this post was submitted on 09 May 2024
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The trick is to use "exponentially greater" when referring to nebulous or generally immeasurable concepts. Since they're impossible to quantify, it allows a term like that to imply a grand difference without really having any real details.
The issue is that something growing exponentially means that it is best approximated by something like f(x) = c_1*e^(c_2*(x-x_0))^+c_3, or, where appropriate, by something from the class O(e^x^) in the relevant topological base.
With just two points of comparison, you can claim any sort of growth. You can fit a polynomial growth there, just as you can fit an exponential one, just as you can fit factorial growth there. Saying that there is exponential growth when all we have are just two points is nonsensical if we go by what the relevant expressions mean in math.
I think you're missing the increased utility that describing something as exponential has though. Compounding and increasing with greater intensity isn't really an easy concept to explain. Sure you can't "prove"that something is exponential with just two points of data, but the demonstration of those concepts within one word is highly useful and effective as an intensifier.
Although, the more I work through the term the more I can see why it could be frustrating.
I guess this is the benefit of not studying math lol.