Universal Probabilty Bound?

[Serious]

To me, as a physicist, there is only one right answer. Having multiple solutions is not an answer, it is not even an option.

Ok. An old joke is in order. A mathematician, a physicist and an engineer are asked to prove the theorem: “all integers are prime numbers”. The mathematician says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is not a prime… so the theorem is false”. The engineer says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is a prime, 5 is a prime, 6 is a prime… so the theorem is true”. The physicist says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is not a prime, 5 is a prime… the 4 is just a measurement error… so the theorem is true”.

We are talking about creating a function of F(x), which will yield the values from a random source. There are infinitely many of such functions. All of them are right. A right answer is the one which gives the right result.

True these seeds can be further randomized by using seeds based on millisecond clock-time, the prescription for getting those numbers is still not random which is what I wanted to point out. There is a formula, so obtaining a formula post facto should not conflict with the design of the RNG.

I am sure that some numerical modelers would disagree with you on it being truly random. All of these lottery ball’s motions work on Newtonian mechanics, so if one were sufficiently inclined and saw how the balls were placed initially, it is possible to predict which balls would be selected. In this scenario, assuming the prediction (which comes first) from the numerical model is correct (matches the subsequent lottery drawing), is this then a design and not randomness?

Random is as random does. The edge of Mandelbrot set is truly random. You cannot “predict” if an (x[sub]1[/sub], y[sub]1[/sub]) value will belong to the Mandelbrot set or not, unless you actually calculate it. Not even God can predict the answer. He would have to actually go through the calculation to find it out.

 

[Alberti_Devoveo]

Ok. An old joke is in order. A mathematician, a physicist and an engineer are asked to prove the theorem: “all integers are prime numbers”. The mathematician says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is not a prime… so the theorem is false”. The engineer says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is a prime, 5 is a prime, 6 is a prime… so the theorem is true”. The physicist says: “1 is a prime, 2 is a prime, 3 is a prime, 4 is not a prime, 5 is a prime… the 4 is just a measurement error… so the theorem is true”.

It has been a while since I have heard that one.

We are talking about creating a function of F(x), which will yield the values from a random source. There are infinitely many of such functions. All of them are right. A right answer is the one which gives the right result.

An infinite number of solutions is hardly an answer, you do not know which one produced the result. It is a non-answer to the question.

Random is as random does. The edge of Mandelbrot set is truly random. You cannot “predict” if an (x[sub]1[/sub], y[sub]1[/sub]) value will belong to the Mandelbrot set or not, unless you actually calculate it. Not even God can predict the answer. He would have to actually go through the calculation to find it out.

Way to avoid the hypothetical question: suppose that all the factors that could possibly affect the Newtonian mechanics of the ~50 balls in the box are known before the air is turned on–such factors as mass, density, and spatial distribution of balls, rotation of wind machine, timing of ball capture, etc. These are fed into a computer that calculates precisely 5 numbers. Suppose, then, that these 5 numbers from the computer are exactly what comes out of the lottery machine when it is run, is the lottery machine then designed or random?

I do not know enough about Mandelbrot sets, but I do not buy that it is truly random. Mostly because there is a defined function and prescribed method for determining it. If there is a pattern for it, it is not random by definition.

 

[Gorgias]

Random is as random does.

I was hoping you would get here, eventually.

OK, then: what you call ‘randomness’ really isn’t an assertion about true random behavior; rather, it’s just an assertion that something has the appearance of random behavior! Therefore, as long as something looked random to you (and remember – you make the claim of existentialism: there is no intrinsic meaning; any meaning is subjective and proceeds from the observer), then you’d call it ‘random’. You’re not making a claim about the true nature of the thing, but rather, how you perceive it. You would look at two systems – a truly random system and a determinate system sufficiently complex that it appears random – with exactly the same conclusion: random is as random does.

So, in the context of the greater discussion here, you’re saying that the true nature of the universe isn’t relevant – as long as it’s sufficiently describable by you in probabilistic terms, that’s what its ‘meaning’ is to you. whether or not there’s a design behind the universe, it appears to be able to be described by probability, and therefore, that’s the basis for your claim that it’s only an expression of non-directed behavior. However, if someone made a claim that it’s designed, you’d just shrug, right? After all, random is as random does.

This doesn’t mean that you eschew the notion of the UPB – it’s just that, as we reach the threshold at which it attempts to assert ‘design’, you’ve already decided ‘randomness’. Fair enough?

The edge of Mandelbrot set is truly random. You cannot “predict” if an (x[sub]1[/sub], y[sub]1[/sub]) value will belong to the Mandelbrot set or not, unless you actually calculate it.

I’m having a hard time with this assertion. If you can calculate it, then it’s not random. If it’s (truly) random, and not just pseudo-random, then it shouldn’t be able to be calculated out…

Not even God can predict the answer. He would have to actually go through the calculation to find it out.

I’d embark on the Thomistic explanation of what it means for God to ‘know’ or ‘predict’ something, in a way that would refute this assertion, but I’m afraid it would take us too far afield. If I had a visualization of a Mandelbrot set, and you asked me whether a point was in the set or not, I’d be able to simply observe and tell you, right? Well, Inasmuch as calculation happens within the bounds of time, and God is outside those bounds, there’s no need to assert that God would have to ‘calculate’; God would simply have the perspective to ‘observe’ and ‘know’…

 

[Serious]

An infinite number of solutions is hardly an answer, you do not know which one produced the result. It is a non-answer to the question.

As a matter of fact, we know that neither one of them did. We just impose our view and say: “either one COULD have done it”.

Way to avoid the hypothetical question: suppose that all the factors that could possibly affect the Newtonian mechanics of the ~50 balls in the box are known before the air is turned on–such factors as mass, density, and spatial distribution of balls, rotation of wind machine, timing of ball capture, etc. These are fed into a computer that calculates precisely 5 numbers. Suppose, then, that these 5 numbers from the computer are exactly what comes out of the lottery machine when it is run, is the lottery machine then designed or random?

The machine is designed to be as random as possible. And then comes the final randomizing factor: a human being, who stops the machine at a random moment,and another human being who reaches into the machine and randomly selects one ball. Unless you wish to assert and prove that we are deterministic beings, you have true randomess. The same principle applies to the roulette, the blackjack, and anything else. Since there is a human involved, and humans are (assumed to be) non-determinstic, the result of these actions is truly random. And of course in the quantum world the events are truly random. When will the next uranium atom decay cannot be predicted.

I do not know enough about Mandelbrot sets, but I do not buy that it is truly random. Mostly because there is a defined function and prescribed method for determining it. If there is a pattern for it, it is not random by definition.

Yes, the formula is there and very simple z[sub]n+1[/sub] = z[sub]n[/sub][sup]2[/sup] + c, where “z” starts with 0, and “c” is a number of the complex plane. If, after infinitetely many iterations the abs(z) is still under 2, the point “c” belongs to the Mandelbrot set. I repeat, after infinitely many iterations. And since the starting number of “c” resides on the continuous complex plane, and even a miniscule change will have an unmeasurable impact on the final result - chaos theory, anyone? - we can say that the boundary of the Manderbrot set is truly random, unpredictable. But you don’t have to dig yourself into chaos theory, even though it is fun. The undeterminsitc nature of humans is sufficient to create random results.

 

[Serious]

I was hoping you would get here, eventually.

OK, then: what you call ‘randomness’ really isn’t an assertion about true random behavior; rather, it’s just an assertion that something has the appearance of random behavior! Therefore, as long as something looked random to you (and remember – you make the claim of existentialism: there is no intrinsic meaning; any meaning is subjective and proceeds from the observer), then you’d call it ‘random’. You’re not making a claim about the true nature of the thing, but rather, how you perceive it. You would look at two systems – a truly random system and a determinate system sufficiently complex that it appears random – with exactly the same conclusion: random is as random does.

You got it right. Let me illustrate it with a thought experiment. Suppose we have a perfect copy machine, which can “read” each particle of an object on its (name removed by moderator)ut tray, and places an identical particle in the output tray at the corrresponding position. Since the elementary particles are interchangable, the end result will be a perfect copy of the original. At that moment it becomes nonsensical to ask: “which is the original and which is the copy?”. If there is no way to tell the copy and the original apart, then the very question of “which is the copy?” loses its meaning.

So, in the context of the greater discussion here, you’re saying that the true nature of the universe isn’t relevant…

Let’s stop right there. What is the “true” nature of the universe? If we were just emulations in a Matrix, then that would be the true nature of the universe. What is the “true” color of the snow? White? And if it is illuminated by a red lamp? The question: “what is the true nature of the universe” cannot be answered in a sensible way, and therefore it is a nonsensical question.

– as long as it’s sufficiently describable by you in probabilistic terms, that’s what its ‘meaning’ is to you. whether or not there’s a design behind the universe, it appears to be able to be described by probability, and therefore, that’s the basis for your claim that it’s only an expression of non-directed behavior. However, if someone made a claim that it’s designed, you’d just shrug, right? After all, random is as random does.

Yes, I would.

This doesn’t mean that you eschew the notion of the UPB – it’s just that, as we reach the threshold at which it attempts to assert ‘design’, you’ve already decided ‘randomness’. Fair enough?

Since there is no way to decide if it is random or designed, the question is not worthy ro contemplate. The assumption of “designed” always gives a false positive.

I’m having a hard time with this assertion. If you can calculate it, then it’s not random. If it’s (truly) random, and not just pseudo-random, then it shouldn’t be able to be calculated out…

I gave the answer to Alberti.

I’d embark on the Thomistic explanation of what it means for God to ‘know’ or ‘predict’ something, in a way that would refute this assertion, but I’m afraid it would take us too far afield. If I had a visualization of a Mandelbrot set, and you asked me whether a point was in the set or not, I’d be able to simply observe and tell you, right? Well, Inasmuch as calculation happens within the bounds of time, and God is outside those bounds, there’s no need to assert that God would have to ‘calculate’; God would simply have the perspective to ‘observe’ and ‘know’…

There is nothing to “observe”. The Mandelbrot set is a mathematical abstraction. It can only be “observed” after it has been calculated. I am not saying that there is any kind of problem for God to calculate infinitely many points for infinitely many iterations. But even God cannot know which points of the complex plane belong to the Manderbrot set, UNLESS he calculates it FIRST. Once it is done, then God can hold the result somewhere in his memory, and consult it whenever he pleases, and does not need to do the calculation again. But the calculation must happen first.

To talk about God being out of time should happen elsewhere. It would totally derail this thread.

 

[Alberti_Devoveo]

As a matter of fact, we know that neither one of them did. We just impose our view and say: “either one COULD have done it”.

At which point you are agreeing with me that an infinite number of solutions is not a solution. Thank you.

The machine is designed to be as random as possible. And then comes the final randomizing factor: a human being, who stops the machine at a random moment,and another human being who reaches into the machine and randomly selects one ball. Unless you wish to assert and prove that we are deterministic beings, you have true randomess. The same principle applies to the roulette, the blackjack, and anything else. Since there is a human involved, and humans are (assumed to be) non-determinstic, the result of these actions is truly random. And of course in the quantum world the events are truly random. When will the next uranium atom decay cannot be predicted.

I most certainly included “timing of ball capture” in the hypothetical of knowns. I am not asserting that we are deterministic in any fashion, all I am asking is that if one could use a computer model to predict the correct outcome of a lottery, is the lottery random or designed?

Yes, the formula is there and very simple z[sub]n+1[/sub] = z[sub]n[/sub][sup]2[/sup] + c, where “z” starts with 0, and “c” is a number of the complex plane. If, after infinitetely many iterations the abs(z) is still under 2, the point “c” belongs to the Mandelbrot set. I repeat, after infinitely many iterations. And since the starting number of “c” resides on the continuous complex plane, and even a miniscule change will have an unmeasurable impact on the final result - chaos theory, anyone? - we can say that the boundary of the Manderbrot set is truly random, unpredictable. But you don’t have to dig yourself into chaos theory, even though it is fun. The undeterminsitc nature of humans is sufficient to create random results.

I do not buy this either. There is no such thing as a physical infinity, so there cannot be an infinite set or infinite iterations. Sufficiently large, yes; infinite, absolutely not. The fact that we have images of Mandelbrot set suggests that it is hardly infinite, since it can be contained on a single image.

 

[Serious]

At which point you are agreeing with me that an infinite number of solutions is not a solution. Thank you.

Whether one or two or infinitely many - it does not matter. Do you understand the problem? We have a random event, and AFTERWARDS we create a formula (or many formulas), which COULD have been used to create that event. It is not the number of the formulas which matters. If you knew calculus, you would already know that. The integral of a function is not unique, there are infinitely many functions f(x), for which the derivative f’(x) is the same.

I most certainly included “timing of ball capture” in the hypothetical of knowns. I am not asserting that we are deterministic in any fashion, all I am asking is that if one could use a computer model to predict the correct outcome of a lottery, is the lottery random or designed?

You cannot create a proper computer emulation, which will predict the outcome, no matter how complex the progam might be and no matter how many variables you feed into the (name removed by moderator)ut. Chaos theory proves that. Just like you cannot - EVER - create a computer emulation to have an accurate weather forecast - even if you could (name removed by moderator)ut all the air molecules, their positions, directions, momentums, etc. Obviously you have no idea about chaos theory. The solution: go and study it.

I do not buy this either. There is no such thing as a physical infinity, so there cannot be an infinite set or infinite iterations. Sufficiently large, yes; infinite, absolutely not. The fact that we have images of Mandelbrot set suggests that it is hardly infinite, since it can be contained on a single image.

I will try one more time. It is getting frustrating, so one more attempt is all I am willing to waste. The Mandelbrot set is an abstract, mathematical idea. In mathematics we deal with infinities all the time. If you wish to say that the actual computations all must stop at a certain point, then that is correct. But we talk about the principle that an exact formula is being used to create a truly random set. If you cannot understand it, that is fine. I could not care less.

 

[yppop]

And a simple answer is coming. As Gorgias pointed out, the simplest polynomial is always unique. A few words about linear equations in general.

First, if you have more unknowns than equations, you will have infinitely many solutions. If you have more equations than unknowns, it may very well happen than there is no solution. Three simple examples:

1. One equation, 2 variables: x + y = 10 … x = 5 and y = 5, or x = 3 and y = 7, or x = -4 and y = 14… and so on - infinitely many solutions.
2. Two equations, two variables: x + y = 10 and x - y = 2… x = 6, y = 2 and no more solutions.
3. Three equations, two variables: x + y = 10 and x - y = 2 and 2 * x + y = 100 … no solution.

Now it could be that you have 2 unknowns and two variables, and still have infinitely many solutions, for example x + y = 10 and 2x + 2y = 20. Or you have 2 unknowns and 2 two equations and there is no solution, for example x + y = 10 and 2x + 2y = 30.

Well… now I guess we can go on. For simplicity’s sake let’s consider a simple problem, when 2 number are drawn in a simplified lottery. Using your 2 numbers of 7 and 23 were drawn. Therefore

F(x) = a*x + b

F(1) = a + b = 7
F(2) = 2*a + b = 23

Eliminating B we get a = 16, and thus b = 9.

If we draw 3 numbers
F(x) = ax[sup]2[/sup] + bx + c

F(1) = a + b + c = 7
F(2) = 4a + 2b + c = 23
F(3) = 9a + 3b + c = 11

Elimintaing “c” we get

3a + b = 16
8a + 2b = 4

Eliminating “b” we get

a = -5.6

and thus:

b = 32.8
c = -25.8

The same method applies to 5 numbers, or 10 or 100…

If you need more explanation, just say it.

Serious
Thanks for the refresher course, but I do know how to solve for unknowns with linear equations. My interest was for a situation of (n+1) unknowns in equations, the example you presented in post 54 to make your case that an equation can be created to produce any sequence of numbers. I understand that and we could have saved time if you had used the formulation Gorgias presented in post 65. His formulation always produces equal unknowns and equations. Had you done that you would have been rid of me several posts ago. So, let me ask two more questions:

(1) This is same one I started with: can you solve for (n+1) unknowns with equations using the same “eliminate and substitute” scheme that is used when the number of unknowns match the number of equations? A simple yes or no will satisfy me. Gorgias, if you read this could you answer the same question.

(2) When I requested a function to produce a specific sequence of numbers you answered in post 54 with

" I will do better than that. I will teach you how to do it, and then next time you don’t have to ask, you can simply do it."

And then you “taught” me a method using a polynomial one degree higher than needed as you point out in post 74 when you wrote,

“The simplest possible polynomial is indeed unique. Of course using a higher polynomial would yield infinitely many solutions, but there is no need to complicate the matter beyond necessity”.

My question: was that an oversight or a deliberate attempt to confound me by “teaching me” a method that would complicate the matter beyond necessity?

Yppop

 

[Alberti_Devoveo]

Whether one or two or infinitely many - it does not matter. Do you understand the problem? We have a random event, and AFTERWARDS we create a formula (or many formulas), which COULD have been used to create that event. It is not the number of the formulas which matters. If you knew calculus, you would already know that. The integral of a function is not unique, there are infinitely many functions f(x), for which the derivative f’(x) is the same.

There is a reason C is included in indefinite integrals, because any constant added to f(x) would result in the same f’(x); I deal with gauge transformations often in my classwork that are negated in taking the curl due to the curl of the gradient always being zero. What my problem with this is that I do not believe having an infinite number of solutions is a solution.

You cannot create a proper computer emulation, which will predict the outcome, no matter how complex the progam might be and no matter how many variables you feed into the (name removed by moderator)ut. Chaos theory proves that. Just like you cannot - EVER - create a computer emulation to have an accurate weather forecast - even if you could (name removed by moderator)ut all the air molecules, their positions, directions, momentums, etc. Obviously you have no idea about chaos theory. The solution: go and study it.

Okay, you are not getting the point. Maybe I am not emphasizing the question enough, so for the third time: ignore the fact that it cannot be done and imagine it was done perfectly–that is the essence of a hypothetical question–the computational model predicted exactly the physical result: would you then say that the physical result was designed or random?

I will try one more time. It is getting frustrating, so one more attempt is all I am willing to waste. The Mandelbrot set is an abstract, mathematical idea. In mathematics we deal with infinities all the time. If you wish to say that the actual computations all must stop at a certain point, then that is correct. But we talk about the principle that an exact formula is being used to create a truly random set. If you cannot understand it, that is fine. I could not care less.

I deal with infinities probably more often than you do, but I have the correct notion that infinity is not physical whereas you seem to be implying that it is physical.
I will try one more time too: the Mandelbrot set cannot be random because there is a definite prescription for defining the set. Beyond that, the set is defined before running it, so it is designed and not random.
To show you how wrong you are on the randomness of the set, take your pick of any of the codes at the site below and run it 100 times, you will get 100 images that are identical. According to you, they should be different because they are ‘truly random.’
rosettacode.org/wiki/Mandelbrot_set

 

[Serious]

(1) This is same one I started with: can you solve for (n+1) unknowns with equations using the same “eliminate and substitute” scheme that is used when the number of unknowns match the number of equations? A simple yes or no will satisfy me. Gorgias, if you read this could you answer the same question.

Yes, the same method can be used. You will have one extra variable, and you can choose it as you wish.

(2) When I requested a function to produce a specific sequence of numbers you answered in post 54 with
And then you “taught” me a method using a polynomial one degree higher than needed as you point out in post 74 when you wrote,

My question: was that an oversight or a deliberate attempt to confound me by “teaching me” a method that would complicate the matter beyond necessity?

It was an oversight. It happens…

 

[Serious]

There is a reason C is included in indefinite integrals, because any constant added to f(x) would result in the same f’(x); I deal with gauge transformations often in my classwork that are negated in taking the curl due to the curl of the gradient always being zero. What my problem with this is that I do not believe having an infinite number of solutions is a solution.

That is your personal opinion. I don’t share it.

Okay, you are not getting the point. Maybe I am not emphasizing the question enough, so for the third time: ignore the fact that it cannot be done and imagine it was done perfectly–that is the essence of a hypothetical question–the computational model predicted exactly the physical result: would you then say that the physical result was designed or random?

The problem is that such a computational model cannot be created, not only practically, but theoretically. The reason is that the (name removed by moderator)ut value is an infinitely long non-periodical fraction. You really should read up on chaos theory.

I deal with infinities probably more often than you do, but I have the correct notion that infinity is not physical whereas you seem to be implying that it is physical.

No, I did not imply it.

I will try one more time too: the Mandelbrot set cannot be random because there is a definite prescription for defining the set. Beyond that, the set is defined before running it, so it is designed and not random.

Of course it will be the same, because the starting number is only an approximation, and the number of iterations is not infinite. You do not see the Mandelbrot set, only an finite approximation of it. The set is defined by a quadratic equation. Therefore the boundary of the set is infinitely complicated and unpredictable. It is random in the sense that it cannot be predicted if a certain point will belong to the set or not. Random = unpredictable.

None of this is pertinent for the UPB.

 

[Alberti_Devoveo]

That is your personal opinion. I don’t share it.

You do not share the opinion because you are a mathematician who does not care about physical results. I am a physicist and care about physical results.

The problem is that such a computational model cannot be created, not only practically, but theoretically. The reason is that the (name removed by moderator)ut value is an infinitely long non-periodical fraction. You really should read up on chaos theory.

Are you normally this thick-headed? Whether or not the model could be created in reality has zero bearing on the hypothetical. Please, read the sentence in my previous post again and try to respond to just the hypothetical question and not the reality of the situation.

No, I did not imply it.

Then I inferred incorrectly. Oh well.

Of course it will be the same, because the starting number is only an approximation, and the number of iterations is not infinite. You do not see the Mandelbrot set, only an finite approximation of it. The set is defined by a quadratic equation. Therefore the boundary of the set is infinitely complicated and unpredictable. It is random in the sense that it cannot be predicted if a certain point will belong to the set or not. Random = unpredictable.

But it is not unpredictable, it has a set function for determining it; that is the antithesis of being unpredictable. I do not understand what is difficult to grasp about this concept: if there is a definite way of determining something a priori, it is not random.
I would like to point out that, elsewhere in this thread, you demanded that for something to be designed, one had to know the formula for deriving it a priori and that any a posteriori formulae were not indicative of a design. If you know a priori the formula for the Mandelbrot set, then by your own demand, it cannot be random and is designed, boundary or not.

 

[Serious]

Whether or not the model could be created in reality has zero bearing on the hypothetical.

Listen, it is impossible even hypothetically. Suppose I would ask you a question about going back into the past, and ponder the ramifications of killing your own grandfather, before you father would be born. It would be a nonsensical request, because time travel is impossible. Or ask a hypothetical question of getting into a black hole and coming back out.

The same applies here. There is no reason to ponder something that is theoretically impossible. It would be different if it were merely practically impossible, because that could change.

 

[inocente]

There is no such thing as a physical infinity, so there cannot be an infinite set or infinite iterations. Sufficiently large, yes; infinite, absolutely not. The fact that we have images of Mandelbrot set suggests that it is hardly infinite, since it can be contained on a single image.

The Mandelbrot set is a fractal, with a border that is infinitely baroque in detail. When visualizing it, we only need to calculate a color value for each pixel in the range on view. Google “program mandelbrot” to see examples of the computation.

 

[inocente]

danserr, buffalo - been out all day and ran out of time, sorry, will respond to you mañana.

 

[davidv]

The Mandelbrot set is a fractal, with a border that is infinitely baroque in detail.

Have any of these details been completely realized? If not, there still exists no actual infinity, it is only a potential one.

When visualizing it, we only need to calculate a color value for each pixel in the range on view. Google “program mandelbrot” to see examples of the computation.

Was this computation designed, or did it pop into existance by itself?

 

[Alberti_Devoveo]

Listen, it is impossible even hypothetically. Suppose I would ask you a question about going back into the past, and ponder the ramifications of killing your own grandfather, before you father would be born. It would be a nonsensical request, because time travel is impossible. Or ask a hypothetical question of getting into a black hole and coming back out.

The same applies here. There is no reason to ponder something that is theoretically impossible. It would be different if it were merely practically impossible, because that could change.

There is a point, you are just cleverly avoiding it by refusing to answer it. This shows me something about you already.

 

[Alberti_Devoveo]

The Mandelbrot set is a fractal, with a border that is infinitely baroque in detail. When visualizing it, we only need to calculate a color value for each pixel in the range on view. Google “program mandelbrot” to see examples of the computation.

Perhaps you should read my post where I give the link to the Rosetta Code link that shows about 50 different programs that calculate the Mandelbrot set.

 

[buffalo]

danserr, buffalo - been out all day and ran out of time, sorry, will respond to you mañana.

No problem - we do have lives outside CAF.

 

[Serious]

There is a point, you are just cleverly avoiding it by refusing to answer it. This shows me something about you already.

Whatever. It should show you that I value logic and reason.

By the way, I want to point out the irrationality of your stipulation that if a problem has multiple solutions, then (according to you) it has no solutions. Look that the problem of Pythagoras, x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], where the three unknownd, x, y and z must all be positive integers. Obviously this equation has infinitely many solutions (one of them would be 3, 4 and 5) - but according to you, it has no solutions - because it has more than one. On the other hand the equation of x[sup]3[/sup] + y[sup]3[/sup] = z[sup]3[/sup] really does not have any solutions where x, y and z are all positive integers.