Acclaimed mathematical physicist and natural philosopher Luciano Boi expounds the quantum vacuum, exploring the meaning of nothingness and its relationship with ...

We present a new model of skilled performance in geometry proof problem solving called the Diagram Configuration model (DC). While previous models plan proofs in a step‐by‐step fashion, we observed that experts plan at a more abstract level: They focus on the key steps and skip the less important ones. DC models this abstract planning behavior by parsing geometry problem diagrams into perceptual chunks, called diagram configurations, which cue relevant schematic knowledge. We provide verbal protocol evidence that DC's schemas correspond (...) with the step‐skipping inferences experts make in their initial planning. We compare DC with other models of geometry expertise and then, in the final section, we discuss more general implications of our research. DC's reasoning has important similarities with Larkin's (1988) display‐based reasoning approach and Johnson‐Laird's (1983) mental model approach. DC's perceptually based schemas are a step towards a unified explanation of (1) experts' superior problem‐solving effectiveness, (2) experts' superior problem‐state memory, and (3) experts' ability, in certain domains, to solve relatively simple problems by pure forward inferencing. We also argue that the particular and efficient knowledge organization of DC challenges current theories of skill acquisition as it presents an end‐state of learning that is difficult to explain within such theories. Finally, we discuss the implications of DC for geometry instruction. (shrink)

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff (...) and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions. (shrink)

The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity temporalizes space.

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the proponent of (...) the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

In this two-part essay, we distinguish several senses in which general relativity has been regarded as “locally special relativistic.” Here, in Part 1, we focus on senses in which a relativistic spacetime has been said to be “locally (approximately) Minkowskian.” After critiquing several proposals in the literature, we present a result capturing a substantive sense in which every relativistic spacetime is locally approximately Minkowskian. We then show that Minkowski spacetime is not distinguished in this result: every relativistic spacetime is locally (...) approximately every other spacetime in the same sense. In Part 2, we will consider “locally specially relativistic” matter theories. (shrink)

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff. The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim (...) is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice. (shrink)

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization (...) establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models. (shrink)

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in (...) the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry. (shrink)

We show that the exterior algebra bundle over a curved spacetime can be used as framework in which both the Dirac and the Einstein equations can be obtained. These equations and their coupling follow from the variational principle applied to a Lagrangian constructed from natural geometric invariants. We also briefly indicate how other forces can potentially be incorporated within this geometric framework.

We face with the question of a suitable measure theory in Euclidean point-free geometry and we sketch out some possible solutions. The proposed measures, which are positive and invariant with respect to movements, are based on the notion of infinitesimal masses, i.e. masses whose associated supports form a sequence of finer and finer partitions.

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...) rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon – Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely, objectivity as faithfulness to facts and objectivity as intersubjectivity. (shrink)

Let us start by some general definitions of the concept of complexity. We take a complex system to be one composed by a large number of parts, and whose properties are not fully explained by an understanding of its components parts. Studies of complex systems recognized the importance of “wholeness”, defined as problems of organization (and of regulation), phenomena non resolvable into local events, dynamics interactions in the difference of behaviour of parts when isolated or in higher configuration, etc., in (...) short, systems of various orders (or levels) not understandable by investigation of their respective parts in isolation. In a complex system it is essential to distinguish between ‘global’ and ‘local’ properties. Theoretical physicists in the last two decades have discovered that the collective behaviour of a macro-system, i.e. a system composed of many objects, does not change qualitatively when the behaviour of single components are modified slightly. Conversely, it has been also found that the behaviour of single components does change when the overall behaviour of the system is modified. There are many universal classes which describe the collective behaviour of the system, and each class has its own characteristics; the universal classes do not change when we perturb the system. The most interesting and rewarding work consists in finding these universal classes and in spelling out their properties. This conception has been followed in studies done in the last twenty years on second order phase transitions. The objective, which has been mostly achieved, was to classify all possible types of phase transitions in different universality classes and to compute the parameters that control the behaviour of the system near the transition (or critical or bifurcation) point as a function of the universality class. This point of view is not very different from the one expressed by Thom in the introduction of Structural Stability and Morphogenesis (1975). It differs from Thom’s program because there is no a priori idea of the mathematical framework which should be used. Indeed Thom considers only a restricted class of models (ordinary differential equations in low dimensional spaces) while we do not have any prejudice regarding which models should be accepted. One of the most interesting and surprising results obtained by studying complex systems is the possibility of classifying the configurations of the system taxonomically. It is well-known that a well founded taxonomy is possible only if the objects we want to classify have some unique properties, i.e. species may be introduced in an objective way only if it is impossible to go continuously from one specie to another; in a more mathematical language, we say that objects must have the property of ultrametricity. More precisely, it was discovered that there are conditions under which a class of complex systems may only exist in configurations that have the ultrametricity property and consequently they can be classified in a hierarchical way. Indeed, it has been found that only this ultrametricity property is shared by the near-optimal solutions of many optimization problems of complex functions, i.e. corrugated landscapes in Kauffman’s language. These results are derived from the study of spin glass model, but they have wider implications. It is possible that the kind of structures that arise in these cases is present in many other apparently unrelated problems. Before to go on with our considerations, we have to pick in mind two main complementary ideas about complexity. (i) According to the prevalent and usual point of view, the essence of complex systems lies in the emergence of complex structures from the non-linear interaction of many simple elements that obey simple rules. Typically, these rules consist of 0–1 alternatives selected in response to the input received, as in many prototypes like cellular automata, Boolean networks, spin systems, etc. Quite intricate patterns and structures can occur in such systems. However, what can be also said is that these are toy systems, and the systems occurring in reality rather consist of elements that individually are quite complex themselves. (ii) So, this bring a new aspect that seems essential and indispensable to the emergence and functioning of complex systems, namely the coordination of individual agents or elements that themselves are complex at their own scale of operation. This coordination dramatically reduces the degree of freedom of those participating agents. Even the constituents of molecules, i.e. the atoms, are rather complicated conglomerations of subatomic particles, perhaps ultimately excitations of patterns of superstrings. Genes, the elementary biochemical coding units, are very complex macromolecular strings, as are the metabolic units, the proteins. Neurons, the basic elements of cognitive networks, themselves are cells. In those mentioned and in other complex systems, it is an important feature that the potential complexity of the behaviour of the individual agents gets dramatically simplified through the global interactions within the system. The individual degrees of freedom are drastically reduced, or, in a more formal terminology, the factual space of the system is much smaller than the product of the state space of the individual elements. That is one key aspect. The other one is that on this basis, that is utilizing the coordination between the activities of its members, the system then becomes able to develop and express a coherent structure at a higher level, that is, an emergent behaviour (and emergent properties) that transcends what each element is individually capable of. (shrink)

We argue that current constructive approaches to the special theory of relativity do not derive the geometrical Minkowski structure from the dynamics but rather assume it. We further argue that in current physics there can be no dynamical derivation of primitive geometrical notions such as length. By this we believe we continue an argument initiated by Einstein.

The consequences for the theory of sets of points of the assumption of sets of sets of points, sets of sets of sets of points, and so on, are surveyed, as more generally are the differences among the geometric theories of points, of finite point-sets, of point-sets, of point-set-sets, and of sets of all ranks.

Geary's emphasis on hunting ignores the possible importance of other human activities, such as scavenging and gathering, in the evolution of spatial abilities. In addition, there is little evidence that links spatial abilities and math skills. Furthermore, such links have little practical importance given the small size of most differences and girls' superior performance in mathematics classrooms.

The existence of fields besides gravitation may provide us with a way to decide empirically whether spacetime is really a nonflat Riemannian manifold or a flat Minkowskian manifold that appears curved as a result of gravitational distortions. This idea is explained using a modification of Poincaré's famous 'diskworld'.