Robert: What happens when we worry about what lies behind the axioms of a system is illustrated by Lewis Carroll's 'What the Tortoise said to Achilles' http://www.mathacademy.com/pr/prime/articles/carroll/index.asp
Thanks for this. It led me to consider The Muenchhausen-Trilemma. The article Robert cited ends with this:
"...by reducing logic to uninterpreted symbols, all semantic content is removed from the conclusions of formal logic. In other words, what we would ordinarily consider meaning is lost. How to restore meaning to systems of inference while still avoiding difficulties such as Carroll’s Paradox remains a thorny question for philosophers of mathematics."
Whew! It's a good thing knowledge isn't confined to things we can prove.
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