For any observed data an infinite number of models will fit it perfectly. How do you choose between them? Illustrated with an IQ test logic puzzle.
Video
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HOMEWORK: Given below; see end of lecture.
Lecture
The logic puzzle is here. The jury nullification post is here.
This is an excerpt from Chapter 7 of Uncertainty.
Per the Duhem-Quine thesis, scientific theories are under-determined and through any finite collection of facts you can always draw multiple theories. And explanatory power is not a guarantee a true cause has been discovered. A contemporary political example might be best in displaying the difficulty under-determination poses for discovering causes because of the importance people place on social questions. Chess Grandmaster Nigel Short said publicly that men and women are different and that men are better at chess than women. He said this was because the two sexes are “hard-wired very differently”. According to a newspaper account, Short said:
quotation
Why should they [men and women] function in the same way? I don’t have the slightest problem in acknowledging that my wife possesses a much higher degree of emotional intelligence than I do…we just have different skills. It would be wonderful to see more girls playing chess, and at a higher level, but rather than fretting about inequality, perhaps we should just gracefully accept it as a fact.
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A chess player and writer named Amanda Ross said in response to Short that it is “incredibly damaging when someone so respected basically endorses sexism”. Without being especially careful, we can define sexism as when a disparity of some kind exists between males and females and which is caused by men or society itself “pushing down” or otherwise limiting females. Short argues for biological differences; Ross claims egalitarianism is true. Now Ross begins her argument with the observation that a female once beat Short in a game, which to Ross proves males and females are equal. And she would be right if “equal at chess” meant “some woman somewhere can beat some man at chess.” Evidently, this is not what Short meant when he said men and women were “unequal at chess.” He meant something like, “In any list of top players, the majority will be men.”
There is a list of Grandmasters, the highest title chess players can earn (awarded by the World Chess Federation). As of this writing, there are 1413 men and 33 females on the list. This evidence bolsters Short’s claim that men are better in the sense he meant. But it also boosts Ross’s theory that sexism is rampant. The same data supports both theories. Short says men are superior chess players and here is a list showing they are. But Ross says mankind (and presumably culture) is sexist and keeps women from reaching top levels, and here is a list showing her prediction is right. The data cannot decide which theory is true. The theories are under-determined. And, of course, other theories might also explain the data. Men and women might be in essence equally skilled, but men like playing more. Or are allowed to. And so on.
There is no use bringing in Bayes’s Theorem and asking about “prior” probabilities on the truth of each theory, because the holders of both theories start by believing they are true. The data we have can’t shake either Short or Ross free from the conviction he or she is right. The data wouldn’t help us (from our perspective) either if we are indifferent between the theories. It is true that different data might support Short and Ross differently. Suppose on the list were 722 men and 722 non-men. Ross, claiming the triumph of equality over sexism, is upheld. But then it would be Short’s turn to claim that men really are superior, but the culture is pushing them down.
There is no general solution. The under-determination of the contingent is a fact. Collecting more data wouldn’t work, either. What we have to do, and even this is not a complete solution as is by now clear, is to look outside the data. For instance, we might argue that chess is an abstract analytical activity. If Short is right, men should be better than women at other abstract analytical activities. What’s more analytical than mathematics? The list of Fields medalists contains only one woman.
For Short to be right, only one thing must be true: men must have different brains, i.e. the essence of men must be different than the essence of women. For Ross to be right, many more things must be happening in more places and at more times. Sexism under Ross’s scheme must operate like the nervous say the Trilateral Commission does: a worldwide occult top secret network with strange unstoppable powers. If we accept the premise that fewer premises are more often associated with true theories, than there is good evidence Short is right. But it’s doubtful we’d get Ross to agree. Or perhaps some of you have thought of other objections to Ross; just as some of you might have thought of criticisms of Short. Jaynes’s Chapter 5 on the Queer Uses of Probability is not to be missed on this subject; Jaynes has several numerical examples that will be of interest.
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