Friday, February 28, 2014

Hempel's paradox of the ravens

To: Stuart B.
From: Geoffrey Klempner
Subject: Hempel's paradox of the ravens
Date: 2nd November 2011 11:24

Dear Stuart,

Thank you for your email of 24 October, with your essay for the University of London BA Methodology module, in response to the question, 'What is the best response to the paradox of the ravens?'

This is an excellent piece of work with which I can find no real objections.

My main problem (which isn't an objection) is that I just don't believe in 'probability theory'. Not even 'subjective probability' (whatever that means). As an undergraduate, my first ever reading on this topic was the excellent book by A.J. Ayer 'Probability and Evidence', which considers the various pros and cons of the frequency theory, logical relation theory (Keynes) etc. I'm mentioning this as it would be a good book to read. Back then, I wasn't a sceptic but I have become one.

When do we calculate probabilities? and to what purpose? When, and in what way, do we raise a question how much 'support' a theory has? My view would be broadly and loosely Popperian, that any theory which hasn't been decisively refuted is on the table, including theories we do not much like. I would add that preferences for this theory over that theory cut no ice at all. Maybe the choice determines which research project you join, but even there other considerations may override your judgement of the likelihood that the research will prove 'successful' (of course, you can 'succeed' in refuting a theory too). Grant money is always an important factor.

One basic intuition I have is that it is just patently absurd that a white shoe confers any increased probability, however small or even infinitesimal, on the hypothesis 'all ravens are black'. I can see that it would be OK to accept this conclusion as the price for accepting a theory which works well in other ways. No theory is perfect. But it is a flaw, a minor absurdity. It's not a result that we want or can 'justify'. The idea that one could justify the conclusion by some formula seems ridiculous. I'd rather say that if the formula is useful to us in other ways, then we will accept that it occasionally delivers odd or absurd results, but that's a different position.

I agree with you that the question of the nature of probability is a 'major' difficulty. I would add that it's one you can't just sweep under the carpet and say, 'Well, just assume we have a way to calculate probabilities, then...'. One just comes back to the question what probabilities are FOR.

Skill at counting cards in Blackjack (as in 'Rain Man') is one way, objectively verifiable, to tip odds in your favour and make a nice steady income. That's an undeniable fact. We know, in a sense a priori, that we are not going to discover that counting cards actually reduces your chances at winning in the long run (ceteris paribus, one always has to insert that, and provided one can avoid the cameras).

The problem is, in methodology we are tempted by this kind of model -- calculating the odds in order to gain an 'edge' -- where it doesn't really apply. There are too many unknowns, and always will be. What probability theory does do is confer a gloss of rigour and objectivity which adds zero value to the scientific enterprise. There are plenty of opportunities to calculate and apply formulae, but deciding which theory to prefer isn't one of them. As I stated before, all the theories -- even those which look rather implausible -- are on the table. Be prepared to be surprised.

I can therefore totally understand Stephen Hawking's impatience with the philosophy of science and methodology. Bayes' theorem may be useful for all sorts of practical purposes. Of course. But it is practice (or 'praxis') which takes the lead. Not philosophers in armchairs debating how science should be pursued, or what standards a theory 'must' meet.

I guess this will sound rather Feyerabendian, but I'm not really sympathetic to his views either. I appreciate order and rigour in science, and deprecate anarchy. Sociological observations about how science is practiced aren't a substitute for standards and rules. Luckily the editors and editorial boards of professional journals maintain a keen interest in maintaining standards. It's just a pity when these laudable aims are corrupted by the kind of fake 'rigour' which I mentioned earlier. (One thing Feyerabend is good on is observing the slavish way researchers attempt apply 'proper rules and procedures' to situations that don't warrant them, e.g. Masters and Johnson!)

Evidence that is anecdotal, unquantified or unquantifiable, can be valuable too -- in its appropriate place.

I don't see that I am saying anything different from what Aristotle said, that rigour is commendable when applied in the appropriate circumstances. I strongly suspect, therefore, that the whole debate over the ravens paradox is a case of inappropriate application of standards of rigour, which are justified in other circumstances.

All the best,

Geoffrey