On Apr 10, 6:03 pm, apoorv <skj...@gmail.com> wrote: > - show quoted text - > "If I make a program that gives for each number n as input the output 1- > (n%2) , I can be reasonably sure it's the desired sequence . > The information is finite. You've got it backwards . We can use a > finite amount of information to generate an infinite string . > You could say we're never interested in the structure of the infinite > string per se (in the sense that we never directly observe the > infinite string in its entirety ), rather , in the structure and > morphology of the different 'shapes of information' that describe the > string . > > Anyway, there are different 'shades' of mathematics along the line > between 'finitism' and 'platonic realism'. > What I said is only not realistic for some extreme forms of finitism . > It works from 'effective computability' upwards. " > > The information is finite if we have a program that outputs the given > sequence. However, for the Universal Sentence > 'Every natural number has a successor' the information content is > clearly infinite.For we can have axiom systems where,for example > 1 has no successor or 2 has no successor or 3 has no successor etc. > It is this sentence ,along with the analogous sentences of Geometery > 'Every line segment can be subdivided' > And 'Every segment can be extended indefinitely ' > That bring in the infinite into Maths and logic . > That is what needs to be reconciled with reality. > Consider the Turing Machine with the infinite tape. > The very assumption of 'infinite tape' presupposes > Infinite information, because verifying that the tape is > Infinite would need infinite number of steps. > Apoorv
You have a very bleak view of what reality is . I can make a program that , for every number n I put into it, outputs (n+1), its successor. How is that any different, in essence, from my previous program?
Why do you need to verify the tape?After all, when imagining the Turing machine, didn't you make the tape with your own mind? You have a faulty mind if you need to verify the tape .
Mathematics has always had the infinite, in some form or another . I'm sensing it won't do any good to attempt to justify the infinite directly , so , let's examine the opposite side of the infinite , namely :
The FINITE : Let's try to ban the infinite from mathematics . That means, we're going to have to pick some finite number , and ban everything that comes after that . The question is , what number do we pick?
Is 2 large enough ? 2 + 2 = 4 . Guess not . Is 400 large enough? 21 * 21 = 441 . Even 5th graders should know that . So guess not . Is 1000000 large enough? Come on, the sun's been alive for a larger amount of years than that .
This 10,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000. is a googol . It has 100 zeroes . You know who thought of this number? A 9-year old . " Mathematics and the Imagination."
How about 10^(10^100) ? This has "1 googol zeroes" . Surly this is large enough ... right? Skewes' number is an upper bound for the smallest integer x such that pi(x) > li(x) , where pi(x) is the prime counting function, and li(x) is the logarithmic integral . It's far greater that 10^(10^100) .
I could go on, but I've made my point . The problem with the finite , is that there's always something bigger . Finitude is like a prison, and the mind needs to escape .
> That is what needs to be reconciled with reality.