Paradoxes Examined

2005-12-29 15:59:00

Larry writes:

There is one other thing that I wanted to point out. Perhaps this will make more sense to some than the abstract mathematical approach.

JJ defines "perfect" when used to describe "logic," that is, "perfect logic," to contain no "ASCERTAINABLE flaw in reasoning." (emphasis mine). In other words one cannot presently discern, or ascertain any flaw in the reasoning. The evidence or premises seem certain to the best ability of anyone to determine at the present time. The logic applied to the evidence is irrefutable. Therefore the conclusion cannot be wrong, per JJ.

Then how about the case where a man is convicted of murder and condemned to death by a jury of twelve peers who have been presented with overwhelming and incontrovertible evidence of his guilt? Then some time later after his conviction and execution it is found that through an almost unbelievably complex and improbable set of circumstances that he was actually innocent?

Would that meet your criterion of people using "perfect logic,"that is, using the highest possible logic and the best available evidence but ultimately reaching a wrong conclusion, a conclusion that led an innocent, but extraordinarily unlucky man, to the gallows?

No. This example does not conflict with my challenge in the slightest.

Let's take a juror who uses flawless logic. He is presented with overwhelming evidence that a man is guilty. What does he conclude? He does not conclude with 100% surety that the man is guilty, but he concludes "beyond a reasonable doubt" that the man is guilty.

Beyond a reasonable doubt would mean to me (if I was a juror) that there is strong evidence that the man is guilty but there is still a possibility that he is innocent. If I felt there was a 95% chance that the man is guilty I would vote to convict. Anyone who thinks he is 100% sure is experiencing a fallacy in logic.

If it turns out the man is innocent then this outcomes still fits within the parameters of my original assumption.

It should be obvious here that the highest possible logic of a reasonable juror would conclude that there could be unknown evidence to support the man's innocence, but he has to make a decision based on what he has.

You are looking at what I have said here with too narrow of a view without seeing the picture I painted. Your example really supports my premise.

"Where a wrong conclusion is reached a flaw in logic ALWAYS can be seen."

(emphasis mine - lk)

Often, as I just pointed out a minute ago in my post "They Hung The Wrong Man," it can be the case that one has what appears to be a mass of overwhelming evidence and incontrovertible logic that does lead to a wrong conclusion.

It did not lead to a wrong conclusion. Part of the logical juror's conclusion was that there was a small chance the man was innocent. A decision is different than a conclusion. A decision is black and white but a conclusion can include many subtleties.

Larry:

"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."

Incorrect. As Judes pointed out earlier, mathematicians DO CLAIM that PI is an EXACT number. They also claim that it would take an INFINITE regress of the decimal expansion to EXACTLY represent it. In regards to that latter claim, the claim that PI is a transcendental irrational number, they have exact and irrefutable proofs, as has been explained before.

I think you're nit picking here. To say that you can write or come up with an exact number for something that will go on forever is sophistry.

Jude's post did not say that PI was a number that could be represented exactly with a calculated number but indicated that it was exact in the abstract and could be represented with a assigned number just as any point on a ruler can.

Larry:

"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."

The formula for the area of a circle is PI * R^2.

If the radius of a circle is exactly "2 over the square root of PI" then the area is EXACTLY 4.

But you do not know the exact square root of PI so what you say cannot be calculated even though it is true in theory. Also there is no such thing as a physical circle with an exact radius of 2 ft.

Larry:

Speculation is not logic.

So Einstein was not logical when he began speculating as to the nature of relativity. Of course one can be logical and still speculate. To not ever speculate is not logical.

Larry:

As to different numbering systems, PI is irrational in any and all conceivable numbering systems (different bases than 10) per the mathematicians. This is not speculation, they claim to have proven it with "perfect" mathematical logic.

I was not referring to different numbering systems as used today, such as binary. I am saying that it is possible that a future creation could be built where PI could be represented as a whole number. I intuitively perceive this is possible but beyond our present ability to conceive because of the current warp and woof of the universe.

Larry:

"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."

Because human technology and perception cannot find the exact point does not mean that it doesn't exist, or is somehow unreal.

They are real in the abstract, but not in time and space. The human race has acknowledged billions of points but so far not one person out of the billions in the history of the world has been able to find even one of them. I'd say logic and reality is on my side on this one.

Larry:

Here you are making a totally unprovable assertion.

The overwhelming evidence is on my side. Why don't you find just one point so you can have some evidence?

Larry:

Your conclusion simply does not follow from your premises. Human fallibility in finding the center point does not prove it is not real.

It is real in the abstract, but not in time and space. It's the same principle as trying to find a still point in time. The only way it could exist is if there was no time - a paradox.

Larry:

It does not prove it in EXACTLY the same way that saying no one has disproven an argument makes it true. This is a classical case of a logical fallacy.

I am talking about two things you are resisting here.

(1) Evidence. The fact that no one or ever has found a point gives overwhelming evidence that a point does not even exist in time and space. (2) Logic. Logic tells us that any point with dimension can be blown up as big as a beachball (see my latest book). Then if you create another point on the beachball and blow that up then it too will become as big as a beachball ad infinitum. A thing that cannot be found by human consciousness cannot exist in time and space, but only in the atmic (and higher) world of ideas. A point is an idea and an idea cannot be found in the physical. It can only be represented.

Larry:

Again your argument is based on claims about human abilities. Ultimately what you are saying is that BECAUSE man cannot precisely measure anything with absolute precision then nothing with absolute precision can exist (or that nothing has absolute precision).

Nothing with absolute precision does exist in time and space. Time and space are created by motion. Without these points that are moving targets there would be no physical creation. This is the same principle I presented earlier when I have taught that no particle is 100% neutral. Where positive and negative charges are completely in balance the form disappears and is no more. All creation happens because of form seeking balance and joining with other forms. A point is formless, but a point in motion is the beginning of form through the principle of illusion. Because motion always surrounds a point the point will never be discovered.

Larry:

You say that any "measurement will always be off by some small amount" but you offer NO PROOF of that assertion other than the demonstrable fact that human measurement and perception has limits.

That's pretty good evidence. If you look for a millions years for an elephant in your room that is pretty good evidence there is no elephant in the room despite the fact you are not perfect. There are trillions of points in every room and no one can find any of them.

Larry:

Your conclusion simply does not follow from your premises. It may very well be that things do have an exact dimension but that dimension may not be discoverable (consider the theorems of quantum mechanics for more on that).

If it is a thing, you are correct, but a point is not a thing. It is an idea that does not exist in time and space just as a point of "no time" does not exist in time.

 

Larry:

"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."

There is no principle here other than a demonstration of a repeated logical fallacy.

And you authoritatively say this because...?

Larry:

"Exact PI" can be precisely demonstrated in mathematics as an infinite series (although obviously no one can make a rule to measure it).

Let me get this straight. You are sure exact PI exists in physical reality but no one can find it, demonstrate it, write it or calculate it. Where is your evidence? Where is your logic that exact PI is any more than an abstract idea in this reality.

 

Larry:

"The first irrationality appears in form; the second (PI) appears in the abstract in math."

There is no irrationality in form that you allude to. You have arrived at it purely by fallacy (as previously demonstrated).

NOT!

Larry:

"No one has ever produced an object exactly one foot in length."

How do you know?

Simple math. There is one chance in infinity that one could produce a perfect foot. Let us say that all six billion people on earth were trying to do it. Infinity divided by six billion is still infinity. Therefore there is only one chance in infinity anyone has done it. One chance in infinity amounts to zero.

This tells us that any exact measurement is an abstract idea that does not even exist in time and space. Infinity itself is an idea that does not exist in physical reality.

Larry:

It is difficult to figure out which fallacy you are invoking. Is it the argument from ignorance (no one knows such and such), or the argument that my theorem has not been disproven (hence making it true)?

See my proof just noted.

Larry:

"I would call it the paradoxical principle. Why? Because measurements and points do exist in principle, but we can never measure or find them."

There is nothing paradoxical here. The only thing being demonstrated is the very rational question of the limits of human knowledge (this is covered in philosophy in the field of epistemology).

The limits of human knowledge have absolutely nothing to do with my point and if you think it does you have completely overlooked what I have been saying. No matter how fine tuned human knowledge and ability gets it cannot measure an idea with a ruler.

Also note that I call this The Paradox Principle because of an apparent paradox from the point of view of the average observer.

Larry:

Here is another principle that apparently you have not yet evaluated.

It is the Law of Non-Contradiction.

It states that contradictions (paradoxes) cannot truly exist.

I agree with this. My teaching here has resolved the paradox that a point exists but it cannot be found. It appears though that you still accept this paradox which shows an error in your thinking according to Ayn Rand.

Here is just one of the numerous paradoxes I see in your thinking.

On one hand you say a point exists and can be measured.

If this is true then that measurement can be divided in two. That which can be divided in two is not a point.

Larry:

1. Mathematics does not say that there is a "great probability" that the exact number for PI cannot be found.

2. Mathematics says that the EXACT number for PI can ONLY be found at an infinite regress.

And since infinity does not exist, except as an idea, the PI will not be found for the infinite can never be reached. It cannot be reached in form because it is an idea.

Larry:

3. Since we cannot perform an infinite regress the exact value CANNOT be found.

That which cannot be found and is impossible to find does not exist in physical reality.

Larry:

Mathematics is saying with ABSOLUTE CERTAINTY that it cannot be found short of an impossible infinite regress.

If you are so certain find just one point. You cannot.

Larry:

5. Logic does NOT tell us that any actual point does not exist.

What logic is this? I see none.

Larry:

6. Logic does tell us that human ability to measure it either cannot find it, or if it did find it, could not confirm that it had found it.

Sounds like another paradox that goes against the philosophy of Ayn Rand.

Larry:

Here is my opinion. You have not discovered any new principle. What you claim to have discovered is nothing more than the product of logical fallacy compounded.

What in fact you have pointed out is the already well known limitations of human knowledge which are the subject in the philosophical study of epistemology. However your conclusions simply do not follow from your premises. Hence they are invalid.

Authoritatively declaring my argument is invalid does not make it so. The principle I pointed out is a new insight on the apparent paradox and has nothing to do with human limitations.

Larry:

The decimal expansion for PI has a beginning at 3 (the integer 3) and proceeds with decimal digits to infinity on the right had side of the decimal point (or the "point" in any other numbering system or base conceivable).

One can BEGIN to calculate PI but one can NEVER END it.

Consider this one statement. If infinity can never be reached then there is always an end.

People can calculate the value until the end of the universe but sooner or later those calculations will end and the final number from those calculations, though large, will be an end number.

Then when a new universe is re-created calculations will begin anew.

Larry:

No ideas or principles have much impact on me UNLESS, and UNTIL they touch upon intuition in some way. Finally, and let me say this for those who are totally sick and tired of this debate; I would not have stuck to this debate for so long, and in the face of so much resistance, disapproval, and criticism if it were not the fact that I felt very strongly that there was something very important at stake - not just a fight over the definition of a word or two.

I think your intuition may be leading you correctly, but it is sometimes difficult to see that to which the intuition is leading because it usually goes against our preconceived notions.

I have gained several new insights through this discussion.

(1) I previously thought that there may be an exact number for PI. Now I see that there is not (at least in this universe) for the same reason that a board cannot be cut exactly one foot in length. I see that PI is a mathematical idea just as a foot, an inch and a point are. I see that all forms in the universe that represent ideas are created just slightly distorted from that idea, but follows the principle of the idea.

(2) I see the Paradox Principle. I can understand that a reader may see nothing new in what I presented, but I do see a new principle, which will lead to many new facts. Principles are a language unto itself and are difficult to portray clearly in our black and white words. Those who see the principle will realize that they have not seen it explained in print before, even though the separate ingredients have been.

Because infinity does not really exist then the universe itself is not infinite, but unlimited, and all things having form within it can be numbered.

Larry:

Perfect reason will ultimately lead you to wrong conclusions, and when you reach that point, all that is left is God. This is the lesson of the paradox. The mystics have always known it. We who are logical have not always believed it.

How do you think we arrive at intuitive perception? Do you suppose that one day it just comes to us?

No.

First we use our hearts, minds and reason. Finally, we conclude that the mind can only take us so far. Then we ask those all-important questions:

Is there more? How do I find it?

Thus does logic and reason lead the seeker to pay attention to the higher until the intuition is seen.

Then when the disciple brings down principles from the intuitive level his logical mind will say. "Yes! Why did I not see that before?"

Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.  Nikola Tesla (1857 - 1943), Modern Mechanics and Inventions, July, 1934