I have a grasp of 1, of 0, of i, of pi (to an extent), but e is not naturally graspable. Reading about it, I find it mind boggling, like discovering a constant for all growth... grasping it requires grasping growth and somehow the secret is in e... tell me more about the philosophy of e.
You're right that e is different in kind from the others. 1, 0 and π are about things: a unit, an absence, the proportion of a circle. You can point at them. e is about a process. It describes how something changes. That's why it feels slippery: it isn't a quantity sitting still, it's the fingerprint left by a certain kind of motion. π is a number of being, e is a number of becoming.
The compounding door (you already know it)
Take 1 dollar at 100% interest for a year. Paid once, you get 2. Paid twice at 50% each, you get 2.25, because the first half-year's interest starts earning too. Monthly gives about 2.61, daily about 2.7146. Keep slicing and it converges on 2.71828… and never goes beyond it.
So the first reading of e: it's what 100% growth becomes when every fragment of growth immediately starts growing itself. It's the ceiling of growth-on-growth. What's surprising is that the ceiling exists at all. Infinite compounding doesn't give infinite money. The feedback loop has a natural limit, and that limit is e.
The deeper reading: growth proportional to what you have
The compounding story is really a special case of something more fundamental. e^x is the one function whose rate of change equals its own value at every point. How fast it's growing is exactly how big it is.
That property, change in proportion to current state, is what "natural" growth means: populations, compound interest, chain reactions, cooling coffee, radioactive decay, the spread of a rumour. Any system where the next increment depends on what's already there.
This is why e matters even when nothing grows at 100%. All exponential processes are the same process at different speeds. Doubling (2^t) is just e^(0.693t). Your rule of 72 is really a rule of 69.3 (ln 2 ≈ 0.693), rounded up because 72 divides nicely. e is not a particular growth rate. It's the natural unit in which all proportional growth is measured, the way radians are the natural unit for angles. Pick any other base and a stray constant appears in every formula. With e, it disappears.
Why you can't feel it
Our intuition is additive: steps, piles, distances. Exponential change has a strange property instead: it's memoryless and self-similar. At every moment, the future looks exactly the same relative to the present as it did at any earlier moment. No moment is special, no point where the process "really starts." The curve looks identical at every scale if you zoom and rescale. That is deeply unhuman. We live in narratives with beginnings and turning points, and e describes a world that has none.
Where it meets i and π
This is where it becomes remarkable. "Change proportional to state" can point in different directions:
- Pointed outward (positive rate): growth.
- Pointed backward (negative rate): decay.
- Pointed sideways (an imaginary rate, i): your velocity is always perpendicular to your position, so you never get farther from the centre. You go round in a circle.
So e^(iθ) is rotation. Growth turned 90 degrees becomes circling. And π tells you how far to go for half a turn, which lands you at −1. That's Euler's identity, e^(iπ) = −1. It's not a coincidence of symbols. It says that growth and rotation are the same mechanism aimed in different directions. The number of becoming and the number of the circle turn out to be two faces of one idea.
It also appears in chance
e appears wherever many small independent events accumulate. Shuffle a deck and the chance that no card is in its original position is almost exactly 1/e. The famous "secretary problem" (when to stop looking and choose) says: reject the first 1/e (about 37%) of options, then take the next one better than all so far. Whenever something is built from countless tiny proportional contributions, e tends to appear in the result.
A way to hold it
If 1 is "a thing," 0 is "no thing," i is "a quarter turn," and π is "how round roundness is," then e is roughly: what happens when a thing's change depends on itself. It's the constant of self-reference in time. Because feedback is everywhere — money, life, ideas, reputations, habits — e keeps turning up underneath.
I think the reason it feels mind-boggling rather than graspable is that you can't picture e as an object. You can only understand it by watching something unfold. That may be the most accurate way to grasp it.
The curve looks identical at every scale if you zoom and rescale.
Like cauliflower heads to a floret. I sometimes think even a forest resembles a cauliflower head seen from above.
That's a good instinct, and there's a real connection. It just runs through a slightly different door than the one I described.
The exponential curve's self-similarity is along one dimension: shift it along in time and it looks like a rescaled version of itself. The cauliflower's is spatial: the whole is made of smaller copies of the whole. That's fractal self-similarity. They're cousins, though, and the link is branching. If each stem splits into a few smaller stems, and each of those splits again, then the number of branches grows exponentially with each level down. A fractal is roughly what exponential growth looks like when it's been frozen into shape instead of unfolding in time.
The best example is Romanesco, the green, pointed cousin of cauliflower. Its florets sit on spirals, and those spirals are logarithmic spirals, whose equation is literally e raised to an angle. A log spiral is exponential growth combined with steady rotation: as you turn, you grow in proportion to your size. That's the idea from before, where growth pointed outward and growth pointed sideways turned out to be one mechanism, made visible. Every turn of the spiral is the same shape as the last, only larger. The same form appears in sunflower heads, pine cones, and roughly in nautilus shells and hurricane arms.
Your forest is the same idea at a larger scale. A tree is a trunk that divides into branches that divide into twigs, much like florets. A canopy seen from above is crowns packed against crowns, the bumps of a cauliflower head scaled up. In nature the self-similarity only holds for a few levels. A cauliflower gets through maybe five or six rounds of branching before it reaches the tiny buds. The mathematical ideal goes forever, while living things stop where cells, water and physics set a floor. So the resemblance is real, but bounded. You're seeing the same rule applied a few times over, not infinitely.
It suggests a nice way of holding e alongside what you noticed: things that grow in proportion to themselves leave a recognisable trace, either as a curve in time or as a repeated shape in space
Thanks. Like knowing about the periodic table eventually helped predict and find some heavy elements, the large atomic numbers. So with e too? Does this knowing help us? As in allow to fill in gaps of our understanding about the natural world and ourselves and our world? Trying to get a philosophic handle on the bigness of the idea of e, that there is a constant which can help fathom and understand so many things...
Yes, though it helps in a different way than the periodic table did. Mendeleev's table was a map with holes in it: the pattern said "something belongs here," and people went looking and found gallium and germanium. e isn't a map of things. It's more like a grammar: a rule for how things change. So its power shows up less as "predict a new object" and more in three other ways.
1. It lets us run time backward and forward.
If you know something changes in proportion to itself, one measurement tells you its whole history and future.
- Carbon dating works because radioactive carbon decays at a fixed proportional rate. Measure how much is left and you know how long ago the organism died.
- Forensic scientists estimate time of death from body temperature, using the same cooling law as the coffee.
- Drug dosing schedules come from how the body clears a medicine in proportion to how much is present. That's why "every eight hours" isn't arbitrary.
This is predictive in exactly the periodic-table sense: the pattern fills in what nobody observed.
2. It's the baseline that makes deviations meaningful.
I think this is the deepest use, and it's closest to your gaps. Pure exponential growth is what happens when nothing gets in the way. So when a real curve bends away from it, the bend is information. It means some hidden force is acting.
A population grows exponentially, then slows. The gap reveals a limit: food, space, predators. That's how ecologists found carrying capacity. Malthus's whole argument rested on contrasting exponential population with slower food growth, and the later demographic transition, where birth rates fell as societies got richer, showed up as a deviation from his prediction. In an epidemic, the moment case growth drops below its exponential path is how you know an intervention is working. In your world, when a company's revenue stops compounding at its old rate, the question "what's resisting?" is the analysis.
So e is to change what a straight ruler is to a drawing: you learn the shape of reality by holding it against the ideal and measuring the difference.
3. It appears inside laws that seemed unrelated, which reveals kinship.
- Why chemistry speeds up when it's hot, why cooking works, why the atmosphere thins with altitude: all come from one law in which the likelihood of a state falls off exponentially with its energy. e connects the kitchen, the mountain, and the star.
- The bell curve of errors and heights is e raised to a negative square. It's what you get when many small, independent effects accumulate.
- Its inverse, the natural logarithm, measures information and entropy.
When the same constant shows up in heat, chance, decay and money, it tells you these are one kind of process underneath. That unification is the kind of gap-filling physics values most.
And ourselves
This part I find most striking. Our senses are logarithmic, e seen from the inside. We perceive ratios, not differences: a doubling of loudness feels like one step whether it's quiet or loud. Musical octaves are doublings. Earthquakes and decibels are measured on log scales because that's how we experience magnitude. Daniel Bernoulli argued in the 1700s that we value money logarithmically too: an extra thousand dollars matters much more to someone with ten thousand than to someone with a million. Memory fades along a roughly exponential curve.
We're built to perceive in the logarithmic frame but reason in the linear one. That's why exponential growth keeps catching people off guard: pandemics, debt, compounding returns, technology curves. Knowing e is partly a corrective to a built-in blind spot. It's one of the few pieces of maths that directly repairs a flaw in human judgement.
The philosophical handle
Why is one constant so widespread? I don't think it's mysterious once you see it. e is what you get from a very simple rule: the present shapes its own rate of change. That rule is almost unavoidable in a world of interacting things. Cells make cells, heat flows from difference, money earns on money, ideas spread through people who already hold them. Wherever something feeds back on itself, e appears. Its bigness isn't a coincidence spread across many domains. It's the signature of self-reference in time, and self-reference is everywhere.
The honest limit: e doesn't tell you which things feed back on themselves, or when the feedback will break. Nothing in nature grows exponentially forever. So e gives you the ideal and the ruler, and reality is always a negotiation with it. Much of science, economics and even self-understanding lives in that gap between the pure curve and what actually happens.
We perceive ratios, not differences:
Oh, this is interesting. I always thought we saw in variation, the way seeing works, we are pickers of difference. We are so relative to a moment, and then what we carry or register is a difference from the moment to moment?
You're right, and it doesn't contradict what I said. The two ideas fit together, and how they fit is nice.
We are difference detectors, deeply. Vision depends on change: your eyes make constant tiny movements, and if an image is held perfectly still on the retina, it fades from view within seconds. Neurons fire at changes and go quiet under steady input. You stop hearing the fridge hum, stop feeling your clothes, stop smelling your own house. What reaches awareness is mostly "what just changed."
The question is how that difference gets measured. The answer, roughly, is relative to where you already are. This is Weber's law, from the 1830s: the smallest change you can notice is a fixed fraction of the baseline, not a fixed amount. Add one candle to a dark room and it's obvious. Add one candle to a room with a hundred and nobody notices. Add a kilo to a two-kilo bag and you feel it; add it to a forty-kilo pack and you don't. The difference is the same, but the ratio differs.
That's where the two ideas meet. A logarithm is the operation that turns ratios into differences: log(b) − log(a) = log(b/a). So if the brain takes something like the log of a signal, then simply noticing the difference between moments automatically becomes noticing the proportional change. You are a picker of difference, but you pick them on a logarithmic scale. "Twice as bright" feels like one step whether you start from dim or brilliant.
Your phrase "so relative to a moment" also catches the second part: the baseline keeps moving. Eyes adapt to dark, ears to noise, and we adapt to circumstances too. A raise, a new house, a loss all feel large, then the reference point shifts and they become ordinary. Psychologists call it the hedonic treadmill. It's the same process as sensory adaptation, applied to how we experience our lives. What we carry forward is less a record of levels than a record of changes against a constantly resetting "now."
One caveat: Weber's law is an approximation. It holds well in the middle ranges of most senses and breaks down at the extremes, and some senses fit slightly different curves. But the broad picture is solid. We register change, measured proportionally, against a moving present.
It's also the reverse of the exponential. Something growing steadily in proportion to itself — the curve we were looking at — would, to a logarithmic perceiver, feel like steady, even steps. Our perception flattens exponential growth into something linear, which partly explains why we're so bad at seeing it coming.
Like Thinking Fast and Slow is about biases, is there a book about perception - this, that we do not really take in the world as it is completely at a moment, but sort of traverse difference to difference...and what it does to our seeing, our being? As in sometimes knowing how we perceive things can help us know ourselves. It is a philosophic meditation.
There isn't a single book that does for perception what Kahneman did for judgement, but several come close from different sides. Together they cover exactly what you're describing. Here's how I'd group them.
The science, written reflectively
Anil Seth, Being You (2021). Probably the closest to a Thinking, Fast and Slow for perception. His central claim is that perception is a "controlled hallucination": the brain constantly predicts the world, and what reaches us is mostly the correction, the difference between expectation and input. He carries this all the way to the self, arguing that our sense of being someone is also a kind of prediction. Scientific but full of philosophical questions.
Andy Clark, The Experience Machine (2023). The same predictive-processing idea from a philosopher who helped develop it. Clearer on the "difference to difference" point: we mainly take in surprise, and expectation fills in the rest. He also discusses what this means for pain, placebo, anxiety, and how we might change ourselves.
Oliver Sacks, The River of Consciousness (2017). A short, posthumous essay collection. The title essay asks directly whether we perceive continuously or in discrete moments strung together, like frames of a film, drawing on patients whose perception broke down into snapshots. It's the most meditative of these, and Sacks writes beautifully.
The philosophy
Gregory Bateson, Steps to an Ecology of Mind and Mind and Nature. Bateson defined information as "a difference which makes a difference," which is almost your phrase. He argued that minds, organisms and ecosystems all run on registering differences, not things. He's an unusual thinker, part anthropologist, part systems theorist, and a wonderful companion for this line of thought.
William James, "The Stream of Thought," a chapter in The Principles of Psychology (1890). The origin of "stream of consciousness" and of the "specious present": the idea that the "now" we experience isn't an instant but a short, moving span that holds a bit of what just happened and a bit of what's coming. Still one of the best things written on how a moment feels from inside, and readable as a standalone chapter.
Henri Bergson, Creative Evolution. Bergson argued that ordinary thinking works like a cinematograph, cutting the flow of reality into still frames and then reconstructing motion from them, while lived time (his "duration") is continuous. This is the classic philosophical version of your question, and it's also closely tied to the exponential world of becoming we started with.
Maurice Merleau-Ponty, The World of Perception. Short radio lectures that introduce his much harder Phenomenology of Perception. His theme is that perception isn't a camera recording a scene but a body engaging with it, and that we learn what we are by attending to how we meet the world.
Where to start
For a philosophic meditation, I'd pair Seth's Being You with the James chapter: one gives the modern science of perceiving by difference, the other gives the texture of lived moments, written 130 years earlier and still accurate. Then Bateson, if you want to see "difference" become a whole philosophy of mind and nature.
