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Weber's law, or why you're middle aged at 9

Humans experience everything relatively, and tend to make comparisons on a logarithmic scale 2. For instance, if something seems only a little bit more painful, it's likely actually a lot more painful (think twice as many nerves firing). This is known as Weber-Fechner's law 1.

\[ \begin{aligned} \underbrace{\frac{\Delta S}{S}}_{\text{the smallest change you notice}} &= \underbrace{k}_{\text{a fixed fraction}} \end{aligned} \]

You only notice a fixed fraction of what you already feel.

If we consider all experiences to be felt on a log scale, then consider the span of our perceived lives. If our life is 100% of the time we've experienced, and we'll likely live to 84, then by Weber's law, we were middle aged when we were 9 3.

\[ \begin{aligned} \underbrace{\operatorname{weber}(\text{years})}_{\text{felt intensity}} &= \underbrace{\frac{\ln(\text{years})}{\ln(\text{life span})} \times 100\%}_{\text{life experienced}} \\ \implies \underbrace{\frac{\ln(\text{years})}{\ln(\text{life span})}}_{\text{share of life felt}} &= \frac{1}{2} \quad\Longrightarrow\quad \text{years} = \underbrace{\sqrt{\text{life span}}}_{\sqrt{84}\;\approx\;9} \end{aligned} \]

Perhaps we don't truly start having experiences until 4, in which case middle age is 18 4

\[ \text{years} = \underbrace{\sqrt{\text{first year} \cdot \text{life span}}}_{\sqrt{4 \cdot 84}\;\approx\;18} \]

or if we consider the "unit" year to be at 23 (that was a fun year) 5

\[ \underbrace{\text{felt years}}_{\text{how old you feel}} = \text{unit year} \cdot \ln\!\left(\frac{\text{years}}{\text{first year}}\right), \qquad \underbrace{\frac{\text{unit year}}{\text{years}}}_{\text{how long this year feels}} \]

At 78% of life lived by 32, existence still seems desperately short. However, at death, with a perceived time of 102 years, you will have lived more than your years! 6

I've made the following graph for you to play with. Just play with the sliders.

lifespan: how long you expect to live. first year: the age your memories start. unit year: the year that felt exactly a year long. age: you, now.
The dashed line is calendar time, where every year is the same. The solid line is Weber time, where every year feels a little less than the one before it. Perceived time at 32: 80 of 102 years.

This is a rather morbid way of saying Happy Birthday to my mother (70!! She's 96% of the way there), my cousin, and my darling sister — who are all Virgos, marking another trip around the sun.

I'll leave on another optimistic note. Weber's law applies only when the stimulus is comparable. A life where every year was new, exciting, and lived to the fullest would be a life fully lived

So wishing you all another year of new experiences, and grateful you're alive.

  1. This comes from the field of "psychophysics", and to my untrained ear, it sounds like it belongs in an Asimov book. ↩

  2. I first wrote a variation of this post in 2019 in a love letter (love blog?). I'm not exactly romantic. It's also heavily featured in a great blog post and Numberphile video. ↩

  3. Weber's observation was that the smallest change you can notice is a fixed fraction of what you are already feeling:

    \[ \frac{\Delta S}{S} = k \]

    Fechner added those "just-noticeable" steps up. Felt intensity \(p\) grows with the logarithm of the stimulus \(S\),

    \[ p = k \ln \frac{S}{S_0} \]

    where \(S_0\) is the faintest stimulus you register at all. If the stimulus is your age \(a\), with year one as \(S_0\), the fraction of a lifespan \(L\) that you have felt by age \(a\) is

    \[ f(a) = \frac{\ln a}{\ln L} \]

    and middle aged (halfway), \(f = \tfrac{1}{2}\), lands at \(a = \sqrt{L}\) and \(\sqrt{84} \approx 9.2\). ↩

  4. If the first year that counts is \(a_0\) rather than year one, it becomes the scaling value.

    \[ f(a) = \frac{\ln (a / a_0)}{\ln (L / a_0)}, \qquad a_{\text{mid}} = \sqrt{a_0 L} \]

    With \(a_0 = 4\), middle age moves out to \(\sqrt{4 \cdot 84} \approx 18\). At 32, with \(L = 84\), the share of life already felt goes from 78% to 68%. ↩

  5. \(k\) is "felt time" converted back into standard years. Given a standard year at age \(a_1\), set \(k = a_1\):

    \[ \tau(a) = a_1 \ln \frac{a}{a_0}, \qquad \frac{d\tau}{da} = \frac{a_1}{a} \]

    \(\tau\) is your felt age, and each calendar year adds \(a_1 / a\) of a felt year. For instance, with \(a_1 = 23\), a year at 9 feels like two and a half years, while a year at 46 feels like six months. ↩

  6. From the quirk that years feel longer than your selected unit year when you're a youth. ↩

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