In today’s society, mathematics underpins almost everything we do. It is vital to the ways in which we communicate with each other and the methods we use to navigate from place to place. It has completely altered how we buy and sell and it has revolutionised the manner in which we work and relax. Its influence can be felt in almost every courtroom and every hospital ward, in every office and every home. Maths might even explain why time seems to speed up as we get older.
Do you remember, when you were younger, that summer holidays seemed to last an eternity? For my children, who are four and six, the wait between consecutive Christmases seems like an inconceivable stretch of time. In contrast, as I get older, time appears to pass at an alarming rate, with days blending into weeks and then into months, all disappearing into the bottomless sinkhole of the past.
It seems that perceived time really does run more quickly the older we get, fuelling our increasing feelings of overburdened time-poverty. In an experiment carried out in 1996, a group of younger people (19–24) and a group of older people (60–80) were asked to count out three minutes in their heads.
On average the younger group clocked an almost-perfect three minutes and three seconds of real time, but the older group didn’t call a halt until a staggering three minutes and 40 seconds, on average.
Read your Q&As about time:
- How does my computer know what time it is?
- How long is a moment?
- When I look to the stars, how far back am I seeing?
This acceleration in our perception of the passage of time has little to do with leaving behind those carefree days of youth and filling our calendars with adult responsibilities. In fact, there are a number of competing ideas that provide explanations for why, as we age, our perception of time accelerates.
One theory suggests that our perception of time’s passage depends upon the amount of new perceptual information we are subjected to from our environment. The more novel stimuli there are, the longer our brains take to process the information. The corresponding period of time seems, at least in retrospect, to last longer.
This argument can be used to explain the movie-like perception of events playing out in slow-motion in the moments immediately preceding an accident. So unfamiliar is the situation for the accident victim in these scenarios that the amount of novel perceptual information is correspondingly huge.
It might be that rather than time actually slowing down during the event, our recollection of the events is decelerated in hindsight, as our brain records more detailed memories based on the flood of data it experiences. Experiments on subjects experiencing the unfamiliar sensation of free fall have demonstrated this to be the case.
This theory ties in nicely with the acceleration of perceived time. As we age, we tend to become more familiar with our environments and with life experiences more generally.
Our daily commutes, which might initially have appeared long and challenging journeys full of new sights and opportunities for wrong turns, now flash by as we navigate their familiar routes on autopilot.
It is different for children. Their worlds are often surprising places filled with unfamiliar experiences. Youngsters are constantly reconfiguring their models of the world around them, which takes mental effort and seems to make the sand run more slowly through their hour-glasses than for routine-bound adults. The greater our acquaintance with the routines of everyday life, the quicker we perceive time to pass and, generally, as we age, this familiarity increases.
This theory suggests that, in order to make our time last longer, we should fill our lives with new and varied experiences, eschewing the time-sapping routine of the everyday.
Neither of the above ideas manages to explain the almost perfectly regular rate at which our perception of time seems to accelerate. That the length of a fixed period of time appears to reduce continually as we age suggests an ‘exponential scale’ to time. We employ exponential scales instead of traditional linear scales when measuring quantities that vary over a huge range of different values.
The most well-known examples are scales for energy waves like sound (measured in decibels) or seismic activity. On the exponential Richter scale (for earthquakes), an increase from magnitude 10 to magnitude 11 would correspond to a ten-fold increase in ground movement, rather than a 10 per cent increase as it would do on a linear scale.
At one end, the Richter scale was able to capture the low-level tremor felt in Mexico City in June 2018 when Mexican football fans in the city celebrated their goal against Germany at the World Cup. At the other extreme, the scale recorded the 1960 Valdivia earthquake in Chile. The magnitude 9.6 quake released energy equivalent to over a quarter of a million of the atomic bombs dropped on Hiroshima.
If the length of a period of time is judged in proportion to the time we have already been alive, then an exponential model of perceived time makes sense. As a 34-year-old, a year accounts for just under 3 per cent of my life. My birthdays seem to come around all too quickly these days. To my four-year-old son, the idea of having to wait a quarter of his life until he is the birthday-boy again is almost intolerable.
Under this exponential model, the proportional increase in age that a four-year-old experiences between birthdays is equivalent to a 40-year-old waiting until they turn 50. When looked at from this relative perspective, it makes sense that time seems only to accelerate as we age.
It’s not uncommon for us to categorise our lives into decades – our carefree 20s, our serious 30s and so on – which suggests that each period should be afforded an equal weighting. However, if time really does appear to speed up exponentially, chapters of our life spanning different lengths of time might feel like they are of the same duration.
Under the exponential model, the ages from 5 to 10, 10 to 20, 20 to 40 and even 40 to 80 might all seem equally long (or short). Not to precipitate the frantic scribbling of too many bucket lists, but under this model the 40-year period between 40 and 80, encompassing much of middle and old age, might flash by as quickly as the five years between your fifth and tenth birthdays.
The Maths of Life and Death: Why maths is (almost) everything by Kit Yates is out now (£19.99, Quercus)