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I came across the idea that the same statement can be true in one model while not true in another, while both models are being consistent. An example is "Is the sum of the angles of a triangle equal to 180 degrees?" While true in Euclidean geometry, it is not in non-Euclidean. Thus, this introduces relativity to mathematics.

So, my question is what are some philosophical implications of this? Where can one go from here? For example, will a model that assumes both Euclidean and non-Euclidean geometry be inconsistent even both are consistent individually? Can both coexist consistently?

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  • I suspect this might mean that we will need more than mathematics, or rationality, to tell us what is real. Welcome to this SE! – Frank Hubeny Sep 30 '18 at 9:32
  • See Mathematical structuralism : "Some philosophers postulate an ontology of structures, and claim that the subject matter of a given branch of mathematics is a particular structure, or a class of structures." – Mauro ALLEGRANZA Sep 30 '18 at 10:10
  • This page does not discuss non-Euclidean geometry, as far as I know, however I think you may find this website interesting. Prof. John Norton pitt.edu/~jdnorton/teaching/HPS_0410/chapters/… – Gordon Sep 30 '18 at 13:59
  • Why does your example introduce "relativity" instead of showing that starting with a general statement like "all triangles" may be too general? – Cell Sep 30 '18 at 18:21
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    A model that incorporates both Euclidean and non-Euclidean geometries is called Kleinian geometry. It is not inconsistent, they just appear as special cases. Mathematical pluralism simply means that there is a multiverse of mathematical systems, none truer than others (some do include inconsistent systems, but this is not popular). For some discussion see Balaguer's Mathematical Pluralism and Platonism – Conifold Sep 30 '18 at 22:35
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All mathematics is basically taking certain axioms and deriving interesting theorems from them. It should be no surprise that taking different axioms can result in different theorems. There is no relativity here, just differing axioms.

Euclidean geometry includes the parallel postulate, which is that, given a line and a point not on that line, precisely one line parallel to the original line goes through the original point. Typically, the geometries we call non-Euclidean have other variations of it. Riemann created a geometry with no parallel lines, and Lobachevsky created one with multiple parallel lines going through a point.

These are frequently modeled by constructing a 2D geometry on a flat plane, a rounded surface, or a saddle-shaped surface, respectively. A geometry can easily include all possibilities, by allowing for possible curvature of a plane or space or whatever.

The major philosophical effect was when non-Euclidean geometries became known, and it was realized that geometry didn't really determine anything about the world, but was just a description. Philosophers that predate these developments sometimes took geometry as an example of finding things out about the world just by reasoning, and philosophers since realize that doesn't work.

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  • "All mathematics is basically taking certain axioms and deriving interesting theorems from them." Only if your knowledge of math was learned in philosophy class, and not a very good one. In the real world the theorems are discovered first and then the axioms are developed to support them. For evidence see the entire history of mathematics. Gauss never heard of set theory, was he not doing math in your view? – user4894 Oct 2 '18 at 1:31
  • @user4894 Intuition is surely important for a professional mathematician, but in the end mathematics is proving theorems. And the only way to prove a theorem in a mathematically acceptable way is deriving it from axioms, and from previously proved theorems based on such axioms. This has always been the case, even if the study of foundations of mathematics and metamathematics is fairly recent (in the past, some statements that are now theorems were taken as axioms). – yuggib Oct 2 '18 at 9:01
  • @user4894 In addition, there have been serious drawbacks in relying too much on intuition, see for example the (partly in)famous italian school of algebraic geometry – yuggib Oct 2 '18 at 9:01
  • @user4894 "Only if your knowledge of math was learned in philosophy class, and not a very good one. ... Gauss never heard of set theory." That may be a bit harsh. In Gauss's day, for example, the associativity of addition was just "common sense" -- implicitly an "axiom" of basic arithmetic among several others. I'm sure he would have agreed with each of Peano's Axioms, and that one could indeed prove associativity from these axioms. With the axiomatization of set theory, one could, in turn, derive each of Peano's original axioms. – Dan Christensen Oct 2 '18 at 15:31
  • I got my answer with existence of Kleinian geometry. (but I cannot upvote! as I am a new member) But, reformulating my question would be "Can we have two consistent models that disagree on a statement and they cannot coexist in any supermodel?" So if this is possible, then what axioms we choose matter, otherwise we can end up with two incompatible models, and maybe someone can exemplify this situation with a real world case if this case is really possible. – yigoli Oct 2 '18 at 18:24

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