Philosophy Now Forum

Fallacy of Euclidean Axioms

1. To draw a straight line from any point to any point.

All lines are defined by points, all points are defined through lines. As the line as "straight" effectively exists as a continuation of points, considering as progressively "projecting" lines are always between points (for example in drawing them) so that each line exists as a continuum of the point. The line effectively is a point.

But the points only exist it there is a line between them. Even marking two points in any given object results in them being in a straight distance from each other as 180 degrees always, or "half" a circle, with each degree and pair of degrees existing as a line between two points.



2. To produce [extend] a finite straight line continuously in a straight line.

All straight lines, as a continuum of points, in themselves are infinite lines; hence no "finite" line exists.

All lines, as straight in the respect they are defined through to points with all point effectively existing as composed of other points and hence infinitesimal lines, necessitates all straight lines can effectively be angles (3 points) as infinitesimal degrees as well as any extended geometric object.

All straight lines can exist as angles and extended geometric objects defined by the angle. These extended geometric objects, such as a septagon, effectively result in frequencies. The line,angle and frequency (as repetitive angles) exist as 1 and 3 and 3 and 1.







3. To describe a circle with any centre and distance [radius].

The radius, as a line between two points composed of infinite points, observes each point in the radius as the center of the circle resulting in infinite circles in all directions thus equating to "point space" in itself where only a point exists and this point is an undefinable boundless field of infinite dimensions.

All lines as existing between two points results in a center point, thus necessitating all points to exist between two points and as such the center of a circle. An infinite progression of circles in 1 relative direction results in a line. An infinite progression of circles in infinite directions as "all" or "1" results in a point.






4. That all right angles are equal to one another.
All right angles are not equal to each other. If a right angle exists at 90 degrees of 1 unit by 1 unit and the next right angle is .1x.1 progress to .0000....1 of the original size, the final angle effectively exists as a point smaller than the degree with composed the ninety degree of the Right Angle.






5. [The parallel postulate]: That, if a straight line falling on two straight lines make 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 the angles are less than two right angles.


The continuation of the lines meeting results in the lines as infinite in length where "infinity" results in the end point, of the lines as the lines themselves hence no finite line exists that constitutes the base definition of a Euclidian Line.




All geometric objects are an approximation of a point, resulting in a dual 1 dimensional and 0 dimensional point. But this dualism itself results in a contradiction.

euclidean axioms work perfectly in euclidean space...

we don't live in euclidean space

-Imp

Eodnhoj7 wrote: ↑Mon Jan 28, 2019 9:37 pm Fallacy of Euclidean Axioms (...) All lines are defined by points, all points are defined through lines. (...)

Me, I define neither lines nor points. I have an intuition of them. I don't need any definition to understand what is a line or a point.
I take Euclidean axioms to be the definition of our intuitive notions of the geometry of space. A proper definition should be a formal theory meant to represent and express our intuitive notions by defining clear and rigorous concepts that should be logically articulated with each other and should cover as much as possible of the semantic field targeted. I think Euclid 2,300 years ago did an amazing job.
The question therefore is not to question the foundation of Euclid's axiomatic. Axioms, in this context, are to be understand as evident, i.e. as reflecting our intuition. This is enough.
The first job that remains to be done, however, is "logical validity", i.e. to check the logical consistency of the theory. You're welcome to do it but 2,300 years after the event, you're very unlikely to discover anything wrong that isn't already well understood as wrong, if anything.
The second job to be done is "consistency with empirical evidence", for example whether space is really flat as assumed by Euclid. Oh, well, you came too late for that as well.
Nothing else that would make sense.
Perhaps question the existence of reality?
EB

Last time I cared to do any carpentry for my house, Euclidian geometry worked just fine.
Last time I had to calculate the escape velocity for a rocket, Euclidian geometry would have ended in a flamboyant disaster.

My 1.8 meter coffee table needs not concern itself with the curvature of the Earth. A rocket does.

The precision required for any particular task is a function of worst-case counter-factual reasoning.

Logik wrote: ↑Tue Jan 29, 2019 2:55 pm Last time I cared to do any carpentry for my house, Euclidian geometry worked just fine.
Last time I had to calculate the escape velocity for a rocket, Euclidian geometry would have ended in a flamboyant disaster.

My 1.8 meter coffee table needs not concern itself with the curvature of the Earth. A rocket does.

The precision required for any particular task is a function of worst-case counter-factual reasoning.

Yes, I am a carpenter. Euclidian Geometry does work. But it is not necessary though either. Usefullness does not equal "necessity".

Besides, upon any in depth examination of the product, Euclidean Geometry is often approximated...never fully applied physicalized...as evidenced by faults within the product itself. Euclidian Geometry is a projection of intentionality, where the observer individuates reality by folding materials that consist of space/time. However the axioms of Euclidian Geometry, as the localization of space, is subject to this same nature of projection of intentionality.

Eodnhoj7 wrote: ↑Tue Jan 29, 2019 8:02 pm Yes, I am a carpenter. Euclidian Geometry does work. But it is not necessary though either. Usefullness does not equal "necessity".

Besides, upon any in depth examination of the product, Euclidean Geometry is often approximated...never fully applied physicalized...as evidenced by faults within the product itself. Euclidian Geometry is a projection of intentionality, where the observer individuates reality by folding materials that consist of space/time. However the axioms of Euclidian Geometry, as the localization of space, is subject to this same nature of projection of intentionality.

The Purist Euclidian geometry was that of the ruler and the compass.

The idealised (conceptual) forms of points and lines is actually disconnected from the pragmatic roots of Euclidians.
They insisted on procedural/imperative knowledge.

It somewhat maintains the scientific spirit of reproducibility, but in the 21st century a few more tools can be found in our toolboxes...

The ancient greeks didn't have CAD.

Logik wrote: ↑Tue Jan 29, 2019 8:23 pm

Eodnhoj7 wrote: ↑Tue Jan 29, 2019 8:02 pm Yes, I am a carpenter. Euclidian Geometry does work. But it is not necessary though either. Usefullness does not equal "necessity".

Besides, upon any in depth examination of the product, Euclidean Geometry is often approximated...never fully applied physicalized...as evidenced by faults within the product itself. Euclidian Geometry is a projection of intentionality, where the observer individuates reality by folding materials that consist of space/time. However the axioms of Euclidian Geometry, as the localization of space, is subject to this same nature of projection of intentionality.

The Purist Euclidian geometry was that of the ruler and the compass.

The idealised (conceptual) forms of points and lines is actually disconnected from the pragmatic roots of Euclidians.
They insisted on procedural/imperative knowledge.

It somewhat maintains the scientific spirit of reproducibility, but in the 21st century a few more tools can be found in our toolboxes...

The ancient greeks didn't have CAD.

You are going have to elaborate on "CAD", I am unfamiliar with that term.

The pragmatic roots of the Euclidian where premised on there ability to merge a dualistic notion of abstraction/empiricism or "concept" and "reality". A concept is deemed as "pragmatic" if it is able to be "joined" to an empirical phenomenon or "diverged" from it; hence pragamaticism (originating from Pierce's Triadic interpretation of Logic) is an extension of Hegelian/Fichte Synthesis which is an extension of Aquinas's synthesis of cataphatic/apophatic truth's, which extend back to the Pythagorean triad and is reminiscient of the triadic nature of many religions.

These extensions are merely a brief definition of a progressive change, however the nature of "sythesis" itself is subject to a form of recursive definition and is synthesized in and of itself.


The synthetic nature of relation exists as a core foundation of recursion where any "new variable" is strictly an extension of the old as it adapts to "nothingness".

Logik wrote: ↑Tue Jan 29, 2019 2:55 pm Last time I cared to do any carpentry for my house, Euclidian geometry worked just fine.

Only because you're doing Euclidean carpentry.

Logik wrote: ↑Tue Jan 29, 2019 2:55 pm Last time I had to calculate the escape velocity for a rocket, Euclidian geometry would have ended in a flamboyant disaster. My 1.8 meter coffee table needs not concern itself with the curvature of the Earth. A rocket does. The precision required for any particular task is a function of worst-case counter-factual reasoning.

Two very simple questions:
1. Where exactly did Euclide address kinetics?
2. What is the relevance of the curvature of the Earth to Euclidean geometry?
EB

Speakpigeon wrote: ↑Wed Jan 30, 2019 10:28 am 1. Where exactly did Euclide address kinetics?

*yawn* You can approximate kinetics with angular motion.

https://en.wikipedia.org/wiki/Euclidean ... isometries

The Euclidean group for SE(3) is used for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant of the transformation cannot jump from +1 to −1.

Speakpigeon wrote: ↑Wed Jan 30, 2019 10:28 am 2. What is the relevance of the curvature of the Earth to Euclidean geometry?[/b]

Helllooo, pay attention! I am trying to build a coffee table. Do I need to take the curvature of the Earth into account or not? If not - why not?

Logic wrote:
I am trying to build a coffee table . Do I need to take the curvature of the Earth into account or not ?

No because the two have nothing to do with each other [ obviously ] and the reason is because the
curvature of the Earth can only be seen from space which is not where your coffee table is going to

surreptitious57 wrote: ↑Wed Jan 30, 2019 1:10 pm

Logic wrote:
I am trying to build a coffee table . Do I need to take the curvature of the Earth into account or not ?

No because the two have nothing to do with each other [ obviously ] and the reason is because the
curvature of the Earth can only be seen from space which is not where your coffee table is going to

So if I can’t see it it doesn’t exist?

Still no closer to deciding whether I can use Euclidian geometry for making my coffee table...

Logic wrote:
Still no closer to deciding whether I can use Euclidian geometry for making my coffee table

Yes you can use Euclidian geometry for making your coffee table because you are not actually mapping four dimensional spacetime
You are just making a coffee table so no knowledge of non Euclidian geometry is required so you go ahead and make one right now

surreptitious57 wrote: ↑Wed Jan 30, 2019 1:39 pm Yes you can use Euclidian geometry for making your coffee table because you are not actually mapping four dimensional spacetime
You are just making a coffee table so no knowledge of non Euclidian geometry is required so you go ahead and make one right now

But Euclidian geometry only deals with 2 dimensions. If the Earth is not flat how can I use it?!?

Logic wrote:
But Euclidian geometry only deals with 2 dimensions

You can make a three dimensional object from a two dimensional plan
then you can very easily make a coffee table from Eucildian geometry

surreptitious57 wrote: ↑Wed Jan 30, 2019 2:02 pm You can make a three dimensional object from a two dimensional plan
then you can very easily make a coffee table from Eucildian geometry

Surely it's better to be more precise though?

It will be a much better table if it follows the curvature of the Earth precisely!