Cornelius Lanczos is professor emeritus at the Dublin Institute for Advanced Studies. Hungarian-born he made significant contributions to the development of relativity and quantum theory in the earlier decades of this century, and during the past twenty years he has established himself as an authority on numerical analysis. In addition he has a deep appreciation of literature from the Old Testament onwards, of music and of the arts. His writings in recent years have consisted largely in philosophical speculations on aspects (...) of physics. The present book is an historical account of man’s investigation of the nature and properties of space. Lanczos’ gift of clear exposition and his broad culture have made it a readable and valuable review of the development of the notion of space throughout the ages. (shrink)

We develop the theory of the nonadiabatic geometric phase, in both the Abelian and non-Abelian cases, in quaternionic Hilbert space.

Why is a red face not really red? How do we decide that this book is a textbook or not? Conceptual spaces provide the medium on which these computations are performed, but an additional operation is needed: Contrast. By contrasting a reddish face with a prototypical face, one gets a prototypical ‘red’. By contrasting this book with a prototypical textbook, the lack of exercises may pop out. Dynamic contrasting is an essential operation for converting perceptions into predicates. The existence of (...) dynamic contrasting may contribute to explaining why lexical meanings correspond to convex regions of conceptual spaces. But it also explains why predication is most of the time opportunistic, depending on context. While off-line conceptual similarity is a holistic operation, the contrast operation provides a context-dependent distance that creates ephemeral predicative judgments that are essential for interfacing conceptual spaces with natural language and with reasoning. (shrink)

Gordon Belot investigates the distinctive notion of geometric possibility that relationalists rely upon. He examines the prospects for adapting to the geometric case the standard philosophical accounts of the related notion of physical possibility, with particular emphasis on Humean, primitivist, and necessitarian accounts of physical and geometric possibility. This contribution to the debate concerning the nature of space will be of interest not only to philosophers and metaphysicians concerned with space and time, but also to those interested in (...) laws of nature, modal notions, or more general issues in ontology. (shrink)

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from (...) first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship $$\rho \sim L_{\rm Planck}^{-2}R^{-2} = L_{P}^{-4} (L_{\rm Planck} / R)^{2} \sim 10^{-122}M_{\rm Planck}^4$$, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl (1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles,....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, (...) electromagnetism, and strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown (...) that the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed. (shrink)

We propose a new approach to describe quantum mechanics as a manifestation of non-Euclidean geometry. In particular, we construct a new geometrical space that we shall call Qwist. A Qwist space has a extra scalar degree of freedom that ultimately will be identified with quantum effects. The geometrical properties of Qwist allow us to formulate a geometrical version of the uncertainty principle. This relativistic uncertainty relation unifies the position-momentum and time-energy uncertainty principles in a unique (...) relation that recover both of them in the non-relativistic limit. (shrink)

The modern use of algebra to describe geometric ideas is discussed with particular reference to the constructions of Grassmann and Hamilton and the subsequent algebras due to Clifford. An Einstein addition law for nonparallel boosts is shown to follow naturally from the use of the representation-independent form of the geometric algebra of space-time.

One cannot expect an exact answer to the question “What are the possible structures of space?”, but rough answers to it impact central debates within philosophy of space and time. Recently Gordon Belot has suggested that a rough answer takes the class of metric spaces to represent the possible structures of space. This answer has intuitive appeal, but I argue, focusing on topological characterizations of dimension, examples of prima facie space-like mathematical spaces that have pathological dimension (...) properties, and endorsing a principle of plenitude for possibility, that we get a different rough answer: either the class of metrizable spaces or the class of separable metrizable spaces are possible structures of space. (shrink)

Le Poidevin has recently presented an argument that gives rise to a serious problem for relationist theories of space. It appeals to the simple geometrical fact that if A, B and C are three points lying in a straight line, then AB and BC together entail AC. He suggests that an ontological relationship of supervenience must be appealed to to explain this entailment. Given this thesis of supervenience, relationism is implausible. I argue that the problem that Le Poidevin (...) raises for relationism should be rejected, because the thesis of supervenience is false. The latter rests upon what Le Poidevin refers to as ‘The explanatory principle', a principle which he claims to be a natural extension of the truthmaker principle. Contrary to this, I argue that given any plausible theory of truthmaking, the explanatory principle is false. With the rejection of this principle, Le Poidevin's argument against relationism collapses. (shrink)

This concise book introduces nonphysicists to the core philosophical issues surrounding the nature and structure of space and time, and is also an ideal resource for physicists interested in the conceptual foundations of space-time theory. Tim Maudlin's broad historical overview examines Aristotelian and Newtonian accounts of space and time, and traces how Galileo's conceptions of relativity and space-time led to Einstein's special and general theories of relativity. Maudlin explains special relativity using a geometrical approach, emphasizing (...) intrinsic space-time structure rather than coordinate systems or reference frames. He gives readers enough detail about special relativity to solve concrete physical problems while presenting general relativity in a more qualitative way, with an informative discussion of the geometrization of gravity, the bending of light, and black holes. Additional topics include the Twins Paradox, the physical aspects of the Lorentz-FitzGerald contraction, the constancy of the speed of light, time travel, the direction of time, and more.Introduces nonphysicists to the philosophical foundations of space-time theory Provides a broad historical overview, from Aristotle to Einstein Explains special relativity geometrically, emphasizing the intrinsic structure of space-time Covers the Twins Paradox, Galilean relativity, time travel, and more Requires only basic algebra and no formal knowledge of physics. (shrink)

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure (...) of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations. (shrink)

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of (...) the law of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

It is shown that classical Dirac fields with the same couplings obey the Pauli exclusion principle in the following sense: If at a certain time two Dirac fields are in different states, they can never reach the same one. This is geometrically interpreted as analogous to the impossibility of crossing of trajectories in the phase space of a dynamical system. An application is made to a model in which extended particles are represented as solitary waves of a set of (...) several fundamental, confined nonlinear Dirac fields, with the result that the same mechanism accounts both for fermion and boson behaviors. (shrink)

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by (...) Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

It is argued that quantum mechanics is fundamentally a geometric theory. This is illustrated by means of the connection and symplectic structures associated with the projective Hilbert space, using which the geometric phase can be understood. A prescription is given for obtaining the geometric phase from the motion of a time dependent invariant along a closed curve in a parameter space, which may be finite dimensional even for nonadiabatic cyclic evolutions in an infinite dimensional Hilbert space. Using (...) the natural metric on the projective space, we reformulate Schrödinger's equation for an isolated system. This metric is generalized to the space of all density matrices, and a physical meaning is proposed. (shrink)

Space, Time, and Stuff is an attempt to show that physics is geometry: that the fundamental structure of the physical world is purely geometrical structure. Along the way, he examines some non-standard views about the structure of spacetime and its inhabitants, including the idea that space and time are pointless, the idea that quantum mechanics is a completely local theory, the idea that antiparticles are just particles travelling back in time, and the idea that time has no (...) structure whatsoever. The main thrust of the book, however, is that there are good reasons to believe that spaces other than spacetime exist, and that it is the existence of these additional spaces that allows one to reduce all of physics to geometry. Philosophy, and metaphysics in particular, plays an important role here: the assumption that the fundamental laws of physics are simple in terms of the fundamental physical properties and relations is pivotal."--P. [4] of cover. (shrink)

The overreliance on verbal models and theories in psychology has been criticized for hindering the development of reliable research programs (Harris, 1976; Yarkoni, 2020). We demonstrate how the conceptual space framework can be used to formalize verbal theories and improve their precision and testability. In the framework, scientific concepts are represented by means of geometric objects. As a case study, we present a formalization of an existing three-dimensional theory of emotion which was developed with a spatial metaphor in mind. (...) Wundt posits that the range of human emotion can be represented along three axes of basic emotions (pleasure/displeasure, excitement/inhibition, tension/relaxation), just as color vision can be represented using three basic colors. We use dimensions to represent basic emotions, points to represent emotional states, and regions to represent broader emotional concepts. We then compare our formalization to an existing structuralist formalization of Wundt’s theory. Further, we discuss the empirical predictions that our formalization generates, such as comparisons of similarity and intensity. We conclude by demonstrating how the tools developed in the conceptual space framework can be used to formulate a theory of emotion based on empirical observation. (shrink)

The purpose of this paper is to generalize Kripke semantics for propositional modal logic by geometrizing it, that is, by considering the space underlying the collection of all possible worlds as an important semantic feature in its own right, so as to take the idea of accessibility seriously. The resulting new modal semantics is worked out in a setting coming from Riemannian geometry, where Kripke semantics is shown to correspond to a particular case, namely, the discrete one. Several correspondence (...) results, established between variants of well-known modal systems and corresponding geometric properties, illustrate the import of this new framework. (shrink)

It is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists an N-dimensional linear vector space with N≥5. This space is decomposed into a four-dimensional tangent space and an (N - 4)-dimensional internal space. On the basis of geometric considerations, one arrives at a number of fields, the field equations being derived from a variational principle. Among the fields obtained there are the electromagnetic field, Yang-Mills gauge fields, and (...) fields that can be interpreted as describing matter. As a simple example, the case N=6 is considered. (shrink)

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational (...) contribution.Nonlinear field equations for the torsion result. In a “low-energy” approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector. (shrink)

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication (...) via spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category $${\mathcal {S}}$$ S, provided that $${\mathcal {S}}$$ S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication. (shrink)

This study presents a new type of foundational model unifying quantum theory, relativity theory and gravitational physics, with a novel cosmology. It proposes a six-dimensional geometric manifold as the foundational ontology for our universe. The theoretical unification is simple and powerful, and there are a number of novel empirical predictions and theoretical reductions that are strikingly accurate. It subsequently addresses a variety of current anomalies in physics. It shows how incomplete modern physics is by giving an example of a theory (...) that is genuinely unified. In doing this, it radically alters the metaphysical interpretation of the nature of time, space and matter currently interpreted from modern physics. It also profoundly challenges materialist expectations about a naturalistic account our own existence. I contend here that there is sufficient evidence to support this theory as a leading paradigm for a unified foundational theory. (shrink)

In his argument for the possibility of knowledge of spatial objects, in the Transcendental Deduction of the B-version of the Critique of Pure Reason, Kant makes a crucial distinction between space as “form of intuition” and space as “formal intuition.” The traditional interpretation regards the distinction between the two notions as reflecting a distinction between indeterminate space and determinations of space by the understanding, respectively. By contrast, a recent influential reading has argued that the two notions (...) can be fused into one and that space as such is first generated by the understanding through an act of synthesis of the imagination. Against this reading, this article argues that a key characteristic of space as a form of intuition is its nonconceptual unity, which defines the properties of space and is as such necessarily independent of determination by the understanding through the transcendental synthesis of the imagination. The conceptual unity that the understanding prescribes to the manifold in intuition, by means of the categories, defines the formal intuition. Furthermore, this article argues that it is the sui generis, nonconceptual unity of space, when takenas a unity for the understanding by means of conceptual determination, that first enables geometric knowledge and knowledge of spatially located particulars. (shrink)

The aim of this paper is to give a geometric interpretation of quantifiers in the intutionistic predicate calculus. We obtain it treating formulae withn free variables as functions withn arguments which run over an abstract set whereas the values of functions are open subsets of a suitable topological space.

It is shown that a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference (WARP) implies the strong axiom of revealed preference (SARP), can be reduced to a Hamiltonian cycle problem: A set of bundles allows a preference cycle of irreducible length if and only if the convex monotonic hull of these bundles admits a Hamiltonian cycle. This leads to a new proof to show that preference cycles can be of arbitrary length for more than (...) two but not for two commodities. For this, it is shown that a set of bundles satisfying the given condition exists if and only if the dimension of the commodity space is at least three. Preference cycles can be constructed by embedding a cyclic $(L-1)$ -polytope into a facet of a convex monotonic hull in $L$ -space, because cyclic polytopes always admit Hamiltonian cycles. An immediate corollary is that WARP only implies SARP for two commodities. The proof is intuitively appealing as this gives a geometric interpretation of preference cycles. (shrink)

The overreliance on verbal models and theories in psychology has been criticized for hindering the development of reliable research programs (Harris, 1976; Yarkoni, 2020). We demonstrate how the conceptual space framework can be used to formalize verbal theories and improve their precision and testability. In the framework, scientific concepts are represented by means of geometric objects. As a case study, we present a formalization of an existing three-dimensional theory of emotion which was developed with a spatial metaphor in mind. (...) Wundt posits that the range of human emotion can be represented along three axes of basic emotions (pleasure/displeasure, excitement/inhibition, tension/relaxation), just as color vision can be represented using three basic colors. We use dimensions to represent basic emotions, points to represent emotional states, and regions to represent broader emotional concepts. We then compare our formalization to an existing structuralist formalization of Wundt’s theory. Further, we discuss the empirical predictions that our formalization generates, such as comparisons of similarity and intensity. We conclude by demonstrating how the tools developed in the conceptual space framework can be used to formulate a theory of emotion based on empirical observation. (shrink)

The spinor covariant derivative through which the equations of quantum fields are generalized to include gravitational coupling has a direct and simple geometric significance. The formula for the difference of two spinor covariant derivatives taken in different order is derived geometrically; and the geometric proof of the covariant constancy of the spin-1/2 γ-matrices in curved space is given.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space (...) over a finite field. (shrink)