So yes, I would be an advocate of teaching formal logic in grade school and high school and would require some proficiency before letting someone graduate! :)
I think this is a good idea. Even though I do not see much value in me taking a logic course I see significant value in the youth taking such a course and would have been interested in it myself when I was young.
Now let us analyze your last post
Larry (quoting me)
"Instead of: PI has a beginning. But apparently it has no end. It should read: The attempt of the human race to calculate PI had a beginning and the attempt will have an end just as the human race will eventually have an end. It is unknown whether or not the final attempt will discover the exact number representing the perfect circle. This does not mean the exact number does not exist. There are no means available to prove that the exact number for PI does or does not exist."
Just for the record, the value of PI has been calculated out to SIX BILLION PLACES so far. How much farther scientists and mathematicians will (or can) pursue that calculation I don't know.
PI belongs to a group of very special numbers called transcendental irrational numbers. Another famous number (for mathematicians) of the same type is the square root of 2. There are many others, but these two numbers are famous in history (the Hebrews in II Chronicles 4 appear to have not been very good mathematicians as they gave the value of PI as 3).
"There are no means available to prove that the exact number for PI does or does not exist."
As I have shown above, and is shown in advanced mathematics textbooks, there are available irrefutable proofs that PI is irrational.
So, assuming JJ is ultimately right about the nature of PI, then here is an example of PERFECT LOGIC that leads to a wrong conclusion.
I rest my case.
You pointed out a probable flaw here Larry, but it may be a different flaw than you intended.
What you disagreed with was: "Perfect logic will never lead to a wrong conclusion."
Where a wrong conclusion is reached a flaw in logic always can be seen. There was a flaw in my logic, but not in the above statement.
Now mathematicians claim to have proven that there can be no exact number for PI, that calculations would virtually go on forever or is irrational. I'll have to take their word on this as I do not have the training to logically follow their formulas wherein the proof is claimed. It is most probable that the proof is mathematically sound.
Let us look at the principle behind why PI, the square root of two and other formulas that are irrational. This is important for this was left out of my original premise and is a missing piece needed to perfect the logic.
PI is used to find the amount of square area within a circle, but because we cannot find the exact value for PI neither can we find the exact square feet, inches etc within the circle.
It is interesting that we have formulas to find the exact square area of a square, rectangle and triangle, but not for a circle. On the other hand the circle has an exact amount of square feet just as a square does. This reveals a prejudice of assigning values in our current mathematical system that is beyond the ability of human consciousness to presently correct.
It is also interesting that life in the previous solar system was built upon the square and in this one it is the triangle (according to DK). We have exact formulas for finding the square of these two shapes, but not the circle, which will be the foundation of life in the future solar system.
Could it be that an exact formula will present itself when the future life on the future system manifests? Perhaps it will manifest through an entirely different type of numbering system.
The reason PI seems impossible to discover is the same reason that the exact center of a circle or the midway point of a line is impossible to pinpoint. The paradoxical thing is that you know the midway point is there, but it cannot be exactly pinpointed. For instance, the midway point of a foot long ruler is at the six-inch point. Mathematically that point is there, but if you take the sharpest instrument available you will not be able to put it on the exact 6-inch point. If you try and place a needle on it you may hit a point 6.001 inches from one edge. No matter how fine-tuned the instrument is you can never hit the exact middle. You may even get a powerful microscope and try to see the exact middle but you will still miss it. Now you may be off by .0000000002 inches, but you will always be off.
As you make the middle larger and larger the point still evades you. Why? Because the actual point only is real in the world of math and does not exist in time and space.
This irrational principle also exists with the square and triangle even though math seems to have their formulas neatly packaged. A square two feet by two feet seems to have an area of four feet according to the formula. It appears we can neatly figure the exact square of a square but not a circle. This is illusion on the same level that PI cannot be found because in the world of form there is no square in existence that is exactly one foot in length. No matter how precise one tries to draw or construct a square of two feet in length the measurement will always be off by some small amount.
Therefore, a form with the square of a two-foot square does not exist in the real world. By that same principle the exact PI does not exist in our math.
The first irrationality appears in form; the second (PI) appears in the abstract in math.
We presently have a system of numbers and math that allows us to assign an exact figure (say one foot) to a distance that no one has been able to measure as existing in the real world. Logically a foot in length does exist for we can see a measurement longer than a foot and smaller than a foot and we know that the point for one foot exists between the two. It exists, but we can't find it. No one has ever produced an object exactly one foot in length.
Similarly, we know a perfect circle exists, but we can't construct one or even find the value of it mathematically through PI.
These points cannot be discovered in our reality even though they exist, but they exist beyond time and space. It is ironical though that such points create time and space, yet they elude us.
Let us name this principle which creates the "irrational" or illusive measurement in our reality. I do not think the mathematical name "irrational" is the best key word to use. I would call it the paradoxical principle. Why? Because measurements and points do exist in principle, but we can never measure or find them.
Perhaps this makes more sense when we consider that all form is created by illusion. Because this is so it only makes sense that no single point can ever be discovered with our current consciousness.
Taking this additional insight into consideration let us re-examine a key phrase to my original premises:
"There are no means available to prove that the exact number for PI does or does not exist."
This was not true according to the paradoxical principle or to mathematical proof as established in our current reality as Larry pointed out. For those of you who grumble I do not admit to mistakes I will admit that this premise was flawed and lacked the highest logic.
There does exist mathematical means to show with great probability that the exact number for PI shall not be found, at least in our reality. This is reasonable when logic tells us that no actual point can ever be discovered because it does not exist in time and space.
Now let us correct the flaw in my original premise and present it again.
My original premise read: "It is unknown whether or not the final attempt will discover the exact number representing the perfect circle. This does not mean the exact number does not exist. There are no means available to prove that the exact number for PI does or does not exist."
Even though it may be argued that this premise could be true it is not solid enough to be flawless.
I believe we can now write a flawless one, which reads: "Under our present construct of math and physical reality PI exists as an ever illusive point that cannot be identified with an exact number, at least not a number that can be written using current mathematical principles.
If we now use this new accurate premise and follow through with logic we will see that my original premise was correct which was.
The truth is always logical. Any false conclusion always involves flawed logic. Perfect logic will never lead to a wrong conclusion.
You revealed a false conclusion Larry, which led me to discover a flaw in my logic along with the teaching of a new principle I had not before identified which I have named "The Paradoxical Principle."
It is a paradox that a point as well, as PI exists, but we cannot find them. Logic leaves us open to the fact that in a future solar system or universe there may be an exact number for PI. Perhaps it will be 42.
PI is not just irrational. It is a transcendental and irrational. That means that when expressed as a decimal expansion that it does not terminate or repeat. That means it goes on and on to infinity, and I believe this is what JJ is objecting to.
Therefore it has a beginning, but no end.
That I think is the real issue for JJ, and it is initially what I claimed.
I guess this point you are making passed over my head. As far as I can see both PI and the circle never had a beginning and will never have an end. Whatever the value of PI is, as well as where the midway point of a line is, are concepts without beginning and end.
If PI has a beginning then what or when is that beginning?
In mathematics you don't understand things. You just get used to them. Johann von Neumann (1903 - 1957)
Copyright © 2005 by J J Dewey, All Rights Reserved