Values and Axioms

The danger arises when a culture takes its own story as the absolute truth, and seeks to impose this truth on others as the yardstick for all knowledge and belief. We should never forget that, at their deepest level, all questions, all searches for knowledge, by they “scientific”, “mystical”, “philosophical”, or “religious” point to the same truch, but often in profoundly different ways. From Certainty to Uncertainty - F. David Peat

There is a long-running debate about hierarchies of purity in science. At the top is mathematic, the cold, hard science of logic and numbers. Physics sits below mathematics, being founded in the rigour of mathematics and focused on the very small and very large scales, but is constrained by an uncertain reality. Chemistry and biology follow physics because they study the world at (or close to) scales observable by humans. Engineering and the social sciences are ersatz shadows of the physical sciences. But is the purity of mathematics an illusion? If so, what does it suggest for other sciences and fields of study?

The philosopher Bertrand Russell believed in the certainty of mathematics. Along with his colleague A.N. Whitehead, he set out to establish its logical foundations with the results summarized in their two volume Principia Mathematica. They defined their proofs in terms of a system of signs, a gammar (i.e., set of rules) for combing signs into formulae, transformation rules, axioms, and proofs involving a finite set of formulas. Axioms are the fundamental building blocks of mathematics, with their roots in ancient Greece. Euclid compiled a set of fundamental units - points, lines, planes, etc. - that defined the basis for geometry. Axioms should be self-evident and not rely on any other assumptions. Proofs are then built from these axioms, and nothing else. Following Russell and Whitehead, mathematicians were undecided as to whether the field had been proven complete and consistent. However, a 25 year old Wiener named Kurt Gödel showed the limits of the axiomatic method in his paper “On Formally Undecided Propositions of Principia Mathematica and Related Systems”. He was able to show that true statements exist in mathematics that cannot be proven: that mathematics is incomplete. One example of an unprovable truth termed Goldbach’s Conjecture states that “every even number is the sum of two primes”. For example, 20 = 17 + 3 and 8 = 7 + 1. No exception has yet been found to this conjecture, but no one has been able to prove it. Even should someone find a proof, the result would be the number of axioms being increased by one and further conjectures remaining by Gödel’s Theorem.

So why does any of this matter to applied and social scientists? My point is to highlight that, if even the pure white cloth of mathematical logic has shades of grey, then how much more will the messier applied natural and social sciences face the spectre of uncertainty? Laws of nature are immutable but constantly evolving as we gain new knowledge. Gravity is a law of nature defined by mathematical terms developed by Newton but expanded upon and refined by Einstein. Laws of social behaviour are by their nature constructs of a particular social (i.e., moral) system. For illustration, let us consider the consumption of goods central to microeconomic analysis. The classical economic calculus says that a good has a monetary price, which an individual considers subject to a constraint imposed by their income. Their goal is to maximize the utility of goods consumption subject to this income constraint. Let us examine the assumptions of this statement. First, do all goods have a monetary price? Certainly not! I can take a tree branch and transform it into a walking staff, fence post, or any number of other useful goods without any expenditure of money. Until the last 300 years, most households produced the majority of their own goods and services outside the economic market. I recall submitting a grant proposal on consumption subject to natural resource constraints. The response from reviewers was that individuals do not consider resources when making consumption decisions, as though this behaviour were an immutable feature of human decision-making and not a feature of the prevailing social system being prefaced on monetary markets. For most of human history, resource considerations were the prevailing constraint on consumption as goods could not be shipped over long distances at low cost. Within the blip of human existence on Earth, the centering of income constrained consumption is a sub-blip of inperceivable duration. Not a fundamental law of nature - just a social construct that may be altered through a change in moral perspective. In the Dawn of Everything, David Graeber and David Wengrow discuss the historical and cross-cultural instability of social axioms. They note the presence of “a mythical substructure to our ‘social sciences’- what appear as unassailable axioms, the stable points around which our self-knowledge is organized, are scattering like mice.”. However, as they note, myth is not a problem in its own right; it is “the way in which human societies give structure and meaning to experience”. The problem is the assumption that the current myths are correct and axiomatic. Again citing Graeber and Wengrow, “Max Planck once remarked that new scientific truths don’t replace old ones by convincing established scientists that they were wrong; they do so because proponents of the older theory eventually die, and generations that follow find the new truths and theories to be familiar, obvious even”. This is a reminder that research as an objective and value-free process for observing and understanding natural and human systems is taken for granted by many researchers. What then is the purpose of new knowledge (or re-discover old knowledge), if not to create new and (or re-create) different forms of social reality?

I was struck by Linda Tuhiwai Smith’s framing of the problem around methods in “Decolonizing Methodologies: Research and Indigenous Peoples”. In the same way that different cultures interpret good and moral behaviour in their diverse ways, different fields of study have their own methods for approaching research problems. As noted many times on this blog, I am trained as an engineer but conduct research on social problems. Some social scientists might (and have) say that I am referencing the wrong literature or using the wrong method. My perspective is that I conduct social inquiry through an engineering lens; at times using different methods but always grounded in the objective of a better society from the intersectional perspective of my Western cultural and engineering background. Tuhiwai Smith brings an Indigeneous New Zealand perspective to the discussion by elevating Indigenous ‘ways of knowing’ that may differ from Western science. While the scientific method takes ‘on faith’ its axioms as certain and foundational, other knowledge systems may rely on other assumptions (termed ‘myths’ by Graeber and Wengrow). At its core, the debate is over the lens through which one should view human society and its position within broader ecosystems, and the corpus of knowledge that validates that perspective. Quoting Tuhiwai Smith, it is a dispute over “the validity of scientific methods within the positivist paradigm, and whether this is an appropriate paradigm for understanding human society.”