On Feb 17, 10:31 am, fom <fomJ...@nyms.net> wrote: > On 2/17/2013 12:49 AM, Dan wrote: > > <snip> > > > > > The problem of 1 = 0.(x) appears for any possible base of > > numeration . > > If you're bothered by the representation being 'lossy' , you can > > always try continued fractions for the numbers in the interval [0,1] : > > Each real number is represented by a (possibly infinite) sequence of > > strictly positive integers : > > You represent r by [a1,a2,a3 ..... an ...] meaning that > > r = 0 + 1/ (a1 + 1 / (a2 + 1/ (a3 + .... ))) > > I'm pretty sure you can build up the whole of analysis this way , > > though nobody's bothered to do it, so it must be tedious. > > That being said, I was never really bothered by the whole 0.(9) = 1 > > business , it's just a quirk in notation . > > I am aware of continued fractions. And, there is > nothing about the particular statement of equality > that bothers me. > > You seem to be focused on the wrong part of the > post. That is fine. I know that most mathematicians > are not accustomed to the kind of logic that comes > from Frege, Russell, Carnap, Lesniewski, Wittgenstein, > Tarski and others. But, in fact, what do most mathematicians > intend when they say that mathematics is "logical" but > then ignore the presumptions and opinions upon which that > is based (in the modern sense)? > > That is a rhetorical question. In your arena, there is an > entirely different set of people such as Turing, Kolgomorov, > Markov, Church, Curry, Kleene, etc. > > > What seems far more troublesome is the representation of finite > > fields , you always have to 'choose' one of many irreducible > > polynomials if you want to work with them . > > I have recently run into that problem. I have been fascinated > by a particular presentation of the elements of the Galois field > over 2^4 generated by > > p(x)=x^4+x+1 > > I doubt that it is unique. My interest, however, is not > computational.
"But in fact all the propositions of logic say the same thing, to wit nothing" - Wittgenstein I shouldn't even dignify Wittgenstein with a response to his "philosophy" .The names 'Derrida' ,'Leibniz' and 'Godel' offer sufficient refutation . Any "wittgensteinian" axiomatic theory can have "unprovable" propositions. *cough* continuum-hypothesis *cough* . No "wittgensteinian" theory can have only "true but unprovable" propositions . (peano arithmetic, godel numbering) A theory with only "true but unprovable" statements already points to an ontological substrate invisible to Wittgenstein. It doesn't say nothing , it says everything .
To quote a rather obscure book: 'Wittgenstein said, ?An equation merely marks the point of view from which I consider the two expressions; it marks their equivalence in meaning.? Is this statement valid? It?s certainly true that 3 + 5 = 4 + 4, but are both sides equal in meaning? In fact, this is precisely what they are not equal in. They are equal in objective outcome/ result but not in their organization and meaning.'
So perhaps you're "looking from the wrong direction" . But this is mostly off topic . Perhaps I'll try again to comprehend the post later . Good luck .