I BELIEVE I HAVE EXTENDED HILBERT'S PARADOX

I post this with the intention to have a kind of record in terms of ‘internet date’, for I might need this for some claiming purposes in the future; an alibi saying “I have derived (or proposed) it first which proof is the date of the posting”, something like that (I also have sent an email with the same content for the same purpose). Why internet? 1) I want to do it quick 2) I might send it somewhere formal but at the moment, I am not sure whether it is technical enough to be accepted, but we will see about this, who knows, right? In fact, I am going to see Dr Zainal, Head of Mathematic Dept. Fac. of Science first thing tomorrow for any possibility of having this in Matematika, a journal published by the Math Department.

What I am to propose is a paradox which I believe, up to this time of writing, is originally mine (I have been Google it up for quite some time now and still have not find any contender). This paradox, let’s call it as Oneness’s Paradox, is actually an extension of Hilbert’s Paradox (or call it as another version of Hilbert’s).

Hilbert’s paradox, also known by Grand Hotel paradox is about the problem of dividing unity by infinity (or large number) which ‘seems’ to be equal to zero. How 1 can be equal to zero? That is the paradox. Mathematically, this paradox can be described as follows:

0 = 0 + 0 + 0 + 0 + ...
0 = (1 – 1) + (1 – 1) + (1 – 1) + ...
0 = 1 – 1 + 1 – 1 + 1 – 1 + ...
0 = 1 + (-1) + 1 + (-1) + 1 + (-1) + ...
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
0 = 1 + 0 + 0 + 0 + ...
0 = 1 ? (weird hence the paradox).

The above is Hilbert’s. In establishing my own paradox, I will go with quite a different approach. First, I will use the division of unity by zero (or a very small number) and second, I will use Newton’s Law of Gravitation for the argument. I believe my attempt of associating Hilbert’s paradox with a physical law what makes my paradox unique and original (but I will continue making literature review to verify such a claim).

Now, based on Newton’s Law of Gravity, the pulling force, F between two masses, m and M can be given by the inverse law as:

where G is the universal gravitational constant and r is the distant between the two masses. Next, let’s make a thought experiment or Gedanken. Say, there exists two masses which are spherical in shape, let’s call them balls due to my engineering background. These balls each have a mass of a unit kilogram as shown in figure below.

As can be seen, from the figure above, these two balls are separated at a distance r. Imagine further that these two balls exist in a universe that has G as equal to unity. This then results in:

Now, if the two balls get closer and closer towards each other to the extreme, r = 0, and the fact that each ball is 1 kg each, Equation 3 will become:

It can be observed that Equation 4 is exactly like Equation 1. The question now (hence the paradox) is how, two existences (referring to the balls which state does not change throughout the argument) can be described as a unity by any means? To make it graphical, refer to figure below:

This is the reason why I propose the name of Oneness’s paradox. And I think it should not be taken as similar to the concept of Singularity because based on what I am going to argue, this Oneness concept may really imply that the two balls fused into one entity.

Let me restate the paradox, “How two existences (referring to the two balls) can ever be described by a single existence?” As mentioned earlier, except for the changes in the distance r which lead to the changes of F, there are no changes whatsoever in regards to the system. The two balls, each remains as 1 kg mass throughout the process, should thus remain as two entities with separate existence. But the mathematics tells us that somehow these two existences are equal to something that is known to us as one or unity.

If this is not a paradox then there must be something physical about it. And, one thing that comes to my mind is that, it might have some relation (I really wish it does) with quantum mechanics. I have this crazy idea that, this concept of ‘oneness’ can be the state of an electron before it is split and such an ‘intimacy’ added by the reversed nature of the argument of the paradox (reversed in the sense that, it is argued from separated state to fused state which in contrast to the usual argument, for example, of an electron which is split into many) would therefore explain why one ‘part’ of the split electron would ‘feel’ the measurement conducted on the other ‘part’ of the electron although they are some distant apart; because they used to be in the state of 'oneness'.

Or maybe it has some relation with black hole theory (or big bang theory) since these theories also involve inverse law. May be with this argument, one can make sense of the state of infinite forces, 'F' (gravity, weak interaction, strong interaction, electromagnetic) contained in zero 'region', 'r'; made sense by the concept of 'oneness', that is everything are fused into one. Wallahuallam.

P/S:

1) I will let you know the outcome of my meeting with Dr Zainal tomorrow. I myself can’t wait.

2) Please do not, at least not yet, drag this paradox into Sufism or tasawwuf argument

LATEST FINDING ON PREVIOUS WORKS

I have found out that what I have written above resembles quite closely the Kalam cosmological arguments championed by Islamic theologians of Kalam tradition such as Al-Kindi and Al-Ghazali. For your information, in present days, Kalam cosmological arguments are promoted by William Lane Craig who also used Hilbert's paradox in his argument but rather in a philosophical way. Kalam cosmological arguments thus Craig himself are now under attack by atheist physicists. May be my 'oneness paradox' can be another defense against these attacks.

SO far, I still have not found any association of Hilbert's paradox to Newton's Gravitational Inverse Law the way I did. The closest would be the association with the Newton's Law of Cooling but since this law deals with heat as the variable (although discussion on infinities and singularity did appear) , it does not sum to my paradox "how two balls can be described as one".

I will keep you updated of other progress.

NOTE: The above paradox has been found as false. To read on the explanation, click the entry MY 'ONENESS' PARADOX IS WRONG.