Evidence that confirms a scientific hypothesis is said to be ‘novel’ if it is not discovered until after the hypothesis isconstructed. The philosophical issues surrounding novel confirmation have been well summarized by Campbell and Vinci [1983]. They write that philosophers of science generally agree that when observational evidence supports a theory, the confirmation is much stronger when the evidence is ‘novel’... There are, nevertheless, reasons to be skeptical of this tradition... The notion of novel confirmation is beset with a theoretical (...) puzzle about how the degree of confirmation can change without any change in the evidence, hypothesis, or auxiliary assumptions... There have not yet appeared any obviously satisfactory solutions to these problems Much of the literature on novel confirmation relies on the Bayesian analysis of conditional probabilities. Let H represent a hypothesis, E an event that confirms the hypothesis, and B some relevant background information. Denote by Pr the conditional probability of x given y. There are various plausible measures of the degree of support that E lends to H. Among these are: Degree of support = Pr Degree of support = Pr – Pr Degree of support = Pr – Pr Degree of support = Pr – Pt In the work cited, Campbell and Vinci offer a somewhat more involved Bayesian interpretation. Formula is discussed by Gardner [1981] who points out that under this formulation there can be no role for novelty. 'The function Pr contains no third slot in which to insert a temporal relation between the invention of H and the inventor's learning E. Obviously, then, this relation could not possibly affect E's support of H.' Formula, on the other hand, suggests a role for novelty. Bayes's Theorem allows us to rewrite the formula as Degree of support = Pr x [Pr – 1] We can use 1/Pr as a measure of the novelty of E. Then shows that the degree of support increases with novelty of E. In the paper already cited, Campbell and Vinci discuss shortcomings of this analysis. Formula expresses an alternative offered by Howson [1984]. In that formula, Pr represents the probability of H assuming that only B – {E} is known. This allows for the possibility of non-novel facts generating support for hypotheses. Niiniluoto [1984] argues for a variant along the lines of, in which we account for the possibility that the theorist was unaware that his hypothesis entails E. One problem in deciding among these approaches is that the choice of a definition for the degree of support appears arbitrary. What kind of argument could justify the choice of one definition over another? It is our position that there can be no basis for addressing this question in the absence of an explicit model of the process by which hypotheses are generated. Only in the presence of such a model can the various conditional probabilities be given meaningful interpretations. We provide such models in Sections 1, 2 and 4. The simple model of Section 1, incorporating strong assumptions, yields the conclusion that novelty is irrelevant. When these assumptions are relaxed in the later sections, novelty becomes relevant for a variety of reasons. It is at least potentially the case that scientists have more information about their own abilities than is publicly available, and this information might influence their decisions about whether even to attempt novel prediction. If this is so, then it should be incorporated into the model of hypothesis generation. This requires an explicit discussion of how scientists respond to incentives and how the incentives themselves evolve, which in turn takes us into the realm of economic theory. We have addressed these issues in another paper, written for an audience of economists. The results of this research are summarized in Section 3. (shrink)

The beginning of the journey -- What this book is about : using ideas from mathematics, economics, and physics to tackle the big questions in philosophy : what is real? what can we know? what is the difference between right and wrong? and how should we live? -- Reality and unreality -- On what there is -- Why is there something instead of nothing? the best answer I have : mathematics exists because it must and everything else exists because it (...) is made of mathematics, with an excursion into artificial intelligence -- Unfinished business -- Unfinished business from chapter two : the nature and purpose of economic models -- How Richard Dawkins got it wrong -- Why Dawkins's argument against intelligent design can't be right and a mathematical analysis of the arguments for the existence of God -- Belief -- Daydream believers -- Most beliefs are ill-considered because most false beliefs are costless to hold -- The next several chapters will explore the consequences of this observation before we return to the question of where our beliefs and knowledge come from -- Unfinished business -- Unfinished business from the preceding chapter : how color vision works, sound, and water waves, the sheer craziness of economic protectionism -- Do believers believe? -- Our ill-considered beliefs about religion : why I believe that almost nobody is deeply religious -- On what there obviously is -- Our ill-considered beliefs about free will, ESP, and life after death -- Diogenes's nightmare -- How is legitimate disagreement possible if you're arguing with someone who is as intelligent and informed as you are, shouldn't you put just as much weight on your opponent's arguments as your own? -- The fact that we persist in disagreeing is strong evidence that we don't really care what's true -- Knowledge -- Knowing your math -- Where mathematical knowledge comes from and logic and why evidence and logic are not enough -- Unfinished business -- Unfinished business from the preceding chapter : the tale of hercules and the hydra, with an excursion into the lore of very large numbers -- Incomplete thinking -- Godel's incompleteness theorem and what it doesn't say about the limits of human knowledge -- The rules of logic and the tale of a Potbellied pig -- The power of logical thought, with excursions into the most counterintuitive theorem in all of mathematics and the tale of a potbellied pig -- The rules of evidence -- What we can and can't learn from evidence, with excursions into the value of preschool and how internet porn prevents rape -- The limits to knowledge -- What physics does and doesn't tell us about what we can and cannot know -- Understanding Heisenberg's uncertainty principle -- Unfinished business -- The oddness of the quantum world and why it matters to game theorists -- Right and wrong -- Telling right from wrong -- Some hard questions about right and wrong and about life and death -- The economist's golden rule -- A rule of thumb for good behavior -- How to be socially responsible -- Putting the rule of thumb into practice -- On not being a jerk -- Goofus and gallant on immigration policy -- The economist on the playground -- Our ill-considered beliefs about fairness in the market place and in the voting booth, contrasted with our carefully considered beliefs about fairness on the playground -- Unfinished business -- How ancient talmudic scholars anticipated modern economic theory -- The life of the mind -- How to think -- Some basic rules for clear thinking, mostly about economics, but also about arithmetic, neurobiology, sin, and eschewing blather -- What to study -- Advice to college students : stay away from the English department and approach the philosophy department with caution, with an excursion into the remarkable life of Frank Ramsey. (shrink)