> ## Content Index > Fetch the complete content index at: https://pasqs.com/llms.txt > Use this file to discover other available public pages before exploring further. # How old do you feel? Bruce’s Aging Counting System - URL: https://pasqs.com/physics/how-old-do-you-feel-bruces-aging-counting-system/ - Published: 2026-10-28T23:00:00.000Z - Updated: 2026-10-29T03:35:44.000Z - Description: Why do years fly faster with age? A few axioms on how novelty fades give your perceived age, and turn two classic laws of time perception into consequences. - Author: Pasquale Trinchese - Tags: Science, #Physics When I was a kid, the summer holidays lasted forever. Now a whole year goes by and I barely notice. Almost everybody feels it: time speeds up with age. A while ago I tried to turn that feeling into a formula, and I called it **Bruce's Aging Counting System**, BACS for short: a way of counting your age not by the calendar, but by the experience you have gathered. This is its second version, built like a [physical theory](https://pasqs.com/physics/what-is-physics-really/): a few axioms, and everything else derived from them. Along the way, two old acquaintances will show up. In 1877 the philosopher Paul Janet [\[1\]](#ref-1) proposed that a year feels shorter the older you are; in 1975 Robert Lemlich [\[2\]](#ref-2) corrected him with a slightly different law. Both simply assumed their laws. In the BACS they come out as consequences: depending on how you think novelty fades, you get one or the other. *Not interested in the math behind it? [Jump straight to the calculator](#bacs-calculator) and find out your BACS age.* ## Not every event is new Every event can bring experience in some form: a notion, a sensation, a conversation, a book. But not every event teaches you something. Some repeat themselves, others look a lot like ones you have already lived, and the more you have lived, the less likely the next one is to be new. Call $E$ the number of events lived since birth, and $\\xi$ your experience: the number of those that brought you something new. Here experience is new by definition, and an event you have already lived adds nothing to it. How fast it grows with events, $$P = \\frac{d\\xi}{dE} \\tag{1}$$ is the probability that the next event will teach you something. Like any probability, it can't go above 1\. Keep this in mind: it will matter. ## The three axioms **Axiom 1: events flow at a steady pace.** You live $\\alpha$ events per year: $$E = \\alpha t \\tag{2}$$ A simplification, of course, but a reasonable one on the scale of a whole life. **Axiom 2: time is measured in experience.** The perceived length of a period is proportional to the experience it brought. A week full of firsts is remembered as long, a week of routine disappears, as William James had already noticed in 1890 [\[3\]](#ref-3). **Axiom 3: novelty fades.** But fades with what? Here the theory splits in two. *Version A: with what you have lived.* The probability of something new is inversely proportional to the events behind you: $$P = \\frac{\\lambda}{\\xi\_h + E} \\tag{3A}$$ *Version B: with what you know.* It is inversely proportional to the experience behind you: the more you know, the more likely the next thing is something you knew already. $$P = \\frac{\\lambda}{\\xi\_h + \\xi} \\tag{3B}$$ Here $\\lambda$ is how slowly novelty fades for you: your learning rate. And $\\xi\_h$ is the experience you were born with, because a newborn brain doesn't start from nothing. Version A adds it to events, but for a newborn experience and events are the same thing: nothing has happened twice yet. That inheritance isn't a decoration. Without it, (3A) and (3B) would be infinite at birth. Asking $P \\le 1$ from the very first event sets a minimum: $$\\xi\_h \\ge \\lambda \\tag{4}$$ With the equal sign, it's a newborn for whom every single event is new. ## From axioms to equations Thanks to (1), axiom 3 is already a pair of differential equations: $$\\frac{d\\xi}{dE} = \\frac{\\lambda}{\\xi\_h + E} \\tag{5A}$$ $$\\frac{d\\xi}{dE} = \\frac{\\lambda}{\\xi\_h + \\xi} \\tag{5B}$$ both starting from $\\xi = 0$ at birth. Equation (5A) integrates into a logarithm: $$\\xi(E) = \\lambda \\ln\\left(1 + \\frac{E}{\\xi\_h}\\right) \\tag{6A}$$ In (5B), separate the variables, $(\\xi\_h + \\xi) d\\xi = \\lambda dE$, integrate, and you get $\\left( \\xi\_h + \\xi \\right)^2 - \\xi\_h^2 = 2 \\lambda E$, that is: $$\\xi(E) = \\sqrt{\\xi\_h^2 + 2 \\lambda E} - \\xi\_h \\tag{6B}$$ Put (2) into them, and here are the two general forms of the BACS: $$\\xi(t) = \\lambda \\ln\\left(1 + \\frac{\\alpha t}{\\xi\_h}\\right) \\tag{7A}$$ $$\\xi(t) = \\sqrt{\\xi\_h^2 + 2 \\lambda \\alpha t} - \\xi\_h \\tag{7B}$$ ## When everything is new Take the limit case of (4), $\\xi\_h = \\lambda$. Both forms then depend on a single time scale, $$\\tau = \\frac{\\lambda}{\\alpha} \\tag{8}$$ roughly how long the world stays new. It depends on you, through $\\lambda$, and on the world around you, through $\\alpha$: $$\\xi(t) = \\lambda \\ln\\left(1 + \\frac{t}{\\tau}\\right) \\tag{9A}$$ $$\\xi(t) = \\lambda \\left(\\sqrt{1 + \\frac{2t}{\\tau}} - 1\\right) \\tag{9B}$$ At first, for $t \\ll \\tau$, both grow as $\\alpha t$: every event is new. Then they part ways, and (9A) slows down like a logarithm, (9B) like a square root. ## Your BACS age Axiom 2 now turns experience into time. Divide the experience gathered since birth by $\\alpha$, the pace at which a newborn gathers it, and you get your perceived age, in years as you felt them at the very beginning: $$a(t) = \\frac{\\xi(t)}{\\alpha} \\tag{10}$$ That's your BACS age, and it depends on $\\tau$ alone: $$a(t) = \\tau \\ln\\left(1 + \\frac{t}{\\tau}\\right) \\tag{11A}$$ $$a(t) = \\tau \\left(\\sqrt{1 + \\frac{2t}{\\tau}} - 1\\right) \\tag{11B}$$ And how long a year feels at age $t$, compared with the first ones, is how fast $a(t)$ grows. For (12B) I've written the square root of (11B) as $1 + a/\\tau$: $$\\frac{da}{dt} = \\frac{\\tau}{\\tau + t} \\tag{12A}$$ $$\\frac{da}{dt} = \\frac{\\tau}{\\tau + a} \\tag{12B}$$ To put some numbers on it, say $\\tau$ is two years: | Age | BACS age (A) | BACS age (B) | A year feels like (A) | A year feels like (B) | | --- | ------------ | ------------ | --------------------- | --------------------- | | 5 | 2.5 | 2.9 | 104 days | 149 days | | 10 | 3.6 | 4.6 | 61 days | 110 days | | 20 | 4.8 | 7.2 | 33 days | 80 days | | 40 | 6.1 | 10.8 | 17 days | 57 days | | 80 | 7.4 | 16.0 | 9 days | 41 days | At 80, you have lived 7 or 16 years, as you would have felt them at the very beginning. The BACS also explains something everybody has noticed. A 50-year-old and a 60-year-old speak the same language and talk about the same things; a 10-year-old and a 20-year-old hardly do. The gap is 10 years in both cases, but not in the BACS. With the same $\\tau$ of two years, the 10- and the 20-year-old are 1.2 BACS years apart in version A, and 2.5 in version B; the 50- and the 60-year-old only 0.35 and 1.3. Try it with your own age: Your BACS age Age years τ years τ = λ/α grows with λ, how fast you learn, and shrinks with α, how many events you live each year. The larger it is, the longer your years feel. Two years is just an example. Version A, Janet years A year now feels like days Version B, Lemlich years A year now feels like days ## Janet and Lemlich, for free Read (12A) aloud: the length of a year is inversely proportional to the years you have lived. Read (12B): it is inversely proportional to the years you *feel* you have lived. Long after childhood $\\tau$ becomes negligible, $$\\frac{da}{dt} \\simeq \\frac{\\tau}{t} \\tag{13A}$$ $$\\frac{da}{dt} \\simeq \\frac{\\tau}{a} \\tag{13B}$$ and these are exactly the two laws from the start. The first is Janet's [\[1\]](#ref-1): a year is a tenth of a 10-year-old's life, but a fiftieth of a 50-year-old's, so perceived age grows like the logarithm of real age. The second is Lemlich's [\[2\]](#ref-2): what counts are the years you feel, not the real ones, and perceived age grows like the square root of real age. Janet and Lemlich assumed their laws. Here they follow from axiom 2 and from the two versions of axiom 3, and the choice between them comes down to a single question: does novelty fade with what you have lived, or with what you know? There's a bonus, too. Both laws blow up at birth, giving the first moment of life an infinite length. In the BACS, (12A) and (12B) start from 1, because of (4). Vladimir Shiltsev [\[4\]](#ref-4) had already noticed the problem with Janet's law, and fixed it by hand with a formula of the same form as version A. Here it isn't a fix: it comes out of the axioms, and every parameter has a meaning. ## Two axioms, two lives The two versions also disagree on how much the world around you matters. In version A, once childhood is over, the experience gathered between two ages $t\_1$ and $t\_2$, $\\Delta \\xi = \\xi(t\_2) - \\xi(t\_1)$, follows from (9A): $$\\Delta \\xi \\simeq \\lambda \\ln\\frac{t\_2}{t\_1} \\tag{14A}$$ So each time your age doubles you gain the same experience, $\\lambda \\ln 2$: as much between 20 and 40 as between 40 and 80, whatever $\\alpha$ is. A life twice as intense uses up novelty twice as fast, and the two effects cancel out. In version B, from (9B), $$\\Delta \\xi \\simeq \\sqrt{2 \\lambda \\alpha} \\left(\\sqrt{t\_2} - \\sqrt{t\_1}\\right) \\tag{14B}$$ and twice as many events give $\\sqrt{2}$ times more experience, for your whole life. So the two versions can be told apart. Take people of the same age who live at very different paces, and ask them how long their past decades feel. If the gap between them closes after childhood, version A is right; if it stays, version B is. ## The mind as a formal system I'll close with a thought that has stayed with me since the first version of the BACS. The rational part of the brain can be seen as a formal system: a set of axioms, and rules to infer theorems from them. Maybe that's why mathematics is the way it is: it has the shape of the machine that does it. In this picture, experience is made of new theorems. Some come from inside, inferred from the axioms you already have: that's *production*. Others come from outside, with external events, and the system checks them against what it already holds: that's *validation*. This also explains version B: the more theorems you hold, the more likely a new statement is one you could already derive. In my opinion there are two layers: a fundamental one, where the axioms of the human being lie, and a higher one, where consciousness probably arises and new theorems and axioms are formed. The higher layer doesn't have to follow the fundamental one, and can even overrule it. So they are not a single formal system, where a contradiction between them would make every statement provable, but two. In the higher layer, a statement that comes from outside, can't be derived from either layer, yet seems true, can be taken as a new axiom. That's exactly what mathematics does with arithmetic. In 1931 Kurt Gödel [\[5\]](#ref-5) proved that any consistent formal system able to do arithmetic contains true statements it can't prove. Add one as a new axiom, and the larger system has unprovable statements of its own. The supply of new axioms never runs out, just as, in the BACS, experience keeps growing for your whole life. A machine, then, whose main purpose is neither survival nor reproduction, but inferring new theorems. And its age, the one it feels, is the count of the theorems it has found. ## References 1. P. Janet, "Une illusion d'optique interne", *Revue philosophique de la France et de l'Étranger* 3, 497–502 (1877). 2. R. Lemlich, "Subjective acceleration of time with aging", *Perceptual and Motor Skills* 41, 235–238 (1975). 3. W. James, *The Principles of Psychology*, Vol. 1 (Henry Holt, New York, 1890). 4. V. Shiltsev, "On soft mathematical models of subjective time acceleration with age", *Nonlinear Dynamics, Psychology, and Life Sciences* 27, 249–258 (2023). 5. K. Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", *Monatshefte für Mathematik und Physik* 38, 173–198 (1931).