The mathematician Felix Klein: “The abstract formulation is excellently suited for the elaboration of proofs, but it is clearly not suited for finding new ideas and methods; rather, it constitutes the end of a previous development.” Such a view, Dirk Schlimm argues, is common among philosophers of science as well as mathematicians and other practitioners of axiomatic science (like economic theory). But is axiomatics limited to formalization, to consolidation, or can the axiomatic method be a creative act, one that opens up new venues and suggests new ideas? Given the emphasis on this site of the explanatory value of theory, it will come as no surprise that I see axiomatics as fundamentally creative. The author of the present paper agrees, diagramming the interesting history of the mathematic idea of a lattice.
Lattices are wholly familiar to economists at this stage, but it is worth recapping that they can be formulated in two identical ways: either as a set of elements plus two operations satisfying commutative, associative and absorption laws, which together ensure the set of elements is a partially ordered set (the standard “axiomatic” definition), or else as a set in which each subset has a well-defined infimum and supremum, from which the meet and join operators can be defined and shown to satisfy the laws mentioned above. We use lattices all the time in economic theory: proofs involving preferences, generally a poset, are an obvious example, but also results using monotone comparative statics, among many others. In mathematics more generally, proofs using lattices unify results in a huge number of fields: number theory, projective geometry, abstract algebra and group theory, logic, and many more.
With all these great uses of lattice theory, you might imagine early results proved these important connections between fields, and that the axiomatic definition merely consolidated precisely what was assumed about lattices, ensuring we know the minimum number of things we need to assume. This is not the case at all.
Ernst Schroder, in the late 19th century, noted a mistake in a claim by CS Peirce concerning the axioms of Boolean algebra (algebra with 0 and 1 only). In particular, one of the two distributive laws – say, a+bc=(a+b)(a+c) – turns out to be completely independent from the other standard axioms. In other interesting areas of group theory, Schroder noticed that the distributive axiom was not satisfied, though other axioms of Boolean algebra were. This led him to list what would be the axioms of lattices as something interesting in their own right. That is, work on axiomatizing one area, Boolean algebra, led to an interesting subset of axioms in another area, with the second axiomatization being fundamentally creative.
Dedekind (of the famous cuts), around the same time, also wrote down the axioms for a lattice while considering properties of least common multiples and greatest common divisors in number theory. He listed a set of properties held by lcms and gcds, and noted that distributive laws did not hold for those operations. He then notes a number of interesting other mathematical structures which are described by those properties if taken as axioms: ideals, fields, points in n-dimensional space, etc. Again, this is creativity stemming from axiomatization. Dedekind was unable to find much further use for this line of reasoning in his own field, algebraic number theory, however.
Little was done on lattices until the 1930s; perhaps this is not surprising, as the set theory revolution hit math after the turn of the century, and modern uses of lattices are most common when we deal with ordered sets. Karl Menger (son of the economist, I believe) wrote a common axiomatization of projective and affine geometries, mentioning that only the 6th axiom separates the two, suggesting that further modification of that axiom may suggest interesting new geometries, a creative insight not available without axiomatization. Albert Bennett, unaware of earlier work, rediscovered the axiom of the lattice, and more interestingly listed dozens of novel connections and uses for the idea that are made clear from the axioms. Oystein Ore in the 1930s showed that the axiomatization of a lattice is equivalent to a partial order relation, and showed that it is in a sense as useful a generalization of algebraic structure as you might get. (Interesting for Paul Samuelson hagiographers: the preference relation foundation of utility theory was really cutting edge math in the late 1930s! Mathematical tools to deal with utility in such a modern way literally did not exist before Samuelson’s era.)
I skip many other interesting mathematicians who helped develop the theory, of which much more detail is available in the linked paper. The examples above, Schlimm claims, essentially filter down to three creative purposes served by axiomatics. First, axioms analogize, suggesting the similarity of different domains, leading to a more general set of axioms encompassing those smaller sets, leading to investigation of the resulting larger domain – Aristotle in Analytica Posteriora 1.5 makes precisely this argument. Second, axioms guide the discovery of similar domains that were not, without axiomatization, thought to be similar. Third, axioms suggest modification of an axiom or two, leading to a newly defined domain from the modified axioms which might also be of interest. I can see all three of these creative acts in economic areas like decision theory. Certainly for the theorist working in axiomatic systems, it is worth keeping an open mind for creative, rather than summary, uses of such a tool.
http://axiom.vu.nl/cmsone/SchlimmOnline.pdf (2009 working paper – final version in Synthese 183)
A Fine Theorem 