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Call me uneducated, but isn't the rock solid foundation for math the real world? To the extent that math corresponds to the real world, it is correct. The "ha

Call me uneducated, but isn't the rock solid foundation for math the real world? To the extent that math corresponds to the real world, it is correct. The "halting problem" sounds like an attempt to predict the future with math, which seems like a futile effort. Maybe the conclusion from the "randomness" in trying to predict the future is that such predictions are simply not possible and a waste of effort?

eru 18y ago
Math has absolutely nothing do with the real world. ;) At least it's not constructed that way.
emmett 18y ago
You are Not Even Wrong, in this instance. Math is not a simulation of the real world; to take a simple example, both Euclidean and hyperbolic geometry are self-consistent and useful, and yet clearly different. So "the rock solid foundation" of math is not the real world, but rather sets of self-consistent axioms. The halting problem is not an attempt to predict the future with math; it's a description of a property of algorithms which is inherently non-computable. You cannot, in the general case, prove whether an algorithm halts or not. So as to your the question of whether the "randomness" comes from trying to predict the future...the answer is not yes, nor is it no. The question is not sensible. The article was very poorly written though, so I'm not surprised it causes confusion. If you are interested in actually learning about Omega, I recommend http://www.scottaaronson.com/writings/bignumbers.html http://www.scottaaronson.com/writings/bignumbers.html which is an amazing introduction to all these ideas.
- cousin_it 18y ago
  So "the rock solid foundation" of math is not the real world, but rather sets of self-consistent axioms. You're mostly right, but there's a fun twist. The self-consistency of e.g. Peano arithmetic (natural numbers) ultimately rests on our intuitions about the real world - it can't be proven starting from nothingness.

eru 18y ago

Math has absolutely nothing do with the real world. ;) At least it's not constructed that way.

emmett 18y ago

You are Not Even Wrong, in this instance. Math is not a simulation of the real world; to take a simple example, both Euclidean and hyperbolic geometry are self-consistent and useful, and yet clearly different. So "the rock solid foundation" of math is not the real world, but rather sets of self-consistent axioms. The halting problem is not an attempt to predict the future with math; it's a description of a property of algorithms which is inherently non-computable. You cannot, in the general case, prove whether an algorithm halts or not. So as to your the question of whether the "randomness" comes from trying to predict the future...the answer is not yes, nor is it no. The question is not sensible. The article was very poorly written though, so I'm not surprised it causes confusion. If you are interested in actually learning about Omega, I recommend http://www.scottaaronson.com/writings/bignumbers.html http://www.scottaaronson.com/writings/bignumbers.html which is an amazing introduction to all these ideas.

cousin_it 18y ago
So "the rock solid foundation" of math is not the real world, but rather sets of self-consistent axioms. You're mostly right, but there's a fun twist. The self-consistency of e.g. Peano arithmetic (natural numbers) ultimately rests on our intuitions about the real world - it can't be proven starting from nothingness.