@augustusmax (a cynical take is that this extra layer of precision is specifically constructed to exclude outsiders from asking these questions. This isn't 100% imo, but it is *suspicious* how it conveniently muddles the discussion between those trained in math and those trained in critique)
@augustusmax in math, "2+2=4 is true" in any cultural context because we think of 2, 4, +, =, and "true" as having certain very precise definitions. We can now be critical at this level: why are these the ones that are valued? But our field usually considers these questions out of our scope.
@smtilson FWIW: I suspect this "all we are supposed to do" thing, (re: the weird feeling in your body) is more like *analytical* thinking. Which is quite useful, obviously! It's just different :)
@smtilson and often with much attention paid to the meta-analysis. We're not really trained to do this. And the things we are trained to do, don't transfer well to this mode of thinkingโ imprecise and conflicting definitions, dualism, and comparatively wide base of relevant knowledge.
@anton_hilado Buzzkill answer to this joke(?) alert, you were warned ๐
The latter has simpler rigorous explanations, and thus does not effectively serve the same cultural purpose as the formerโ namely, knowledge hoarding and smug gatekeeping (which, sadly, remains valuable in our community).
These are really good questions about what math *is* in a very deeply probing way. Unfortunately the haters are piling in... I have transcribed the questions and typed up my instant answers here. https://t.co/KdKX7enCMh
@Rodorongo This is not as nice of an object as R, certainly. But this is "because" I insisted 0.99...9 in Q. If we instead think of decimals as elements of ZxZ^infinity, e.g. (0; 9,9,...,9,0,0,0,...) then we can do honest-to-goodness formal power series, a topologically complete local ring.
@Rodorongo (The ? in tweet 1 is b/c I checked the triangle inequality only hastily.) This turns the set of Q-sequences (or Cauchy Q-sequences, if you prefer) into a metric space in which 0.99... and 1.00... both have meaning but are different objects.