June 24, 2015 2 Comments
Reflecting on my recent contribution to the economics ejournal special issue on uncertainty (comments invited), I realised that from a purely mathematical point of view, the current mainstream mathematical view, as expressed by Dawid, could be seen as a very much more accessible version of Keynes’. But there is a difference in expression that can be crucial.
In Keynes’ view ‘probability’ is a very general term, so that it always legitimate to ask about the probability of something. The challenge is to determine the probability, and in particular whether it is just a number. In some usages, as in Kolmogorov, the term probability is reserved for those cases where certain axioms hold. In such cases the answer to a request for a probability might be to say that there isn’t one. This seems safe even if it conflicts with the questioner’s presuppositions about the universality of probabilities. In the instrumentalist view of Dawid, however, suggests that probabilistic methods are tools that can always be used. Thus the probability may exist even if it does not have the significance that one might think and, in particular, it is not appropriate to use it for ‘rational decision making’.
I have often come across seemingly sensible people who use ‘sophisticated mathematics’ in strange ways. I think perhaps they take an instrumentalist view of mathematics as a whole, and not just probability theory. This instrumentalist mathematics reminds me of Keynes’ ‘pseudo-mathematics’. But the key difference is that mathematicians, such as Dawid, know that the usage is only instrumentalist and that there are other questions to be asked. The problem is not the instrumentalist view as such, but the dogma (of at last some) that it is heretical to question widely used instruments.
The financial crises of 2007/8 were partly attributed by Lord Turner to the use of ‘sophisticated mathematics’. From Keynes’ perspective it was the use of pseudo-mathematics. My view is that if it is all you have then even pseudo-mathematics can be quite informative, and hence worthwhile. One just has to remember that it is not ‘proper’ mathematics. In Dawid’s terminology the problem seems to be that the instrumental use of mathematics without any obvious concern for its empirical validity. Indeed, since his notion of validity concerns limiting frequencies, one might say that the problem was the use of an instrument that was stunningly inappropriate to the question at issue.
It has long seemed to me that a similar issue arises with many miscarriages of justice, intelligence blunders and significant policy mis-steps. In Keynes’ terms people are relying on a theory that simply does not apply. In Dawid’s terms one can put it blunter: Decision-takers were relying on the fact that something had a very high probability when they ought to have been paying more attention to the evidence in the actual situation, which showed that the probability was – in Dawid’s terms – empirically invalid. It could even be that the thing with a high instrumental probability was very unlikely, all things considered.