# Are the truths of euclidean plane geometry contingent truths?

Existence of non-euclidean geometries does not seem to imply an affirmative answer to that question. It might be possible that such geometries are formal constructs to abstract our primitive and intuitive notions on space. Such axiomatic formal constructs would still exist independent of our intuition of space, because non-euclidean geometries are primitively based on notions of numbers and set theory.

For example truths of arithmetic seems to be necessary truths. 1+1=2 is necessarily true under natural interpretation of 1,2,and +. (We can perhaps construct a formal system where 1+1 is not equal to 2, but then we would have changed the natural interpretation) Now let us say we take the words point and line in their natural intuitive interpretation(Not how we take them in modern mathematics, as tuples of real numbers). Then everybody will agree that from any point to another there exists a line. This is euclids first postulate. What I am asking is that: Under the same interpretation is it possible that this sentence may not be true? If it is so then euclidean geometry can be called contigently true. If not, it can be called necessarily true.

This must be taken as a conceptual problem. How well non-euclidean geometries do well in explaining physical phenomena is irrelevant.

• Welcome to Philosophy.SE! Great question, but it may be too broad as currently written. We welcome focused questions and discourage discussion, see the faq: "If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here." My suggestion: delete the last paragraph ("We can discuss:…") and your question should fit the format just fine.
– DBK
Jun 28, 2014 at 19:29
• I'm not exactly seeing the question. Could you break it into its own paragraph and explain it more clearly? I don't see why / how you're maintaining it could be false that a line exists that crosses two points... Jun 29, 2014 at 0:45
• Is it possible that a universe can exist where it is false that from two points a line crosses. Or for example can there exist a universe where it is possible that 4 lines can be drawn that intersects at the same point such that those lines are mutually perpendicular. (More than 3 we say impossible) My point is that is it possible that a universe exist where the inhabitants do not find euclidean axioms intuitive Jun 29, 2014 at 0:55
• I maintain it because physicists(in string theory) suggest that 3 dimensional structure of our universe is somehow accidental , not a conceptual necessity. Maybe they are making a conceptual mistake. Jun 29, 2014 at 1:03
• The generality of arithmetical truths are much easier to grasp.Under their familiar meanings aritmetical truths can never change and is not accidental. It was never possible that there can exist a universe where current arithmetical statements are false or meaningless. 1+1=2 is not an accident or a property of our universe, but rather a conceptual necessity. Does the same apply for geometry? Jun 29, 2014 at 1:09

A necessary truth is one that follows from logic. A contingent truth follows from happenstance.

Consider a theorem T of Euclidean geometry. It has no truth values about the real world; because we have not specified an interpretation. Theorems are syntactic entities; but truth is a semantic one. The best we can say about a theorem is that it is or isn't provable from the axioms.

So when we assert T, we are really making the assertion "T is a theorem of E" where E represents the axioms of Euclidean geometry. "E proves T" is a true statement; and moreover, it's a necessary truth, because it follows directly from logic. In fact you could write a computer program that can verify the validity of the derivation from E to T. Theorem checking is algorithmic procedure. [Theorem finding, especially the finding of interesting theorems, requires a human.]

Notice that we can say nothing of the truth value of T. T is a formal statement, it has no meaning. If we interpret T relative to modern physics, it might be false. If we interpret T in the Newtonian universe, it might be true. Truth is always relative to an interpretation.

Now, consider the statement, "I am left-handed." In this world, that's true. But what about other possible worlds? I could have been born right-handed and my life would be pretty much the same. "Socrates is a philosopher" is true in this world; but in some other world, Socrates was a fishmonger.

Now you might say, What do you mean, "possible world?" I admit that this is a very deep subject with an extensive literature in philosophy; and that I'm ignorant of all of it. I do know that it's a murky notion. Is there a possible world in which 2 + 2 = 5? We tend to think not, because 2 + 2 = 4 is a necessary truth. But why is that?

I can imagine a world in which I'm righthanded. But I can't imagine a world in which 2+2=5. So my handedness is a contingent truth; and 2+2=5 is a necessary truth.

Summary: Each theorem T of Euclidean geometry is neither true nor false by itself. If you give me an interpretation, I'll tell you if it's true or false.

But, the statement "E proves T", where E is the collection of Euclidean axioms, is in fact a necessary truth; because it's provable by logic.

"Euclid wrote The Elements" is a contingent truth. In some other possible world, Euclid herded goats and somebody else wrote the first modern math book.

• A great response. The best way to think of "possible worlds" in the context of the logic of necessity and possibility is as an assignment of truth values to all of the atomic sentences there are. Suppose there are only three atomic sentences S1, S2, S3, each of which has only two truth values, T or F. Then there are 2^3=8 "possible worlds" formable from these three sentences. Now suppose we have some "molecular" sentence like "if S1, then S1". Now if this sentence is true in every possible world, then we say it is a "necessary truth".
– user5172
Jul 30, 2014 at 22:48

Consider two points on the surface described by `z = 1/x * 1/y`. Is there a line on that surface that connects them?

Imagine two airplanes flying in the same direction 500 miles apart. If they both go straight, will their flight paths be parallel?

If I have a drop of water, and I add a drop of water to it, how many drops of water do I have?

Our "natural intuitive interpretation" of mathematical concepts is highly context-dependent. If you're talking about apples, then 1+1=2, but if you're talking about drops of water 1+1=2 doesn't make any sense. If you're talking about flat surfaces, Euclidean geometry follows naturally, but not all surfaces are flat--and I don't just mean that we live in three dimensions instead of two. 3D space itself can be curved!

Euclidean geometry, like Newtonian physics, seems intuitive and natural to us because it's simple, and it's a good approximation of how the world actually behaves most of the time. However, that behavior is not a necessary fact. In a universe where physics worked quite differently, I doubt the inhabitants would find such things as obvious as we do.

• Can you please explain your example z = (1/x) * (1/y) = 1/(xy)? It's a 3D surface that blows up on the x- and y-axis. Is there a particular idea I'm supposed to be getting from this? wolframalpha.com/input/?i=z++%3D+1%2F%28xy%29 Jul 30, 2014 at 1:31
• It's essentially a 3D version of the hyperbola y = 1/x. The "shelves" on the graph's surface are asymptotes, so if you picked two points on the colored surface that were on opposite sides of a "shelf," no line could sensibly be drawn on that surface connecting them. Jul 30, 2014 at 1:47
• I don't follow, my apologies if I'm being dense. There is a line between any two points on the surface, even if they lie on different connected components of the graph of the function. Just as there's a line between any two points on the different connected components of the hyperbola y = 1/x. Or (maybe this is what you mean) there's a line in Euclidean space, but there is no line all of whose points are on the surface. Which would be the same idea for any set consisting of multiple connected components. Is that what you meant? Jul 30, 2014 at 2:03
• Oh I see your point. But now I don't see how this bears on the question. Your space is non-Euclidean, since as you note, two points don't necessarily determine a straight line. So are you arguing that Euclidean geometry is contingent? Now I'm not understanding how this addresses the question. Jul 30, 2014 at 6:26
• The question is about whether Euclidean geometry (and other mathematical concepts) is a necessary or contingent truth; the example is intended to show that it is contingent, even in the context of 2-dimensional surfaces (in this case, a manifold). This is relevant because most real-life surfaces are actually non-Euclidean manifolds, including both the surface of the Earth and 3-dimensional space. Jul 30, 2014 at 19:52

The law that asserts 1+1=2 is essentially the law of identity - that is this is this; this is why it is neccessarily true. To specify a context for this is to be more particular about what we are identifying and how.

In Liebniz's ontology it is the law of indiscernibles which he first wrote about in the Discourse on Metaphysics; it is typically understood as:

no two [distinguishable] objects can have the same properties

This seems eminently reasonable: if two objects are distinguishable then they must differ in one or more properties. We can say then:

an object is uniquely determined by its properties

The question then becomes is this law universally valid? Are there conceptual schemes in which it can be weakened?

On the face of it, 1+1 is not equal to 2; they look different; however the context supplies a proof that the first can be altered into the second, and back; this is why I say essentially rather than exactly in my first sentence. One could say in a certain language that they are homotopic (to make this precise - we are moving into the realm of homotopy type theory from type theory)

Hence Leibnizs law though essentially true, is not exactly true; and this can be made more formal. In Set Theory, choosing ZF for precision, Leibnizs law is translated into the axiom of extensionality:

A set is determined uniquely by its members.

The correspondance between this and my restatement of Liebnizs law is obvious.

It is possible to take a different line, and this grew out of the notion of isomorphism in algebra; that is two objects can be essentially indistinguishable for all purposes yet they may be different; this notion finds its correct context in category theory; and in fact, ideally higher category theory.

Essentially all of mathematics (as currently understood) can be geometrised: Geometry and algebra are dual. For example, the integers, the archetypical algebra is geometrised (by Grothendieck) through his notion of schemes - and in this language one can talk about coverings, bundles and curvature.

Given this insight it is not surprising that there are non-euclidean geometries; they just happened to be the first to be discovered.

Hence the truths of Euclidean Geometry are neccessarily true as one has specified this geometry in the space of all geometries; but when think of the modality of neccesity in Kripke or Lewis's possible world semantics where a truth is neccesarily true when it is true in every world - then one should say they are contingently true.

I would also say that this duality (that of geometry to algebra) is natural to us as we see (thus geometry) and touch (thus counting). In this sense the physical immediacy of the world, the way it is present to us - that is phenomenologically - is what determines these two categories of mathematical understanding that are so intimately related to each other; in this line of thought it is a philosophical error to remove both the world and consciousness to understand number and geometry. In this sense, geometry is contingent on this world and on the structure of consciousness. This isn't surprising here - as this view grew out of Kantian philosophy where it is an essential starting point.

I think there is a bit more than there should be, by my taste, in the others answers. It is a straightforward answer for a person with a bit of mathematical work. The fifth postulate does not stand in 3D curved space, as our world is(see the angles on the spheres for example). Carl Friedrich Gauss once stated in a letter that he doubts the fifth postulate and not the other 4 axioms. Afterwards, János Bolyai, created a new geometry, where the fifth postulate did not hold (called the hyperbolic geometry).

As a conclusive answer, if I understood the question correctly, the axioms are not contingent truths, since another geometry could have been created based on the euclidean geometry.