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> Now a funny think I said was "pick a random real number". Since the computable numbers are also a measure 0 subset of the real numbers, it's literally impossi

> Now a funny think I said was "pick a random real number". Since the computable numbers are also a measure 0 subset of the real numbers, it's literally impossible to randomly pick a real number with a computer... I dispute that. See this: http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/ http://math.andrej.com/2008/02/06/representations-of-uncompu... [edit] The downvoters can't argue with facts. Let me hear them. I am not deleting this comment.

gpm 8y ago
That blog post was a good read. I also wanted to say that I certainly believe that you've done real math and got interesting results from the sort of model we were discussing above. It does seem like, as you say, a rich and natural theory. That said I wanted to make another counter argument here. Particularly based on the meaning of the phrase "pick a random real number". To pick a number at random is a necessarily randomized process, however it should give a classical output. After picking the number I should be able to make two copies of it, put them on ships flying away from eachother at the speed of light, and both ships should be able to perform the same computation (without entangling particle pairs beforehand). Moreover any reasonable representation of a real number should admit operations like "what is its n^th digit". Under these two constraints, you would have to flip an infinite number of coins, and it would take infinite time and space to pick a real number. Or in other words, you can't really.
- man-and-laptop 8y ago
  The way your spaceship analogy works in TTE, is that the random real number would be some kind of radio source that gets listened to by the two ships. Also, the way TTE works, there are diminishing returns from listening to the radio source for longer and longer time - this is because all functions computable in TTE are continuous, and therefore have diminishing returns from knowing their input to greater and greater accuracy (this is exactly the difference between continuous functions and discontinuous functions). Therefore you only have to listen to the radio source for some time before saying "Yep, I'm happy, I know the output of my function to within epsilon". Even the value of Pi in physics only needs to be known to a small number of decimal places - which suggests that all functions that are "physically computable" are continuous, further suggesting how natural TTE is. Again, my comments are getting downvoted by ignoramuses. This is not the first time.

gpm 8y ago

That blog post was a good read. I also wanted to say that I certainly believe that you've done real math and got interesting results from the sort of model we were discussing above. It does seem like, as you say, a rich and natural theory. That said I wanted to make another counter argument here. Particularly based on the meaning of the phrase "pick a random real number". To pick a number at random is a necessarily randomized process, however it should give a classical output. After picking the number I should be able to make two copies of it, put them on ships flying away from eachother at the speed of light, and both ships should be able to perform the same computation (without entangling particle pairs beforehand). Moreover any reasonable representation of a real number should admit operations like "what is its n^th digit". Under these two constraints, you would have to flip an infinite number of coins, and it would take infinite time and space to pick a real number. Or in other words, you can't really.

man-and-laptop 8y ago
The way your spaceship analogy works in TTE, is that the random real number would be some kind of radio source that gets listened to by the two ships. Also, the way TTE works, there are diminishing returns from listening to the radio source for longer and longer time - this is because all functions computable in TTE are continuous, and therefore have diminishing returns from knowing their input to greater and greater accuracy (this is exactly the difference between continuous functions and discontinuous functions). Therefore you only have to listen to the radio source for some time before saying "Yep, I'm happy, I know the output of my function to within epsilon". Even the value of Pi in physics only needs to be known to a small number of decimal places - which suggests that all functions that are "physically computable" are continuous, further suggesting how natural TTE is. Again, my comments are getting downvoted by ignoramuses. This is not the first time.