trust in language. It has been said that Pictures can say a thousand words, but what if the picture you're looking at is from a movie? We all know that movies are not real. They are a human depiction of 'reality' as a form of enterainment. This picture is speaking a thousand words, but not about a reality. It is speaking a thousand words about someone's interpretation through language.
The concept of language is so powerful, yet so vague. Honestly, I find it hard to believe how children can pick up on ideas and actions. I don't find it as hard to understand how they learn nouns like the names of objects. They are 'concrete' objects, even though 99.999% of the objects are empty space, at least at the atomic level. They have a visual representation. Think about verbs though. Running. Jumping. Thinking. Loving. Going. Doing. How do you explain the meaning of these words? There is no concrete object to identify them with. They are actions. I honestly don't see how you can explain the definition of run to a child without actually demostrating the action over and over again. At least, not to a child who lacks an extensive vocabulary.
But even so, we all know what 'running' is. But don't we all run differently? Some people may consider 'running' to be 'a pace slightly faster than a jog.' Yet others may define it in terms of greater speed.
This goes back to my first quote, about complex deductive systems. Language is one such system. The principles and basis of communication is language, which is open to interpretation. Each and every individual person can interpret things differently. No one has the same perspective as anyone around them. Therefore, communication itself has no logical basis because the concept of language is not lucid.
This is once again shown in the concept of a mathematical proof. Most educated peoples know that a mathematical proof is the only way that certain concepts become mainstream. But what is a proof? Is it simply mathematically representing a concept we believe to be true? On a very small scale, yes. Think back to 5th grade, when you first started doing long word probelems. Interpreting the language into mathematical equations seemed more difficult then. This is because mistakes are often made through misinterpretation of the written language. Therefore, we assign symbols certian ideas. For example, 9 + 6 = 15. To someone who did not know what '+' means, this statement would be totally illogical, since it is beond their comprehension. Godel himself proved that any formal system complex enough to support extensive number theory had at LEAST one undecidable statement in it, which is a statement that can neither be proven or disproven in a formal system. (This excludes situations in which the statement can neither be proven or disproven, but the opposite of the statement CAN be disproven or proven. For Example, I can't prove that someone did or did not construct 'object A', but I can prove that there is no way that the person is capable or incapable of constructing 'object A.')
The title of Godel's first theorm, "the incompleteness theorm" comes from this concept. Undecidable statements can not be proven or disproven, no matter who much we KNOW the statement is true. Logic can imply it's validity all day long. The statement can be tested and found to work again and again, but without the proper axiom (ideas which are too obvious to be proven, such as subtracting one from any number creates a number smaller than the first), it cannot -no matter how hard you try- be proven. Godel's conclusion was that because this known fact could not be proven, it was considered incomplete.
Godel's second theorm is closely related to his first. It proves that provability is not as strong as truth, no matter what axiomatic system used.
(This is, of course, based on the assumption of an infinite system. Humans do not have the ability to fully understand the concept of infinity. If we were to base it on a finite system, we would be able to prove or disprove anything and everything we cared to. This gives way to the theory that because Godel's ideas are 100% sound, that whatever governs our world is indeed infinite. It essentially means that no matter how hard we try, we can never fully understand anything, because everything is infinite. This then relates back to 'energy is neither created nor destroyed,' which cannot be proven or disproven, because we assume energy is infinite, which we don't understand. This is where the scientifically accepted fact that Humans do not exist comes from. We are considered an illusion because we represent something that is finite, which contradicts the fact already proven that everything is indefinite.)
It seems so strange that a mathematical Proof concludes that some truths cannot be proven. What fascinates me even more is how he deduced this proof.
Consider this paradox, first introduced by Bertrand Russell.
Let 'N' represent normal sets, which DO NOT contain themseves.
Let 'V' represent non-normal sets, which DO contain themselves.
(to illustrate: The set of all dogs is not a dog itself. It is considered normal. However, the set of all imaginary things is itself imaginary, so it is considered non-normal.)
Let O'(N) represent the set of all normal sets. Is O'(N) a normal set? If it is, then by definition of a 'normal set,' it cannot contain itself. That would make it a non-normal set, but we just defined O'(N) as the set of all normal sets. If it were concluded to be a non-normal set, then it would contain non-normal sets. This once again creates a problem, because non-normal sets contain themselves, and O'(N) is defined to be the set of all normal sets.
No matter how you approach this problem, you can't prove or disprove either fact or it's opposite. Godel used Russell's paradox as an illustration of a something he wanted to prove. Then he formulated a very complicated but totally precise proof that involved mapping prime numbers onto statements.
The theory is there. The proof exixts ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems" by Kurt Godel).
And yet, no matter how complicated the prime number system he used is, there is not an undecidable statement in Godel's proof. This is in total violation of his 'incompleteness theory.'
He proves that nothing involving a complex number system can be proven, even though he used one of the most extensive number systems known to modern mathematics to prove it.
It is the ultimate paradox.
You have no idea how much I want a copy of that damn paper.
15.
I found some fabulous quotes online today. It's math related, I know. I'm a math nerd remember? These quotes are so exciting to me!
"It is impossible to establish the logical consistency of any complex deductive system except by assuming principles of reasoning whose own internal consistency is an open question as that of the system itself."
-Kurt Godel's Incompleteness theorm.
"Our conception of substance is only vivid so long as we don't face it. It begins to fade when we analyze it. ...It's amazing how complimentary Godel's [incompleteness] theorm and the [Heisenberg] Uncertainty principle are - they both devistate the idea of a solid physical world filled with 'truths.' There are no objects, no people, no truth. It seems like you can never find ultimate truth peace, and the purest of love because you are trying to get these things under the false assumption that they exist in some real way."
-Ken Korczak, commenting on the mathematically proven fact that reality is simply an illusion.
It's an interesting concept really. This is how I understood it, but as you will see, it is only an illusion, as is everything we know and 'understand.'
The only reason mathematical proofs are so important is because they transcend language. To quote Mean Girls, "It [math] is the same in every language." According to Godel, language is the first method of human illusion. It's the gateway to believing that the world is real when in fact it's not. For example, think about the civil war. We've all read in text books about it, we've all seen movies about it, etc. We all know what happened. But what if some of recorded history isn't true? The only basis for our belief of what happened is language-written or expressed. Expressed language includes artwork and later photograpy and videos. We believe what we do about the past even though we have not seen it with our own eyes through our belief and trust in language. It has been said that Pictures can say a thousand words, but what if the picture you're looking at is from a movie? We all know that movies are not real. They are a human depiction of 'reality' as a form of enterainment. This picture is speaking a thousand words, but not about a reality. It is speaking a thousand words about someone's interpretation through language.
The concept of language is so powerful, yet so vague. Honestly, I find it hard to believe how children can pick up on ideas and actions. I don't find it as hard to understand how they learn nouns like the names of objects. They are 'concrete' objects, even though 99.999% of the objects are empty space, at least at the atomic level. They have a visual representation. Think about verbs though. Running. Jumping. Thinking. Loving. Going. Doing. How do you explain the meaning of these words? There is no concrete object to identify them with. They are actions. I honestly don't see how you can explain the definition of run to a child without actually demostrating the action over and over again. At least, not to a child who lacks an extensive vocabulary.
But even so, we all know what 'running' is. But don't we all run differently? Some people may consider 'running' to be 'a pace slightly faster than a jog.' Yet others may define it in terms of greater speed.
This goes back to my first quote, about complex deductive systems. Language is one such system. The principles and basis of communication is language, which is open to interpretation. Each and every individual person can interpret things differently. No one has the same perspective as anyone around them. Therefore, communication itself has no logical basis because the concept of language is not lucid.
This is once again shown in the concept of a mathematical proof. Most educated peoples know that a mathematical proof is the only way that certain concepts become mainstream. But what is a proof? Is it simply mathematically representing a concept we believe to be true? On a very small scale, yes. Think back to 5th grade, when you first started doing long word probelems. Interpreting the language into mathematical equations seemed more difficult then. This is because mistakes are often made through misinterpretation of the written language. Therefore, we assign symbols certian ideas. For example, 9 + 6 = 15. To someone who did not know what '+' means, this statement would be totally illogical, since it is beond their comprehension. Godel himself proved that any formal system complex enough to support extensive number theory had at LEAST one undecidable statement in it, which is a statement that can neither be proven or disproven in a formal system. (This excludes situations in which the statement can neither be proven or disproven, but the opposite of the statement CAN be disproven or proven. For Example, I can't prove that someone did or did not construct 'object A', but I can prove that there is no way that the person is capable or incapable of constructing 'object A.')
The title of Godel's first theorm, "the incompleteness theorm" comes from this concept. Undecidable statements can not be proven or disproven, no matter who much we KNOW the statement is true. Logic can imply it's validity all day long. The statement can be tested and found to work again and again, but without the proper axiom (ideas which are too obvious to be proven, such as subtracting one from any number creates a number smaller than the first), it cannot -no matter how hard you try- be proven. Godel's conclusion was that because this known fact could not be proven, it was considered incomplete.
Godel's second theorm is closely related to his first. It proves that provability is not as strong as truth, no matter what axiomatic system used.
(This is, of course, based on the assumption of an infinite system. Humans do not have the ability to fully understand the concept of infinity. If we were to base it on a finite system, we would be able to prove or disprove anything and everything we cared to. This gives way to the theory that because Godel's ideas are 100% sound, that whatever governs our world is indeed infinite. It essentially means that no matter how hard we try, we can never fully understand anything, because everything is infinite. This then relates back to 'energy is neither created nor destroyed,' which cannot be proven or disproven, because we assume energy is infinite, which we don't understand. This is where the scientifically accepted fact that Humans do not exist comes from. We are considered an illusion because we represent something that is finite, which contradicts the fact already proven that everything is indefinite.)
It seems so strange that a mathematical Proof concludes that some truths cannot be proven. What fascinates me even more is how he deduced this proof.
Consider this paradox, first introduced by Bertrand Russell.
Let 'N' represent normal sets, which DO NOT contain themseves.
Let 'V' represent non-normal sets, which DO contain themselves.
(to illustrate: The set of all dogs is not a dog itself. It is considered normal. However, the set of all imaginary things is itself imaginary, so it is considered non-normal.)
Let O'(N) represent the set of all normal sets. Is O'(N) a normal set? If it is, then by definition of a 'normal set,' it cannot contain itself. That would make it a non-normal set, but we just defined O'(N) as the set of all normal sets. If it were concluded to be a non-normal set, then it would contain non-normal sets. This once again creates a problem, because non-normal sets contain themselves, and O'(N) is defined to be the set of all normal sets.
No matter how you approach this problem, you can't prove or disprove either fact or it's opposite. Godel used Russell's paradox as an illustration of a something he wanted to prove. Then he formulated a very complicated but totally precise proof that involved mapping prime numbers onto statements.
The theory is there. The proof exixts ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems" by Kurt Godel).
And yet, no matter how complicated the prime number system he used is, there is not an undecidable statement in Godel's proof. This is in total violation of his 'incompleteness theory.'
He proves that nothing involving a complex number system can be proven, even though he used one of the most extensive number systems known to modern mathematics to prove it.
It is the ultimate paradox.
You have no idea how much I want a copy of that damn paper.