On the Unity of Science: Law, Finance and the Philosophy of Category Theory

1.  We should measure our progress as a species of knowledge by how well individual disciplines meet the criterion of the unity of science.  This criterion was first stated as an almost urgent request by Edward O. Wilson with his concept of consilience.

2.  There are so many analogies between one subject and another, the vocabularies (as objects may be different), but the way in which these vocabularies are used (the morphisms) are so similar that they might as well be said to be the same.  What do we mean by "same" or better, if we capture the water colourist wash, and call it by its technical name, "weakened equivalence"?

3.  Mazur (June 12, 2007) in "When is one thing equal to some other thing," htt://www.math.harvard.edu/%7Emazur/preprints/when_is_one.pdf writes in tribute to one of the founders of category theory, Mac Lane, sets out an approachable essay on the question of the meaning of equivalence.  This is the deep point where all our equations and assertions in science sink to.  He sets out three approaches of how we have answered this question.

4.   The first is the "bureau of standards" where by convention we can point to something in a designated office that is an equivalent exemplar. [Id @ 4-5.]  The second is a type of universal quantification as in Frege's definition of cardinality. [Id @ 5.]  And the third is a compromise where "we indicate what we do rather than what we say we do when quizzed about our foundations." [Ibid.]  I call this third method a promiscuous stitching, using the same needle and thread or glue may be all we need to make appropriate connections between subjects, disciplines and fields of knowledge. 

5.  In mathematics, you can "package" entire mathematical theories either as (1) formal systems a la the David Hilbert programme or as (2) categories. [Id @6.]  On the one hand, the formal systems go all the way back to Euclid and are much admired under the rubric of axiomatization.  On the other hand, categories are a relatively recent invention (1945 with a paper by Samuel Eilenberg and Saunders Mac Lane was more an announcement of new technique than a new view of mathematics) and its method of a sparse vocabulary and sketches of arrows betrays its deep goal which is to reveal structure.  Mathematicians, like others engaged in doing or performing in their particular discourse, don't really "axiomize" but rather "play games with conviction." 

6.  Somewhere in "Tool and Object, A History and Philosophy of Category Theory," (2007), Kromer quotes Bill Lawvere (my preferred radical category theorist) for saying something to the effect, "the point is not to achieve maximal abstraction, but an optimal abstraction, a just-right abstraction that works appropriately at the level where it is most needed and used."  Of course, I am attributing a certain line of argument to Lawvere which I do not think he would disagree with.  Lawvere was motivated to find a theory of physics, to explain how things worked, but his work on the philosophy of category theory takes him on exoduses into Hegel.  I believe it is this urge to find "synthesis" with simple tools that motivates him.  He has been accused of being both revolutionary and idiosyncratic.  Revolutionary for advocating that all of mathematics can be thought of as a category of category theory.  And idiosyncratic because for such a great mathematician, he and Schanuel wrote a best selling book entitled, Conceptual Mathematics, A First Introduction to Category Theory (2009 2nd Edition) wherein you don't need any university level mathematics to understand.  In fact, I recommend this book to all my law and finance students who are interested in pursuing the application of category theory to the field of law and finance.

7.  The main points about the test for the unity of science (consilience) is that the most appropriate method for pursuing a rigorous apprehension of science (i.e., the three approaches stated by Mazur: bureaucratic standard, universal axiomization or balanced compromse (I call "promiscuous stitching") may be something so natural and simple that even our high school students can be engaged in this entreprise.

8.  For theories of law and finance, we see that there has been an influence of the latest trend or fashion from other fields that have filtered into the vocabulary of the legal theorist or financial theorist.  For example in the last 5 to 10 years, in both fields there is an emphasis at least in the titles to papers on the concept of "complexity" and "behaviours."  This is not to say these concepts are  red-herrings.  From my view, they are just another batch of ideas that come from a few equations.  Another example, what would Hart's programme of primary and secondary laws be without the notions of first order and second order logics emanating from the Cambridge logicians in the early part of the 20th century?  Not that Hart genuinely meant to implement the same programme, but the inspiration for an orderly resolution of the definition of the meaning of law was certainly intended to take the script from the philosophy department--and these were the ideas pre-Wittengstein.  In finance, the initial idea of covariance goes back to Bachelier's PhD dissertation (1905) and then developed as various methods for "curve fitting" against time horizons.  Very little work has been done on how the various theories of law and finance might be approached in a unified way.  But here the stumbling block may have been the limited view on the number of approaches to reach unification.  I do not mean by "unification" a form of axiomatiion or foundational premises evolved and expanded in a universalistic sense that may have endeared Spinoza.  Rather we have a very powerful alternative which is a kind of Kantian insight that the intellectual revolution begins with a recognition that

"There are only two possible ways in which synthetic representations and their objects...can meet one another.  Either the object (Genenstand) alone must make the representation possible, or the representation alone must make the object possible." [quoted from Mazur supra @ 20.]

9.  In law and finance theory, I would (and will) argue that one of the significant leaps in our imagination of how law and finance work together is to recognize the structure of something called "default invariance."  This is captured by or very conveniently set out with a category theory approach.  Default invariance permeates all financial contracts, and all states of the financial-regulatory-political system.