Re: Making money from Java
- From: docdwarf@xxxxxxxxx ()
- Date: Fri, 16 Dec 2005 19:39:18 +0000 (UTC)
In article <ScEof.2084$ic1.1633@edtnps90>,
Oliver Wong <owong@xxxxxxxxxxxxxx> wrote:
>
><docdwarf@xxxxxxxxx> wrote in message news:dnuf3f$p7g$1@xxxxxxxxxxxxxxxxxxxx
>> In article <b2kof.1248$lv3.1229@clgrps12>,
>> Oliver Wong <owong@xxxxxxxxxxxxxx> wrote:
>>>
>>><docdwarf@xxxxxxxxx> wrote in message
>>>news:dnqo50$iq3$1@xxxxxxxxxxxxxxxxxxxx
>>>> In article <Jk_nf.256568$ir4.37132@edtnps90>,
>>>> Oliver Wong <owong@xxxxxxxxxxxxxx> wrote:
>>>>
>>>> [snip]
>>>>
>>>>>For example, we have a whole system of geometry
>>>>>called "Euclidean geometry" that, for example, tells us (among other
>>>>>things)that the sum of all the angles in a triangle is 180 degrees.
>>>>
>>>> With all due respect, Mr Wong, the Euclid I studied (Heath edition) made
>>>> no mention of degrees... it may be that somewhere in the Geometry it is
>>>> concluded that the sum of the angles of a triangle is equal to the sum
>>>> of two right angles but that is, I believe, a proposition which is
>>>> demonstrated.
>>>
>>> Euclid contributed an axiomatic system which we now call "Euclidean
>>>geometry". Not sure if Euclid himself mentions what the angles in a
>>>triangle
>>>add up to in his text, but it IS using his system that we have derived the
>>>above fact.
>>
>> Once again, Mr Wong... as I was taught it Euclid gave (in the sense that
>> 'Homer' gave the Illiad and the Odyssey) a set of works beginning with
>> Definitions, Postulates and Common Notions; using these he generated
>> Propositions and the whole of these constitute the Geometry. I am not
>> sure how this is to be reconciled to what you are calling 'an axiomatic
>> system' but to state 'since Euclid's system is used to derive facts about
>> sums of angles in a triangle the source of these facts is Euclidean
>> geometry'; this appears similar to a logical extension along the lines of
>> 'Joe builds walls; since within these walls a plumbing system is
>> constructed the source of the plumbing system is Joe's walls'.
>>
>
> I'm assuming you say "*the* Geometry" (emphasis mine) because at the
>time, Euclid was only aware of this one geometry (because as we now know,
>there is more than one geometry).
Actually it is a merely a labelling of the text. Just as I refer to
Ptolemy's Almagest, Herodotus' Histories, Plato's Republic, Lucretius' De
Rerum Natura... so do I refer to Euclid's Geometry.
>My greek may be a bit rusty, but I'm
>assuming geometry comes geo (world) and metry (measure), so Euclid may have
>intended to be measuring the world (where "world" in his understanding
>corresponds with "space" or "reality" in our understanding of the terms);
>that is, assuming that Euclid himself actually refers to what he came up
>with as "geometry" (I'm not sure if he does).
Measuring of the earth may be what the Geometry can be applied to,
certainly, but Euclid does his best to avoid 'worldly' applications. Some
say that the Geometry is constructed so that it can exist without humans
to interpret it... but I am unaware of such a universe in my neighborhood.
>
> As for reconciling, "system" is such a vague term, I think it can be
>applied to almost anything without problems of reconciliation. I call the
>system that Euclid came up with "axiomatic" because of his emphasis ensuring
>that his geometry be built from axioms. Whether we want to call them
>"axioms" or "postulate", I think Euclid intended for there to be as few of
>these things as possible.
Heath says of the Postulates 'Each postulate is an axiom - which means a
statement which is accepted without proof - specific to the subject
matter, in this case, plane geometry.' He also labels the Common Notions
as such... but not the Definitions.
(http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#guide)
>>From what I heard, he struggled for most of his
>life trying to reduce their number from 5 down to 4, but never succeeded. If
>you prefer, I could call what Euclid gave us a "postulatory system", but
>"axiomatic system" has already been coined, and is a more popular term. I
>think we are talking about the same thing.
In that the Geometry is based on Postulates and Common Notions you are
correct... in that the Geometry is the result of Definitions and the
Propositions following those three, perhaps not. A foundation of stone
can underly a house made of wood; this does not make it a 'stone house'.
>
> If I said "since Euclid's system is used to derive facts about sums of
>angles in a triangle the source of these facts is Euclidean geometry", then
>I apologize, for I was being imprecise. What I probably meant was "The fact
>that the sum of the angles in a triangle is 180 degrees, was derived from
>the axiomatic system known as Euclidean Geometry. It is not true of so
>called non-Euclidean Geometry, so this fact can be said to be a
>distinguishing feature of Euclidean Geometry."
Gah... once again, Euclid didn't deal with degrees. What you are trying
to say here might be rephrased in accord with Euclidean constructs as
'The sum of the angles in a triangle equals two right angles in Euclidean
geometry, in other geometries this is not always the case.'
[snip]
>> [snip]
>>
>>>> (And... my memory is, admittedly, porous but I recall that if one
>>>> attempts
>>>> to conctruct a geometry on the surface of a hypersphere then the
>>>> Parallel Postulate holds... but that's for another time, perhaps.)
>>>
>>> In Euclidean geometry, the parallel line postulate says (or is
>>>equivalent to):
>>>
>>> Given a line L and a point P which is not on L, there exist exactly
>>> one
>>>line which crosses P but which does not cross L. This line is said to be
>>>"parallel" to L.
>>
>> Equivalences are one thing, Mr Wong, texts are another. The Heath
>> translation renders Postulate 5 as 'That, if a straight line falling on
>> two straight lines makes the interior angles on the same side less than
>> two right angles, the two straight lines, if produced indefinitely, meet
>> on that side on which are the angles less than the two right angles.'; it
>> may be beneficial to view this in light of a preceding Definition (23),
>> which he translates as 'Parallel straight lines are straight lines which,
>> being in the same plane and being produced indefinitely in both
>> directions, do not meet one another in either direction.'
>
>>From http://en.wikipedia.org/wiki/Parallel_postulate
>
><quote>
>Several properties of Euclidean geometry are logically equivalent to
>Euclid's parallel postulate, meaning that they can be proven in a system
>where the parallel postulate is true, and that if they are assumed as
>axioms, then the parallel postulate can be proven.
'(L)ogically equivalent...' ... once again, Mr Wong, equivalencies are one
thing, texts are another.
[snip]
>It should be noted, however, that the alternatives which employ the word
>"parallel" cease appearing so simple when one is obliged to explain which of
>the three common definitions of "parallel" is meant - constant separation,
>never meeting or same angles where crossed by a third line - since the
>equivalence of these three is itself one of the unconsciously obvious
>assumptions equivalent to Euclid's fifth postulate!
></quote>
'(U)nconsciously obvious assumptions'? I'll leave such musings to The
Professionals in matters of 'the unconscious'... as I've demonstrated here
I have enough trouble with the text.
DD
.
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