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You don't blindly believe axioms, it is just that certain sets of axioms happen to allow us to make useful predictions about the world. While you could have phi
by fela 13y ago
You don't blindly believe axioms, it is just that certain sets of axioms happen to allow us to make useful predictions about the world. While you could have philosophical discussions about weather certain axioms are true (whatever that means), and what their exact relationship with the world is, you don't need to do this to see that axioms are useful to arrive at mathematical conclusions, which then can be used to make predictions about the world.
Why do we believe that if we take 1111 apples and then add 2222 more we will get 3333? Because we have seen from experience that a certain way of applying mathematics to the real world gives great results. The same for conclusions following known axioms, they seem to generally work well for describing our world.
- VLM 13y agoA good post although I suggest rephrasing "useful" to something like "experimentally falsifiable". With a pretty wide definition of "experimentally" and "world", especially for non-applied math. If you have no way to do a falsifiable test, and can't just substitute in a X and do abstract analysis of X instead of worrying about X, then its just a belief. I'd be careful categorizing stuff eternally as beliefs not axioms. Blindly only permitting geometry to be Euclidean meant some exciting non-euclidean results couldn't happen until it was "allowed" to be thought about somewhat recently (well, a century or two ago...). Also physicists have this amazing ability to turn pure math into applied math, at least over a long enough historical scale. And physicists have a pretty good ability at coming up with crazy experimental methods to test, look at the last 75 yrs or so of quantum physics... so let me get this straight, you do what with two geiger counters and a truly giant magnet and some radioactive atoms or what? Or shine a light beam or ion beam thru that weird magnet?