Learning to understand a single causal system can be an achievement, but humans must learn about multiple causal systems over the course of a lifetime. We present a hierarchical Bayesian framework that helps to explain how learning about several causal systems can accelerate learning about systems that are subsequently encountered. Given experience with a set of objects, our framework learns a causal model for each object and a causal schema that captures commonalities among these causal models. The schema organizes the (...) objects into categories and specifies the causal powers and characteristic features of these categories and the characteristic causal interactions between categories. A schema of this kind allows causal models for subsequent objects to be rapidly learned, and we explore this accelerated learning in four experiments. Our results confirm that humans learn rapidly about the causal powers of novel objects, and we show that our framework accounts better for our data than alternative models of causal learning. (shrink)
The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract (...) mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content. (shrink)
Is language understanding a special case of social cognition? To help evaluate this view, we can formalize it as the rational speech-act theory: Listeners assume that speakers choose their utterances approximately optimally, and listeners interpret an utterance by using Bayesian inference to “invert” this model of the speaker. We apply this framework to model scalar implicature (“some” implies “not all,” and “N” implies “not more than N”). This model predicts an interaction between the speaker's knowledge state and the listener's interpretation. (...) We test these predictions in two experiments and find good fit between model predictions and human judgments. (shrink)
A view of the sources of mathematical knowledge is sketched which emphasizes the close connections between mathematical and empirical knowledge. A platonistic interpretation of mathematical discourse is adopted throughout. Two skeptical views are discussed and rejected. One of these, due to Maturana, is supposed to be based on biological considerations. The other, due to Dummett, is derived from a Wittgensteinian position in the philosophy of language. The paper ends with an elaboration of Gödel's analogy between the mathematician and the physicist.
Classification of the boys in 48 families of three or more brothers according to their Moray House Test (VIQ) scores awarded at age 11±1[fraction one-half] years in 1947–75 confirms the postulated existence of a total of nine male phenotypes for this X-linked trait in the score range <70 to 140 points. The phenotypic means lie close to the sequence 69(8)133. Recombination is shown to occur. An additive effect of three alleles at each of two X-chromosome loci is the most likely (...) simple scheme. An iterative approach matching the observed mixtures of proposed genotypes in the families with probabilities for variable levels of homozygosity, non-crossover/crossover heterozygosity, and maternal meiotic recombination yields a best goodness of fit where this last is 20%. (shrink)
This paper reports the preliminary findings of an analysis of the Moray House (verbal IQ) results of a base sample of 4383 Orkney children due to be tested in the schools at age 11±1[fraction one-half] years in 1947–75. Girls enjoy a 3·63% premium relative to phenotypically equivalent boys at test age. Relationships within the sample, which includes some members of the parental generation, are known. The trait is found to be X-linked, with nine phenotypes in the boys, and lyonization is (...) involved. (shrink)