May 11th, 2017, 10:46 AM  #11 
Senior Member Joined: Aug 2012 Posts: 1,414 Thanks: 342  As one of the Wiki pages on the subject mentions, Kolmogorov himself was unhappy about this situation. I've never said that you're not raising a valid philosophical point. You're just not raising a valid mathematical one. Personally I regard $\int_0^1 dx = 1$ as a great philosophical mystery. How can we add up infinitely many infinitesimally small things to get exactly $1$? And adding up the same number (by bijection) of infinitesimally small things can give us $2$ or $3$ or $\infty$ or any number that we like. How does that work? But any high school calculus student can knock down the symbology with no problem and without giving it a second thought. Most mathematicians don't give it a second thought. Because when they're doing math they're not doing philosophy. Perhaps your concerns are along philosophical lines and would be better addressed on a philosophy site. But of course the mystery of how dimensionless points give rise to dimension is a question that nobody knows the answer to. 

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cantor, distribution, set, ternary, uniform, vitali way 
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