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When we do nothing for a while we say that some "time" has passed.
In terms of actions, doing nothing is a special type of action.
If we denote it by t, then
tt is an action of waiting for two times longer than with t.
When we measure time, we take some repetative process, like one swing of
a pendulum, for a model of other processes. We may say, for instance, that
John needes 80 'pendulums' of time to smoke a cigarette. In terms of
the homomorphism picture, the state when John is lighting his cigarette
is w_1; the state when he extinguishes it is w_2; the language L is
the pendulum, with some kind of counter of swings; the mapping M is
registration of the current value of the counter. The process
M must be a real physical process, not just a mental association
of some states of the counter with some states of cigaret smoking -
the truth which has been dramatically demonstrated
by Einstein's relativity theory.
We often say that all real processes take place in space and time.
The meaning of such statements is that in addition to what really goes on,
we imagine some reference actions of consecutive shifts ("in space")
and waits ("in time") and esatblish relationships between these actions
and actual objects and processes. Thus, in accordance with Kant's view,
space and time are not observable realities, but our ways to organize
Henri Bergson was first to notice and emphasize the difference between
real time, in which we live and act,
and the objectified time of history and physics.
Imagine a pendulum which at each swing puts a mark on a
moving tape. We have a historical record of "the time moving". This
historic record is an object at every moment we look at it. We use it
as a part of our model of reality. We shall refer to the marks on the tape
as representing a model time. It is very much different from
the real time.
Real time is such that two moments of it never coexist. In model time
the moments coexists as different objects in some space. Thus Bergson
calls model time a projection of real time on space. Bergson's real time
is irrreversible. Model time, the time of Newton's mechanics, is reversable:
we read historical records equally well from left to right and from right
to left. The seemingly inconceivable feature of Feynman's diagrams, the
movement in the direction opposite to time, is explained simply by the fact
that the time of physical theories is model time, i.e. a spacial
phenomenon. Real time shows up in probability theory and statistical
physics. We are dealing there with real acts of choosing from a number
of possibilities. Hence this time is irreversible. In mechanics, to every
action there is an inverse action which brings back the original state.
So, when we project time on space the projection has an additional property
of reversibility. But the act of choosing has no inverse. If you drew
ticket No.13, you drew it. You can return it to the pool, but the fact
will still remain that it was No.13, and nothing else, that was drawn first
and then returned. You can choose, but you cannot "unchoose".