Friday, 9 August 2013

Why a mathematician will be talking to a theologian

As part of the Edinburgh Fringe's Cabaret of Dangerous Ideas I will be interviewed by a theologian.  My wife was a bit perplexed by this and wanted an explanation.

When I was preparing to teach derivative pricing to "sophmore" actuarial students, I thought it might be useful to talk a little about the early development of probability theory.  I think I was aware that there were links between probability and commerce, but I did not know the details.  When I read Anders Hald's history it struck me that the canonical origins of mathematical probability, in the Pascal-Fermat solution to the Problem of Points was formally equivalent to the Cox-Ross-Rubenstein model, this I discovered is Whiggish.  There was one question relating to the different approaches, the way the probabilities are calculated.

Conventionally people assume probabilities are calculated by counting frequencies of events (the chance of rolling a six with a dice in 1/6, because there is one six with six alternatives, etc.), in financial mathematics we use probabilities without reference to history.  This is difficult for many who learn(t) derivative pricing late in life after years of education about "what" conventional probability "is".  My initial assumption was that Pascal-Fermat were calculating probabilities in the conventional way, but I thought it worth checking.

At this point I was a typical Atheist British Empiricist with little interest in ethics or theology, and then I found Edith Dudley Sylla's work on the early development of probability where she argues that probability was developed in an ethical context, it represented fairness (equality) in the context of Aristotelian "restorative" justice.  This was significant in that Sylla argues that the frequentest interpretation of probability emerged later in the eighteenth century, along side Sylla,  James Franklin provides a comprehensive account of the early development of Probability in the context of the Humanities (law and theology).  The question that occurred to me, was was there an ethical interpretation of contemporary financial mathematics.  I have concluded that there is, and it rests of the equivalence of the existence of a "risk neutral pricing measure" with an absence of arbitrage, the potential to make riskless profits, i.e. scholastic usury.

I had absorbed a lot of material in arriving at this conclusion, and the next question to appear in my mind is what had changed in the three hundred years between Pascal-Fermat and Cox-Ross-Rubenstein that meant today it is unorthodox to think there is an ethical basis to a key mathematical theory, which was something as a surprise to me.  This is less Whiggish and led me to do a (shallow) review of the development of ethical theory and epistemological theory in the intervening centuries.

Deirdre McCloskey's advocacy of Virtue Ethics, which dominated ethical thinking up until the nineteenth century, in the context of economics proved useful as a starting point on the first question.  From this perspective I explored Realist and Empiricist epistemological perspectives.

The Realist/Empiricist categorisation appeared incomplete as it did not fit with my Intuitive approach to mathematics (Intuition relates to experience, but not in the same way as Empiricism), which follows principally from Poincare:
The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.
These ideas remained a bit of a stew until I returned to Pragmatism through Hilary Putnam.  Putnam  argues that ethics can be "objective" and incorporated into "science", it is a "false dichotomy" to separate the two.

This whole thesis is presented in A Context for Financial Economics: Ethics in the Face of Uncertainty

The topic of the discussion (chosen independently)  is not ethics in finance but "Rethinking the Rules" of contemporary science, essentially how my ideas of ethics in finance can inform contemporary science.  without prejudicing Prof Fergusson's interview, I think the starting point is the Pragmatic idea that ethics in science should be explicit, not implicit as at the moment.

Sociologists make this point because they see science as embedded in culture, and so morality is implicit in all science.  An example is the implicit foundation of the Fundamental Theorem of Asset Pricing in the moral concept of Justice, it is so "hidden" that people do not appreciate it out of a long historical context.  My work is interesting to sociologists because I have "stumble" on this point, rather than constructing the argument.

I think my arguments are relevant to contemporary scientific debate in based on uncertainty.  I regard the significant problems facing science are about an uncertain future.  The nineteenth century developed science to solve (broadly) deterministic problems.  This started looking precarious at the end of the nineteenth century and was undermined completely during the First World War.  Science responded and showed itself capable of addressing "randomness" in the Second World War, exemplified by code breaking: transforming meaningless streams of letters into information.  However, since the reordering of global politics at about the same time as the collapse of Bretton-Woods, science has looked less capable to actually solve problems in the face of uncertainty.

I think finance provides a laboratory for science in the face of uncertainty because randomness is the key feature of the domain.  Developing "good science" in the context of finance should help us develop methodologies for dealing with problems of uncertainty in other domains.  I am not suggesting that financial economics is better than other sciences, but as an  environment to develop the science of the future it is probably better than super-facilities such as the LHC or conventional laboratories.  By focusing on what sort of science needs developing in regard to finance, I think the "certainties" of Realist and Empiricist approaches will have to be abandoned, and a more inclusive approach based on Pragmatic principles will emerge.  In particular there will be an acknowledgement of the contingency of knowledge and so an emphasis on risk management in the process of achieving objectives, rather than identifying the ideal objective.  Putnam notes that Pragmatism emphasises that ends need to be flexible, not just the means to an end.

For example, finance/economics is riven into neo-classical and behaviourist approaches, these can be categorised (respectively) as Realist and Empiricist camps.  In the context of finance Pragmatism might conclude that there is no single model to value assets, but there will be different approaches and tools in different circumstances.  One issue is being conscious of the weaknesses of a particular methodology, and being prepared to abandon it or to not engage in an activity that is "uncertain".  Because knowledge is contingent these decisions are fundamentally ethical.

Similarly the climate change debate is more explicitly divided into climate change advocates and sceptics, which can again be seen as representing Realists and Empiricists respectively.  Because there is a deep philosophical divide between the two camps it might prove difficult for a consensus to emerge.  Acknowledging the contingency of knowledge challenges the principle assumptions of Realism, that "Truth" exists out there, and Empicism, that "Truth" will be discovered through data.  Accepting that the "Truth" of climate change cannot be established should shift the debate onto one not of what should be done but "how lucky do you think you are", i.e. do we wish to risk the possibility of environmental catastrophe for immediate cheap fuel.  I don't know the answer to this question, but I do think it requires the participation of all of society to arrive at a consensus, not just scientists.

Early in my exploration of the origins of probability I chanced (literally) on Joel Kaye's argument that western science originates in the scholastic analysis of the ethics of markets.  The figure of Pierre Jean Olivi stands out, for me, in this narrative.  As a Spiritual Franciscan, arguing for the limited use of resources,  he might today be described as a "Red-Green" socialist environmentalist, yet he was positive towards contemporary commerce and instrumental in the early development of probability.  Also, one of the key differences between the Franciscans and Dominicans was that while the Dominicans might regard the universe as a machine designed by God, who left it to run according to immutable laws of nature, the Franciscans argued that God could interfere with nature (through miracles) and that nature was not immutable, and this feature needed to be incorporated into science.

1. I think the problem science with probability is that it treats time as a vector from past to future, rather then the process by which future becomes past, ie. tomorrow becomes yesterday because the earth rotates. Past and future do not ontologically exist, so neither are really "cut in stone." Events are subject to interpretation as they are occurring and this only becomes more pronounced the more they recede into the past. All that can really be said is "somthing happened," but details are relative. As for what does happen, while the laws deciding outcome are deterministic, total input cannot be known prior to the event occurring. While one could argue all this input is ontologically real and thus the outcome is determined, it should be remembered that prior and succeeding states are not physically real, so the very notion of determinism is based on an epistemic concept. There is no such thing as a "God's eye view," or objective knowledge, as knowledge is a function of distinction and distinction is a consequence of limitation. The theological presumption is that a spiritual absolute would be an intellectual and moral ideal from which we fell, but absolute is basis, not apex, so a spiritual absolute would be the essence of being from which we rise, not an ideal from which we fell. This elemental awareness cycles through expanding perception and concolidating knowledge, slowly bulding to where we are now.
On phone, so leave it at this....

2. How did it go?

I agree with the agenda, but I fear that a narrowly pragmatic approach wouldn't have been of much help 2005-2010.

As I see it, the rules of science and economics are that a dominant theory is developed, which it is very hard to question. (Depending on the editors of the relevant journals). Science and economics being empirical, it is inevitable that sooner or later one comes across an exception, which can confuse the established experts. For technology-focused science, this doesn't matter too much: academia has a few exciting years and then establishes a new dogma. But, just sometimes, it can be really important.

An alternative approach to science would be to encourage the development of the widest range of theories consistent with evidence. One could then have a variant of the precautionary principle, in which one seeks to act in ways that no credible theory regards as risky.