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If a mathematical statement did not have a corresponding experimental test would you dismiss it? What about those?
by abrezas 10y ago
If a mathematical statement did not have a corresponding experimental test would you dismiss it? What about those?
- jules 10y agoAll mathematical statements in constructive logic have an experimental test (actually since nonconstructive logic can be embedded in constructive logic via a double negation translation, they do too). Of course this isn't always the most interesting test, since usually you are modeling some higher level concept inside the logic, and you would have a more high level test. For example for the calculation of pi you would measure the ratio between the diameter and perimeter of a circle.
- hyperpape 10y agoIt sounds like we just stopped knowing a lot of the digits of pi if what you're saying is accurate: http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals...
- jules 10y agoPi isn't defined by how we measure it. It's defined mathematically. Predicting how a measurement of the ratio between the perimeter and the diameter of a circle comes out is just one of the predictions we can make having calculated pi inside mathematics.
- abrezas 10y agoBut if at some point further decimals cannot be experimentally proven to be right, at best you can say "pi is such and such for such precision" and any definition that gives the same number up to that precision must be accordingly accepted. Otherwise you need to demonstrate an experiment that uses further precision. But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.
- hyperpape 10y ago"why mathematics don't have to do with experience" is too strong. Nothing you just said implies that math isn't intimately connected to empirical predictions in some way. What it shows is that contrary to Jules' earlier statements, there's no statement by statement correspondence--for any given mathematical statement, you can't find an interesting empirical prediction you associate with it.
- abrezas 10y agoI don't believe they are entirely independent, but there's nothing you can look outside your window that will change any mathematical theorem.
- jules 10y agoI agree with this statement (depending on how you interpret "change": if the world was different we may have been lead to prove different theorems), and nothing I've said previously contradicts it.
- jules 10y agoI never claimed that, in fact I said precisely the opposite: > Of course this isn't always the most interesting test
- hyperpape 10y agoSubstitute non-trivial for interesting. For instance, a calculation of pi to 50 digits will map to the same test as a calculation of pi to 51 digits. But the claims have different content so they should map to different tests!
- jules 10y agoSo? (By the way, they do map to different tests if you view it as a statement in constructive logic, rather than geometry.)