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date: ‘2025-06-11’

summary: ‘Timing is Everything’

draft: true

tags: []

title: ‘Exponential Growth is an Illusion’

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Instead of saying “x grows exponentially”, it always means: “x appears to grow exponentially as long as all possible causal effects don’t change as well (a froth of uncertainty)”

The Exponential Illusion

Exponential/geometric growth

$y = ae^{bt}$ (continuous)

$ y = ar^n$ (discrete) “geometric”

examples of when it happened in reality

  • computer operations in an algorithm

    • still subject to real constraints which will prevent the ideal form from being instantiated

  • Malthus thought population grows geometrically

    • x? y?
    • but we can hypothesize several things that will change the exponential: food, space, time, oxygen, temperature, lightning strike, neutrino collision

  • Moore’s Law

    • transistor count per chip size
    • transistor density per time

  • Kurzweil bullshit / any random thought in some pundit’s head about exponential

in each case, the real result is bounded by something. An algorithm terminates in its end condition or runs out of memory or energy, or time.

Power Laws

$y = ax^b$

examples

  • frequency vs magnitude of quakes
    • earthquake magnitude per frequency Gutenberg-Richter Law
    • animals’ metabolic rate per body mass
      • $rate_{metabolic} \approx mass^{3/4}$

Linear/Arithmetic Growth

$ y = a + bn$

  • Malthus thought “agriculture” grows arithmetically
    • what is x and y though? is it the same as for population?

Logistic Growth

Exponential plus logarithmic?

$y = K / (1+ae^{-bt})$

the more parsimonious and generic curve (closer to reality)

All of these are approximations

All models are wrong, some are useful.

It can be useful is we have some idea of how much “room to run” any given trend has. Moore thought a certain trend (exponential) would continue for 20 years. we saw in hindsight that slightly different trends (logistic)

This gets to what it means to “believe” a trend. Belief only matters insofar as you DO something differently because of the belief. If you change your mind and do something differently, that is functionally equivalent to changing your belief.