Sunday, May 15, 2011

A short observation about limits

Some scientists need a better understanding of the concept of limit as in
\[
\lim_{\begin{array}{l}x \rightarrow 0 \\ y \rightarrow +\infty \end{array} } f(x) g(y)
\]
where $f(\cdot)$ and $g(\cdot)$ are increasing functions with $f(0)=0$.

This thought was motivated by Leslie Valiant's talk at the Lens 2011 Conference (video at the link). He was trying to determine the computational feasibility of evolution by random mutation and natural selection. In his case the question was simply whether the rates of mutation and the incidence of beneficial mutations could evolve the set of specific biochemistry cycles that control many functions of life.

The standard biologist's answer is that the rate of mutations is small and the probability is small, but when they accumulate over 4.5 billion years and integrate over all possible planets, they add to a big enough number. Clearly they don't understand that Small $\times$ Big is an undefined quantity, as the limit above can be anything. Their hand-waving argument is sloppy thinking.

You can't have that in science.

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Addendum: Just in case it's not obvious, my issue is not with the theory of evolution per se, but with the lack of good numerical models and complexity evolution models which would allow for rate of evolution calculations.