Subjective Life Span
When I was 7 years old, a year seemed a very long time. And indeed it was -- it was 1/7 of my life. Now that I'm 39, a year seems much shorter. But of course now a year is only 1/39 of my life. When I was 7, 30 minutes seemed a long time; now it doesn't seem nearly so long.
Let's suppose that subjective time is inversely proportional to life span. The subjective time of any period is then the integral of 1/x, which is to say the difference between the natural logs of the end and the beginning of the period.
(Most recent psychological work about "subjective time" tends to be about subjective estimations of clock time, or about comparisons of periods close together in time as seeming to go relatively more quickly or more slowly. These are completely different issues than the one I'm contemplating here. They don't get at the fundamental question of whether the clock itself seems to speed up over the life span -- though see Wittmann & Lehnhoff 2005.)
On this model, since 1/x approaches infinity as x approaches 0 (from the positive direction), it follows that our subjective life-span is infinite. We seem, to ourselves, subjectively, to have been alive forever. (Of course, I know I was born in 1968, but that's merely objective time.)
There's something that seems right about that result; but an alternative way of evaluating subjective life span might be to exclude the earliest years -- years we don't remember -- starting the subjective life span at, say, age 4.
Adopting that second method, we can calculate percentages of subjective life span. Suppose I live to age 80. At age 39, I've lived less than half my objective life span, but I've already lived 76% of my subjective life span ([ln(80)-ln(39)]/(ln(80)-ln(4)] = 0.76). At what age was my subjective life half over? 18. Whoa! I feel positively geriatric! (And these reflections about philosophers peaking at age 38 don't help either.)
Regardless of whether the subjective life span begins at age 0 or age 4, we can compare the subjective lengths of various periods. For example, the four years of high school (age 14-18) are subjectively 25% longer than the four years of college (age 18-22). Doesn't that seem about right? Similarly, it wasn't until I was teaching for 9 years (age 29-38) that I had been a teacher as long, subjectively, as I had been a high-school student; and it will take 'til age 60 for my subjective years of teaching to exceed my subjective years of high school, college, and grad school combined.
If we throw in elementary school, objective and subjective time get even more out of synch. Those 7 years (age 5-12) will be subjectively equivalent to the 42 years from age 30-72! And unless I teach until I'm 168 years old, I'll always have had more subjective time as a student than as a teacher. Is that too extreme? Maybe so. But I don't really know what it's like to be 168 years old; and I'm not sure how stable and trustworthy my judgments now could be about how long 3rd grade seemed to take. Is 6 times as long as a middle-aged year so unreasonable?