The Corner

More Catching Up

Some more feedback on last week’s March Diary:

Reader O [On the stupefying boredom of that DMV Defensive Driving course I’ve been dragging myself through]: Dear Derb, I feel your pain…at least the pain that comes from waiting for your computer to do something. You mentioned that reading to fill those empty spaces doesn’t work for you. Perhaps you are trying the wrong books. I have an 8-volume set of Who’s Who in British History in British history, and as of this writing I have finished 6 of them whilst waiting for my computer to warm up, load web pages, etc.

I doubt it would have worked for me. The course is a grand total of six hours. I got into the neurotic obsession that if I didn’t keep hitting the NEXT button the very instant the clock ticked down to zero, I’d be prolonging the course to seven hours. So I couldn’t concentrate on reading anything for fear of getting absorbed in it, forgetting to watch the clock, and stretching out the agony. I’m glad to report I have now finished the thing. I am a fully certified Defensive Driver, so keep the hell out of my way on the turnpike if you don’t want a bit fat scrape upside your paintwork.

On the intersection of math and religion:

Reader P: If you haven’t read Arthur C. Clarke’s “The Nine Billion Names of God,” you should. It’s the final word in onomatodoxy. Religion and mathematics overlap just enough to create – I dunno, a singularity?

Are you kidding? I grew up on that stuff. And not only was I a fan of ACC, he was a fan of mine.

Reader Q: The book Naming Infinity is worth buying for its beautiful illustrations as well as for its compelling story. The authors do not try as hard as I would have liked to make precise the technicalities, but overall their treatment of the math is sound.

It seems that theological reflection can indeed assist in the development of higher set theory, which translates down into the very concrete theory of first-order arithmetic via consistency statements.

My own view on this is somewhat radical… I believe there to be a sharp philosophical difference between statements about arithmetic and higher-order statements involving infinite sets. Viewing Kronecker’s dictum “God made the integers, and the rest is the work of Man” from a different angle, I regard the integers as the domain of Man in the epistemological sense. Real knowledge is possible here. On the other hand, although I am sure that God knows whether the Continuum Hypothesis is correct, I do not think we can ever know the truth value of that statement (though I do not deny that it has a truth value, a question with theological implications).

The empirical observation underlying this is that there are all sorts of plausible but incompatible higher axioms for mathematics, but none of the axioms that have been seriously proposed and are believed consistent have ever been found to have incompatible arithmetical consequences. You can make a good case for real-valued measurable cardinals; fine, this contradicts he Continuum Hypothesis (not to be taken seriously) but has the arithmetical consequence that ZFC is consistent (a serious advance). You can make a good case for the Axiom of Constructibility (which imples GCH), but that doesn’t imply any new arithmetical facts.

Another way of thinking about this: an independent mathematical civilization (e.g. on some distant planet) may well disagree with us about basic axioms for infinite sets (they may reject the Axiom of Choice, accept Inaccessible cardinals, deny CH, accept GCH, reject the Power Set Axiom, accept the Axiom of Determinacy, etc.) but I think it practically inconceivable that we could permanently disagree with such a civilization about an arithmetical statement. We may think we have proved an arithmetical statement that they don’t accept as proved, but we will never think we have proved one that they think they have dis-proved, assuming we’ve checked each others’ proofs for ordinary logical mistakes.

You have me reaching for my Kneebone, Sir.

Finally, there’s always one pedant in the email bag:

Reader R: Is havoc “wreaked,” do you think? Or is it “wrought”?

Webster’s Third says “wreaked.” So does the OED; though it also gives an archaic form “wroke” which I am going to use in future, in a spirit of pure reactionary obfuscation.

 

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