| Dr.Matt wrote: Quote:Look. Indeed, you're confusing mathematics with science here. "2+2=4" is a theorem of arithmetic because the equality relationship is defined to be transitive, reflexive, and commutative, 2 is defined as 1+1, and 4 is defined as 1+1+1+1. You regularly use Z12, a ring in which "9+6=3" is a theorem (look at your left wrist for a reminder of how you use it). But interposing, e.g., Z12 with arithmetic to get "9+6 is sometimes 15 and sometimes 3" is a fallacy of equivocation--between the meanings of "+" and "=" in arithmetic and their meaning in Z12. Got it? "There's a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms." (Wiles) Well maybe that has nothing to do with modular arithmetic, but it's still cool. In the above example, I wonder if you can write the equals symbol as a triple bar congruence symbol to prevent the fallacy. michael |