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Posts:
1,097
Registered:
12/4/12
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Re: distinguishability - in context, according to definitions
Posted:
Feb 16, 2013 4:21 PM
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On 2/16/2013 5:55 AM, Dan wrote:
> This comes as a consequence that all computable functions are
> continuous ,while equality is not.
> http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html
Nice post.
Ok. So, the trivial topology
d(x,x)=0
d(x,y)=1,-(x=y)
is always continuous.
So that puts you in a "nice" place to begin with. The
general notion of a mathematical function (objected to
by some) evolved from functions based on a "rule" to
functions based on arbitrary collections of ordered pairs
meeting certain well-definition requirements.
I have not found any direct text to confirm, but the book
I have that examines Wittgenstein's theories attributes
this definition of function to Dirichlet.
Naturally, you do not get to the theory of algorithms in the
constructive sense of Markov and Sanin until you stop
defining functions according to rules. Furthermore, as
more creative people started to consider what kinds of
rules could be used to formulate functions, odd kinds
of mathematics had to be given an account.
Your example of 1/3 is actually a rational number. And,
my post explicitly does what you do with that example. I
simply used the representation of state tables so that
I could contrast the situation with a real, irrational
state table and so that I could address the
lossless/lossy character of deciding between two
names, each of which could be representable.
But, I am thrilled to see that others are finally
looking at questions to which I have been directing
my attention for years.
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