Tuesday, December 30, 2008

The Counterintuitive Time: 1. Time and Determinism

This is a series of posts about the counterintuitive nature of time in Physics. This first post tries to identify the reasons why people may not agree with the deterministic view, and with the block spacetime view, in our sense of free-will and of flowing time.

The paradigmatic example of a world governed by determinism is provided by a system of differential equations (DE), or, more generally, of partial differential equations (PDE). In Physics, the PDE are required to have as solutions functions of position and time. In Newtonian Mechanics, the position is a point in the 3-dimensional Euclidean space, and the time is a real number. The positions and the instants form a 4-dimensional real vector space. The solutions of the PDE are functions $f(x, t)$ defined on this space. The state of the system at a moment of time $t$ is given by $f_t(x) = f(x, t)$.

The PDE systems appearing in mathematical physics have the nice property that, by knowing the state at a moment of time $t_0$, and the values of some additional partial derivatives of $f$ (in general the first order ones are enough), we can determine uniquely the states for another time $t$. For the solution to exist at $t$, the initial states and the derivatives appearing in the initial conditions are required to be well-posed, but I will not detail here. What is important is that we can extend the state at a time $t$ to contain not only $f_t$, but also the partial derivatives involved in the initial condition. This way, all the information about the system and its time evolution is contained in the extended state at each instant.

The fundamental equations of Physics are PDE, and they satisfy these conditions. One important exception seems to be provided by the Quantum Mechanics, where the indeterminism seems to be fundamental, but for the moment I will concentrate on the deterministic situation.

In such a deterministic world, the extended states contain all the information about the system. There is no physical property which is not contained in the extended state. Is the world we live in, of this type? It may be, or it may be not. If the world is like this, then it is a block world. The solution of the PDE is defined on the spacetime, and together they form a timeless, frozen entity, the function $f(x, t)$.

Many biologists and neurobiologists believe that, at least in principle, life and consciousness can be explained by making use of the deterministic properties of atoms and molecules, and perhaps more complex systems only, and not appealing to the indeterminism. Many persons understand what a deterministic world is, and even believe that our world may be of this type. Yet, they hardly accept the block world. A deterministic world contains all the information regarding all moments of time, at the extended state at each instant. There is no need to “play” this world, like playing a pick-up disk. If our world is deterministic, and if the minds are reducible to configurations of matter, then the extended state contains also the mind state of a possible observer. Are the observers just states depending on a real number (which is interpreted as time)? If it would be so, then there will be no change, in the sense that, at any instant, the 3d-observer at that instant will contain in its state the impression that he or she perceive a time flow, and a dynamic evolution of his/hers state. There will be only timeless 3d observers containing in their states the impression of time evolution. For a person, there will be one such 3d timeless observer, associated to each instant (which is just a real number). Considering all the instants making a “lifetime”, there will be an infinity of such 3d timeless observers, connected.

It is easy to understand why such view is rejected by many. Determinism leads to a block world view. Some may accept determinism, and reject block view. When they understand the relation between them, they may continue rejecting the block view, and therefore reject also the determinism. The main problem seems to be the block view. If our world is such, then we are also reduced to parts of a set of timeless states.

Perhaps the main parts of our intuition contradicted by this view are the following. First, the feeling of subjectivity, the sense of “I”. Second, the feeling that we have free-will. A block view seems to make everything frozen, predetermined.

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