Re: Circular reasoning examples

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 10/07/04 

Date: Thu, 07 Oct 2004 12:34:27 -0600

In article <6caa1e8b.0410070235.7aeb9df1@posting.google.com>,
 gtsavdar@auth.gr (George) wrote:

> federico1983@gmail.com (F. Olmedo) wrote in message
> news:<6982a677.0410061841.71d75531@posting.google.com>...
> > I've been looking for examples of wrong mathematical proofs due to a
> > circular reasoning and I haven't found much. In the example I found,
> > the flawn was really obvious. Does anybody know of a wrong
> > mathematical proof (NOT a logic fallacy) where the circular reasoning
> > is rather hidden?
>
> I don't know any really good example, but with your post i want to ask
> a similar question (VERY IMPORTANT IN MY OPINION) too.
>
>
> The question is simple:
> Suppose we have to prove something. And we use some theorems or axioms
> for this.
>
> First Question: Do we have to show in our proof, the proof of the
> theorems too? ***(The reason for this is explained at the example-1
> below and it's one circular logic problem).

If there are reference works available containing proofs of those
theorems you use, usually a reference to some work containing such a
proof is sufficient.
>
> Second Question(and the most important): Suppose we are constructing a
> proof and we suppose that the answer to the first question is YES.
> So to prove our problem we have to
> prove some theorems one by one, so to prove each theorem of these we
> should prove some other theorems or axioms-definitions(for ex. 1=1)
> and since we accept as true all axioms-definitions, we just have to
> prove these new theorems, so again we have to prove another set of
> theorems or axioms-definitions,and since we accept as true.....we just
> have to prove these new theorems.........., so we end that we have to
> prove a set of axioms-definitions and since we accept
> axioms-definitions as true, we solved the problem.
> The question is what is the number of axioms-definitions exist?
> I know that this can't be answered but it should be. It should be a
> page, a book that would have all definitions human beings have made
> (can i say them axioms?). How the mathematical community works without
> having such a system of definitions?
> For example:
> 1)0=0
> 2)In an equation a=b that is valid we can go to the a+1=b+1 and still
> be valid or the opposite (a,b natural numbers).
> 3)E=>R means that if E is true then R is true also.
> 4)a^(b+c) => a^b·a^c (a,b,c reals and a>0)
> 4)i^2=-1 (i the imaginary unit)
> ..................
> ..................
>
> ------------------------------------------------------------------
> ***Example-1:
> Once i wanted to prove that Lim Sin(x)/x = 1.
> x->0
>
> And i tried the De'Hospital's rule that:
> Lim Sin(x)/x = Lim Sin'(x)/x' = Lim Cos(x)/1 = 1
> x->0 x->0 x->0
>
> But one "clever" teacher corrected me that this is not true as in our
> book
> the proof for Sin'(x) = Cos(x) included in the solution that
> Lim Sin(x)/x = 1.
> x->0
>
> I disagreed because i said that if there is another proof that
> Sin'(x) = Cos(x) which will NOT include in the solution that
> Lim Sin(x)/x = 1
> x->0
>
> then my solution would be right. So i said since we don't prove the
> theorems on our proofs, we should not care for things like that. And
> if it's not the case, then only when we prove EVERYTHING until we come
> to an axiom(definition), our proof MUST BE VALID.
> ------------------------------------------------------------------------
> A proof that Sin'(x) = Cos(x).
>
> For every x at R and h<>0 it is true that:
>
> (Sin(x+h) - Sin(x)) 2·Sin((x+h-x)/2)·Cos((x+h+x)/2)
> ------------------- = ------------------------------ =
> h h
>
> Sin(h/2)
> -------- · Cos(x+ h/2).
> h/2
>
>
> Sin(h/2)
> And since Lim -------- = 1 and
> h->0 h/2
>
>
>
> Lim Cos(x+h/2) = Cos(x) then ..................
> h->0
> ------------------------------------------------------------------------

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