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My purpose in this chapter is to argue that we have inconsistent intuitions about the structure of space (or spacetime, or some stuff, the aether, that fills space or spacetime). I obtain a contradiction from eleven premises, each of which is either directly intuitive or supported by intuitions. The use of so many premises results from the desire to exhibit as clearly as possible the places where readers might decide some intuition is to be undermined. Nonetheless the argument for inconsistency is, in outline, straightforward: our intuitions support the existence of a 'supersponge', namely a region of less than the total volume but not disjoint from any connected part of space of positive diameter. But the complement of a supersponge is an intuitively impossible region. Yet, intuitively a region of less than maximal volume must have a complement. Some have complained that the technical aspects of this chapter require more effort on the part of readers than the result warrants. I have three things to say about that complaint: First, I am claiming that the eleven premises are literally jointly inconsistent. If this is wrong I need correcting, but that I turn out to be wrong, if I am, will not be controversial. So the reader who does not want to invest much time on this project may simply leave the task of checking to others, and provisionally concede the inconsistency, pondering which are the less firm intuitions. Second, the inconsistency result is not a mere curiosity, for we may generate a family of hypotheses by abandoning just one of the least firm intuitions. These hypotheses should be taken seriously in spite of the clash with the intuition in question.

Despite an extensive literature on the nature and origins of mathematical truth, few if any studies exist of the everyday practices through which the adequacy of mathematical argumentation is cultivated and assessed. The work of a novice prover afforded insight into these practices and, in particular, into the disciplined character of discovering and proving mathematical theorems.

We classify the ordinary differential equations that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.

This paper introduces an anthropological approach to the foundations of mathematics. Traditionally, the philosophy of mathematics has focused on the nature and origins of mathematical truth. Mathematicians, however, treat mathematical arguments as determining mathematical truth: if an argument is found to describe a witnessably rigorous proof of a theorem, that theorem is considered—until the need for further examination arises—to be true. The anthropological question is how mathematicians, as a practical matter and as a matter of mathematical practice, make such determinations. This paper looks first at the ways that the logic of mathematical argumentation comes to be realized and substantiated by provers as their own immediate, situated accomplishment. The type of reasoning involved is quite different from deductive logic; once seen, it seems to be endemic to and pervasive throughout the work of human theorem proving. A number of other features of proving are also considered, including the production of notational coherence, the foregrounding of proof-specific proof-relevant detail, and the structuring of mathematical argumentation. Through this material, the paper shows the feasibility and promise of a real-world anthropology of disciplinary mathematical practice.

The homotopy ∏-algebra of a pointed topological space, 'X', consists of the homotopy groups of 'X' together with the additional structure of the primary homotopy operations. We extend two well-known results for homotopy groups to homotopy ∏-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to ∏-algebra and beyond. We also describe an abstract ∏-algebra and give three abstract ∏-algebra structures on the homotopy groups of the loop space of 'X' which can be realized as the homotopy ∏-algebras of three different spaces