“One must be able to say at all times – instead of points, straight lines, and planes – tables, chairs, and beer mugs. “David Hilbert [4]

Axioms serve as fundamental bricks in the foundations of mathematics. Given a small collection of statements assumed to be true, a universe of subsequent truths may spring forth, grounded in those assumptions and constructed using mathematical logic. Change the axioms even slightly and you change that mathematical world.  In geometry, the collective reconsideration of Euclid’s Fifth Postulate and the alternatives offered up in its place led to the “strange new universe” of hyperbolic geometry created by János Bolyai and others. [2]

Students at the onset of a GeT course typically have at least a passing understanding of proof, and many have taken a university course focusing on sets, logic, and proof structure.  However, students generally have little experience with axioms beyond the definition of the term.  Nevertheless, developing a sense of axioms and axiomatic systems is essential to understanding the nature of mathematics. A GeT course provides a natural place in the undergraduate curriculum to address this important concept.

One problem that I have used in my GeT course for many years to introduce the idea of axiom systems was developed by a group of mathematicians at the University of Wisconsin in the early 1990s:

The axioms here are essentially those for a finite affine geometry with “people” and “clubs” playing the respective roles of points and lines. Notably, Axiom H4 is a disguised version of Playfair’s postulate, logically equivalent in Neutral Geometry to Euclid V.  Disguising the axioms serves a pedagogical purpose.  Students’ familiarity with Euclidean notions of points and lines comes with some baggage;they often hold tacit assumptions about the relationship between points and lines (e.g., lines are “straight,” and if two lines look like they intersect, then they do, at a point).  These assumptions tend to impede their ability to grapple with the consequences of the axioms for geometry on a purely logical basis.  Framing the problem in terms of “people” and “clubs” counteracts that tendency and encourages students to prove statements by relying solely on the axioms.

I should note that while the GeT course that I teach centers around axioms and the Euclidean archetype [3], this activity neither requires nor expects students to possess a deep understanding of axioms.  In fact, I typically use it as a “first day” problem to introduce them to axioms and the desirable properties of axiom systems.  I have also used the problem successfully in a lower-level geometry course for non-majors.  On the other hand, the problem allows for further deep investigation; the clubs in the town of Hilbert have a rich structure!

“People and Clubs” offers students opportunities to model, to construct viable arguments, and to reason abstractly, processes highly valued by our community and advocated by educators. [1] Students model the problem in several different ways.  Some use letters (or numbers) to represent the n people in Hilbert, and sets of those letters as clubs (e.g. “BDE” as a three person club).  Others provide visual representations with people as dots, containing ovals showing club membership.   

Regardless of representation, however, students are generally able to dispose of the n=2 population rapidly since determining its viability does not require the use of Axiom H4.  Students tend to find that fourth axiom thorny, in part because the statement involves both the existence and uniqueness of a particular club.  When they understand this language, they are better able to see the logical difficulties with a 3-person town.  In our class discussion, I challenge them with several different examples of impossible 3-person systems (see the picture below).  For example, the system consisting of strictly two person clubs illustrated in the third example violates the “existence” clause of Axiom H4: Given club AB and person C, there is no club that contains C that has no members in common with AB.  Challenging students with examples such as this encourages them to make explicit their use of the axioms—a skill that becomes useful when investigating larger Hilbert populations and that is critical when considering the extreme cases of n=0 and n=1. 

Perhaps the most interesting situation that we consider in class is a 5-person Hilbert. A careful analysis of the smaller population situations pays off here because a direct proof of the impossibility of the n=5 population requires several cases.  In my classes, students are often initially uncertain about whether a town of this size is possible.  Many believe that a system consisting entirely of 2-person clubs is possible, as it is in an n=4 person town.  Others may think that at least one of the other 5-person systems is valid.  The identification and disposal of the cases offers opportunities for an intellectually satisfying discussion.  For example, a system of 2-person clubs violates the “uniqueness” clause of Axiom H4:  Given club AB and person C, there are two clubs (CD and CE) that contain C and have no members in common with AB.  The table below illustrates some (but not all!) of the possible 5-person club systems and where they go wrong:

Discussion of the “People and Clubs” problem typically takes about one class period, though I often ask my students to spend an evening considering the 5-person Hilbert situation.  The activity works well as a standalone problem, and GeT instructors who do not teach an axiom focused course may wish to stop here.  On the other hand, a deeper analysis of the problem in a GeT course can pay dividends throughout the semester in several ways.  First, it can help students identify important general properties of axiomatic systems.  For example, adding the axiom “(H5): There are five people in Hilbert,” quickly leads us to the notion of consistency.  On the other hand, “(H5): There are not five people in Hilbert,” creates a redundant axiom system since we can prove that statement.

Second, the problem offers meaningful examples of models of an axiomatic system.  This is particularly helpful when discussing independence of axioms: an axiom “(H5) There are four people in Hilbert” is independent since we can find models of (H1)-(H4) where the (H5) holds (the n=4 Hilbert case), and other models (e.g., n=0, 1) where (H5) fails.  Finally, “People and Clubs” supports further mathematical exploration and conversation. The trivial case n=0, for example, provides an occasion to highlight the importance of precise language and interpretation of quantifiers.  For deeper exploration, I will often have students try to prove (or disprove) statements that rely heavily on Axiom (H4).  For example, if clubs X and Y have no members in common and Y and a third club Z have no members in common, then can we conclude that X and Z have no members in common?  A series of questions such as this, perhaps posed as true/false statements, affords opportunities for students to practice making arguments within the system and in refining general proof skills.  Other potential questions are included below.

References

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
2. Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman and Company, 2008. 
3. Grover, B.W., Connor, J. Characteristics of the College Geometry Course for Preservice Secondary Teachers. Journal of Mathematics Teacher Education 3, 47–67 (2000). https://doi.org/10.1023/A:1009921628065.
4. Reid, C. (1970). Tables, Chairs, and Beer Mugs. In Hilbert (pp. 57-64). Springer, Berlin, Heidelberg.