Damn, my paradox is wrong.
Yesterday, I went to see Dr Zainal (Head of Math. Dept) and Prof. Norsarahaida (Deputy Dean, Fac. of Science) to discuss about my ‘Oneness’ paradox. We met in Prof Sarah’s office and in no time they showed me where I got it wrong. By the way, Prof Sarah was quite upset because I only mentioned Dr Zainal in my previous post but not her (of course she was joking). For your information, Prof. Sarah and I have written six papers together and I am co-supervising two of her PhD students. Herein, I would like to express my gratitude to Prof. Sarah for being my mentor and it is my greatest honor to work with her for the past three years. I have posted the list of our papers in this blog for your viewing, just click here.
So, where I got it wrong? It was as early as Equation 1, as shown below.
Why is it wrong? It is wrong because zero multiplied by infinity is not zero, but undefined instead. So since this equation is wrong, what follow are therefore wrong.
You see, infinity is not a number but instead, it is a concept or notion, created by us human so as to comprehend something that is never end. So, since infinity is not a number, you can not equate it to be equaled to anything except under the strict rule of limit expression as below:
This is therefore the collapse of my paradox. But having said this, the same argument also collapses Hilbert’s Paradox since he also argued by equating infinity to a number; only that I have overlooked such an argument before.
But, both Prof Sarah and Dr Zainal agreed that my attempt to imagine the two balls as tool for argument might help in the understanding of the concept of ‘undefined’ itself. What is it mean to be ‘undefined’? Another way to put it, what ‘defined’ means? As you going to see, we will be talking about ‘meaning’ itself.
By using the balls analogy, it is easier for technical men like me to imagine the whole process of the argument. Let me rewrite the Newton’s Inverse Law of Gravitation as follows:
What Equation 3 tells us is that, as the distance between the two balls got smaller, the force got bigger and bigger. These changes are all still comprehensible until the distance become zero at which the magnitude of F is conceptually prescribed as infinite. This is a typical explanation; there is nothing new about it. In fact, you might have heard it thousand times.
But, by imagining the two balls, we can have an alternative explanation. What we can say is, as long as there is a finite distance between the two balls, the concept of a ball remains valid, comprehensible thus defined. But when there is no more distance, a ball loses it concept. We can no longer define what happened to something which we used to call a ball. If we extend this discussion into higher dimensions, say into our familiar three dimensions, what we can say is that, a ball can be defined if there is a space for it (please take note that our previous discussion of a distance corresponds to a space in 3-dimensional region). If the space seizes to be finite, thus zero, although the ‘ball’ still exists (zero times infinity is not zero remember so the ‘ball’ does not diminish), it becomes ‘undefined’. We lose the ‘concept’ of meaning.
What we can deduce or learn from this are:
1. Our mathematics are handicapped thus valid only in a finite space (or space-time) which we currently live in, so do not ‘believe’ in mathematics, just use it
2. Since we possess the concept of meanings, then the universe must be finite (of course this is in agreement with present physics). This is because, if we have an infinite distance between the two ball, we will go back to Equation 1, where things are similarly undefined or become meaningless
3. There is existence ‘beyond’ the limit of infinity only that, this existence is not comprehensible through our mathematics. By now, we should have perceived the word ‘beyond’ the same way we perceive ‘infinity’; it should not be a ‘distance’ or anything like that, else we would have meaning out of it.
As conclusions, I hope that my writings somehow able to shed some light on some aspect of mathematics especially on the ‘lacking’ and ‘incompleteness’ of the mathematics itself. But, this does not mean that mathematics is evil or anything like that. How can a man like me say that? Instead, I promote everyone to learn mathematics to the deepest degree so that we can learn ‘how stupid we are’ actually as a human being. And as I said in point 1 we must use or exploit or manipulate mathematics to our advantages because somehow we still have to live in this world although temporarily. So while we are waiting for another life, exploitation of mathematics for the sake of goodness is what we have to do. As an analogy, a man who used to have a knee problem can now pleasantly ‘sujud’ and can be more ‘khusyuk’ in solat after a knee operation; a technology evolved due to the advert of ‘mathematics. You know what I mean?
Finally, I must say this, when I said ‘meaning’ I refer to those concepts comprehensible through the ‘exploitation’ of our five senses (aren’t our mathematics are built on these senses, are they?) and not the ‘TRUE MEANING’. For the ‘true meaning of existence beyond the realms of our mathematics’, we as a Muslim must therefore resort to the ultimate truth, sent to us through prophet Muhammad SAW, that is Al-Quran. Al-Quran is therefore THE ONLY AND THE ULTIMATE TRUTH, granted to us by ALLAH SWT so that we can know what we are allowed to know about the things that we can never know through our handicapped mathematics.
REFINEMENT OF DISCUSSION (made on Saturday, 22 October)
Actually, my line of argument in the previous Oneness paradox is not all wrong. Yes, the final part of Equation 1 is wrong, that is
as it should be below
What this means is, whilst the previous Figure 2 is wrong (given again as below):
the following below is however correct:
So, although in the Oneness paradox we have wrongly argued (derived from Equation 4 and Figure 2):
“How two existences (referring to the two balls) can ever be described by a single existence?”
as far as our new discussion on meaning is concerned, the following statement is valid (as it is derived from Equation 5 and Figure 3):
“How two existences (referring to the two balls) can ever be described as undefined (or meaningless) ?”
hence the ground for our argument on ‘meaning’ as outlined in this entry. And, if we still want a paradox, let's call it 'MEANINGLESS PARADOX'. Huh, how about that?