The word “Evolution nowadays suggests “evolution of organisms by natural selection as per Darwin & modern population genetics”.

But what about other kinds of evolution? Any unary endomorphism from a system onto itself, applied over and over to generate “time”, could be considered a kind of “evolution”.

  • Crystals and quasicrystals evolve naturally.
  • Caves and stalagmites evolve naturally.
  • History evolves artificially … although no-one knows the mapping.
  • Art evolves artificially … again, no-one knows the mapping. (but we know there is cross-pollination. Could we call it “Art sex”?)
  • Proto-biological chemical compounds, like basic amino acids, “evolved” by a method similar to natural selection.
  • Businesses (and entrepreneurs) that grow through trial-and-error, evolve their ideas and their business processes by artificial selection.
  • Romantic relationships evolve. Any human relationship evolves. Sometimes “a” relationship can evolve with a group of people as-a-unit. But here I’ve found the selection to be more exogenous than endogenous. Friendships seem to pick up exactly where they left off γᵢ(Tᵢ) rather than be pitched to the dustbin of 0. Even romantic relationships seem to hang mostly where they were, even after a breakup. Perhaps the breakup zeroes the romantic part ⌊γᵢ⌋ᵣ, but everything else—sexual chemistry, personality dynamics, humour dynamics, and history—remains stubbornly unbudged by a breakup per se.
  • And like I said, any endomorphism, repeatedly applied to a system, could count as “evolution”. (An endomorphism draws both the input and the output from the same domain, i.e. ƒ: X→X.) If there is a throw-away criterion (mapping ↦ 0), we could call that “selection”. Any fixed point of the mapping ƒ(p)↦p is an endpoint of evolution.

In this video, John Baez talks about how the inherent interestingness of the number 5 has made itself apparent to us through several processes:

  • artists (mosque designers) discovered it. God speaks to us in the language of mathematics, remember?
  • crystals and quasicrystals discovered 5 by evolution — but not the biological kind
  • soot and space dust found , also by natural non-bio evolution
  • BTW, unrelated but some Scots (perhaps Picts) carved some Platonic solids out of stone centuries before Plato … so perhaps they should be called Scottish solids.
  • the Pariacoto virus found by biological evolution
  • Roger Penrose (a mathematical physicist) discovered & described 5-way symmetry in modern mathematical (group theoretic) terms

In each case, logic is the canvas. Art — nature — mathematicians are the painters.


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