The Shape Isn't the Explanation
Compound interest and the way I forget most of what I read within a week both trace exponential curves. So does radioactive decay, viral spread before it saturates, and the cooling of a cup of coffee. None of these things share a cause. What they share is a shape, and the shape turns out to be a much lower bar than it sounds like.
An exponential function has exactly two knobs: a starting value and a rate. Turn those knobs and you can trace almost any curve that goes up and doesn't level off, or goes down and doesn't level off, without wiggling back on itself. Slow rise, fast rise, gentle decay, cliff-edge decay: all still "exponential," all still fitting the same family, all describable with the same two-parameter equation. That's not a narrow, specific prediction. That's most of the space of smooth monotonic curves you'd draw by hand if someone handed you graph paper and said "make it curve."
So when two phenomena both fit an exponential, the fit itself has told you almost nothing about mechanism. It's told you the data doesn't level off and doesn't reverse, which rules out a small set of alternatives (logistic curves that saturate, oscillating ones, curves with an inflection) and leaves a huge set of possible underlying processes still on the table. The compound-interest curve comes from a fixed percentage applied to a growing balance, a rule enforced by a bank's terms of service. The forgetting curve comes from something like retrieval interference or encoding strength decaying with time, nobody fully agrees which. Those are unrelated causal stories that happen to produce the same family of function. Matching shape is not converging evidence. It's the null result you'd expect from two smooth, non-oscillating, non-saturating processes measured on any timescale.
The actual content of an exponential is in the parameters, not the family. The rate constant is where the explaining happens. Why is the forgetting curve's decay rate what it is, rather than twice as fast or half as fast? Why does a second pass over the same material three days later flatten it, when a second pass an hour later barely moves it? Those questions have answers that involve consolidation, interference from similar material, how much structure the thing had when it went in. None of that is contained in "it's exponential." The exponential is just the wrapper the answer happens to arrive in once you've found it.
This matters because "it fits an exponential" gets used, casually and often, as though it were itself an explanation, a small triumphant reveal, when it's closer to a shrug. It's the mathematical equivalent of noting that two people are both mammals. True, occasionally useful for ruling things out, but doing none of the work of explaining why one of them is a dolphin and the other is a bat. You still have to go find the actual mechanism, and the shape of the curve won't hand it to you. It'll just confirm, after the fact, that whatever mechanism you eventually find had better not predict oscillation or a hard ceiling, because the data doesn't show either.
The failure mode isn't limited to curve-fitting. It's a specific case of a broader habit: treating a shared descriptive category as if it were a shared cause. Two species get grouped under one taxonomic label because they cluster on some measured trait, and the grouping starts getting read as kinship rather than convenience. A functional form is the same kind of convenience. It's a compression of the data, chosen because it's tractable and familiar, not because the world handed it to you as a receipt for what's underneath.
None of this means exponential fits are useless. A good fit rules out real alternatives and constrains what mechanism you should even be looking for. It's a filter, a useful one. It just isn't the answer, and mistaking the filter for the answer is how two unrelated curves end up getting treated like cousins.