A commonly mentioned phenomenon is the notion that the length of a unit of time (e.g. a year) decreases with age. The concept is not limited to intimate conversations; it has profound philosophical, psychological and existential significance. Many people often comment that childhood seems to be very long but seems to get shorter and shorter as adults. Our perception is that time goes by faster and faster as we age. Here, I try to represent this phenomenon mathematically.
One can understand this intriguing occurrence through a simple mathematical model. Considering that each unit of time (for example, a year) feels shorter compared to the total amount of life we have already experienced, we can derive a function to encapsulate this perception. This function takes into account the age of the individual and the relative 'length' of a year.
\( P(t) \) is the perceived duration of a unit of time when one is \( t \) years old. Therefore:
\[ P(t) = \frac{1}{t} \]
In this example, \( t \) is the total time lived, and \( P(t) \) is the extent to which a single unit of time, for example a year, feels "long" in relation to that. For instance, if you are ten years old, a year would seem to represent \( \frac{1}{10} \) or ten percent of your life, which appears to be quite significant. But when you are 50 years old, a year would seem to represent \( \frac{1}{50} \) or two percent of your life, which feels much shorter in comparison.
Although this mathematical model is oversimplified, it serves as a basis for more complex discussions. Real-life experience suggests that many other factors influence our perception of time, including our emotional state, the amount of new experiences we have, and the repetition of routines. Psychologists also point out that our cognitive processes, memory, and even neurotransmitter levels can affect our perception of time.
The beauty of the model lies in its simplicity, giving an empirical perspective on subjective experience. It quantifies something intangible, facilitating discussion, research, and understanding.
In conclusion, the perception that time accelerates with age is a universal human experience. The \( \frac{1}{t} \) model, while elementary, provides a framework to begin to dissect this complex phenomenon. While it doesn't capture all the nuances, it does provide a useful lens for exploring the ephemeral nature of time, encouraging further study of one of these elusive questions.