It’s been a great week on the blog with thoughtful and thought-provoking comments cropping up everywhere. Several threads have touched on the question phrased most succinctly by Al. V. on the Unreasonable Effectiveness of Physics thread:
Are the laws of mathematics inherent in our universe, and therefore really a product of physics (and not the other way around), or are they supra-universal?
This question, of course, plays a starring role in The Big Questions , where I’ve explained why I believe that the supra-universality of mathematics (thanks for that word, Al!) gives the most coherent explanation of why anything exists at all.
The same issue arose—with many comments well worth rereading—on the What Are You Surest Of? thread. There, Bill T. raises a provocative question: If (as I’ve suggested in The Big Questions), the physical world is in some sense “made of mathematics”, why can’t we take this one step further and speculate that mathematics is “made of logic”? The answer is that Godel’s theorems make that a very difficult step to take. Starting with the standard axioms of arithmetic, and armed with the full power of logic, there remain true statements about arithmetic that cannot be proven. This means that arithmetic must be more than just logic.
(Logicians in the audience will want to quibble about what constitutes the “full power of logic”; I will duck that question to avoid a long technical digression right now.)
I offered the first in a planned series of posts on attempting to see Darwin through 19th century eyes; Snorri Godhi checked my math carefully and made a valuable correction. And we got into a debate about whether cellphones really would have resolved the plot tension in a substantial fraction of 20th century movies.
Thanks for the fun and the enlightenment. I’ll be back on Monday with more.