Mechanics: Are 'Newton's laws of motion' laws indeed?

As schoolkids, and even afterwards, all of us have started our mechanics with the three 'Newton's laws of motion'. And, for them who ever scrutinized these laws, doubts arose. Doubts whether these are actually laws or mere definitions. In the following I will present the reasons for these doubts.

An equivalent statement of 'the first law' is - Every body, in absence of action of forces, moves with a constant velocity (rest being a special case). What difficulties, if any, arise from this statement? For this statement to be a law, one should be able to assert the following two facts independently. Firstly, it should be possible to distinguish the case that a body has a constant velocity from when it has not. Secondly, it should be possible to tell whether the body is being acted upon by some external force(s). Assuming that both these can be independently determined,the statement can be said to be a statement of law. This law would be true if we empirically discover that constant velocity indeed appears only in the absence of external forces. Otherwise, the law is falsified.

To determine the absence of force, it is essential to know certain characteristics of force with which to determine its absence/presence. But the very concept of force is not known prior to these laws. Moreover, without a clear prior definition, 'force' occurs in all the three laws. If, then, we depend upon these laws to know what force is, we are led to the conclusion that force is that which causes acceleration of the bodies it acts on. This conclusion can be derived from both the first and second laws. Then, if the absence of the force is determined by the absence of acceleration, the first law is a tautology. It is perhaps better to say that the 'first law' is itself the definition of force. However, the 'second law' says even more. It is therefore a better definition in which case the 'first law' is simply a special case of the second. But we should not forget that both these 'laws' are mere definition.

The second law however does assert that acceleration is of fundamental importance in writing the equations of motion of any system. The second law also prompts us to find a cause of the acceleration in a force which must necessarily depend upon the properties of the environment of the system and also upon the properties of the interaction of the system with the environment. Surely, one need not employ the fiction of force. The laws of motion can, of course, be written without such a notion. Although superfluous, it is harmless to call some terms of these equations by the name force.

The first two laws are perhaps a definition of an 'independent system'. A system in which the total dp/dt = 0 is said to be an independent system. As a result of this definition, whenever dp/dt != 0 the system is said to be independent, otherwise it is being acted on by external 'forces'.
Let us now assume that we are given an independent system which can be considered as sum of two distinctly identifiable systems. If p1 and p2 are the momenta associated with two parts of the system, and if p is the momentum of the whole given system then p = p1 + p2, or dp/dt = dp1/dt + dp2/dt = 0 (since we are given an independent system by assumption). Therefore, dp1/dt = - dp2/dt. This, as one can readily recognize, is 'the third law'. If we regard a system as made of three parts or more rather than two, we would have other laws like the third. For example, for n parts the law would look like dp1/dt + dp2/dt + ... + dpn/dt = 0. Given that p = p1 + p2 + ... + pn, we can state that dp/dt = 0 which, in turn, is true by assumption and therefore the source of the equation with n terms.

As to how we assert that the momentum of a system is equal to the sum of momenta of the parts of that system, the answer is in kinematics. The vector sum of momenta follows from the possibility of the vector sum of displacements and its derivatives. This, we shall not pursue here.
From the above, it will become evident that 'Newton's three laws' are not laws at all. Apart from asserting the importance of the time derivative of momentum, they merely define an independent system. It remains to examine whether there are any independent systems at all.

Science: Quantum Immortality

First, let us consider the many-worlds interpretation of quantum mechanics because this is essential to understanding the concept of quantum immortality.

In what follows I will presume that you have a basic acquaintance to the concepts of quantum mechanics.

In the many-worlds interpretation of quantum mechanics, the wave function collapse mechanism is substituted by a splitting of the observed system - each new branch corresponding to one possibility. Suppose the wave function of a system S is the superposition of two measurable states A and B. According to the Copenhagen interpretation, on measurement the wave function collapses and the system evolves from being in a mixed state to the measured state. In the many-worlds interpretation, the system splits into two: one yielding the value A and the other B. If we apply this interpretation to all the systems i.e., to our world itself then the this universe is continuously splitting and branching so that all quantum-theoretically possible combinations of measurements are realized. Thus the splitting leads to a tree structure in which following one line of branches from the root yields one possible world. One may also state that mixed quantum mechanical states result from superposition of more than one world.

Secondly, let us come to the idea of Quantum Immortality.

The many-worlds interpretation implies that any likely event, however improbable, does get realized in some world. Now imagine the process by which a man X is led to his death dependent upon factors each of which may or may not happen. Note that even if one of the contributing factors to death is such that the quantum theory predicts it with a 100% probability then the death is a must no matter how and how many time the observer's world splits and branches. Otherwise, of course, there will be worlds in which none of the factors contributing to death may happen. Consequently, these will be the worlds in which X will not die. All the worlds being considered together, X will continue to live in some worlds and thus X will be immortal.

But is this really the sense in which a living being can be said to be immortal? This leads us to the third part of our argument.

Assuming that we understand the meaning of consciousness, it is not as if when the man dies in one world his consciousness is transferred to another. Like all other attributes, consciousness too should split along with X as should his identity. On death, few worlds will inherit consciousness while others won't.

If the above assumptions hold (including absence of a 100% probable factor leading to death) then I (the first person) should also continue to live forever. But the "I" is also splitting. And the likelihood that the "I" that is writing all this is the same "I" that is immortal (or say, the "I" which will live for 1000 years) is just as much as the fraction of worlds in which this "I" is immortal (or, has a life span of 1000 years). To get an idea of this likelihood we can do our maths in this very world if we could calculate the probability of this "I" surviving forever (or for 1000 years). I have included this '1000 years' alternative so that you wouldn't at once assert impossibility of the referred event.

Maybe, one could argue, the probability of observing a 1000 year old man can be increased if we include in our survey all the men on the earth. Yes, the likelihood will increase roughly six billion times but how significant that increase will be depends upon the result that we arrive at (which, for all you know, could be too low to expect to yield a positive outcome in an experiment. Thus it must be extremely unlikely for any of the experiments to witness a 1000-year old man.

We have so far assumed that the identities across the various worlds do not interact. Suppose, in some way, they did. In other words, suppose the various worlds interfere with each other. Then, for any given man, in the long run, we should expect a certain concoction of many of his dead versions and a very small fraction (relative to the number of the dead versions) of his immortal or near-immortal versions. Although I have not given due thought to this aspect I think the identity as we assert it in terms of "I" cannot be a superposition of the dead and immortal "I".

I will try to explain why I think so. Call it the splitting of the world into more than one or the collapse of the wave function, the end result is that you, as the observer capable of asserting an identity, finally record only one possible state of a system even though others were likely. This record becomes a part of your history. The other possibilities which were potential until the measurement had occured do not become history. At least, not in your world. If these possibilities are part of the records in other worlds, then those other worlds also have a different copy of "you" than the one in this world. Since you are unable to see those other possibilities as interfering with your records; and also the same quantum theory that leads you to hypothesize the other worlds gives you no clue as to their existence, I believe that the various worlds, if they exist, do not interfere. Thus there cannot be a trans-world identity of an individual. As a result, few of your copies (formed as a result of splitting of the world) would be immortal, but the majority will not be. Whether you are immortal or not depends upon which particular copy you are.

Let us remember that the idea of quantum immortality makes sense only if all your copies can somehow interact or interfere or superpose etc. (choose your verb) and the resultant "you" carries a single consciousness (whatever that means) that spans across the many worlds.

I hope the above is useful in clarifying the ideas and issues related to quantum immortality.