he grounds the arithmetical succession of elements relative to an "unfindable" last line according to an analogy with the progression of time
================ Everything that is in the list 1 1, 2 1, 2, 3 ... 1, 2, 3, ..., n is in the last line. Alas as soon as you try to fix it, it is no longer the last line.
Think of the time. What is "now"? As soon as you try to fix it, it is past. ================
From the first set of quotes -- unless WM's philosophical position on physics is non-material physicalism -- one must interpret the second quote as referring to "empirical, physical time".
I have asked WM to clarify his theories on time so that his readers can judge the felicity of his remarks. In view of the statements above, one might expect his answer to reflect on current trends in physics. So, to assist his readers, I have prepared these resources.
The excerpted remarks by Craig Callender appeared in the JUne 2010 issue of Scientific American. They discuss the implication of reconciling general relativity with quantum mechanics.
The first possible resolution mentioned in those remarks is a "timeless" universe based on relational quantum mechanics. It can be found at
"Newton's time may seem old hat to us now, but a moment's reflection reveals how astonishing it is. It's many features -- order, continuity, duration, simultaneity, flow, and the arrow -- are logically detachable, yet they all stick together as the master clock that Newton dubbed 'time'. This assembly of features succeeded so well that it survived unscathed for almost two centuries.
"Then came the assaults of the late 19th and early 20th centuries. The first was the work of Austrian physicist Ludwig Boltzmann, who reasoned that, because Newton's laws work equally well going forward or backward in time, time has no built in arrow. [...]
"Einstein mounted the next assault by doing away with the idea of absolute simultaneity. According to special relativity, what events are happening at the same time depends on how fast you are going [...]
"And things only get worse in 1915 [...]. Gravity distorts time, so that a second's passage here may not mean the same as a second's passage there. Only in rare cases is it possible to synchronize clocks and have them stay synchronized. [...]
"Yet even in relativity, time retains a distinct and important function: namely, that of locally distinguishing between 'timelike' and 'spacelike' directions. Timelike events are causally related. [...] Spacelike events are causally unrelated. [...]
"In a very precise sense, time is the direction within spacetime in which good prediction is possible. [...]
"Quantum mechanics says that objects have a much richer repertoire of behaviors than we can possibly capture with classical quantities such as position and velocity. [...]
"The theory's probabilistic predictions require time to have certain features. First, time is that which makes contradictions possible. [...]
"Second, the temporal order of quantum measurements makes a difference. [...]
"Third, a quantum state provides probabilities for all of space at an instant of time. [...]
"Canonical quantum gravity emerged in the 1950's and 1960's, when physicists rewrote Einstein's equations for gravity in the same form as the equations for electromagnetism, [...] The equation (dubbed the Wheeler-DeWitt equation) utterly lacked a time variable. The symbol t denoting time simply vanished.
"If you take this result literally, time does not really exist.
"[...] although general relativity lacks a global time, it still manages to describe change. In essence, it does so by relating physical systems directly to one another rather that to some abstract notion of global time. [...] Time becomes redundant. Change can be described without it.
"Getting rid of time has its appeal but inflicts a good deal of collateral damage. For one, it requires quantum mechanics to be thoroughly rethought.
"[...] Because time is so basic, banishing it would transform physicist's worldview.
"Even if the world is essentially timeless, still it seems as though it does have time in it. [...] But canonical quantum gravity already offers a more developed idea.
"Known as semiclassical time [...]
"[...] The universe may be timeless, but if you imagine breaking it into pieces, some of the pieces can serve as clocks for the others. Time emerges from timelessness. [...]
"As interesting and startling as this idea is, it leaves us wanting more. The universe cannot always be broken up into pieces that serve as clocks, and in those cases, the theory makes no probabilistic predictions. Handling those situations will take a full quantum theory of gravity and a rethinking of time."