Quantum Minds

A New Scientist Cover Story (No. 2828 3 Sept 2011) opines that:

‘The fuzziness and weird logic of the way particles behave applies surprisingly well to how human thinks’. (banner, p34)

It starts:

‘The quantum world defies the rules of ordinary logic.’

The first two examples are The infamous two-slit experiment and an experiment by Tversky and Shamir supposedly showing violation of the ‘sure thing principle’. But do they?

Saving classical logic

According to George Boole (Laws of thought), when a series of assumptions and applications of logic leads to a falsehood I must abandon one of the assumptions of one of the rules of inference, but I can ‘save’ whichever one I am most wedded to. So to save ‘ordinary logic’ it suffices to identify a dodgy assumption.

Two-slits experiment

The article says of the two-slits experiment:

‘… the pattern you should get – ordinary physics and logic would suggest – should be ..’

There is a missing factor here: the classical (Bayesian) assumptions about ‘how probabilities work’. Thus I could save ‘ordinary logic’ by abandoning common-sense probability theory.

Actually, there is a more obvious culprit. As Kant pointed out the assumption that the world is composed of objects with attributes and having relationships with each other belongs to common-sense physics, not logic. For example, two isolated individuals may behave like objects but when they come into communion the sum may be more than the sum of the parts. Looking at the two-slit experiment this way, the stuff that we regard as a particle seem isolated and hence object-like until it ‘comes into communion with’ the apparatus, when the whole may be un-object-like, but then a new steady-state ’emerges’, which is object-like and which we regard as a particle. The experiment is telling us something about the nature of the communion. Prigogine has a mathematization of this.

Thus one can abandon the common-sense assumption that ‘a communion is nothing but the sum of objects’, thus saving classical logic.

Sure Thing Principle

An example is given (pg 36). That appears to violate Savage’s sure-thing principle and hence ‘classical logic’. But, as above, we might prefer to abandon out probability theory rather than our logic. But there are plenty of alternatives.

The sure-thing principle applies to ‘economic man’, who has some unusual values. For example, if he values a winter sun holiday at $500 and a skiing holiday at $500 then he ‘should’ be happy to pay $500 for a holiday in which he only finds out which it is when he gets there. The assumptions of classical economic man only seem to apply to people with lots of spare money and are used to gambling with it. Perhaps the experimental subjects were different? 

The details of the experiment as reported also repay attention. A gamble with an even chance of winning $200 or losing $100 is available. Experimental subjects all had a first gamble. In case A subjects were told they had won. In case B they were told they had lost. In case C they were not told. They were all invited to gamble again.

Most subjects (69%) wanted to gamble again in case A. This seems reasonable as over the two gambles they were guaranteed a gain of $100. Fewer subjects (59%) wanted to gamble again in case B. This seems reasonable, as they risked a $200 loss overall. Least subjects  (36%) wanted to gamble again in case C. This seems to violate the sure-thing principle, which (according to the article) says that anyone who gambles in both the first two cases should gamble in the third. But from the figures above we can only deduce that – if they are representative – then at least 28% (i.e. 100%-(100%-69%)+(100%-59%)) would gamble in both cases. But 36% gambled in case C, so the data does not imply that anyone would gamble for A and B but not C.

If one chooses a person at random, then the probability that they gambled again in both cases A and B is between 28% and 100%. The convention in ‘classical’ probability theory is to split the difference (a kind of principle of indifference) yielding 64% (as in the article). A possible explanation for the dearth of such subjects is that they were not wealthy (so having non-linear utilities in the region of $100s) and that those who couldn’t afford to lose $100 had good used in mind for $200, preferring a certain win of $200 to an evens chance of winning $400 or only $100. This seems reasonable.

Others’ criticisms here. See also some notes on uncertainty and probability.

Dave Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

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