Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 03/18/05 

Date: Fri, 18 Mar 2005 11:53:18 -0500

Tony Orlow (aeo6) wrote:
> Wolf Kirchmeir said:
>
>>Tony Orlow (aeo6) wrote:
>>[..]
>>
>>>Sphere: The 3D geometrical figure that minimizes surface area per volume. Do
>>>you like that definition? Circloids can be defined as the family of shapes that
>>>minimize boundary per contained space for the number of dimensions they
>>>contain. That's nice and general.
>>
>>Those are theorems.
>
> Theorems as opposed to definitions? Is there necessarily a distinct difference?
> I wonder if you could derive the definition from the theorem?

One can agree that the proven expression will be notated a certain way,
and in that sense the notation is a definition derived from a theorem.
Simple example: x^4 is defined as x*x*x*x. NB that x*x*x*x* follows
from the axioms of ordinary arithmetic: ie, these axioms imply that
[x*{x*(x*x)}] = x*x*x*x*, where the different bracket pairs denote the
order of operations. (BTW, in some arithmetics [x*(x*x)] != [(x*x)*x],
so it matters to show that in "ordinary arithmetic" the order of
multiplication is irrelefvant.)

>>>One could probably define it as the smoothest geometrical shape too.....
>>
>>Yes, but "mooth" needs to be defined (axiomatised) to make this useful.
>>"Smooth" does have a mathematical meaning or two, BTW.
>
> Yes, I realize that was an almost entirely vague suggestion. At least I didn't
> say "least bumpy". Or, would that have been better...?

A bumpy figure is something else againa. Some bumps ar smooth, and some
aren't, you see...

>>>Circles are special.
>>
>>Um, yes, they are ellipses/ellipsoids with coincident foci. Among other
>>things.
>
> Many other things. In my opinion, circles (or cycles) are so widespread in the
> universe, both physically and conceptually, that their properties (almost)
> attain the status of universal truths. But, that's kind of a religious
> statement, I admit, not really mathematical.

It's the kind of thinking behind Ptolemaic astronomy, actually. The
Church supported Ptolemy not only becasue his was a geocentric system,
but also because the movements of teh stars were all circle, which is a
perfect shape, and god would not create anything less than perfection,
right? Of course, they forgot about the imperfection in Man, which led
to his disobedience... but theology has never shied away from
inconsistencies.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... Circloids can be defined as the family of shapes that ... Theorems as opposed to definitions? ... circles are so widespread in the ... attain the status of universal truths. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... Circloids can be defined as the family of shapes that ... Theorems as opposed to definitions? ... circles are so widespread in the ... attain the status of universal truths. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... Circloids can be defined as the family of shapes that ... Theorems as opposed to definitions? ... circles are so widespread in the ... attain the status of universal truths. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... > Theorems as opposed to definitions? ... > attain the status of universal truths. ... Church supported Ptolemy not only becasue his was a geocentric system, ... perfect shape, and god would not create anything less than perfection, ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... > Theorems as opposed to definitions? ... > attain the status of universal truths. ... Church supported Ptolemy not only becasue his was a geocentric system, ... perfect shape, and god would not create anything less than perfection, ...
    (sci.cognitive)
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