A few years ago I attended a meeting on the applications of mathematics to "energy problems" sponsored by the Engineering and Physical Sciences Research Council and the London Mathematical Society. During one of the presentations a British Nuclear Engineer noted that in France they do not employ probabilistic arguments in favour of nuclear power because probabilistic arguments are too advanced for the public. Two people who did not snigger at the ignorance of the French were myself and the senior probabilist at the University of Oxford. We understood the French position: mathematical probability is a branch of mathematical analysis. The British, by and large, associate probability with statistics, counting things and calculating relative frequencies.
The difference is captured in the different words we use for the English mathematical term 'expectation', the French use the word espérance, with the literal translation of hope. Since the Latin root of expectation is associated with waiting, and in the past 'expect' and 'wait' have been used synonymously (i.e. Dickens's Great Expectations), we can infer that there is a tendency for the English speaker to anticipate a mathematical expectation, where as a French speaker merely hopes for it. Similar issues exist within English for the words 'risk' and 'uncertainty'. An economist will usually interpret a risk as calculable chance, a physicist might view an uncertainty as an error rather than something undependable.
I sometimes distinguish mathematical probability from statistics by using the analogy of hope (Spes, Elpis) and faith (Fides, Pistis). Most theologians agree that faith is necessary because there is doubt: statistics is a necessary part of science because our results are doubtful. Statistics provides a justification for our doubtful claims based on what has happened in the past. Probable also implies provable and true, but where as statistics has an empirical angle, probability has a more metaphysical aspect. This was exemplified in the medieval genesis of probability in the context of jurisprudence where conscience and moral certainty were key issues, issues that were still important in Bernoulli's The Art of Conjecture and through the eighteenth century discussions of the Petersburg game. They disappeared in the post-Laplacian conception of probability.
One of the original purposes of probability was to price contracts and it did this by considering ratios, on which justice is based, this is present in Aristotle's Nicomachean Ethics. The word 'rational' is derived from 'ratio' and is associated with reasons and accounts: justification. Mathematical probability emerged in the mid-seventeenth century as a result of centuries of discussion around the morality of commercial contracts. When Huygens coined the term expectatio in the first treatise of probability he did so in to context of establishing the fair, or just, price of a contract. If we win £10 on a head and £2 on a tail the expectation of the coin toss, its fair price, is £6. We can not 'expect' to win £6 because it will never happen. I argue that this ethical dimension to probability is still implicit in Financial Mathematics, but not recognised.
My main point is, the mathematics of probability and statistics exist because we need to justify our claims. My secondary point is that historically, there was an explicit moral dimension to the process of valuation that became obscured in the nineteenth century, at the same time as episteme came to dominate discourse (before Foucault).
If the question was "Is it episteme or make believe?" I think the question would be wrong. Firstly the dichotomy is false. It is not a choice between "scientific" or "true knowledge" and "make believe" it is a problem of deciding on the best course of action in the uncertain, undependable, world outside a controlled laboratory. Aristotle distinguished episteme, passive knowledge from phronesis, active thinking and I think the issue pertinent to "Research, politics, media and impact" is robust thinking rather than just robust knowledge. The antonym of "make believe" is not "true belief" but "justified belief" and, it is my opinion, that science should focus on the manner of its justifications rather than its results.
Re-emphasising phronesis, with the aim of good, virtuous, living, might be worth considering in light of our experience of the Financial Crisis, which highlighted that scientific knowledge is not as robust as many of us would like to believe. For example, most of the research in Financial Mathematics has been focused on establishing the "true" price of a contract, some of my work is about re-orientating the discipline to focus on the principles that make thinking robust, given that we cannot rely on certain knowledge in an uncertain world. For example, I argue that the origins of Financial Mathematics are in the synthesis of the virtues Faith, Hope and Charity (Caritas, Agape). I am not alone in thinking this way, Rethinking Economics argue that
It is clear that maths and statistics are crucial to our discipline. But all too often students learn to master quantitative methods without ever discussing if and why they should be used, the choice of assumptions and the applicability of results.I interpret this as it is not the tools of mathematics that are important, but how they are employed. This observation is as pertinent to most students and their teachers.
My motivation for these comments originates in a conversation I had before the crisis about the use of mathematics in finance. Basically mathematical models were being used as rhetorical devices: if a trader had a "better" model than the risk manager, they could do what they wanted to; the trader didn't necessarily believe the model but justified their actions. This idea emerged in the Bank of England's testimony to the Parliamentary Commission for Banking standards
unnecessary complexity [of mathematical models] is a recipe for […] ripping off […], in the pulling of the wool over the eyes of the regulators about how much risk is actually on the balance sheet, through complex models.The existence of a model was enough, there is no understanding of how it works, how it justifies.
More generally I sense a connection here to issues around, for example, climate science. As a bystander I feel that the climate debate is one of which camp has the larger collection of peer reviewed papers wins. There seems to be little rational discourse of making a claim, challenging a claim, justifying a claim. The claim, if published, stands as a fact. The problem is that while many academics are judged on their ability to create facts, a collection of facts is no more science than a collection of bricks is a house. Le savant doit ordonner, the role of the academic is to turn the facts into `science'.