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While I can appreciate the desire to (and need for) stretching the frontiers of our understanding of all things, numbers included, I do think it's wrong to mess with the meaning and value of symbols like + and =, the foundation of mathematics. And infinity is not a number -- that's why it can only be described as ". . . "
1 + 2 + 3 + . . . doesn't "equal" -1/12. It has no value, other than to say "infinite," which is another way of saying we can't represent it.
Just as one can (and should) draw purple unicorns, or postulate a biology based on something other than carbon, or create any number of interesting non-Euclidean geometries, one can devise all sorts of alternative "maths" where there are different rules (where "infinity" is a number, for example), and these rules will lead to all sorts of fun conclusions, like the sum of all natural numbers equaling -1/12. These endeavors might even turn out to be useful, in some cases . . . complex numbers, for example.
But it's dishonest, unintentionally perhaps, to use familiar symbols in an attempt to present fanciful derivations as mathematical truth. And yes, there is a non-relativistic mathematical truth. And part of that truth is that you can't get a negative fraction by adding a series of positive integers.