William Thurston, the Fields Medalist who made major contributions to low-dimensional topology, died last week. He proved that one could turn a sphere inside out without tearing it up, and made the now proven geometrization conjecture which states that every 3-manifold has a prime decomposition. Among other things this implies that the Poincaré conjecture is a theorem. Thurston wrote a very interesting tract on mathematics. He says that the business of mathematics is not to prove theorems per se, but to come up with ideas so that we understand why something is true. The key to the trade is these organizing principles, underlying ideas that make it intelligible. The business of mathematics is to come up with more and more refined ideas of making sense of the structure we discover.
Although he was talking about math, this is literally true of what used to be called natural philosophy. In this post I want to highlight the key organizing principles in the disciplines I am familiar with. We will be concerned with the hierarchy of the schemes of explanation. Girdle up for a preposterously ambitious post.
The intelligibility of nature
How do we know that the sun will rise tomorrow? Mere observation that this is what the sun does everyday is insufficient. The turkey being fattened for Thanksgiving can say that its human captors will continue to feed and take good care of it, until the fateful day when it is slaughtered. No, we know that the sun will rise tomorrow because we have a model of the planetary system. Regularity itself is insufficient. We need an explanation. The meta assumption that all phenomena is in principle explicable is foundational. It is not even clear what it would mean to give up this premise. Philosophers have given fancy names to it but the idea is simple. If there were any doubts about this–for a while there pure empiricists had people fooled–they have been laid to rest by the cumulative achievements of science.
What does it mean to give a coherent explanation of a phenomena? If one has accounted for the entire history of all the particles in the universe, could one predict the flash crash on the New York stock exchange? The dynamics of the market are not independent of the physics but physical principles cannot explain them: the explanation lies at a higher level. There is a hierarchy of explanatory schemes. Different models and frameworks apply at different levels. They are irreducible to lower levels. What is this hierarchy?
In the first half of the last century, mathematicians tried to reduce math to logic, culminating in the famous Russell-Whitehead project. Gödel’s impossibility theorem put that project to rest forever: math just cannot be reduced to logic. Mathematicians no longer think of their trade as peddling in mathematical objects that exist eternally in a platonic realm. Modern mathematicians realized that what they actually care about is structure, the relationships between things. The notion of equivalence is that of an isomorphism: to say that X is equivalent to Y, one needs to first clarify what category one is talking about: there are isomorphisms of sets, ordered sets, vector spaces, topological manifolds, smooth manifolds, manifolds equipped with a notion of distance (i.e., a metric), groups, and so on and so forth.
The hierarchy of schemata spans the entire gamut of human knowledge and we shall not attempt to map it explicitly. Rather we shall explore organizing principles that connect the edifice and thereby try to gain some sense of how it all fits together.
In math and physics, a key organizing principle is symmetry. Finding symmetries invariably makes an otherwise complex problem tractable. The reason why black holes were predicted was because these are highly symmetrical exact solutions to Einstein’s field equations. They were studied because it was relatively easy to analyse them. Our best models of cosmology–the FLRW spacetimes–are constructed almost explicitly on the symmetries of the observable universe: that the universe appears to be spatially homogeneous (evenly filled with matter) and isotropic (all directions look more or less the same). When Paul Dirac predicted the existence of antiparticles it was because he thought the symmetrical splitting of the second order equation for the electron into a product of first order ones natural. [The first governs the electron and the second governs the positron.]
Symmetries also give rise to conserved quantities and invariants. One can’t hope to solve any system without exploiting these things. No wonder that modern theoretical physics is trying out higher order symmetries to come up with a theory of everything. If there is a central field in math, it is group theory, which is nothing but the study of symmetry. One of the crowning achievements is Galois theory, which reduces hard questions in terms of fields to tractable questions in terms of groups.
Symmetry may be the core organizing principle of math and physics but it is the notion of equilibrium that extends up the hierarchy to economics and political science. A long time ago in a mathematical economics class, my professor wrote a general equilibrium model of an economy on the board. It was a set of four simultaneous equations. He turned to us and declared, “I don’t understand what it means for this economy to not be in an equilibrium.” He was making a crucial point. When we make models of complex systems, all we are doing is formalizing our assumptions. Once we have laid out the assumptions mathematically, we have to accept what springs out as a necessary implication of our premises. What one does with models is analyze the effects of exogenous shocks to state of the system. Does the shock move the system to another equilibrium?
The same analytical law prevails in physics. For instance, once one has specified the equations for a particle–given by an operator like the Dirac operator above–one obtains solutions which are eigenstates. These eigenstates are nothing but equilibria. But this principle is subordinate to another, more basic, principle at the higher levels. We turn now to this crucial piece of the puzzle.
When I wrote about realism, I emphasized how Waltz thinks about anarchy: “International-political systems, like economic markets, are formed by the coaction of self-regarding units…[they] are individualist in origin, spontaneously generated, and unintended. In both systems structures are formed by the coaction of units.” This is absolutely central to the neoclassical framework that now dominates economics and political science. The reason why the equilibria–of essentially game-theoretic models of industries, economies, financial markets, ecologies, and political systems–are relevant is because these are systemic theories. A systemic theory is one where system structure itself–how constituent units sit in relation to one another–accounts for the dynamics of the system. In as much as these models add to our understanding of the real world at all whatsoever, it is from what emerges endogenously from the models.
The appropriate response to seeing a terribly simplified version of reality–which is what these models amount to–is not dismay over the absurd nature of the assumptions. According to the standard rationality assumption in economic theory, two players sitting down to play chess would immediately shake hands and not actually bother playing because the game is deterministic. Despite this absurdity, something worthwhile does emerge from these models: predictions. And it is solely by matching said predictions against empirical reality that we can learn the value of the models as analytical tools. Theoretical models are nothing more than convenient fictions in the theorist’s toolbox.
The second law of thermodynamics generalizes to information theory, probability theory, stochastic analysis and beyond. In physics, it extends the principle of least action, and is carried on mutatis mutandis to information theory: a system left alone will tend to a state of maximal entropy. In probability theory, it lurks behind the law of large numbers and the central limit theorem. In fact, the central limit theorem is a special case of a much more general law. Just as a large sample from a population with a fixed and finite mean and variance approximates a Gaussian distribution, one with only a finite mean fixed over approximates an exponential distribution. Similarly, one obtains the beta distribution over as a limiting distribution, and so on.
In the theory of stochastic processes, the key organizing principles are martingales and the Markov property. A martingale is random process that has the property that whatever value you catch it at, its expected value in the future is precisely that value. These are the constants of random processes. Moreover, every continuous martingale is just brownian motion with a different clock. Whence, the entire theory of stochastic calculus is built around brownian motion. We figured out how to integrate that and that is pretty much all we have when it comes to random processes. A Markov process has the property of being memoryless: every time you catch it it starts over again. One can make predictions for the future of the process based solely on its present state just as well as one could knowing the process’s full history.
The scattering of scholars
All the organizing principles outlined above belong to the nomothetic disciplines: those involved in the discovery of general laws. These three social sciences–economics, political science, and sociology–attempted to replicate the methods and epistemology of Newtonian mechanics. They had physics envy. Opposed to them are the idiographic disciplines like history which emphasize the particularity of social phenomena and the limited utility of generalizations. This partition is a recent development.
For centuries, all knowledge was epistemologically unified. All scholars were called philosophers and it was not until the advent of modern science that there was a divorce between science and philosophy in the late eighteenth century. This is the origin of the “two cultures” [C.P. Snow]. The social sciences were pulled in opposite directions. Rising science on the one hand, and philosophy and history et cetera on the other. Till about the mid-nineteenth century, there was a proliferation of proto-disciplines. Later that century and into the twentieth century, “there took place a standardization of disciplines reducing the number to six widely recognizable ones reflecting the three underlying cleavages in the European world view. The split between the past (history) and the present (economics, political science, and sociology); the split between the Western civilized world (the four above) and the rest of the world (anthropology for primitive people and Oriental studies for non-Western high civilizations); and finally the split valid only for the modern Western world, between the logic of the market (economics), the state (political science), and civil society (sociology).” [Immanuel Wallerstein]
This scattering of scholars was the result of a specific historical conjuncture. It is in no way a natural division required by the hierarchy of schemes of explanation. The divide between current affairs and history is superficial. There is no way to understand current events independent of at least recent history: social, political, and economic systems aren’t Markov processes. Moreover, it is the height of absurdity to talk about the dynamics of an economy without an understanding of the security and political underpinnings of the system. In fact, once one becomes familiar with the dynamics of the system, analysis of national economies is immediately seen to be absurd. The system level is that of the world-economy: national economies are merely subordinate sub-systems. Furthermore, the refusal to look at capitalism as a socio-economic-political system with a specific history blinds us to highly relevant issues. Let’s look at a couple of things that the horses with the blinders are missing and how it warps analysis and policy.
The Minksy model
The dominant doctrine in neoclassical economics has something called the Efficient Market Hypothesis (EMH). It says that all the relevant information is already embodied in the price of a security. If not, there would be arbitrage, and someone would make a lot of money while the price immediately gets corrected. So, in theory, there cannot be an asset price bubble. In practice, there have been very many. Economists of course ignore financial history completely. If one were to pay attention however, something stark jumps out. In his book Manias, Panics, and Crashes, Charles P. Kindleberger takes a deep look at the historical record. What he finds is that booms turn into bubbles only when there is a giant inflow of capital due to radical developments elsewhere in the world-economy. This is the essence of the Minsky model.
To take a recent example, the Asian financial crises in 1997 led to a massive flight of international capital to the United States. There it engorged an already booming Nasdaq exchange. The Nasdaq which was at 1500 right before the Asian crises, soared as capital flowed to the United States, and peaked at above 5000 is March 2000, before crashing precipitously when the bubble burst that month. Now, I studied economics for five years, and have been reading about financial history for a decade. I read the New York Times and the Economist religiously. How is it that something so stark has been missed all together by all the thousand observers I have read? And this is not a one off phenomena. This is the case with all the market manias since the Tulip mania in seventeenth century. How is this not explicitly factored into discussions about financial crises and policy making??
The role of the state
The state either does not show up in economic theories or is relegated to a single figure of total government outlay. The fiction of the free market has wrought untold misery on the inhabitants of this wretched planet. The picture of capitalism is one of entrepreneurs and investors meeting in the capital markets, and consumers and producers meeting in the real markets. There is no geography, no organic linkages, no ecology, and certainly no gunships and artillery. The wealth of the capitalist world is then seen as the result of free and open market societies where the role of the state is limited to provision of public goods. Is that so in reality??
As it actually happened, the state has been at the very center of every single story of national economic achievement. From the very beginnings in fourteenth century Italy, the power of the state was necessary to guarantee the capital accumulation of merchant-financiers. We have already documented the nexus of military and capital power in the next cycle of accumulation when Great bankers of Genoa ruled the world-economy. This is the case in every single century and every single power. The industrial revolution in England in the nineteenth century took place under the active protection of the British navy and the protected markets of the colonies. Moreover, Britain imposed tariffs on Indian textiles and protected Lancashire without which the industrial revolution would not have taken off. We can talk about the role of MITI in Japan and the Pentagon in the United States but let me not belabor the point: the nexus of capital and state power is an iron law of capitalism. [More on this in the next post where we consider at some length the transition from Dutch to British hegemony.]
Back to natural philosophy
By way of conclusion, let me just say this: It is not enough to refine our social, economic and political models and make them more realistic. It is not even enough to integrate them into one edifice. We must unify them again with historical analysis as well. Only then, will we find organizing principles that serve us in constructing a more promising hierarchy of explanatory schemes. Most of all we need to climb out of the narrow intellectual crevices we have dug up for ourselves. This will be a herculean effort but the rewards will be ample. Braudel started his doctoral research on Phillip II’s Mediterranean policy. He then spent three decades trying to understand the Mediterranean world-economy in the sixteenth century. This is how it’s done.