Understand the relationships between axioms, theorems, and different geometric models in which they hold.

SLO 4 Summary

Geometry courses are one of the few places where students have opportunities to engage explicitly with axiomatic systems. Mathematical theories can be developed from a small set of axioms with theorems proven from those axioms. However, GeT students may have limited prior experience working with axiomatic systems. Students in GeT courses should gradually develop the ability to: 

(a) recognize and communicate the distinction between axioms, definitions, and theorems, and describe how mathematical theories arise from them, 

(b) construct logical arguments within the constraints of an axiomatic system, and(c) understand the roles of geometric models, such as the plane, the sphere, the hyperbolic plane, etc., in identifying which theorems can or cannot be proven from a given set of assumptions.

SLO 4 Narrative

Most teachers of college geometry classes will be familiar with the idea that it is possible for mathematical subjects to be reduced to a set of axioms and then a set of theorems proven (exclusively) from those axioms. However, even though this can be done in principle, it is something that working mathematicians rarely do outside of geometry and logic classes, so not everyone teaching a geometry class will have a lot of experience working with axiom systems.

In building an axiom system, we begin with undefined terms, as well as with statements that are accepted to be true without proof called axioms. In geometry, the undefined terms include “points” and “lines.” The axioms establish assumptions about undefined terms and the relationships between them [see SLO 5 on definitions]. A world in which we can give meanings to all of those terms is called an interpretation of the terms₁. For example, if we are talking about “points” and “lines,” we could interpret them as points and lines in the Euclidean plane, but we could also interpret them as points and great circles on the sphere. An interpretation is called a model of the axioms if all of the axioms are true in the interpretation. For example, if we have an axiom that says that two points lie on a unique line, the Euclidean plane would be a model of this axiom. The sphere would not, because antipodal points like the north and south poles on the sphere can be connected by many different great circles [see SLO 9 on non-Euclidean Geometry].

Axioms within a system are independent if no axiom in the system is a logical consequence of the others. This means that for any axiom, we should be able to find an interpretation in which that axiom fails, but all of the others are true. Models and independence are intimately tied into the history of geometry [see SLO 7]. Perhaps the biggest question about Euclid’s axiom system was whether his fifth postulate could be proven from his first four axioms. Although an apparently consistent hyperbolic geometry was developed in the early 1800s, it was not until Beltrami presented a model for that system later in the century that the independence of the parallel postulate was established. 

A theorem is a statement that has been proven from the axioms without regard to interpretation. In a college geometry class, proof can be thought of as a convincing deductive argument relying on explicit reference to axioms or previously proven theorems. Since a model of an axiom system is an interpretation of the undefined terms that satisfies the axioms, every theorem translates to a true statement in a model. Therefore, if a mathematical statement turns out to be false in a model, then the statement cannot be a theorem (i.e. it cannot be proved from the axioms). Moreover, just as models can be used to show that a statement cannot be proven, they can show that a statement cannot be disproven. That is, demonstrating a model where a statement holds shows that the negation of the statement is not a theorem. Models, therefore, serve as a sort of laboratory for geometric conjectures and can be a powerful tool for exploring the properties of an axiom system. 

Students need to understand what the axioms mean, and then they can try to convince someone that a theorem is true whenever the axioms are true [see SLO 1 on Proof]. As in other parts of the geometry curriculum, we see a trade-off between trying to be as rigorous as possible and trying to be developmentally appropriate. Because this is the main place for considering axiom systems in the college math curriculum, understanding of the elements of an axiom system (axioms, models, theorems, interpretations, undefined terms) should be an explicit learning goal. Considering interpretations where axioms do not hold is a good starting point. It encourages students to grapple with what the axioms actually mean, their distinction from other axiomatic elements, and their role of being a foundation of mathematical systems. This is therefore a natural place to bring non-Euclidean geometries into the picture [see SLO 9]. 

When choosing how to introduce an axiom system, instructors must balance the need to establish expectations for axiomatic proof with the need to understand significant, non-obvious geometric results in a reasonable amount of time. In many classes, students start out with a very simple axiom system of basic facts true in almost any geometry and then proceed to prove theorems from them. A common approach is to introduce an axiom system consisting of Euclid’s axioms without the parallel postulate, which is used to develop a “neutral geometry.”.This has the advantage of making the axioms simple enough to focus on the logic of building deductive inferences using them. One strategy for motivating students to reason axiomatically about neutral geometry is to introduce problems with rules in which undefined terms “points” and “lines” have been replaced either with nonsense terms (“Every Fo has two Fes.”) or with some other context (“Every club has at least two members.”) as is done in Greenberg (2008). This encourages students to reason from the stated rules rather than using their geometric intuition. 

When considering an axiomatic development of geometry, it is important to consider the developmental readiness of one’s students for reasoning abstractly. Although GeT students should have learned some elements of the Euclidean geometric system in their K-12 geometry curriculum, it can be helpful if the instructor starts with lower Van Hiele level tasks to scaffold students’ development of geometric reasoning and proof, especially with the consideration that GeT students are often from different STEM majors with varied prior knowledge. Varying expectations of rigor and abstraction can also provide opportunities for assessing students’ development of deductive reasoning (e.g., their application of logic) that is not purely axiomatic. Highlighting differences in expectations for justifications can help to solidify students’ understanding of the elements of axiom systems and proofs.

References

Ebbinghaus, H. D., Flum, J., & Thomas, W. (1994) Mathematical Logic. Springer.

Greenberg, M. J. (2008) Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman.

Hodges, W. (2022) Model Theory. The Stanford Encyclopedia of Philosophy (Spring 2022 Edition). Retrieved May 14, 2022 from https://plato.stanford.edu/archives/spr2022/entries/model-theory/.

Kleene, S. C. (1972) Introduction to Metamathematics. Wolters-Noordhof publishing.

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¹The definitions of terms given in this section are standard in mathematical logic. See, for example (Kleene, 1971; Ebbinghaus, Flum, and Thomas, 1994). The branch of logic that deals with models and interpretations is called “Model Theory,” which is not to be confused with “Mathematical Modeling,” which is a completely different subject in which the word “model” is used to mean something else. See (Hodges, 2022) for more on this.