On Time

 Tuesday, November 11, 2003  11:11 AM


“Why do we remember the past
but not the future?”

— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

The flow of imaginary time is at right angles to that of ordinary time.“Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”

Is Time Quantized?



We don’t really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional.  Then the following picture,

from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:

“It seems, as one becomes older,
That the past has another pattern,
    and ceases to be a mere sequence….

I sometimes wonder if that is
    what Krishna meant—
Among other things—or one way
    of putting the same thing:
That the future is a faded song,
    a Royal Rose or a lavender spray
Of wistful regret for those who are
    not yet here to regret,
Pressed between yellow leaves
    of a book that has never been opened.
And the way up is the way down,
    the way forward is the way back.”

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

Tuesday, November 11, 2003  11:00 AM


Sunday, November 9, 2003  5:00 PM

For Hermann Weyl’s Birthday:

A Structure-Endowed Entity

“A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way.”

— Hermann Weyl in Symmetry

Exercise:  Apply Weyl’s lesson to the following “structure-endowed entity.”

4x4 array of dots

What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array’s structure you choose to examine. It could be in the hundreds, or in the hundreds of thousands.)

Friday, November 7, 2003  7:00 PM

A Beautiful Fantasy:

The Secret life of
 John Nash

“Dr. Blind (pronounced ‘Blend’) was about ninety years old and had taught, for the past fifty years, a course called ‘Invariant Subspaces’ which was noted for its monotony and virtually absolute unintelligibility, as well as for the fact that the final exam, as long as anyone could remember, had consisted of the same single yes-or-no question. The question was three pages long but the answer was always ‘Yes’. That was all you needed to pass Invariant Subspaces.”

The Secret History, by Donna Tartt


“…I put my arms around him yes and drew him down to me so he could feel my breasts all perfume yes and his heart was going like mad and yes I said yes I will Yes.


Ulysses, by James Joyce

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