Since I have spent the summer studying mathematics at Harvard University with Jameel Al-Aidroos (Ph.D Berkeley), expect that my next few posts, or at least some of them, will be on topics related to mathematics. I want to take some time, in this blog post, to look at where mathematical thought fits into some of my understandings of I have gleaned from studying the history of science. The upshot of the historical, philosophical, and mathematical content and musings will be pedagogical, just to give the reader some idea of where I am going. An important thing to understand, before reading this post, is the distinction between pure and applied mathematics. “Pure mathematics,” as opposed to “applied mathematics,” is, in its essence, math for its own sake, entirely apart from possible applications. In many cases, pure mathematics initially has no known application. Additionally, pure mathematics deals with abstract entities that have been detached from particular entities —and this will prove to be important to what I will say later.
Part of what made this summer’s studies interesting is that, while educated in advanced mathematics, none of us (save for Jameel) had any exposure to graph theory. What sprung from this was a rather interesting set of realizations. I had begun reading, more for fun than anything, Richard J. Trudeau’s Introduction to Graph Theory, which I can confidently recommend to all, whether mathematically minded or not. In his introduction, he explains that he feels the problem with mathematics education is that most people are overwhelmed by real-world contexts that students of math become alienated; and that “pure mathematics” is what individuals learning need to be introduced to, to avoid being alienated and developing the sentiment of “I am just not a math person.” What I realized in our development of graph theoretic concepts, reconstructing graph theory from the ground up (in a method of study called inquiry-based learning), was that the mathematical process was essentially empirical. In order to figure out what general rules govern planar graphs, for instance, there was no other way than to take markers to the whiteboard, or rubber bands and pins to cardboard, or whatever material construction you choose, and come up with ideas that seem to be rules, establish that they were not merely contingent to the particular set of graphs empirically constructed, and generalize through social intercourse, wherein other mathematicians try to present counter examples. Having studied mathematics as an undergraduate, the process I described above is strikingly different from the process I had previously encountered. The difference is in the packaging and in the metaphysical outlook of those who repackaged the mathematics. The repackaging done is from the ground up, typically referred to as axiomatized mathematics. The metaphysical outlook, quietly pushed up undergrads and some high school students, is that the world has a mathematical structure, which is grounded on first principles, the axioms and the rules of relations. Empiricism doesn’t work that way. The essence of this idea is probably best articulated —and very simply, at that— by Richard P. Feynman, who referred to the messy real-world process of problem solving the “Babylonian method.” As Feynman puts it on page 46 (Chapter 2) of The Character of Physical Law (1985, twelfth printing by MIT Press), “The Babylonian attitude —or what I call Babylonian mathematics— is that you know all of the various theorems and many of the connections between, but you never fully realize that it could all come from a bunch of axioms.”
It’s really not that surprising that “Babylonian” is how Feynman described this mindset, method, what have you. When you do any kind of initial research, which is inherently going to be inquiry based, as all research is, the theorems that you come up with in math will be a mess of theorems that might not even appear to be related. Especially in constructing setups in graph theory, the facts about the nature of a type of graph seem wildly contingent, though they hold for a class of graph. As a result, I have developed this anxious feeling when thinking about math, the feeling that contingency and process reign supreme in mathematics, and that the organization of branches of math into an axiomatic setup is secondary, perhaps artificial. As it is, the further we delve into an area of mathematical research, we often have to revise our axiomatic setup. For example, Euclid’s fifth postulate, qua axiom, disappears from the horizon in the history of mathematics in the most general generalization of geometry. By the lights of the most general arrangement of mathematics, there is something very alien about the empirical nature of mathematics research. We abstract the empirical process from the consequents, in axiomatizing any branch of mathematics. There even seem to be levels of abstraction that go on. Consider the problem of the bridges of Kӧnigsberg. If you are not familiar with it, a brief glance at the Wikipedia page will suffice in bringing you up to speed. The bridges are abstracted into edges (lines) of a graph, the islands and land, vertices (dots). In doing this, I think some mathematicians feel as though all empirical and real-world content has been removed from the mathematical inquiry, but it has not: the process of discovering theorems in graph theory is still empirical, as I described above. The generalization of an idea from a particular graph across all graphs is another level (kind?) of abstraction. This is all still very Babylonian. We have a pile of theorems, and so it is when we reorganize consequents of this process that we must either say that the reorganization is either heuristic or metaphysical, which is to say, for the sake of learning or for the sake of reflecting nature’s real underlying order. Typically, we don’t see “heuristic” appended as the adjectival construction. Maybe we don’t see any explicit qualification noting the reorganization of the mathematical truths and components; but it does arise in commentary elsewhere, as in “the order of nature” or “our mathematical universe.” The reorganization has to have been done for some reason. My concern —and the upshot of this post— is that there has been some confusion between the metaphysical claim that the world and math are axiomatically structured, and the separate claim that an axiomatized ordering of mathematics is heuristically valuable.
There is no reason for the association of these claims, that a metaphysical ordering is heuristically valuable. In my experience, the empirical process is a much smoother educational process than axiomatic learning, the only —and I mean, the ONLY— exception having been Euclid’s Elements. Aside from doing graph theory, I have also done a combinatorics course, back when I was an undergrad at the University of Pittsburgh, under Whitehead, using Kennth Bogart’s “discovery method.” The discovery method was a little lighter on inquiry, but the leading-questions, as one would lead a witness in court, were highly, highly effective. The engaged mind learns, from what I have experienced, and both Al-Aidroos’ and Bogart’s methods do this. The salient point is that these methods are antipodal to the metaphysical bias of axiom-first approaches. All of this leads me to suggest that the nature function of the mind is to deal with messes, pull apart highly tangled situations, solve problems that are messes, and to engage in a way that heightens one’s own interest in the matters at hand. After all, one cannot think on matters that one cannot care a lick about. And so, I leave it to the reader to consider the impact of the discussed metaphysical assumption that has snuck into mathematics and science education. In my mind, Socrates may not have been on the right track in the Meno, in his discussion of where knowledge comes from and its nature, but he certainly knew how to teach, using leading-questions and an empirical process. I think this is the heart of human learning, and the axiomatic process, though useful, perhaps, for other things, has hindered STEM education, particularly mathematics education. Historically, people learn via the messy empirical process, the Babylonian mindset in tow.
 I can also recommend a number of other texts in graph theory. Since most of my readership comes from a more philosophical knowledge context or generally intellectual background, another book I’d immediately suggest looking into is Chartrand’s Introductory Graph Theory. Trudeau’s is a bit less rigorous and approachable, though its approach is from the standpoint of pure mathematics, which may turn off a few, though I think it is super easy to understand; and Chartrand’s approach is applied mathematics, focusing on games and real life situations. I have Amazon reviews for both of these, if you’d like more information.
 There is supporting evidence for these claims in science, by way of the history of science, as well. I am at the initial stages of toying with an article that feature’s the research of Domenico Bertoloni Meli’s Thinking with Objects. That humans learn in a particular way through research in the history of science is philosophically relevant to the educational process. This goes well beyond the mere “recapitulation theory” in education theory.