Re: Making money from Java


<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). 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).

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. 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.

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."

>> Yes, it sounds like what you are calling "Postulates", I would (in a
>>more rigorous context) called "axioms". I believe axioms can be further
>>subdivided into "assumptions" and "tautologies". Tautologies are those
>>statements which are true by their own definition, and I figured none of
>>the
>>Euclid's five "postulates" falls under that category (though I don't
>>actually remember all five of them), so I called them assumptions.
>
> Mr Wong, what I am calling Postulates are also called Postulates in
> various translations of The Elements; the Heath edition begins this
> section with 'Let the following be postulated'. The Definitions, in
> general, concern themselves with nouns ('a point', 'a line', 'an angle',
> 'trilateral figures', etc.) while the Postulates concern themselves with
> actions ('to draw', 'to describe') and conditions (4. 'That all right
> angles are equal one another', 5. 'That, if a straight line falling on two
> straight lines makes... the two straight lines, if produced
> indefinitely...') It is a habit of my training to try and make use of as
> much original material as possible, including the terms used... perhaps I
> should find a laundromat for nuns and try to deal with this unclean habit.



>
> [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. Strictly speaking, some
of these are actually equivalent to the conjunction of Euclid's parallel
postulate and its converse, and thus can be used to distinguish Euclidean
geometry from both elliptic geometry and hyperbolic geometry simultaneously.
One of the most important of these properties, and the one that is most
often assumed today as an axiom, is Playfair's axiom, named after the
Scottish mathematician John Playfair. It states:

Exactly one line can be drawn through any point not on a given line
parallel to the given line.

[...]

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>

- Oliver


.

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