dkv:?So I have managed to move the infinity to different location...
We all agree .. ?
dkv, your infinity is an artifact of improper use of the concept of function. Proper application of the concept of function tells you in fact that if you have a well-behaved function, your infinity does not even exist there.
But it seems you don?t want to use the concept of function. Fine, let?s use the concept of array, and define your array as R(ak)=(ak-b)/ak. Under these circumstances, you are bothered by the fact that in fact you cannot define the array for say a0=0.
dkv: ?Now I can similarly move the infinity to any point of discussion I want ...The most important fact here is that Physical continuity or comprehensibility can be achieved at the so called infinity by doing transformation... without any application of new rule...?
By no means does your ?trick? offer understanding of what happens at the location of the original infinity. The shift that you perform is equivalent to you looking at another point where your array IS defined. You DO NOT look at the point of singularity, but at some other regular point, which does not offer you any fundamental information about the singularity. You have just circumvented the issue of singularity, you have not solved it.
dkv: ?In fact this means that there is no such concept of infinity in reality and it is the limitation of our expression which makes it infinite .. and when we are not able to completely remove it from the system ...then it is the drawback of the tool....?
Nope, your conclusion is invalid. You have just swept your infinity under the rug, but this does not mean that it does not exist anymore. This is not the way to actually remove infinities. You might try another change of variables, like Q(ak)=arctg[R(ak)] on some domain, and this will bring your results in the range [-pi/2, pi/2) with no infinities whatsoever being involved.
dkv: ?I am computer professional and I know how such cases break the flow of logic... the continuity of a simple mathematical process gets broken down into if and then conditions...
And suddenly the Universe appears to have intelligence ....otherwise how it is able to hide the infinity...?
Don?t blame the bloody machine for this, it is just a piece of junk that does what YOU tell it to do, within the limits of what IT can do. Such cases do not brake any flow of logic, because your logic is higher than the machine?s, and before coding a process you can design the process without the limitations imposed by a dumb processor.
As for your conclusion about the intelligence of the Universe, at least in the case of this topic, it is not warranted. Furthermore, if you adopt the previous view that singularities do not actually exist and are just artifacts of the language in which we describe nature, your conclusion in not warranted at all. At the worst, it is based on flawed logic, and at the worst it is a self-serving argument for a, let?s call it mildly, a philosophical agenda.
dkv: ?Now I will like generalize the problem for all such cases where we have possible 0 in the denominator...?
OK, let?s generalize it.
dkv: ?I know science has been struggling with infinities and if there is any one reason why physics has been able to partially conquer the infinity. It is because it enforces quantum limits...?
Well, roughly speaking you are right. However, you don?t know the details of it, so you should be careful when you make such all-encompassing generalizations.
First of all, even at the classical level there are ways to deal with infinities. Some of these ways rely on the fact that the mathematical representation is not appropriate (i.e. by a change of variables as I did above, the singularities vanish, in some cases), others rely on logically introduced bounds (like the infinite-wall interaction potential for the molecules of a gas) which do not require quantum reasoning.
In the quantum case, certain classical singularities vanish, indeed, but other appear and removing them is not a logical step anymore. Introducing so called IR and UV limits/cutoffs is not always natural, and they are mostly introduced by hand and are not emerging from the description. Similarly, renormalization is a rather awkward technicality/principle, which lacks a natural explanation (except for the firm held belief that singularities should not exist!).
dkv: ?In fact the uncertainty principle refuses to recognize zero...?
This is a rather poor interpretation of the uncertainty principle, but so what? This is not a trend in quantum mechanics. Quantum mechanics has no problem in recognizing ?zero?. You have 0-spin states, don?t you? And you also have infinities of your preferred type, i.e. the energy in the Hydrogen atom for n=0, where n is the principal quantum number. In order to remove such an infinity, you have to introduce the assumption that you can offset the energy by an infinite amount and the result is finite, and you actually have no reason to say that (you are back in the renormalization case).
dkv: ?Even Vaccum has something in it...?
You are making the usual confusion between a vacuum state of a system and vacuum. They are slightly different concepts.
dkv: ?The necessity of something inside nothing is again a logical necessity of the Maths itself and not physics...?
No, it is actually a necessity of physics that has to be fulfilled by a mathematical description, and not the other way around. But once again, you are generalizing concepts with no insight in the (already existing) details. Let?s take the vacuum state for a system. You need to have such a state because physically your system may contain no particle. However, this state can be highly unstable, such that the system has a vanishing probability of being in that state. So in reality, a vacuum state exists, but the system almost never uses it. I don?t see how this can be something required by the math and not by the physics. It is physically fundamental for such a state to exist, irrespective of whether the system uses it or not.
dkv: ?Earlier I had discussed why One Unified Theory is a logical necessity of Maths but not reality....(becuase a half truth is still false as per boolean logic)?
Well, in reality the GUT is at this stage nothing more than wishful thinking, based on the argument that in reality nature is highly structured, and the level of complexity is decreasing from classical to quantum (you have fewer distinct processes at the quantum level than at the classical level). So from this viewpoint, GUT are a requirement of the physical reality, as perceived by some. However, there is no actual evidence that such a viewpoint is justified. And since such a GUT has not been implemented yet exactly due to mathematical difficulties, I would say that a GUT is certainly not a requirement of the mathematical description.
Furthermore, there is one more fallacy of your argument. Mathematical description can indeed make predictions, but it never dictates the physical reality. Mathematics is the but a more or less appropriate language in which we express what we learned through observation from nature, and not the other way around.
dkv: ?It is true that I am not very technical in my description but I firmly believe the reality is all about common sense and it should not be confused with the approach used to understand it...?
True, physics should not be confused with math, but believe me when I say that reality is not about common sense. And you should know this, from quantum mechanics. There is no common sense in the uncertainty principle (what do you mean we cannot simultaneously measure any two quantities, when common sense ? i.e. day to day experience - shows the opposite), there is no common sense in saying that Schroedinger cat is half-dead and at the same time half-alive before we check on it, and I could continue. There is a logic of the physical reality, but I am afraid we haven?t learned it yet in totallity.
dkv:?There are many more interesting ideas to share with you but please keep correcting me whenever I go wrong technically...I hope I am able to make at least some sense ...And my end result is not zero...?
Be my guest to share your ideas and discuss them. But I hope to realize that in order to consolidate what you know you will need more than just a conversation on a forum.
dkv: ?As far as functions are concerned the definition has been modified to suit its own limitation ... At least a function should return the same result on passing the same input..f(x)=y should not give me y1 ,y2 etc...for x1.
Well, you only required your function to be a bijection. Which is fine. But beware that there are also functions that are multivalued, i.e, the can return different values for the same argument. They are proper mathematical objects, and they are well-understood.
dkv: ?In case of limits this rule does not apply ... but we circumvent the problem by saying that limit is a finite value 'just' above (or below) x... But no one dares to ask that no matter how close someone is to the x .. it can not be x...?
Of course it applies for limits too. First of all, the limit at a point is the limit at a point, not above the point, and not below. However, when you calculate this limit, you can come from below (limit from below) or from above (limit from above). Both these limits are well defined and have well understood properties.
As for your function, a function of the type f(x)=(x-a)/x is mathematically NOT DEFINED at x=0. You must be very careful about zero and infinity. It does not matter that mathematics includes them, they have a very special statute and cannot be used as the rest of the numbers (for example you cannot divide by zero and infinity IS NOT a number!). So, in attempting to EVALUATE your function at x=0, you disregard the rules for division, and try to extend division in the range where it is not working, where such an operation is not defined. This is not my conclusion, it is the tight logic of the mathematics. Mathematics does not allow you to use undefined operations. Period.
You can however investigate what your function does around that point, by using limits. And in the case of your function f(x) defined above, if you try hard enough you see that as you get closer and closer to x=0, you see that -b is an accumulation point for f (this means that all the values for f are in a smaller and smaller vicinity of ?b as x gets closer and closer to zero). And in this case, you can extend the domain of your function to include x=0 by saying that at x=0 f(0)=-b, and this process is called continuation/extension by continuity. This is your only way in which YOU CAN define a function at a point where its value is ambiguous, or where the function is not defined.
dkv: ?Maths is self enclosed and I am trying to find something in the real world which will validate it and so far I have not found anything which matches perfectly...?
Well, you are looking at the wrong example. Addition of vectors works well in the physical world, within experimental limits, the Fourier transform works well in optics, and I could continue. But exact matches you will never get, because while mathematics is an abstract science, and as such the only exact one, physics is an empirical science, and nature is observed within experimental limits, so perfect matches are principially excluded.
dkv: ?almost everything has been reduced to probability and I wonder that one day we may wake up to see that 1+1= 2 is not working because the there was a small probability that it will not work...?
True, but this may only be a reflection of our poor understanding in very many cases (although there are schools of thought that support the idea that the fundamental level of nature is actually probabilistic). Keep your mind open.
As for your example regarding 1+1=2, let me tell you something (which you should already know, since you work with computers). You can also have 1+1=0, if you use basis 2 or equivalently modulo 2 residual classes. So get use to the fact that things are never what they seem, no matter how much we wish.
