http://www.kafalas.com/Logtime.html
Logtime: The Subjective Scale of Life
The Logarithmic Time Perception Hypothesis
By James Main Kenney
Time scarcer? Years getting shorter? Want an explanation? Logtime is
the cognitive hypothesis that our age is our basis for estimating
time intervals, resulting in a perceived shrinking of our years as we
grow older. A simple mathematical analysis shows that our time
perception should be logarithmic, giving us a subjective scale of
life very different from that of the calendar. Our perception of
aging seems to follow the same (Weber-Fechner) law as our perception
of physical stimuli.
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Ticking away the moments that make up a dull day
You fritter and waste the hours in an offhand way.
...You are young and life is long and there is time to kill today
And then one day you find ten years have got behind you.
...Every year is getting shorter; never seem to find the time...
-- "Time" from The Dark Side of the Moon: Pink Floyd
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When you've grown up, my dears,
tempus and are as old as I, time flies
fugit you'll often ponder on the years
that roll so swiftly by,
my dears,
that roll so swiftly by...
-- "Toyland" from Babes in Toyland:
Glen MacDonough & Victor Herbert
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Problems of Time Perception
It's common knowledge that our perception of the passage of time can
be influenced by psychological factors: time flies when we're busy,
but really drags when we're waiting. (Stare at a clock and wait for a
minute to pass. Or wait for the commercials to end, or for Windows to
load!) These are generally short term experiences, but what of long
periods of time such as years? Is there something other than
transient psychological factors affecting our time perception?
We usually think about the years of our lives in terms of decades:
our teens, twenties, thirties, etc. This is an implicitly linear
view: that all our years are equal; that clock time is our time,
through which we move at a uniform pace.
This simple picture, however, doesn't square with our perceptions as
we age. By our middle years, at least, most of us have become aware
that something is amiss, that a very slow but profound change has
been sneaking up on us: the years that formerly crawled are now
racing by. Where are the long, leisurely summers we knew as children?
If it seemed forever to get through the fifth grade, what happened to
last year? Why do we now seem so rushed by life? Where are all the
things we wanted to accomplish, but never seemed to find the time
for?
There is another clue that our lives are not running in a linear,
clocklike fashion: when we try to remember back to the earliest years
of our childhood, they seem incredibly distant, like a far horizon
that always recedes as we attempt to approach it. Why should we find
it so much harder to remember the first few years of life than to
remember later years, even after a longer time? And why do parents
see their children growing up so much faster than they did?
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Logtime: The Logarithmic Explanation
Since the linear view of time perception seems inadequate, it is
reasonable to look for a non-linear alternative. The observations we
make about the apparent shrinkage of our years as we age strongly
suggest a logarithmic scale: stretched out at the low end and
compressed at the high end.
The fundamental importance of the logarithm has suggested the term
"Logtime" to succinctly refer to the cognitive hypothesis discussed
here. (The reader who cringes at the very mention of the word
"logarithm" need have no fear: no tables of logarithms will be used
here and no math knowledge, except for the optional Appendix , will
be required!)
Another term may usefully be coined here to denote this general field
of study: "psychochronometry", the psychology of time estimation.
Logtime is a psychochronometric model, i.e., an attempted
mathematical simulation of subjective human temporal experiences. The
logarithmic scale of time perception presented by this model may be
only a rough approximation of actual human perception, but is
probably a much closer one than the linear scale usually assumed.
That our time perception should be logarithmic can be easily
rationalized (although proving it is a different matter!). The simple
premise of Logtime, from which the logarithmic relationship can be
derived (see Appendix ), is that the human mind judges the length of
a long period of time, such as a year, by comparing it with current
age. For example, a year adds 10% to the life of a ten-year-old, but
only 5% to that of a twenty-year-old. For the twenty-year-old, two
years are required to add 10%.
The Logtime hypothesis is that it is this percentage that we
perceive, not the years themselves: to the twenty-year-old, two years
will seem to pass as quickly as one year will seem to the
ten-year-old. Similarly, three years to a thirty-year-old and four
years to a forty-year-old, etc., will seem to pass equally fast.
(This argument was recently found to have been used, comparing a
"child of 10" with a "man of 50", by Sorbonne professor of philosophy
Paul Janet, date unknown but quoted in an 1890 book by the eminent
Harvard philosopher and psychology pioneer William James , who seemed
to accept the description but added his own explanation of an
underlying psychological cause which would be difficult to analyze
quantitatively.)
The Logtime hypothesis is consistent with the widely accepted
description of the perception of physical stimuli commonly referred
to as the "Weber-Fechner law" . For time perception, clock time
(calendar age) is the "stimulus". Weber-Fechner has been found to be
only an approximation over a limited stimulus range, and this would
probably be the case for aging perception if objective measurements
were possible.
The older we become, the faster we seem to age or, conversely, the
shorter the years seem to be. Mathematically, this relationship is
said to be either logarithmic or exponential, depending on which
variable is used as the reference: the length of the years seems to
shrink logarithmically if we regard our subjective aging as uniform,
while the speed of passage of these years seems to increase
exponentially if we regard the years as being of equal length. Using
other terminology, our sense of aging follows an arithmetic
progression while the corresponding calendar years follow a geometric
progression.
The Logtime hypothesis probably applies to all time intervals, not
just years and seasons. Our days and hours should be similarly
shrinking as we age, but short-term psychological factors tend to
dominate, making the shrinkage less obvious.
When we are young, the changing nature of our lives tends to obscure
the shrinking years: the twenty-year-old rarely thinks about how life
was at age 10; life at 20 is filled with different activities and
concerns, and it is the future that dominates reverie. (The
twenty-year-old may be loath to admit to even having been 10, let
alone suffer the remembrance of it!)
It is only after life becomes more settled and routine that we become
more retrospective, and only then do we have an easier basis for
comparing the years. However, we tend to adjust to our changing time
scale, and our declining physical abilities tends to conceal the
nature of the underlying change: since we see ourselves as "slowing
down", we accept that life around us seems to go faster. (Of course,
if you accelerate down the road, the passing scenery also
accelerates!*)
Some authors have identified the change in time perception with
changing metabolic rate, but is there enough change in the latter to
account for the former? If not, it may be irrelevant to the "clock"
the human mind uses.
*Analogies between time perception changes and spatial motion can be
confusing and ambiguous since we don't actually move through time
(nor does time move through us): we merely exist as four-dimensional
bodies in a four-dimensional space-time continuum (ignoring the many
new dimensions being added by modern physics). In a sense, only our
consciousness can be said to "move" as it perceives three-dimensional
spatial cross-sections in sequence along the time axis. Actually,
nothing can really move in space-time: things just exist, including
the four-dimensional physical structure underlying our consciousness.
(But what is existence? What is consciousness?... What is this
headache?!)
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The Scale of Life
It will be convenient, though perhaps depressing, to define equally
perceived units of time to replace the decades we conventionally use
to pace our lives. A consequence of the logarithmic function is that
it is the ratio of the years defining an interval of time that we use
to judge the duration of that interval, not the absolute magnitudes
of those years. (See Appendix .)
For example, using the simplest ratio, 2: the years from ages 10 to
20 seem to pass at the same rate as the years from 20 to 40, or 40 to
80. The starting age is arbitrary: 8 to 16, 16 to 32, and 32 to 64
are also of equal subjective duration, and are all perceived as being
the same as 10 to 20!
The term commonly applied to ratios of 2 is the musical one of
"octave" (ignoring its Latin root meaning "8"). (Our perceptions of
the pitch and amplitude of musical tones are logarithmic; so,
apparently, is our perception of time.) The octave is the most
obvious Logtime replacement for the decade, although it has no fixed
connection to our base-10 number system. All octaves are equally
spaced on a log scale, just as decades are equally spaced on a linear
scale.
Going back in time, the octaves from ages 4 to 8, 2 to 4, and 1 to 2,
etc., should have seemed equally long, although the logarithmic
description may no longer be a good approximation near birth, and can
certainly not be quantified by observation!
There is a question of origin: when is "zero"? Is it birth,
conception, sometime in between, or even a time after birth when
long-term memories may actually start to form? (There are cultures
that consider the newborn to be a year old.) Since a true log scale
has no origin, no zero, going back toward age "zero" is a limitless
process: it would require an infinite number of octaves. This offers
a plausible explanation for our difficulties in remembering our
earliest years if our memories decay in proportion to octave
separation. (If forgetfulness was linear, i.e., uniform through the
years, why should a twenty-year old not be able to recall his first
year of life as easily as a forty-year old recalls his twenty-first
year?) (Of course, for those of us with really poor memories, the
difference may not be all that great!)
On a log scale, the reference replacing the zero of a linear scale is
unity. Age "1" is five octaves removed from age "32", six octaves
from "64", etc., however "1" (or "zero") is defined. A mathematically
correct choice for "1" would be possible only if these temporal
effects were measurable and predictable.
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Logarithmic Lifetimes
Since the starting age of an octave is arbitrary, we can scale our
lives logarithmically in various ways. The series of numbers on each
line below all define the end points of (equally-perceived) octave
age intervals. Note that the even spacing of subjective ages on a
logarithmic scale results in an exponential growth of the objective
(clock) age, an octave representing a growth of 100% per step:
1 2 4 8 16 32 64 128
1.0 2.1 4.1 8.2 16.5 33 66 (132)
1.1 2.1 4.2 8.5 17 34 68 (136)
1.1 2.2 4.4 8.8 17.5 35 70 (140)
1.1 2.2 4.5 9 18 36 72 (144)
1.2 2.3 4.6 9.2 18.5 37 74 (148)
1.2 2.4 4.8 9.5 19 38 76 (152)
1.2 2.4 4.9 9.8 19.5 39 78 (156)
1.2 2.5 5 10 20 40 80 (160)
1.3 2.6 5.1 10.2 20.5 41 82 (164)
1.3 2.6 5.2 10.5 21 42 84 (168)
1.3 2.7 5.4 10.8 21.5 43 86 (172)
1.4 2.8 5.5 11 22 44 88 (176)
1.4 2.8 5.6 11.2 22.5 45 90 (180)
1.4 2.9 5.8 11.5 23 46 92 (184)
1.5- 2.9 5.9 11.8 23.5 47 94 (188)
1.5 3 6 12 24 48 96 (192)
1.5+ 3.1 6.1 12.2 24.5 49 98 (196)
1.6 3.1 6.2 12.5 25 50 100 (200)
1.6 3.2 6.4 12.8 25.5 51 102 (204)
1.6 3.2 6.5 13 26 52 104 (208)
1.7 3.3 6.6 13.2 26.5 53 106 (212)
1.7 3.4 6.8 13.5 27 54 108 (216)
1.7 3.4 6.9 13.8 27.5 55 110 (220)
1.8 3.5 7 14 28 56 112 (224)
1.8 3.6 7.1 14.2 28.5 57 114 (228)
1.8 3.6 7.2 14.5 29 58 116 (232)
1.8 3.7 7.4 14.8 29.5 59 118 (236)
1.9 3.8 7.5 15 30 60 120 (240)
1.9 3.8 7.6 15.2 30.5 61 122 (244)
1.9 3.9 7.8 15.5 31 62 124 (248)
2.0 3.9 7.9 15.8 31.5 63 126 (252)
2 4 8 16 32 64 128 (256)
Any (horizontally) adjacent pair of ages defines an octave of life,
and each octave should seem equally long. The numbers in parentheses
represent ages greater than 128, which are not physically attainable
at present; they are the theoretical bounds on the "broken octaves"
that conclude our lives.
On any of the above lines of numbers, find a number closest to your
age. On the same line will be the numbers bounding the octaves of
your life. Think back to the ages at the left: do they seem equally
spaced, i.e., does each interval seem to have been equally "long"? Do
they represent equally important stages in your life? (You may not
remember how important the earliest octaves really were!) How many
numbers are at the right? If your next octave is a "broken" one,
contemplate your fortune in having survived all the octaves at the
left; many have not.
The course octave spacing of the above series can be relieved to any
extent desired by geometric interpolation, which produces numbers
that are still evenly spaced on a logarithmic scale. For example, the
following have adjacent numbers that (before rounding) differ by a
factor of the square root of 2 (1.414...), the "half octave", i.e.,
the exponential growth in clock age is a little over 41% per step:
1 1.4
2 2.8
4 5.6
8 11
16 23
32 45
64 91
128
0.6 0.9
1.2 1.8
2.5 3.5+
5 7.1
10 14
20 28
40 57
80 113
(160)
Interpolating again (before rounding) produces the following "quarter
octave" series (as with the above, one series is based on an exact 1,
and the other on an exact 10):
1 1.2 1.4 1.7
2 2.4 2.8 3.4
4 4.8 5.7 6.7
8 9.5+ 11 13
16 19 23 27
32 38 45 54
64 76 91 108
128
0.6 0.7 0.9 1.1
1.2 1.5- 1.8 2.1
2.5 3.0 3.5+ 4.2
5 5.9 7.1 8.4
10 12 14 17
20 24 28 34
40 48 57 67
80 95 113 (135)
Another interpolation and rounding produces the "eighth octave"
series, with horizontally adjacent numbers differing by a factor of
the eighth root of 2. (On all of these interpolated-octave tables,
vertically adjacent numbers differ by a factor of 2.) Note the
quasi-linear stretches, and the one-year spacing of the teen years
that may serve as a convenient basis for comparison with other stages
of life:
1 1.1 1.2 1.3 1.4 1.5+ 1.7 1.8
2 2.2 2.4 2.6 2.8 3.1 3.4 3.7
4 4.4 4.8 5.2 5.7 6.2 6.7 7.3
8 8.7 9.5+ 10 11 12 13 15
16 17 19 21 23 25 27 29
32 35 38 41 45 49 54 59
64 70 76 83 91 99 108 117
128
0.6 0.7 0.7 0.8 0.9 1.0 1.1 1.1
1.2 1.4 1.5- 1.6 1.8 1.9 2.1 2.3
2.5 2.7 3.0 3.2 3.5+ 3.9 4.2 4.6
5 5.5- 5.9 6.5- 7.1 7.7 8.4 9.2
10 11 12 13 14 15 17 18
20 22 24 26 28 31 34 37
40 44 48 52 57 62 67 73
80 87 95 104 113 123 (135)
Musicians and music lovers may like to "scale" their lives using
twelve steps per octave to match the chromatic (equal tempered)
half-tone musical scale. Horizontally adjacent numbers differ by a
factor of the twelfth root of 2 (1.059463...), i.e., there is an
exponential growth of just under 6% per step. (Those who are into
numerology as well as music may like to try finding a key with the
accidentals best matching the years when they were particularly
"sharp" or "flat"! Could this become a new occult fad?)
1 1.1 1.1 1.2 1.3 1.3 1.4 1.5- 1.6 1.7 1.8 1.9
2 2.1 2.2 2.4 2.5+ 2.7 2.8 3.0 3.2 3.4 3.6 3.8
4 4.2 4.5- 4.8 5.0 5.3 5.7 6.0 6.3 6.7 7.1 7.6
8 8.5- 9.0 9.5+ 10.1 10.7 11.3 12.0 12.7 13.5- 14 15
16 17 18 19 20 21 23 24 25 27 29 30
32 34 36 38 40 43 45 48 51 54 57 60
64 68 72 76 81 85 91 96 102 108 114 121
128
0.6 0.7 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.1 1.1 1.2
1.2 1.3 1.4 1.5- 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.4
2.5 2.6 2.8 3.0 3.1 3.3 3.5+ 3.7 4.0 4.2 4.5- 4.7
5 5.3 5.6 5.9 6.3 6.7 7.1 7.5- 7.9 8.4 8.9 9.4
10 10.6 11.2 11.9 12.6 13.3 14 15 16 17 18 19
20 21 22 24 25 27 28 30 32 34 36 38
40 42 45 48 50 53 57 60 63 67 71 76
80 85 90 95 101 107 113 120 127 (135)
It's not essential to base an age series directly on the octave,
which can be identified on any log scale. Any number can be used for
the ratio of adjacent entries. For example, we can make both 1 and 10
(and 100, also) exact by using a ratio of the n-th root of 10. Using
the tenth root of 10 produces ten steps per factor of 10. This is
commonly used in engineering (as the "decibel") to express power
ratios. Every third number differs by almost a factor of 2 to a close
approximation. (Power very nearly doubles for three-decibel steps.)
(Ages below 10 can be obtained by moving the decimal point to the
left.)
10 13 16 20 25 32 40 50 63 79
100 126 (158)
Interpolating to give finer resolution (twenty steps per factor of
10) produces ratios which, for voltage or current, etc., correspond
to decibel steps. (Six-decibel steps very nearly double voltage or
current, quadrupling power.)
10 11 13 14 16 18 20 22 25 28
32 35 40 45 50 56 63 71 79 89
100 112 126 (141)
For an even finer resolution, we can use thirty steps per factor of
10:
10 11 12 13 14 15 16 17 18 20
22 23 25 27 29 32 34 37 40 43
46 50 54 58 63 68 74 79 86 93
100 108 117 126 (136)
Using forty steps per factor of 10:
10 10.6 11.2 11.9 12.6 13.3 14 15 16 17
18 19 20 21 22 24 25 27 28 30
32 33 35 38 40 42 45 47 50 53
56 60 63 67 71 75 79 84 89 94
100 106 112 119 126 (133)
Finally, using fifty steps per factor of 10, for an exponential
growth of about 4.7% per step:
10 10.5- 11.0 11.5- 12.0 12.6 13.2 13.8 14.5- 15.1
15.8 16.6 17.4 18 19 20 21 22 23 24
25 26 28 29 30 32 33 35 36 38
40 42 44 46 48 50 52 55 58 60
63 66 69 72 76 79 83 87 91 95
100 105 110 115 120 126 (132)
On any of the above tables, compare the spacing in the preteen years
(moving the decimal point where required) and in the middle years
with the spacing in the teens and twenties: how stretched out is
childhood, and how compressed is middle age! The even-greater
compression of old age makes that period seem very short, which
brings us to a discussion of...
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The Last Judgement
Is that old saying "life is short" starting to make sense? Since we
seem to be hurtling toward oblivion at an accelerating pace, an
obvious question is: how much time do we have left on our subjective
scale, i.e., how much longer will life seem to last? The above age
tables are course for the later years, making estimates difficult.
At the risk of further depressing the reader (who must now be fully
aware of the disturbing nature of this material!), a simple way to
more precisely judge the time remaining is to perform the following
calculations:
1. Assume an age to which you may reasonably expect to live
(e.g., 80).
2. Divide this assumed age of death by your present age (if you
are 40, then 80/40 = 2).
3. Divide your present age by this number (40/2 = 20).
4. The result is a "reference age" (20) as subjectively remote in
your past as your assumed age of death (80) is in your future.
Consider the years from that point in your life (age 20) to the
present (age 40): the time you have left (40 to 80) should seem
about as long.
5. Your present age is the geometric mean of the reference age
and the assumed age of death, and becomes closer to the
arithmetic mean as your present age approaches the latter. In old
age, therefore, you can assume linearity: each future year will
seem almost as long as each past year back to the reference age.
(In other words, the worst is over!)
The basis for this calculation is, of course, the same fundamental
property of a log scale that makes all octaves equal: all equal
ratios of numbers are equally spaced. Since the ages involved will
generally not have integral ratios, use of a calculator is
recommended, especially for repeated calculations:
On a calculator, enter your age and square it. (Multiply it by itself
if a "squaring" key is not provided.) Save the result in memory.
Divide by the age you expect to reach in order to find the past
"reference" age. Recall from memory the stored square of your present
age and divide by a different age of death assumption. Doing this
several times will give you a feel for the subjective time you have
left. This could make a great party game! (Then again...?) (Well,
maybe for an "Addams Family" party!)
You can also reverse the computation by dividing the square of your
age by some past age that you may particularly recall and have a feel
for the time from then till now. Will you live long enough to
experience a similar future interval?
Those who place their greatest trust in computers may like to use the
following BASIC program (using only integers); it will run under
either GW-BASIC or QBASIC in MS-DOS computers (as LOGTIME.BAS), and
should also run under most other versions of BASIC. (For the Tandy
100/102/200 and the NEC 8201A/8300 substitute MENU for SYSTEM and
load as LOGTIM.DO.)
0 CLS:PRINT:PRINT" LOGTIME by James Main Kenney 1996
1 PRINT:INPUT"What is your present age";A
2 INPUT"To what age do you expect to live";L:IF L<=A THEN 2 ELSE
PRINT"The time from now until then will seem about as long as the
time from when you were"(A^2)\L"until now.
3 PRINT:PRINT"Do it again (Y/N)? ";:K=INSTR("NnYy",INPUT$(1)):IF
K>2 THEN 1 ELSE IF K THEN SYSTEM ELSE 3
A compiled version of LOGTIME.BAS, LOGTIME.EXE, runs directly under
all versions of MS-DOS (back at least to version 3) including DOS
mode (or in a DOS window) of Windows 3.x and 9x. LOGTIME.EXE is in
LOGTIME.ZIP along with ready-to-load versions of LOGTIME.BAS and
LOGTIM.DO (and also this document). LOGTIME.ZIP may be downloaded
from
http://ourworld.compuserve.com/homepages/jmkenney/logtime.zip
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Will Life Extension Help?
A popular topic these days is the extension of life span through
nutrition or medical intervention, e.g., by genetic modification to
allow more cell divisions.
Examine the highest ages on the tables presented above, or
extrapolate past them to higher octaves, and you will see how far
life must be extended to give even a small increase in subjective
life span: even an added decade provides only a fraction of an
octave. (Perform the calculations described immediately above,
assuming increasingly higher ages at death, and see how the increase
would be perceived by comparison with your past life.)
Only a truly astounding increase in years, approaching immortality,
could provide a meaningful change (and the social consequences of
such an increase could be devastating). An interesting speculation is
what life would be like for really old immortals: would the sun be
streaking across the heavens and the days flashing like a strobe, as
seen by the time traveller in H. G. Wells' The Time Machine? And
would their old memories be edited to provide room for new memories?
(How: FIFO or selective?) Perhaps a finite memory would provide a
limit to the Logtime shrinkage.
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Intergenerational Relationships
An important consequence of a knowledge of Logtime can be an
appreciation of an important difference between the young and the
old. While the elderly are commonly regarded as slow, by their own
perceptions they are racing past the lives of the young: it is only
the young who experience the "endless summers".
In the tables above, compare the years in your current octave with
those of your children or parents (and grandchildren or
grandparents). Try to see how these differences may affect
communications between you. For example, what the young may think of
as prudent planning for the future may be dismissed by the old as
living for the present; the generations simply scale their lives
differently.
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What is the Clock? (Unanswered Questions)
There are many questions evoked by the Logtime hypothesis: What is
being compared? Is there a real biological clock counting the days,
or is the flow of data into the brain used (analogous to an hourglass
or a water clock)? If the latter, does it use raw or processed data,
or some weighted combination? Are new experiences and familiar ones
weighted differently? This has to involve the nature of memory. Is
there a sense of elapsed time independent of memory?
An interval of time may be judged differently in retrospect than
while being experienced; it can fly when you're short of time, but
may seem long afterward if you accomplished much or had a memorable
experience. How does one's health, particularly relating to memory,
affect time perception? This is a rich, albeit difficult, field for
investigation.
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References
(biographical data added by present author)
William James (1842-1910; professor of philosophy at Harvard;
"one of the founders of modern psychology"; "father of American
psychology"): The Principles of Psychology. New York: Henry Holt
1890.
This monumental work ("Perhaps the most important English-language
psychology text in history.") has been placed online in its entirety
(http://psychclassics.yorku.ca/James/Principles/index.htm) (link
above); Chapter XV: "The Perception of Time" (http://
psychclassics.yorku.ca/James/Principles/prin15.htm) is a lengthy
treatise largely of short-term phenomena, but with an important
discussion of the effect of aging:
"The same space of time seems shorter as we grow older -- that
is, the days, the months, and the years do so; whether the hours
do so is doubtful, and the minutes and seconds to all appearance
remain about the same.
'Whoever counts many lustra in his memory need only question
himself to find that the last of these, the past five years,
have sped much more quickly than the preceding periods of
equal amount. Let any one remember his last eight or ten
school years: it is the space of a century. Compare with them
the last eight or ten years of life: it is the space of an
hour.'
So writes Prof. Paul Janet (1823-1899; from 1864 professor of
philosophy at the Sorbonne) [Revue Philosophique, vol. III. p.
496], and gives a solution which can hardly be said to diminish
the mystery. There is a law, he says, by which the apparent
length of an interval at a given epoch of a man's life is
proportional to the total length of the life itself. A child of
10 feels a year as 1/10 of his whole life -- a man of 50 as 1/50,
the whole life meanwhile apparently preserving a constant length.
This formula roughly expresses the phenomena, it is true, but
cannot possibly be an elementary psychic law; and it is certain
that, in great part at least, the foreshortening of the years as
we grow older is due to the monotony of memory's content, and the
consequent simplification of the backward-glancing view. In youth
we may have an absolutely new experience, subjective or
objective, every hour of the day. Apprehension is vivid,
retentiveness strong, and our recollections of that time, like
those of a time spent in rapid and interesting travel, are of
something intricate, multitudinous, and long-drawn-out. But as
each passing year converts some of this experience into automatic
routine which we hardly note at all, the days and the weeks
smooth themselves out in recollection to contentless units, and
the years grow hollow and collapse."
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Only one publication introducing a logarithmic scale of time
perception has been identified:
Rodney Collin (1909-1956): The Theory of Celestial Influence --
Man, The Universe, and Cosmic Mystery (1948). London: Stuart &
Watkins 1971. New York: State Mutual Book 1981.
This book was cited by
Michael Shallis: On Time. London: Burnett Books 1982. New York:
Schocken Books 1983.
As described by Shallis, Collin devised a logarithmic time scale
based on lunar months, with 1, 10, 100, and 1000 lunar months
(equally spaced on a linear scale labeled 0, 1, 2, and 3) roughly
corresponding to, respectively, the dates of conception, birth, 7
years, and death (77 years), thereby dividing life into three equal
parts: gestation, childhood, and maturity. (Maturity at 7?) After
noting that "time seems to pass about ten times more slowly for a six
year old child as for a sixty year old man", Shallis identifies the
logarithmic scale with "metabolic rate and therefore to some extent
with human experience."
Two relevant books, seen many years ago, could not be located. They
both described a World War I French study of wound healing which
concluded that the time for tissue to grow over a superficial wound
of a given size was directly proportional to the age of the patient.
At least one of these books suggested a common metabolic connection
with changing time perception. The rapid healing of injuries in
children is well known, but are there any modern studies quantifying
this phenomenon?
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Ernst Heinrich Weber (1795-1878; professor at the University of
Leipzig 1818-1871): De Tactu (On Touch) (1834; in Latin); and Der
Tastsinn und das Gemeingef hl (The Sense of Touch and the Common
Sensibility) (1851)("considered to be 'the foundation stone of
experimental psychology'").
Gustav Theodor Fechner (1801-1887; physicist; professor of
philosophy at Leipzig): Elemente der Psychophysik. Leipzig:
Breitkopf und H rtel 1860. Bristol: Thoemmes Press 1999
(translation: Elements of Psychophysics, Volume I only. New York:
Holt, Rinehart and Winston 1966.)
In the nineteenth century, a very general description of sensory
perception was developed by Ernst Heinrich Weber and Gustav Theodor
Fechner, known as "Weber's law", "Fechner's law", or the
"Weber-Fechner law", etc. Weber found that, over a range of stimulus
intensity, the minimum amount by which the stimulus must be changed
for the change to be noticed is directly proportional to the
stimulus. After rediscovering this relationship, Fechner showed that
this indicates that the intensity of sensation is proportional to the
logarithm of the intensity of the stimulus. A translation of Sections
VII and XIV of Fechner's Elements may be found online (http://
psychclassics.yorku.ca/Fechner/). At the same site there is also an
introduction, by Robert H. Wozniak, to Fechner's Elements (http://
psychclassics.yorku.ca/Fechner/wozniak.htm).
There are many sites that discuss Weber-Fechner, such as Dennis Leri:
MENTAL FURNITURE #10/The Fechner Weber Principle
(www.semiophysics.com/menta10.htm).
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Internet Links
(Some web links appear in References, immediately above.)
Generation Gap Calculator, By Edgar Matias
http://edgarmatias.com/gapcalc.html
This site, dated 1997 (earlier than the Logtime site), starts with
the same premise as Logtime, and even uses startlingly similar
phrases (see below), but does not use logarithms. Instead, there is
an interactive program (not requiring script!) which produces tables
of relative ages for a visitor-supplied pair of ages to be compared.
"Aging and Age Perception
Getting older? Feel like the years are flying by faster than they
used to? Ever wonder why?
As you age, your perception of time changes. Every year, 1-year
represents a smaller and smaller piece of your life.
To a two-year-old, 1 year is half their life. To a 10 yr old, a
year is 10%; to a 20 yr old, only 5%.
Thus, for a 20 yr old to experience the age increase of a 10 yr
old's year, (s)he would have to age by 10% or 2 years.
Using the Generation Gap Calculator (below), you can experiment
with aging yourself like someone of a different age."
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Stanford Encyclopedia of Philosophy:
The Experience and Perception of Time
http://plato.stanford.edu/entries/time-experience/
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Time Keepers of the Calendar and Clock
http://www.ernie.cummings.net/time.htm
A colorful site about clocks and calendars and their histories.
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Appendix: Logtime Math
For those readers who have had some calculus, or at least are
familiar with the elementary properties of logarithms, a derivation
of the mathematical relationships of Logtime is presented below. What
will be shown is that a simple, plausible assumption about the
estimation of time intervals by the human mind ("psychochronometry")
implies a mathematical model that can explain common observations, at
least to the extent allowed by the unmeasurable subjective nature of
those observations.
This is not a "proof" of Logtime: the basic premise is not only
unproven but probably unprovable because of its subjective nature and
the difficulty of experimentation. Long time intervals would be
required for test subjects to age, and their aging could not be
reversed! Note that the same problem exists for the linear
alternative model. The best choice among psychochronometric cognitive
models must ultimately depend upon which predicts consequences that
seem the closest to actual human experience. As crudely describable
as it may be, this experience seems at least to enable the
logarithmic model to be selected over the linear one. (Are there any
others?)
Note that the following analysis (developed independently)
recapitulates the derivation of the "Weber-Fechner law" of sensory
stimulation, with aging as the stimulus:
For human age, let
t = objective (clock) time
and its dependent variable
s = subjective (perceived) time
The Logtime hypothesis is that the length of a time interval is
perceived as its proportion of current age. This can be expressed
in differential form as
ds = dt/t
where the constant of proportionality is made unity by proper
choice of the (undefined) unit of s.
Integrating:
s = ln(t) + c
For a finite increment of s, s to s', bounded by corresponding
clock ages t and t':
s' - s = ln(t') - ln(t)
= ln(t'/t)
The perceived increment of subjective age is seen to depend only
on the ratio of corresponding clock ages and not on their
absolute values (or unit). All values of t and t' having a given
ratio (e.g., t'/t = 2) yield the same value of (s' - s). The
natural logarithm function (ln) relating these temporal
increments (objective and subjective) suggested the name
"Logtime".
Note that a consequence of this logarithmic relationship is that
subjective increments add as clock increments (ratios) multiply.
Consider a second subjective increment, s' to s", immediately
following s to s'. The total increment, bounded by t and t", is
s" - s = (s" - s') + (s' - s)
= ln(t"/t') + ln(t'/t)
= ln[(t"/t')*(t'/t)]
= ln(t"/t)
The special case of doubling a subjective increment therefore
requires a squaring of the clock age ratio. If that ratio is 2
(the octave), then two successive octaves are required,
quadrupling the clock age.
Expressing the clock age ratio as a function of the corresponding
subjective age increment:
t'/t = exp(s' - s)
shows the exponential growth in relative clock age required for a
linear (uniformly perceived) passage of subjective time.
Return to Logarithmic Explanation
Return to Scale of Life
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The Author
The author received the degrees of Bachelor of Electrical Engineering
from The City College of New York and Master of Science in Electrical
Engineering from Columbia University, and is now retired from an
engineering career spent largely in the field of microwave
semiconductor measurements.
The author's concept of Logtime, including the basic math, dates from
the 1960's, inspired in part by the logarithmic scales on the slide
rules then in use before the development of electronic calculators
and personal computers, and also by the log scales on some types of
graph paper used to plot data.
Recent searches (occasioned by writing this monograph) have revealed
that the basic elements of Logtime were described by others much
earlier, going back at least to the 19th century, although references
are few and these ideas seem not to have achieved a general
awareness, let alone acceptance.
In particular, Paul Janet may have been first to propose that current
age is the basis for estimating time intervals, and Rodney Collin the
earliest author to introduce a logarithmic subjective time scale
(albeit a somewhat bizarre one), but no reference has been found
making an explicit mathematical connection between these concepts.
Gustav Theodor Fechner, however, used analogous math for physical
stimuli. Logtime may therefore be regarded as an extension of the
Weber-Fechner law, with aging as the stimulus. Has any such
suggestion been published?
Additional references and reader reactions would be greatly
appreciated. Please send to:
jmkenney@kafalas.com
The latest version of this document may be found at
http://ourworld.compuserve.com/homepages/jmkenney/
or downloaded in LOGTIME.ZIP from
http://ourworld.compuserve.com/homepages/jmkenney/logtime.zip
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...The Bird of Time has but a little way
To flutter--and the Bird is on the Wing.
...The Wine of Life keeps oozing drop by drop,
The Leaves of Life keep falling one by one.
Ah, but my Computations, People say,
Reduced the Year to better reckoning?--Nay,
'Twas only striking from the Calendar
Unborn To-morrow and dead Yesterday.
-- Rub iy t of Omar Khayy m: Edward FitzGerald
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...Today will die tomorrow;
Time stoops to no man's lure...
-- The Garden of Proserpine: Swinburne
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Contents
Top
Opening Verse
Problems of Time Perception
Logtime: The Logarithmic Explanation
The Scale of Life
Logarithmic Lifetimes
The Last Judgement
Will Life Extension Help?
Intergenerational Relationships
What is the Clock? (Unanswered Questions)
References
Internet Links
Appendix: Logtime Math
The Author
Closing Verse
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Prose text Copyright 2000-2002 James Main Kenney. All Rights
Reserved.
Permission granted for non-commercial reproduction or reposting of
unaltered page.
Revised 2002-03-16
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