Time is not linear: we’re growing older faster

What are the mathematics behind the years flying by?

Artwork by Victor Lee

Have you ever heard people say time speeds up as you get older? It turns out this common saying is grounded in mathematical truth. Half of our life is, in fact, lived by the time we hit 20.

To be precise, our lives actually follow a logarithmic function. Imagine you are 20. Reaching 80 will feel as long as your entire life to date. How is this possible?

Firstly, the notion that time is linear is false.

Currently, we are convinced that all our years are equal and that we are moving through time at a uniform rate. But this view is mathematically incorrect. Your life is not moving forward, like the hands of a clock, nor does it match the movement of the seasons Ask yourselves why is it that humans find it much more difficult to remember the first few years of life compared to later years? Why do parents experience their children growing up faster than the children themselves do?

This is because time is perceived logarithmically (logtime) and accords to Gompertz Law.

Gompertz Law—life leads on an exponential function

According to the Gompertz Law of mortality, your probability of dying during any given year doubles every eight years. In other words, because your mortality rate is increasing exponentially with age, your probability of surviving to a certain age is actually decreasing super-exponentially.

If you are now 25, your probability of dying during the next year is reasonably minute — about 1 in 3,000 (0.03 per cent). At 33, this will have risen to 1 in 1500; at 42, 1 in 750. Flash forward to 100, your probability of surviving to celebrate your 101st birthday will only be around 50 per cent.

Mortality rate statistics support Gompertz Law. The two graphs on this page use 2015 US census data from the National Center for Health Statistics to plot the probability of death against age. The resulting exponential trend supports Gompertz Law.

Gompertz Law applies to all areas of human life. We can use an analogy of ‘cops and criminals’ inside your body to illustrate how rates of DNA degradation also observe Gompertz This analogy requires us to ignore dangerous environmental factors, and presume that our bodies’ in-built expiration date is responsible for t our rapidly increasing mortality rate.

In your body, ‘cops’ (which represent your DNA repair enzymes) and ‘criminals’ (DNA damage – such as single and double strand breaks, 8-Hydroxydeoxyguanosine residues) are constantly fighting one another. In the beginning, the police win (the enzymes can correctly repair the DNA damage). They patrol and remove any criminals they catch. If the cops do not react fast enough, a criminal can construct a ‘fort’ (e.g. DNA damage beyond repair) so strong it is immune from police intervention. When that occurs, you (your damaged cells) die.

During youth, cops are abundant and, on average, patrol each area of the body 14 times a day (i.e. rate of DNA damage repair). The probability of cops failing to check a particular area is low, and can be represented as t e-14≈8 x 10-7, a number known as the Poisson distribution. With age, this internal police force grows weaker as its ability to repair DNA declines. They only scout each area 12 times per day (i.e. increasing unrepaired accumulation of DNA damage). The probability of missing a spot now increases to e-12≈6 x 10-6. The drop from 14 to 12 may appear insignificant, but this increases your probability of death in any given day by more than seven times. Essentially, though it seems that the strength of your police force is falling linearly over time, in fact, your mortality rate is increasing exponentially.

Logtime —life leads on an logarithmic function

The second concept we must understand is logtime. Essentially, logtime is a mathematical model used to explain psychochronometry, the psychology of time perception.

Because the human mind judges the length of a period of time by comparing it with current age, we actually perceive the length of one year as a proportion of our current age. That means the older we are, the smaller each year is as a fraction of our total time alive—and therefore seems shorter.

That has sobering implications:

The years from ages ten to 20 will 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. Thus, there is a perceived diminishing of time as we grow up.

For instance, one year adds ten per cent to the life of a ten year old, but only five per cent to that of a 20 year old. For the 20 year old, two years are required to add 10 per cent.

The significance of this is that for the 20 year old, two years will seem to pass as rapidly as one year seems to the ten year old. Likewise, three years to a 30 year old and four years to a 40 year old will appear to pass equally as fast. Thus, if we view years as being of equal length, the speed with which time passes will seem to rise exponentially.

To disturb you even further, logtime does not stop at applying to just the years of your life. It applies to all time intervals, meaning that even days and hours also dwindle with age. But because life tends to become increasingly hectic and busy with age and maturity, short-term psychological factors usually overshadow and conceal the obviousness of this loss of time.

Ultimately, the truth is that our life is leading an exponential or logarithmic function, depending on which factor you use. We can dispel the myth that time is linear. If you are 20 now, 80 is going to hit you as fast as it took to arrive where you are now from when you turned 5. Depressing or not, life is shorter than you think. So make the most of it.

survival probability over a lifetime probability of death over lifetime

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