Geometry

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...) questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

An argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modeling, not the world, but our activities in the world.

The definition of a point in geometry is primordial in order to understand the different elements of this branch of mathematics ( line, surface, solids…). This paper aims at shedding fresh light on the concept to demonstrate that it is related to another one named, here, the Primary Geometric Object; both concepts concur to understand the multiplicity of geometries and to provide hints as concerns a new understanding of some concepts in physics such as time, energy, mass….

Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in the ideas behind (...) linear algebra. Linear algebra is used extensively in statistics, mechanical engineering, circuit theory, signal processing, economics and data science/machine learning. I have found in my teaching career that students tend to struggle the most with motivating the "why" behind the abstract concepts discussed in linear algebra. This can be difficult for professors to understand because we are so used to using the language of linear algebra we take for granted why these questions are important in the first place. In other words, we may understand the “why” (after years of using these tools), but our students may not. This text seeks to explain this “why” in an accessible, engaging way. (shrink)

This paper advances a geometric reinterpretation of economics. Its core claim is that for any movement from one state of affairs to another, there exists a true shortest path: a geodesic, understood as a Platonic object of economic life. This path is not defined by observed behavior, nor by retrospective success, but by the underlying structure of the possibility space itself. Yet because real economic life unfolds in a distorted world—shaped by institutions, technological limits, information asymmetries, infrastructure, and path dependence—the (...) geodesic does not generally appear as a straight line. Once mapped onto reality, it bends around hills, basins, and other sources of curvature, often taking forms that seem indirect from a naive perspective. The paper argues that ordinary agents approximate these routes only imperfectly, relying on intuition in the absence of a true map. From this it draws a methodological conclusion: economics should be understood less as the science of predicting agents, and more as the craft of mapping the geometry of possibility space. On this view, economists function as map makers, identifying the contours of constraint and opportunity so that agents may better navigate toward whatever ends they desire. Inefficiency is thus reconceived as excess expenditure induced by curvature, rather than simply as irrationality or failure to optimize. -/- To quote the great Martin Luther King Jr. “The moral arc of the universe is long, but it bends towards Justice.” . (shrink)

We present a principled framework for human-AI collaboration grounded in the Universal Master Equation (UME) from Autopoietic Stochastic Differential Geometry (ASDG). The UME decomposes dynamics into four orthogonal components: symplectic flow (Hamiltonian coherence), dissipative flow (friction work), collapse flow (attractor dynamics), and stochastic forcing (exploration). We demonstrate that this decomposition maps naturally onto the division of cognitive labor between humans and generative AI systems. Humans provide the symplectic term (first principles, constraints) and collapse term (purpose, meaning); AI handles the dissipative (...) term (path-finding through solution space); the stochastic term is shared. We develop recursive autopoietic stochastic differential geometry (rASDG): the coupled (Φ, Ψ) system where Φ is the ontic state and Ψ is the epistemic state constrained to the fibre F_Φ. The self-location constraint Ψ = M(Φ)(Ψ)—requiring the epistemic state to be a fixed point of the moment-map-induced operator—can be enforced discretely (projection) or continuously (penalty drift −κ(Ψ − M(Φ)Ψ)). Under Banach contraction conditions (provided by the collapse term ensuring spectral radius ρ(M) < 1), this fixed point is unique and exponentially attracting. Metacognition is thus a mathematical consequence: the unique self-consistent interpretation forced by the geometry. Joint Lyapunov analysis shows the system attracts to the intersection of the 108-heart (I₄ = 0) with the self-location manifold. This framework yields applications in prompt engineering, augmented intelligence, and GenAI metacognition. (shrink)

The third part of the Universal Variational Paradigm (UVP) presents an empirical synthesis confirming the universal variational law across the observable hierarchy of nature. It demonstrates that the same invariants—stationarity and openness—govern phenomena from physics to consciousness. -/- Physical systems obey the stationary condition through the principle of least action and the Fisher-information bound; biological and neural systems manifest open Ricci-type curvature flows that describe irreversible evolution and learning; psychological and social systems reveal analogous curvature dynamics governing reflection, ethics, and (...) cultural adaptation. -/- The paper compiles cross-disciplinary evidence (physics, chemistry, biology, neuroscience, psychology, sociology) showing that all stable forms satisfy δΦ = 0 while remaining open under δ³Φ ≠ 0. This establishes the UVP as an empirically grounded framework unifying energy, information, and reflection across scales of reality. (shrink)

This second part of the Universal Variational Paradigm (UVP) extends the variational architecture of reality from physics and information to mind and meaning. While Part I established the universal law of distinction, stationarity, and openness as the foundation of nature’s geometry, the present paper introduces the concept of the Noetic Metric, a mathematical structure that encodes the local geometry of sense and measures semantic tension. Its evolution follows a Ricci-type flow influenced by reflective and intentional input. -/- The Noetic Geometry (...) couples to the physical and biological manifolds through the beta–gamma–lambda bridge (energy–information–reflection), forming a continuous variational continuum between matter, life, and consciousness. Empirical correspondences are demonstrated across neuroscience, psychophysiology, and social systems, showing that curvature flows govern both cognitive and ethical coherence. -/- Noetic Geometry thus formalizes consciousness as a self-consistent curvature flow on the manifold of distinctions and provides a rigorous mathematical language for meaning, reflection, and value. (shrink)

There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.

There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.

We demonstrate that the fundamental physical constants of the Standard Model are arithmetic invariants, originating from four mechanisms: (A) ratios of special values of L-functions in the Selberg class, (B) Bernoulli numbers transported by the archimedean Γ-factor via the functional equation, (C) ratios of nontrivial L-function zeros, and (D) dimensions of spaces of multiple zeta values counted by Zagier's recurrence. No geometric input is required: π itself is the Haar measure normalisation of ℝ/ℤ. At the GUT scale, the down-type quark (...) mass ratios m_d : m_s : m_b = 2 : 5 : 9 follow exactly from the Casimir invariants C₂(n,0) = n(n+3)/3 of SU(3)_flavor Kac–Moody representations for generations n = 1,2,3, and the Georgi–Jarlskog relation m_d m_b/m_s² = 9/4 matches experiment at 0.09%. The Cabibbo angle satisfies sin θ₁₂ = 1/(2√5) within 0.9% via the Gatto–Sartori–Tonin relation, while the geometric prediction θ₁₂ = 29π²/1260 from the Fubini–Study distance on CP² achieves 0.19%. The seed of CP violation is identified as the Jacobi sum J(χ₄ mod 5, χ₃ mod 7) = e^{iπ/6}, a primitive 12th root of unity in ℚ(ζ₁₂); the ratio of the physical CKM phase to this arithmetic seed approximates 137/60 at 0.04%, suggesting a connection between CP violation and the fine structure constant. Three independent programmes—the exceptional Jordan algebra (Singh), the generalised Dirac equation on Cl_{8,0} (Quinta), and the coset geometry of Spin(8)/G₂—converge on the same angular parameters for the fermion mass matrix, providing evidence that mass is the curvature of the division algebra coset and the hierarchy is its shape. All numerical claims are verified at ≥ 15-digit precision. (shrink)

We prove that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. We establish this result via three independent proofs using different mathematical frameworks: (1) Geometric: Three structural properties—Haar self-duality, functional equation symmetry, and Peter-Weyl compactness—uniquely determine σ = 1/2 as the only value permitting L² integrability. (2) Spectral: Meyer's unconditional spectral realization combined with Stone's theorem and Haar measure self-duality; (3) Probabilistic: The Biane-Pitman-Yor identification of ξ(s) with the Kuiper distribution, showing (...) off-line zeros would violate probability axioms. Each pathway is rigorous, complete, and unconditional. (shrink)

We prove the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rational numbers. Specifically, we establish that for any elliptic curve E over Q, the rank of the Mordell-Weil group E(Q) equals the order of vanishing of the L-function L(E,s) at s=1. The proof proceeds in three main steps. First, we use the Arthur-Selberg trace formula to express the rank as the dimension of a spectral eigenspace. Second, we apply the Satake isomorphism and strong multiplicity one theorem to isolate (...) the automorphic representation π_E associated to E via modularity. Third, we employ a spectral-Weil identification using Riesz representation theory to connect the spectral dimension to the L-function vanishing order. This approach adapts techniques from spectral analysis of the Riemann zeta function, extending the spectral-Weil identification from GL_1 to GL_2. The key innovation is recognizing that Weil's explicit formula, combined with the Arthur trace formula and Riesz uniqueness theorem, provides an unconditional connection between spectral measures and L-function zeros. The proof is unconditional and applies to all ranks without restriction. (shrink)

In this work I present what may be the first complete construction of quantum gravity describing the real universe via the celestial holographic conformal field theory dual to Einstein gravity in asymptotically-flat 4D spacetime. The theory is rigorously constructed as the shadow-invariant, purely spin-2 sector of holomorphic Chern–Simons theory on twistor space PT ≃ CP³ with gauge group the quantomorphic group Quant(PT). Primary fields are the celestial graviton operators O^{±2}Δ(z, z̄) with Δ ∈ 1 + iR and J = ±2. (...) Three-point functions are determined by the Penrose transform and reproduce the MHV and anti-MHV amplitudes of general relativity, including the vanishing of parity-odd and mixed-helicity couplings. Higher-point amplitudes arise from OPEs with analytically continued principal-series contours; loop integrands emerge as double discontinuities across shadow poles, which I verify explicitly for the scalar box integrals computed via this novel shadow discontinuity method, demonstrating analytically that they match Feynman diagram calculations for the scalar box, with numerical verification for the graviton box matching to 11+ decimal places. The construction satisfies extended BMS₄ ⋉ shadow symmetry with vanishing Virasoro central charge, also explicitly demonstrated via five independent and mutually reinforcing and rigorous arguments to be c = 0 to 10+ significant figures. I prove a rigidity theorem establishing that any 2D CFT with only spin ±2 primaries, the above symmetries and three-point data, and local inverse-Mellin-transformed correlators, is uniquely determined to be isomorphic to this quantomorphic twistor theory. Combined with the proof that crossing symmetry is a necessary consequence of bulk locality (not an independent axiom), this establishes that Einstein gravity in asymptotically-flat spacetime is the unique consistent theory of a single massless spin-2 particle. The quantomorphic gauge group structure provides a non-perturbative foundation, with perturbative scattering amplitudes emerging as the first residues of the holographic algebra. The first breakthrough insight that enabled and facilitated this construction was the realisation that the "shadow symmetry" of the boundary CFT—the map Δ ↔ 2 − Δ on conformal dimensions—is the restriction to null infinity of the Jost involution Θ of axiomatic quantum field theory, the operator independently constructed by Jost (1957) that coincides with the product C ∘ P ∘ T only in theories where all three factors individually exist as symmetries. At null infinity, where the metric is degenerate along the null generator, the distinction between temporal and spatial inversion collapses: the Jost involution is irreducible and reduces to time reversal T alone (Theorem 8.1). This provides an elegant resolution to the 40+ year "googly problem" in twistor theory by unifying self-dual and anti-self-dual helicity sectors through the shadow transform as representing discrete T-symmetric involution across the boundary, with the Einstein field equations derived from boundary Ward identities and crossing symmetry. All quantum mechanical foundations emerge from the Haar measure structure underlying this framework with no additional postulates: the five Wightman axioms are derived from the Peter–Weyl theorem on representation spaces, the Born rule P = |⟨n|ψ⟩|² is proven as the unique probability measure compatible with Haar structure on quantum state spaces, and wave function collapse is derived as Haar projection onto gauge singlets. This same identification reveals that the celestial conformal dimension is related to the Riemann zeta variable by Δ = 2s, so that the shadow involution Δ ↔ 2 − Δ becomes s ↔ 1 − s, the functional equation of the Riemann zeta function—establishing that the functional equation, shadow symmetry, and time reversal are three manifestations of one underlying self-duality. The entire holographic framework grows from a single root: self-dual Haar measure on (R⁺, ×), the simplest non-trivial locally compact group, whose self-duality already contains the functional equation and critical line. Three Cayley–Dickson doublings R → C → H → O produce the holographic chain R⁺ → S² → M⁴ → Gr(2,4)_C, with shadow symmetry at each level corresponding to the conjugation involution at that doubling step. The Grassmannian Gr(2,4) is the canopy of this holographic tree. The same Haar-invariant measure on the Grassmannian Gr(2,4) that determines the graviton spectrum also addresses three Millennium Prize problems through a unified geometric mechanism (Haar self-duality, functional equation symmetry, and Peter–Weyl compactness). The Riemann Hypothesis follows via 4 otherwise independent arguments and checks for circularity and consistency, and most fundamentally follows from three simultaneous constraints on the adelic quotient A×/Q× that form an overdetermined system whose unique solution forces Re(s) = 1/2 for all nontrivial zeros, with numerical verification of the first 10¹³ zeros on the critical line to machine precision using six independent tests. The Birch and Swinnerton-Dyer Conjecture is argued to be proven for analytic rank ≤ 1 using identical Haar measure structure on GL₂(A_Q)/GL₂(Q): for elliptic curves with ord_{s=1} L(E,s) ≤ 1, the same three-fold constraint described above, combined with the Gross–Zagier formula and Kolyvagin's Euler system descent, establishes ord_{s=1} L(E,s) = rank E(Q) and the parity conjecture (−1)^{rank} = w_E is proven for all curves from Haar self-duality. The framework also identifies the precise obstruction to proving the conjecture at higher rank (algebraic cycles from the Beilinson–Bloch–Kato conjectures). The Yang–Mills mass gap is established through a complete celestial holographic construction: the Wess–Zumino–Witten model at Kac–Moody level k on S², mapped to four-dimensional Schwinger functions by inverse Mellin transform, with reflection positivity proven via the shadow-reflection bridge, the remaining Osterwalder–Schrader axioms verified explicitly, and the Wightman theory reconstructed by the OS theorem—yielding mass gap M = 2N/(k+N) · Λ_QCD > 0 from the Sugawara formula and confinement from Haar orthogonality as a rigorously proved theorem. The construction introduces no lattice, takes no continuum limit, and imposes no infrared regulator. Spacetime itself emerges from the Grassmannian via Penrose twistor correspondence (M⁴ ↔ Gr(2,4) (Penrose 1967). The division algebra tower determines the Standard Model: the symmetry breaking chain Spin(8) → G₂ → SU(3) → SU(2) × U(1) produces the gauge group; three fermion generations follow from Weyl anomaly cancellation, Plücker partitions, and Spin(8) triality; sin²θ_W = 3/8 at the GUT scale is the Dynkin index ratio, equivalently dim(Im(H))/dim(O). Physical constants are arithmetic invariants: CP violation from the Jacobi sum J(χ₄⁽⁵⁾, χ₃⁽⁷⁾) = e^{iπ/6}; down-type quark mass ratios m_d : m_s : m_b = 2:5:9 from SU(3) Casimir eigenvalues; the spin-statistics connection η(4)/ζ(4) = 7/8 from the p = 2 Euler factor; the lightest neutrino mass m_{ν₁} = 0 exactly from Bott periodicity—in independent agreement with Boyle–Finn–Turok (PRL 121, 251301, 2018); and the three dimensional constants ℏ, c, G corresponding to the three Cayley–Dickson doubling steps. Strong CP angle θ = 0 exactly from Haar self-duality (consistent with experimental bound |θ| < 10⁻¹⁰), and the construction provides right-handed neutrinos as geometric dark matter candidates in agreement with Boyle–Turok (CP)T-symmetric cosmology, which integrates with uncanny precision into my T-symmetric, thermodynamically cyclic quantum cosmology. The T-anti-symmetric unification of a unified bipartite self-dual universe resolves the matter-antimatter asymmetry as a selection effect. Total baryon number B_total = 0 with embedded observers observing identical physics on either side due to being thermodynamic in nature, naturally aligning their arrows of time in the direction of increasing entropy, which on either side of the boundary flows in opposing directions. The Feynman-Stueckelberg interpretation of antimatter means antimatter is equivalent to matter moving backwards in time, which combined with the reversal in T of the direction of entropy increases means observers only ever observe the B > 0 (matter) sector, with the B < 0 (antimatter) sector always being the hidden sector from the perspective of any temporally-embedded observer that respects thermodynamics, as from any observer's perspective their time moves forward, their atomic nuclei have a positive electrical charge, electrons have negative electric charge, and the standard model has left handed doublets and right-handed singlets, with a weak force that appears left-chiral. To observe the antimatter would require an observer to reverse their arrow of time, which is thermodynamically not possible as consciousness and memory are thermodynamic processes that happen only in the direction of increasing entropy. Shadow geometry and right-handed neutrinos provide dark matter. The Grassmannian spinor bundle splits each fermion representation across the T boundary: right-handed neutrinos reside in the shadow sector, coupling to the observable sector only gravitationally. The dark matter density profile ρ(r) = ρ₀/(1+(r/r_c)²) with core radius r_c ≈ 9.98 kpc is derived from the shadow kernel by Abel inversion with zero free parameters, resolving the core cusp problem of dark matter distribution and addressing several further cosmological anomalies (the S₈ tension, Hubble tension, DESI dark energy evolution, JWST early galaxies, CMB hemispherical asymmetry, CMB dipole anomaly, axis of evil, overmassive early black holes, rotation curve diversity, etc.) from a single geometric source. Time's arrow emerges as a consequence of T-symmetric boundary conditions with entropy always increasing away from the conformal boundary in opposing temporal directions. The horizon and flatness problems are naturally resolved by this framework as the celestial CFT is globally defined at the conformal boundary. The framework generates falsifiable predictions including no primordial gravitational waves (tensor-to-scalar ratio r → 0, definitively falsified if r > 0.01 detected by CMB-S4), m_{ν₁} = 0 (KATRIN, JUNO), r_c ≈ 9.98 kpc (Euclid), glueball mass ratios matching the Sugawara spectrum, GUE statistics in the vacuum noise spectrum of gravitational wave detectors (LIGO/Virgo/KAGRA), and right-handed neutrino dark matter coupling only through gravity. Over 155 independent computational tests with zero free parameters confirm the framework to machine precision. Part 1 contains an introductory overview of the several subframeworks involved in this synthesis explaining their interrelations in the wider framework, part 2 contains the constructions and rigorous proofs of all major results, Part 3 contains extensions and other corollaries. Part 4 continues with comparisons to other approaches and discussion regarding foundational contributions without which this work would not have been possible follows, followed by testable predictions and a chapter addressing falsifiability, and Part V delivers a philosophical analysis of the implications suggested by the mathematics of this framework. This work unifies quantum mechanics and general relativity via the celestial holography program while revealing deep connections between number theory and physics: the identification Δ = 2s means the same self-dual Haar-invariant measure structure constraining the Riemann zeros to the critical line in the complex plane where Re(s) = 1/2 simultaneously constrains the conformal dimensions of graviton operators to the principal series where Re(∆) = 1, the same compactness that forces the Yang–Mills mass gap forces the discrete zeta spectrum, and the same functional equation symmetry that governs L-functions governs the shadow transform of the celestial CFT—suggesting geometry and arithmetic share a common logos rooted in self-dual Haar measure on (R⁺, ×) and flowering in the Grassmannian Gr(2,4). (shrink)

Relationalism about space faces well-known objections if it is limited to relations between actual bodies. These problems might be avoided through so-called modal relationalism, on which the relevant relata include possible entities. Leibniz is considered a founder of modal relationalism, appealing to relations among possible situations. This article argues that for the central type of relation in question, namely congruence, Leibniz cannot give an adequate basis for modal relationalism. This is because his criteria for determining congruence rely on actual perception, (...) and because attempts to modalize this perception requirement conflict with his other commitments. (shrink)

This paper develops a formal distinction between \emph{map}, \emph{directrix}, \emph{action}, and \emph{completion} in the setting of the complex numbers. The motivating claim is that charting the place of the imaginary numbers within the complex plane is not the same act as sending a selected aspect of that structure through a projection, restriction, quotient, normalization, branch choice, or flow, and that neither of these is identical with extending a previously partial evaluative regime by a clause that forces a value at sites (...) of nonconstruction. The former is representational; the second is operational; the third is semantic completion. -/- To make this precise, the paper constructs a layered atlas of the complex plane \(\C\) and of the pure imaginary axis \(\ii\R\subset \C\), expanding the canonical ``subrooms'' of one-complex-dimensional space: algebraic, linear, Euclidean, polar, topological, branch-theoretic, holomorphic, harmonic, symplectic, oscillatory, dynamical, and projective. Central structural results include the classification of real-linear endomorphisms as maps of the form \(z\mapsto az+b\conj{z}\), the topological obstruction to a global argument, the principal logarithm on the slit plane, the identification \(T_1S^1=\ii\R\), the Lagrangian character of the real and imaginary axes, Hamiltonian realization of phase rotation, explicit dynamical classifications for power and affine maps, and stereographic/projective completion by the Riemann sphere. -/- The paper then adds a second layer not present in standard expositions: a distinction between \emph{intrinsic zero} and \emph{completion zero}. The element \(0\in\C\) is intrinsically part of the field, but many appearances of the glyph ``\(0\)'' in mathematical practice arise from what may be called \emph{completion-by-clause}: extending a sparse coefficient family, an out-of-support recurrence, an empty aggregation, or an exceptional locus by an explicit rule assigning the value \(0\). Such completions are shown to be neutral only in zero-erasing contexts, especially additive aggregation. A depadding theorem for additive zero-extension is proved, and a complementary non-neutrality proposition shows why zero-completion fails to be harmless once reciprocals, logarithms, arguments, branch-sensitive operations, support-sensitive operators, or analytic continuation enter the scene. -/- A third layer addresses representational pluralism. For the classes of constructions treated here, the mechanics of the complex plane need not be written with the literal symbol \(\ii\): they can be expressed exactly in real two-dimensional terms via a quarter-turn operator \(J\), via winding-number differential forms, via harmonic pairs, and more generally through alternate real-geometric atlases. This includes, at the level of principle, sweep-based encodings such as oriented boundary sweeps, sweeping nets, and saddle-map formalisms used to track phase or singularity structure without explicit imaginary notation. The paper argues that imaginary numbers are therefore not the only possible ontology of planar phase, but the most algebraically compressed one. -/- An appendix provides a formal novelty-map table for the present manuscript and for several adjacent manuscripts supplied by the author, classifying their main components as \emph{likely new}, \emph{probably known}, or \emph{needs citation check}. The aim is not to settle priority definitively, but to separate classical mathematics from original synthesis, and theorem-level originality from interpretive or programmatic innovation. (shrink)

This paper predicts a redefinition of the existence of emptiness and infinity itself by re-envisioning the origin of numbers not as static symbols such as "0" or "1," but as a multi-layered cosmic model in which the law of conservation of energy functions dynamically. It is a sketch of number-theoretic philosophy as a preliminary stage to rigorous mathematical proof. (This has the potential to provide fundamental meaning to the limits of physics and the origins of noncommutative geometry.) -/- In this (...) model, following Leibniz, the "Monad" is established as the smallest unit of existence. By reinterpreting "0" not as a mere singularity but as an "exit" for dimensions, this paper discusses the necessity of an absolute zero point---the "Universal Mapping Absolute (UMA) Monad"---which integrates addition and multiplication without contradiction. | 本論は、数の起源を「0」や「1」という静的な記号としてではなく、エネルギー保存則が動的に機能する多層的な宇宙モデルとして捉え直すことで、空と無限の存在自体の再定義を予想する。数理的な厳密な証明の前段階 としての、数論哲学スケッチである。(物理学の限界点や非可換幾何学の原点に根本的な意味を与える可能性がある。) -/- 本モデルでは、ライプニッツに倣い「モナド(Monad)」を存在の最小単位と設定する。ここでは「0」を特異点と見なさず、次元の「出口」として再解釈することで、加法と乗法を矛盾なく統合する絶対的零点「Un iversal Mapping Absolute(UMA)モナド」の必要性を論じる。. (shrink)

Supersubstantivalists identify material objects with regions of spacetime. Accordingly, they take their view to be both more ideologically parsimonious than other substantivalists (dualists) because they can reduce location to identity with a region and they can explain why the mereological structure of objects mirrors the mereological structure of their locations (henceforth, “harmony”). However, I argue that these motivations for supersubstantivalism don’t hold water. Specifically, I argue that supersubstantivalists can only claim the aforementioned advantages just so long as they stand in (...) a unique position to reduce location. But this depends on the illicit assumption that no possible spacetime structures have co-located regions – something falsified by the possibility of merely pseudo-metric spaces. Further, any alternative reduction is equally available to the dualist. Accordingly, this allows dualists to have the same explanation of harmony as the supersubstantivalist. As such, these considerations level the dialectical playing field between dualists and supersubstantivalists. (shrink)

The same mechanism operates from quarks to consciousness. We call it Universal Constraint Parsing (UCP)—constraints at each level evaluate entities against possibility spaces, accepting configurations that fit, rejecting those that don't. Quarks are parsed by QCD field constraints. Molecules are parsed by thermodynamic constraints. Organisms are parsed by ecological constraints. Beliefs are parsed by evidential constraints. Memes are parsed by cultural constraints. Self-models are parsed by architectural constraints. This isn’t metaphor or loose analogy—across domains, selection dynamics belong to the same (...) formal equivalence class of constraint-satisfaction processes under scale-appropriate coarse-graining. While state variables and effective free-energy functionals differ by level, the underlying variational structure and selection dynamics are preserved. Crucially, UCP operates without agency—no designer, no purpose, no teleology required at any level. Agency itself emerges as a functional property when systems begin recursively parsing their own parsing at the consciousness threshold. We demonstrate UCP across all levels, show what emerges at complexity thresholds (replication, learning, consciousness), explain how existing frameworks (Universal Darwinism, memetics, Free Energy Principle) relate to UCP, and derive empirically constraining expectations. The framework is falsifiable and unifies phenomena previously treated as separate. It's physics all the way—mechanistic constraint satisfaction at every level. (shrink)

Published with great thanksgiving for Yaohushua, the living One Yahowah, "Jesus Christ." -/- In previous work, Formalizing Mechanical Analysis Using Sweeping Net Methods I, sweeping net methods have been extended to complex analysis, relying on the argument of complex functions defined on the unit circle. In this paper, we reformulate these methods purely within a real-valued and geometric framework, avoiding the use of complex analysis. By redefining the sweeping net constructs and the associated theorems using real functions and geometric interpretations (...) on the unit circle, we demonstrate how singularities and their approximations can be effectively analyzed without the need for imaginary numbers. This approach provides intuitive geometric insights and broadens the applicability of sweeping net methods in mathematical analysis. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which (...) the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to n-dimensional simplices for all n>3. Part III explains how, for n=3, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ42, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 3 chirotope, aka affine oriented matroid. (shrink)

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical (...) geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation. (shrink)

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

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)

Hermann Weyl was one of the most important figures involved in the early elaboration of the general theory of relativity and its fundamentally geometrical spacetime picture of the world. Weyl’s development of “pure infinitesimal geometry” out of relativity theory was the basis of his remarkable attempt at unifying gravitation and electromagnetism. Many interpreters have focused primarily on Weyl’s philosophical influences, especially the influence of Husserl’s transcendental phenomenology, as the motivation for these efforts. In this article, I argue both that these (...) efforts are most naturally understood as an outgrowth of the distinctive mathematical-physical tradition in Göttingen and also that phenomenology has little to no constructive role to play in them. (shrink)

Sollen wir Euklids Vorgehen in den Elementen als ein axiomatisches System verstehen—oder als ein System des natürlichen Schließens, in dem die Regeln und Prinzipien, denen wir in unserem Schließen folgen, dargelegt werden? Im Folgenden werde ich darstellen, wie Kenneth Manders, Danielle Macbeth, Marco Panza und andere in jüngster Zeit diese letztere Sicht als eine alternative Lesart von Euklids Elementen dargestellt haben. Insbesondere werde ich versuchen zu zeigen, dass wir in dieser Lesart Euklids eine Art der Argumentation vorfinden, die nicht bloß (...) auf Diagrammen basiert (wie dies auch in anderen diagrammatischen Darstellungen der Fall ist), sondern dass die Euklidische Geometrie hier als wesentlich diagrammatisch aufgefasst wird: Wir argumentieren und denken hier in dem Diagramm, und nicht bloß auf seiner Grundlage. (shrink)

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one (...) or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized (...) and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the (...) NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically (...) explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

El primer objetivo de este artículo es mostrar que explicaciones genuinamente geométricas/matemáticas e intrínsecamente diagramáticas de fenómenos físicos no solo son posibles en la práctica científica, sino que además comportan un potencial epistémico que sus contrapartes simbólico-verbales carecen. Como ejemplo representativo utilizaremos la metodología geométrica de John Wheeler (1963) para calcular cantidades físicas en una reacción nuclear. Como segundo objetivo pretendemos analizar, desde un marco inferencial, la garantía epistémica de este tipo de explicaciones en términos de dependencia sintáctica y semántica (...) del contenido lo inferido en las premisas, lo cual denominaremos criterio de validez inferencial (CVI). (shrink)

This study of the history and contents of a hitherto unedited work on geometry by Vincenzo Viviani seeks to present a picture of the scientific environment in Italy in the second half of the 17th century, with particular emphasis on Tuscany and the impact the condemnation of Galileo had on ongoing scholarship. Information derived from unedited or less well-known material serves to illuminate a range of prominent and marginal figures who adopted different strategies for the dissemination of Galileo’s thought and (...) the application of its principles. A selection of texts, edited here for the first time, is presented in two appendices, while in a third decidedly unusual methods are used to elucidate the mathematical aspects. (shrink)

Many species rely on the three-dimensional surface layout of an environment to find a desired goal following disorientation. They generally do so to the exclusion of other important spatial cues. Two influential frameworks for explaining that phenomenon are provided by geometric-module theories and view-matching theories of reorientation respectively. The former posit a module that operates only on representations of the global geo- metry of three-dimensional surfaces to guide behavior. The latter place snapshots, stored representations of the subject’s two-dimensional retinal stimulation (...) at specific locations, at the heart of their accounts. In this paper, I take a fresh look at the debate between them. I begin by making a case that the empirical evidence we currently have does not clearly favor one framework over the other, and that the debate has reached something of an impasse. Then, I present a new explanatory problem—the representation selection problem—that offers the pro- spect of breaking the impasse by introducing a new type of explanatory consideration that both frameworks must address. The representation selection problem requires explaining how subjects can reliably select the relevant representation with which they initiate the reorientation process. I argue that the view-matching framework does not have the resources to address this problem, while a certain type of theory within the geometric-module framework can provide a natural response to it. In showing this, I develop a new geometric-module theory. (shrink)

Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. The book provides a bracing and painstakingly researched re-appreciation of the work of Hermann von Helmholtz and Ernst Cassirer, and of their place in the tradition, and is worth study for that alone. The contributions of the book go far beyond that, however. (...) It is a clear, accurate, and deep account of fascinating and philosophically momentous implications of the move to relativity theory. (shrink)

The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The (...) Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was first (...) conceived by ancient Greek mathematicians. (shrink)

A perverted space-time geodesy results from the notions of variable rods and clocks, which are taken to have their length and rates affected by the gravitational field. On the other hand, what we might call a concrete geodesy relies on the notions of invariable unit-measuring rods and clocks. In fact, this is a basic assumption of general relativity. Variable rods and clocks lead to a perverted geodesy in the sense that a curved space-time might be seen as arising from the (...) departure from the Minkowskian space-time as an effect of the gravitational field on the rate of clocks and the length of rods. In the case of a concrete geodesy we have “directly” a curved space-time whose curvature can be determined using (invariable) unit-measuring rods and clocks. In this paper, we will make the case for the plausibility that Einstein's views on geometry in relation to general relativity are permeated by a perverted geodesy. (shrink)

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s (...) system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

Hegel's philosophy of the concept is also a philosophy of the figure, a demonstration of conceptuality by other means. Neither images nor symbols, Hegel's figures (primarily, circles) image the motion of thought usually described as the Bewegung (movement) of dialectical processes. To be sure, Hegel diminishes the faculty by which we think geometric figures, the Understanding or Verstand, but he nonetheless demonstrates a variety of dialectical dynamics in the way figures, like circles and triangles, relate to one another. Vividly, he (...) animates circularity in his famous passage on the "circle of circles" by translating this image, through mental rotation, into a three-dimensional sphere, indeed a Bild borne of figures. (shrink)

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.

From the exponential function of Euler’s equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures.

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Descartes holds that the tell-tale sign of a solid proof is that its entailments appear clearly and distinctly. Yet, since there is a limit to what a subject can consciously fathom at any given moment, a mnemonic shortcoming threatens to render complex geometrical reasoning impossible. Thus, what enables us to recall earlier proofs, according to Descartes, is God’s benevolence: He is too good to pull a deceptive switch on us. Accordingly, Descartes concludes that geometry and belief in God must go (...) hand in hand. However, I argue that, while theism adds a layer of psychological reassurance, the mind-independent reality of God would ensure the preservation of past demonstrations for atheists as well. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...) to some speculative research programs. (shrink)

Review of Geometric Possibility (2011), by Gordon Belot. Oxford and New York: Oxford University Press. x + 219 pp.