Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Re: distinguishability - in context, according to definitions
Posted:
Feb 17, 2013 4:51 AM
|
|
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 .
|
|
|
|
