Re: Epistemology 201: The Science of Science

From: Albert (albertwagner_at_cox.net)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 12:08:12 -0600

Wolf Kirchmeir wrote:
> Albert wrote:
>
>> Wolf Kirchmeir wrote:
>>
>>> Albert wrote:
>>>
>>>> robert j. kolker wrote:
>>>> <snip>
>>>>
>>>>> His [Kant's] assumptions lead to the position that Euclidean
>>>>> geometry is true. But we know that is not the case because
>>>>> consistent non-Euclidean geomtries exist.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> In the real world, or only in mathematiker's minds?
>>>
>>>
>>>
>>>
>>> In the real world. In fact, we live on a non-euclidean surface.
>>
>>
>>
>> Don't spheres exist in Euclidean geometry?
>
>
> Sure, but the surface of a sphere isn't flat. Euclidean geometry is
> about flat surfaces (surfaces of zero curvature.) Spherical geometry was
> developed from Euclidean. However, recognising that E theorems about
> flat surfaces don't hold on curved surfaces is the first step towards
> asking the in-hindsight-obvious question: are all the E-postulates
> necessary? NB that one of the internal-angle-sum proofs relies on the
> parallel postulate. On a sphere, it's possible to have two straight
> lines intersect another line at right angles yet not be parallel. So the
> parallel postulate doesn't hold on a sphere's surface.
>
> This didn't bother anybody, since people believed that space was "flat",
> otherwise you couldn't tell that sphere was curved, right? And anyhow,
> you can run parallel lines tangent to the sphere, so it still holds "off
> the sphere." Just "straighten" those pesky meridians, and there you are.
> So that's all right, then, right? That's what Mercator's projection
> does, but it does so at the cost of turning the N. Pole into a line.
> (That's an example of bob k's point that in projective geometry a point
> can be a projected onto a line.)
>
> But generalise from the surface of the sphere's surface, which is 2D, to
> any space. That's roughly speaking what Riemann did. He showed you could
> make a consistent geometry without the parallel postulate. Then one can
> speak of the curvature of a 3D or higher dimensional space, in which
> lines tangent to a sphere are not necessarily parallel, anymore than the
> meridians on a sphere are parallel. In such geometries, "parallel" in
> the usual (intuitive) sense of the word is a derived, and hence limited,
> relationship between lines - it holds for some but not all cases. (BTW,
> on a sphere, parallel lines in the intuitive sense are not straight, but
> curved: the equator intersects a meridian at right angles, but every
> other parallel of latitude intersects the meridian at less than a right
> angle. As every surveyor knows.)
>
> NB that the sphere's surface is "curved" only when considered in 3D
> space. But its curvature could be detected even by beings who can't
> perceive 3D, because the parallel postulate doesn't hold on it. See
> Abbot's Flatland, and the updated version (whose author I forget, sorry,
> and I don't have a copy of that one). There's a pattern and a clue in
> that observation: we can tell that the Earth's surface isn't flat
> because certain E-geometry theorems do not hold on it. It helps that we
> are 3D creatures, but it's not necessary for us to be so. In fact, in
> relation to the Earth as a whole, we are about as close to
> zero-dimensional points as real objects can get. In relation to the
> universe as whole, even the Earth is merely a point.
>
> So, even though we can't perceive 4D space directly, we can determine
> whether or not the space that we do perceive is curved or flat -- IOW
> whether it's a space within which the parallel postulate holds or not.
> Measurements at astronomical scales indicate that it's probably curved
> overall. (There is still IIRC some argument about the whether the
> measurements have a low enough error.) It's definitely curved in the
> vicinity of massive objects such as the Sun.
>
> (Technically not exact, but I'm using as much NL as possible.)
>
Very impressive, but you still haven't addressed why Euclidean
geometry is rendered false by the existence of non-Euclidean
geometry. You have only explained the efficiency of one geometry
over another when used by science to explain reality.

The statement under discussion was:

"His [Kant's] assumptions lead to the position that Euclidean
geometry is true. But we know that is not the case because
consistent non-Euclidean geomtries exist.

It seems to me that Euclidean geometry remains internally
consistent, and hence mathematically true, regardless of the
invention of additional geometries, or the efficiency of another
geometry in the real world. 

-- 
"Don't you see that the whole aim of Newspeak is to narrow the 
range of thought? In the end we shall make thoughtcrime literally 
impossible, because there will be no words in which to express it."
     -- George Orwell as Syme in "1984"

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