On the Consistency of Non-Euclidean Geometry

Question

I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that :
if non-euclidean geometry is inconsistent, then so is euclidean geometry, and the reverse is also valid.
But the book wasn't as helpful as it doesn't prove, or give some reference or name to this result.

Does anyone have a proof for this or for something similar?
Even a definition of what exactly "inconsistency" means in this geometric context would be welcome.

Answer 1

A relative consistency of geometry was shown by David Hilbert. Hilbert provided an interpretation (in model-theoretic sense) of his axiomatic system of geometry in the real plane, and thus, reduced the consistency question of geometry to the consistency of real number system. Hence, geometry (whether Euclidean or not) is consistent if real number system is consistent and vice versa. That is why we say the consistency proof has been given in a relative sense (i.e., relative to real number system), not in an absolute sense.

A widespread view is that the question of an absolute proof of geometry cannot answered, or alternatively, has been negatively answered by Gödel's incompleteness theorems. As Howard Eves says in his A Survey of Geometry (revised edition, p. 343)

Relative consistency is the best we can hope for when we apply the method of models to many branches of mathematics, for many of the branches of mathematics contain an infinite number of primitive elements. This is true, for example, of plane Lobachevskian geometry. In the next section we shall, however, by setting up a model of plane Lobachevskian geometry within plane Euclidean geometry, show that the former geometry is consistent if the latter geometry is.

A set of axioms is said to be consistent if contradictory statements are not deducible from the axioms in the set. To have an idea about what an inconsistent system of geometry may be like, we can consider a system of equations that have no common solution. Just for the sake of illustration, take for instance, $x – y = 3$ and $4x – 4y = 15$. The lines defined by this system of equations neither intersect at some point nor lie parallel, hence if the equations were the axioms, they would describe an inconsistent geometry.

The Prehistory of Mathematical Structuralism (edited by E.H. Reck and G. Schiemer) is a nice book (freely accessible by the courtesy of OUP) that offers philosophical and historical perspectives on this topic.

Addendum

A visual way to make sense of consistency in geometry may be contrast our ordinary geometrical perception to the impossible objects such as (figure from Wikipedia):

Unsurprisingly, such inconsistent geometry does not go without inquiry, though, probably, more of philosophers' delectation than mathematicians'. Chris Mortensen's book Inconsistent Geometry (College Publications, 2010) is a good example of such effort. The following passage from the book (p. 4) may be helpful to give a view of Mortensen's project:

This book will go about the project of geometrising logic in two broad ways. In the first part, the aim is to find models for paraconsistent logics exploiting geometrical objects, and conversely to show how logics and logical theories develop out of classes of geometrical objects.

In the second part of the book, we will tackle the topic of providing mathematical descriptions of the so-called impossible pictures, drawn by Reutersvaard, Escher, and others. Only a few classical mathematicians have made an attempt at this difficult class of problems; for example, Penrose, Cowan, Francis. But invariably, their otherwise adroit accounts miss something essential. Being classical and thus consistent, they cannot give a sense of the content of the experience of the impossible, I see it but it can't exist. The only way to complete the story, I propose, is to give an inconsistent theory as the content of the experience. It will stand to the experience of the impossible in somewhat the way that projective geometry stands to the experience of perspective, only the theory will perforce be inconsistent.

Answer 2

[[ EDITORIAL COMMENT , since this is getting too many Downvotes :

OP is asking "Does anyone have a proof for this or for (Something Similar)? Even a (Definition) of what exactly ("inconsistency") means in this geometric context would be welcome."

I have given Something Similar & I have shown what Consistency means , in general , through analogies.

I could add more Examples though it will make this long Answer even longer. ]]

Explanation in Simple terms :

Here are my made-up "Axioms" for the age of the Universe :

Set 1 :
(1) Universe is at least 8 Billion years old
(2) Universe is at most 14 Billion years old
(3A) Earth is at least 5 Billion years old
(4) Earth is at most 20 Billion years old

Is this Set of Axioms Consistent ?
Yes , Universe=10 & Earth=8 works fine & all "Axioms" are satisfied.
There is some way we can find some ages which will satisfy all the Axioms.

Suppose I change Axiom (3A) to this , to get Set 2 :
(3B) Earth is at least 15 Billion years old

Is it Consistent ?
NO , "Universe will be younger than Earth" , which is a Contradiction !
There is no way we can find some ages which will satisfy all the Axioms.
These Axioms will make Universe younger than Earth which must be younger than Universe : Hence "Universe is younger than itself" !
We will get all sorts of Contradictory Conclusions , Eg "Earth is younger than itself" , Etc.

That is what we mean by Consistency. There is some way to satisfy all the Axioms. It must not lead to Contradictions.

Moving on , when I add one new Axiom out of the following :
(5A) Earth has only one gas Nitrogen
(5B) Earth has no gas
(5C) Earth has multiple gases Nitrogen , Oxygen , Hydrogen , Helium , ETC

No matter which new Axiom I add to Set 1 or Set 2 , there will be no change to the Existing Consistency , because this new Axiom is Independent of the other 4.

That is what we have in various geometry types :

We have a Set of Common Axioms which may or may not be Consistent.
We are adding a new Axiom concerning Parallel lines (1 Parallel line , no Parallel line , multiple Parallel lines) to that Set of Common Axioms.
This new Axiom will not change the Existing Consistency.

Quasi-Geometric Example : My theory of Squares :
Axiom 1 : When Side of Square is $A$ , all Sides are Equal to $A$.
Axiom 2 : When Side of Square is $A$ , Area is $A^2$
Axiom 3 : There Exists a Square X of Area 100.
Axiom 4A : That Square X has Side 10.
Axiom 5 : That Square X is Purple in Colour ( think that this is the Parallel Postulate in Euclidean geometry ! )

All Consistent till now.

When I change 4A Axiom to :
Axiom 4B : Square X has Side 20.

It is no longer Consistent.
I will get Contradictions like $20^2=100$ & $20=10$ ETC.

That Consistency or lack of Consistency will not change when I change the last Axiom to some other Colour ( think that this is some other Parallel Postulate in Some Non-Euclidean geometry ! )

That is because the last Axiom ( about Colour or about Parallel lines ) is Independent of the other Axioms.

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Interesting titbits :
The Common Axioms ( without Parallel Postulate ) are known as Neutral geometry or Absolute geometry.
Neutral geometry is known to be Consistent.
Neutral geometry + Some Parallel Postulate gives Euclidean geometry & Non-Euclidean geometry.
Absolute geometry theorems are valid & true in Euclidian & Non-Euclidean geometry.
Parallel Postulate has been proven to be Independent of the Neutral geometry by Beltrami.
[[ Basically , this means that when the other Axioms are Consistent ( or not Consistent ) , including the Parallel Postulate will keep it Consistent ( or not Consistent ) ]]

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reference :

https://en.wikipedia.org/wiki/Parallel_postulate
https://blogs.scientificamerican.com/roots-of-unity/chasing-the-parallel-postulate/
https://new.math.uiuc.edu/public402/euclidsgeometry/elements.html
https://www.maths.gla.ac.uk/wws/cabripages/klein/hilbert.html
Proof that the Parallel Postulate is independent from the other four?
https://www.math.hkust.edu.hk/~mabfchen/Math4221/Independence%20of%20parallel%20postulate.pdf