On Thursday, April 11, 2013 12:57:02 AM UTC+5:30, Dan wrote: > 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?
Will your program ever output 'all' the numbers? We can verify that on input 1 it gives 2, but maybe on input 10^(10^100) it runs out of memory(tape); for we could never verify that the memory was actually infinite. So, we abstract, and say , On input 1 ,it gives 2 , on input 3, it gives 4 And So On . We cannot express this in propositional logic, because that needs an infinite conjunction, so we bring in FOL and the universal statement 'For every natural number n, there exists a successor ' and claim that this finite string conveys all the infinite information that we are trying to grasp or convey.
> > > 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 .
A finite mind :-) > > > 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? > > > The problem with the finite , > > is that there's always something bigger . Finitude is like a prison, > > and the mind needs to escape .
The questions 'Is this infinite ? ' and ' Is there something always bigger?' both need an unending verification.Can I always put the next chalk mark? How would I know? The quest for 'escaping the prison of finitude' can only be unending. > > > That is what needs to be reconciled with reality. > > > > Who says reality isn't infinite? Go out in space and travel in any > > direction . You're never going to suddenly hit a wall that says : > > "the Universe is finite . You can go no further" . > > http://4.bp.blogspot.com/-5Olo8-EgrZI/TZBqclcfPqI/AAAAAAAAC5c/920EyWecwiU/s1600/background_brick_wall.jpg > > because the universe has no such walls. That is Euclid :'every line segment can be extended in either direction' Can it be ? Who can verify? Is there always a next cell on the tape? Is the tape infinite? Can these questions ever be answered? > > > Within infinity is harmony.
But can something finite really represent the infinite?