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 | | From: | Don H | | Subject: | Re: proof by induction | | Date: | Wed, 12 Jan 2005 17:27:03 GMT |
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 | Does maths take "values" into account, or is it confined to enumeration?
Arithmetic, for example, is adding and subtracting (multiplication and
division are special cases).
Algebra brings in general "numbers", denoted by letters (a,b,c for
constants; x,y,z for variables).
Set theory is where apples & pears *should* appear, but do they? Or, are
we still dealing with abstract enumeration?
"Proof by induction" is an enumeration, yet again, and values haven't
really been considered. This is where mathematical induction has the same
flaw as logical induction - extension from the known to the unknown can be
dangerous, as it involves an assumption of uniformity which may not be
warranted.
Hence, +1, while numerically alright, may not be so in an actual value
example. "All cats have tails - until we come to the Manx cat."
==============================
"Tan Thuan Seah" wrote in message
news:418cf289$1@clarion.carno.net.au...
> Not quite sure if this should be the way. I was taught usually after we
> proved a base case (n=1), we assume the n case is true, and construct a
> proof that n+1 is true using the base case and n case. From there we can
say
> it's true for all n.
>
> Thuan Seah
>
>
> "Gregory Toomey" wrote in message
> news:2undcmF2bvbj5U1@uni-berlin.de...
> > Bill wrote:
> >
> >>
> >> "Peter Webb" wrote in message
> >> news:4186178e$0$31625$afc38c87@news.optusnet.com.au...
> >>>
> >>> "Bill" wrote in message
> >>> news:ZWmhd.6520$K7.885@news-server.bigpond.net.au...
> >>>> how the hell do you do it?
> >>>>
> >>>> regards,
> >>>>
> >>>> bill
> >>>>
> >>>
> >>> You show its true for 1.
> >>>
> >>> Then you show that if it is true for n, it is true for n+1.
> >>>
> >>> Therefore it is true for all numbers.
> >>>
> >>> Proof:
> >>>
> >>> You have shown it is true for 1.
> >>> You have shown that if it is true for n, then it is true for n+1.
> >>> So if its true for 1 (which it is), it must be true for 2.
> >>> If its true for 2, it must be true for 3.
> >>> If its true for 3, it must be true for 4.
> >>> And so on ...
> >>
> >> how do you show it's true for n?
> >
> > You dont. You show that IF it works for n, it ALSO works for n+1.
> >
> > In other words you end up with two axioms:
> >
> > True_for(1).
> > True_for(n) True_for(n+1).
> >
> >
> > See http://mathworld.wolfram.com/PrincipleofMathematicalInduction.html
> >
> >
> > gtoomey
>
>
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