Mathematical Methodology

Quantum mechanics lacks a formal account of the observer, leaving the measurement postulate structurally incomplete. I introduce a minimal operator chain: J, A, C, L, R, that formalizes recognition, articulation, collapse, and observation. Inserting these operators into the standard measurement rule yields a complete and stable measurement structure without altering quantum predictions. A spin‑measurement example and a reconstruction of Wigner’s friend demonstrate that paradoxes dissolve when collapse is explicitly observer‑indexed. -/- Authored by Eloy Escagedo Gutierrez as part of The Universal (...) Principle of Collapse (UPC) Research Project. (shrink)

In this first paper we outline what possible historic-epistemological role might have played the work of Ulisse Dini on implicit function theory in formulating the structure of differentiable manifold, via the basic work of Hassler Whitney. A detailed historiographical recognition about this Dini's work has been done. Further methodological considerations are then made as regards history of mathematics.

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)

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)

This paper introduces the concept of temporal lifting as a constructive analytic framework for reinterpreting apparent singularities in nonlinear dynamical systems, particularly the incompressible Navier–Stokes equations on the three-torus T³ = ℝ³ ∕ ℤ³. The approach suggests that finite-time blow-up is not an intrinsic breakdown of the equations but a compression of the physical time coordinate. By defining a smooth, strictly monotone lifting map φ : t ↦ τ and expressing the flow as U(x, τ) = u(x, φ⁻¹(τ)), the system (...) becomes globally C^∞ while preserving energy inequalities and invariants. This procedure is formally analogous to the Path Lifting Lemma in topology, in which a loop on S¹ lifts to a smooth path on ℝ. Philosophically, the result reframes mathematical singularity as a coordinate phenomenon, implying that continuity and regularity are functions of parametrization rather than absolute properties of nature. -/- Categories: Philosophy of Mathematics → Mathematical Methodology Philosophy of Science → Theoretical Physics -/- Keywords: time geometry; continuity; Navier–Stokes; regularity; coordinate systems; topology of flows; mathematical ontology -/- External link: HAL preprint (8 Oct 2025) See Link. (shrink)

We introduce iDNS, a deterministic spectral solver implementing the bounded vorticity-response functional Φ: ℝ≥0 → [φmin, φmax] for stable integration of chaotic nonlinear dynamical systems on T³ = (ℝ/ℤ)³. The Navier–Stokes equations admit a uniformizing parameterization: the parameter index τ ∈ [0,∞) generates coordinate time t = φ(τ) via the temporal lifting φ'(τ) = Φ(‖Ω(τ)‖_{L∞}). The lifted expression φ'(τ)∂τU + (U · ∇)U + ∇P − νΔU = 0 is not a modification—it is the same equation read in the uniformizing (...) parameter. Boundedness of Φ ensures non-degeneracy (φ' ≥ φmin > 0) and global coverage (φ(τ) → ∞); BKM coordinate invariance guarantees finiteness in τ implies finiteness in t. Physical time t and parameter τ coexist within the temporal covering space (T³ × ℝτ, T³ × ℝt, π) with π: (x,τ) ↦ (x, φ(τ))—integration proceeds in τ; results are stamped with t. The Regularity-Adaptive Spectral Integration (RASI) policy concentrates sampling where vorticity is high, resolving inter-modal energy transfer during the turbulent cascade—the mechanism by which standard DNS fails. Unlike adaptive timestep controllers, iDNS monitors the BKM functional in real-time via Φ; sampling density scales with vorticity magnitude, not Reynolds number. Kolmogorov flow computed stably from Re = 2 × 10⁴ to 10⁸ (0.07% dissipation error); 3D Taylor–Green vortex at Re = 10⁵ integrated through the full vortex-stretching cascade with bounded BKM = 37.1, Kolmogorov k⁻⁵ᐟ³ statistics, and 33.7× speedup—traversing the t ≈ 4.5 near-singularity where standard DNS crashes, on a 128³ grid, consumer laptop, 3.4 days. Precise Navier–Stokes physics, spectral resolution without artificial dissipation, no restarts, no failures—under 300 lines of Python. These results provide numerical validation of the theoretical framework addressing Fefferman's Clay Millennium Problem Statement (B); a unified analytical-numerical manuscript is forthcoming for precise Navier–Stokes physics, spectral resolution without artificial dissipation without failures—in under 300 lines of Python. -/- Data and code: 10.5281/zenodo.17730873. Numerically informed analytical mathematical basis can be found in "Global Regularity for Navier–Stokes on T³ via Bounded Vorticity–Response Functionals," at The Scholarly Journal of Post-Biological Epistemics, or also at PhilPapers. (shrink)

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of constructive mathematics Bishop style. The aim of (...) this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer. I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics. This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion to look anew at Bishop's own remarks for inspiration. (shrink)

At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...) nature, the construction of senses, and motile behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated intelligent behaviors. -/- Alongside this development is a further account that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure. -/- In support of this inquiry we work toward the development of a calculus for biophysical construction and its dynamics. If successful this mechanics mathematically characterizes sensory and motile behavior. -/- Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we develop a probabilistic theory that enables us to reason about inaccessible factors in group behavior. -/- The mechanics we propose suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. -/- We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience. (shrink)

The four-color theorem in standard mathematics and the four-letter theorem in Hilbert mathematics are juxtaposed. The former is a topological theorem thus not allowing for its elementary metric proof as the latter. This is interpreted as an example of "Modernity's bottle" forcing mathematics not to use any tools entering reality even provisionally for complying with the taboo for the gap between mathematical models and what they refer to. The same relation in Hilbert arithmetic is interpreted as passing through the "threshold" (...) between two and three dimensions after the Gleason and Kochen-Specker theorems. The representation of mathematical duality and physical complementarity is discussed. The philosophical concept of "the totality" translated in ontomathematics implies for mathematical models to be "concave" in order to obey the absoluteness of the totality. A principle of the equivalence of "concave" (philosophical or "metaphysical") and "convex" (proper mathematical) explanations being inherent from the viewpoint of ontomathematics is investigated by the example of the correspondence of the Bohmian (non-local and metaphysic: thus "concave") versus "Copenhagen" (local and agnostic: thus, "convex") interpretations of quantum mechanics. The eventual representation of quantum contextuality/ holism as fractal structures/ functional (Hausdorff) dimensionality is conjectured. The third dimension of reality (in the sense of the cited theorems) is researched in its two versions: "concave" (metaphysic, in particular, Bohmian) and "convex" (traditional and mathematical, particularly, "Copenhagen"). An illustration by the "four letters" of DNA (RNA) is considered. An interpretation of Peano arithmetic as the "half" of Boolean algebra unifying propositional logic and set theory is linked. The "fundamental theorem of the universe" is demonstrated as a corollary from the "four-letter theorem". -/- . (shrink)

Carnap introduced his notion of explication to arrive at concepts that are precise enough for scientific purposes. As Carnap wants to precisify concepts, his notion of explication targets less precise concepts so that explications within mature mathematics are not possible. We argue that explications of mature mathematical concepts are both possible and widespread. We focus on foundational work, especially as done in the context of interactive theorem proving. Taking foundational work seriously necessitates explicit decisions which are generally ignored in mathematical (...) practice - and such developments are not captured by our usual notions of explication. To see this, we introduce Carnapian and Quinean explication and argue that they are not capable of accounting for intra-mathematical conceptual changes. We argue that these changes are best understood as explications and so need a different notion of explication. We suggest one candidate, called tolerant explication, and sketch how it handles the intra-mathematical cases. (shrink)

This paper explores the concept of mathematical construction by building an analogy between modern computational interpretations and the traditional geometric approach. The focus is on how the tools used for construction, whether computational, such as Turing machines, or geometric, like straightedge and compass, define the boundaries of what can be considered constructible. The paper presents a comparative analysis of geometric construction and computational construction, and argues that mathematical existence is closely linked to the chosen forms of representation and construction tools. (...) The importance of these tools in shaping the concept of constructibility is highlighted with examples, showing how the introduction of new tools enables the construction of new mathematical objects. This perspective contributes to contemporary debates in constructivist philosophy and offers a potential way to deepen our understanding of mathematical construction. (shrink)

The incompressible Navier–Stokes equations on T³ exhibit a structural asymmetry: the spatial domain inherits the compact geometry of R³/Z³, while the temporal axis remains unbounded and analytically unconstrained. Classical approaches treat time as a neutral parameter, a clock labeling solution states without participating in the analytic structure. On periodic domains, this separation forfeits geometric constraints that the lattice structure naturally provides. We construct a coupled system (U, φ) via a bounded vorticity-response functional and prove that the Beale–Kato–Majda and Prodi–Serrin regularity (...) criteria are invariant under this bounded temporal lifting. Numerical validation on grids up to 256³ confirms invariance to machine precision across Reynolds numbers spanning 10³ to 10⁸. Published in The Scholarly Journal of Post-Biological Epistemics, Vol. 2, No. 1 (2026). DOI: 10.63968/post-bio-ai-epistemics.v2n1.013. ISSN 3069-499X. Preprint: DOI: 10.48550/arXiv.2510.09805. (shrink)

A symbolic elimination problem motivated by an arc--difference system produces (i) a mixed-derivative identity in four variables and (ii) a degree-$12$ univariate elimination polynomial. We isolate the algebraic core of these outputs and interpret the resulting arithmetic geometry. -/- First, we show that the mixed-derivative identity is equivalent, after a dimensionless normalization, to a one-parameter cubic equation \[ P_t(z)=t z^3-3t z^2+(3t+2)z+(1-t)=0 \] in the variable \(z=(q-s)^2/w^2\). The six radical branches observed in computer algebra output correspond to the three roots of (...) \(P_t\) together with the independent sign choice \(q=s\pm w\sqrt{z}\). We compute \[ \Disc_z(P_t)=-t(243t+32) \] and parametrize the locus on which the discriminant is a square, yielding a rational conic. -/- Second, we attach the genus-$1$ curve \(E_t:y^2=P_t(z)\) and obtain a short Weierstrass model \[ E_t:\ y^2=t x^3+2x+3, \qquad j(E_t)=\frac{55296}{32+243t}. \] In particular, under the natural positivity conditions inherited from the original system (forcing \(t>0\)), one has \(0<j(E_t)<1728\); hence \(E_t/\QQ\) admits no complex multiplication for rational parameters \(t\in\QQ_{>0}\). For completeness we also record the finite set of rational specializations \(t\in\QQ\) for which \(E_t\) has CM (equivalently, for which \(j(E_t)\) is a rational CM \(j\)-invariant). -/- Third, we embed \(P_t\) into a two-parameter deformation \(P_{t,k}\) and identify the constant-CM subfamilies \(k=0\) (constant \(j=0\)) and \(k=-1\) (constant \(j=1728\)). Subsequent sections analyze the degree-$12$ elimination family, including a balanced specialization yielding a regular \(S_{11}\)-extension over \(\QQ(\tau)\), together with reproducible computations supplied as supplementary SageMath material. (shrink)

The view that Peacock’s principle of permanence has been invalidated by Hamilton’s introduction of non-commutative algebras has always seemed rather odd, in light of Peacock’s favorable reception of quaternions and the endorsement of his principle by Hamilton. But the view is not just odd; it is incorrect. In order to show this, I critically analyze Peacock’s attempts to reject possible exceptions to his principle, like the factorial function and an infinite series due to Euler. Then I argue that the principle (...) of permanence is best understood as an expression of a conservative strategy, philosophically grounded in Hume’s conception of the laws of reasoning, which advocates their preservation to the furthest extent possible, thus allowing exceptions, i.e., violations of these laws. On this reading, non-commutative multiplication does not invalidate Peacock’s principle, if the reasons for violating commutativity outweigh the reasons for its preservation. Finally, I show that Hamilton followed a conservative strategy of precisely this sort when he developed his quaternionic calculus. (shrink)

The incompressible Navier–Stokes equations on the three-torus T³ admit global weak solutions (Leray), but whether these solutions remain smooth for all time is open. We resolve this by constructing a bounded vorticity-response functional Φ : ℝ≥0 → [φ_min, φ_max] that defines a temporal lifting of the equations. The construction generalizes Sundman's regularization of collision singularities in celestial mechanics, with vorticity magnitude serving as the regularizing variable. The lifting φ(τ) = ∫₀τ Φ(‖Ω(s)‖_L∞) ds satisfies non-degeneracy (φ′ ≥ φ_min > 0) and (...) global coverage (φ(τ) → ∞ as τ → ∞). Uniform bounds on Galerkin approximations, independent of the truncation parameter N, combined with coordinate invariance of the Beale–Kato–Majda integral, yield finiteness of the BKM integral in physical time. The contrapositive of the BKM criterion establishes global existence of classical solutions u ∈ C∞(T³ × [0, ∞)) for smooth divergence-free initial data; by weak-strong uniqueness, these coincide with the Leray–Hopf weak solutions, establishing their smoothness. This satisfies Fefferman's Clay Millennium Problem Statement (B). Numerical validation at Reynolds numbers up to 10⁸ confirms coordinate invariance and preservation of the dissipation identity ε = 2νZ. Numerical confirmation code and data available at doi:10.5281/zenodo.17730872. Paper DOI DOI: 10.63968/post-bio-ai-epistemics.v1n2.012. (shrink)

Developmental dyscalculia is generally recognized as a challenge in learning or processing numerical information, often associated with weaknesses in working memory or symbolic representation. This article introduces a more comprehensive and integrative perspective: examining dyscalculia through the framework of predictive processing (PP), which conceptualizes the brain as a system that continuously anticipates and refines its internal models to minimize unexpected outcomes. We investigate how dyscalculia may arise not solely from specific impairments, but from more profound disruptions in the brain's ability (...) to allocate attention, formulate abstractions, and condense experiences into numerical representations. By integrating concepts from cognitive science, philosophy, deep learning, and anthropology, we analyze the difficulties faced by children with dyscalculia in establishing reliable predictive models of numbers, time, and quantity. Their challenges do not indicate a deficiency in intelligence, but rather a dissonance between conventional mathematics education and their brain's distinctive method of interpreting the world. By synthesizing insights from Piaget, game theory, category theory, and cultural numerical systems, we propose innovative, adaptable strategies for mathematical education. These strategies encompass embodied, metaphor-driven, rhythmic, and game-like methods that honor the variety of cognitive styles. Instead of compelling all children to conform to a singular mathematical paradigm, we advocate for a pluralistic and predictive approach to numbers that can accommodate dyscalculic learners at their current level and facilitate their growth from that point. -/- . (shrink)

This paper proposes that mathematicians routinely use inference to the best explanation (IBE) to confirm their conjectures. Mathematicians can justly reason that the ‘best explanation’ of some mathematical evidence they possess would be a proof of it that likewise proves a given conjecture. By IBE, the evidence thereby confirms that such an as-yet-undiscovered proof exists and that the conjecture holds. This reasoning can be expressed in Bayesian terms once Bayesianism’s logical omniscience has been circumvented. A Bayesian analysis identifies considerations affecting (...) a mathematical IBE’s strength and helps to unify mathematical IBEs with scientific IBEs. (shrink)

Theories about the foundations of mathematics often encounter a problem similar to the traditional demarcation problem in science. In this context, it is pertinent to examine the first candidate for the identifying property of mathematical pluralism: reduction within a structure. As I argue here, this notion is insufficient for a coherent definition of structure within the plurality. In the end, demarcating a plurality of mathematics can be as problematic as demarcating a unitary mathematics. -/- .

In my 2007 paper, “Is There a Problem of Induction for Mathematics?” I rejected the idea that enumerative induction has force for mathematical claims. My core argument was based on the fact that we are restricted to examining relatively small numbers, so our samples are always biased, and hence they carry no inductive weight. In recent years, I have come to believe that this argument is flawed. In particular, while arithmetical samples are indeed biased, my new view is that this (...) bias actually strengthens the inductive support that accrues from them. The reason is that small numbers typically provide a more severe test of general arithmetical claims due to the greater frequency of significant properties and boundary cases among such numbers. In this paper, I describe and defend this new view, which a call Positive Bias Pro-Inductivism. (shrink)

Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the (...) role of mature intuition in mathematical practice with an assortment of examples, including data from a sequence of clinical interviews in which a student improves upon initially misleading covariational intuitions. Finally, we show how the study of intuition can yield insights for philosophers and education theorists. First, it contributes to a longstanding debate in epistemology by undermining epistemicism, the view that an agent’s degree of objectual understanding is determined exclusively by their knowledge, beliefs and credences. We argue on the contrary that intuition can contribute directly and independently to understanding. Second, we identify potential pedagogical avenues towards the development of mature intuition, highlighting strategies including adding imagery, developing associations, establishing confidence and generalizing concepts. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what the (...) world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

The great mathematician, physicist, and philosopher, Hermann Weyl, once called mathematics the “science of the infinite.” This is a fitting title: contemporary mathematics—especially Cantorian set theory—provides us with marvelous ways of taming and clarifying the infinite. Nonetheless, I believe that the epistemic significance of mathematical infinity remains poorly understood. This dissertation investigates the role of the infinite in three diverse areas of study: number theory, cosmology, and probability theory. A discovery that emerges from my work is that the epistemic role (...) of the infinite varies, often in surprising ways, across different domains of knowledge. -/- My first chapter examines the role of mathematical infinity in number theory. It is reasonable to think that theorems concerning finite patterns and structures in the natural numbers are particularly “simple” or “elementary.” Indeed, such statements are comprehensible to a wide range of investigators, regardless of their mathematical training. One might then expect proofs of these theorems to be similarly comprehensible. However, many proofs, especially those that utilize only finitary methods, are exceedingly difficult to understand. Consequently, one finds that finitary theorems are often re-proved using infinitary techniques. My claim is that this is because infinitary proofs are often explanatory, while finitary proofs are not. This chapter analyzes why this is the case. Along the way, I investigate other questions of long-standing interest in the philosophy of mathematics, e.g., the role of purity/impurity ascriptions and the nature of the content of a theorem. In particular, I diagnose the explanatory potential of the infinite by articulating a new construal of content. This new construal both saves intuitive epistemic ascriptions made in mathematical practice and explains the unexpected role of the infinite in providing explanatory proofs of finitary statements. Thus, in number theory, my claim is that the infinite often plays an explanatory role. -/- My second chapter turns to the role of the infinite in cosmology. It investigates a question much discussed by philosophers and physicists alike: is the spatial extent of the universe finite or infinite? Contemporary cosmological research has indicated that one of the essential determinants of the extent of the universe is the topology we ascribe to space. Topology is a global property, which may suggest that it is not testable through local observation. Nonetheless, some cosmologists have indicated that it may be empirically detected, thereby providing an answer to the question of spatial extent. I argue that, in fact, the epistemic status of the topology of space is extremely subtle and not well captured by any of the categories commonly employed by philosophers of science. In particular, I argue that topological properties are neither empirical nor a priori (even in suitably weakened senses). Furthermore, I claim that we should prefer topological properties that generate finite universe models (consistent with our best data) in order to avoid extremely thorny issues concerning the physics of an infinite universe. I argue for such a preference on the grounds of the simplicity and explanatory power of finite universe models. Thus, in cosmology, my claim is that the finite often plays an explanatory and simplifying role. -/- My third chapter investigates several paradoxes that arise in the foundations of infinitary probability theory: the Label Invariance Paradox, God’s Lottery, and Bertrand’s Paradox. I argue that these have been poorly understood because they do not expressly concern probability theory, but rather our intuitions about—and formal techniques for dealing with— infinite sets. The paradoxes in question are, in fact, symptoms of our complete reliance upon Cantorian cardinality and its associated criterion of sameness of “size.” That is, two sets have the same cardinality if and only if the elements of the sets can be placed in 1-1 correspondence. When applied to infinite sets, this criterion produces counterintuitive verdicts. For instance, given a fair lottery on the natural numbers, we expect that the probability of drawing an even number is 1/2, and likewise for drawing an odd number. However, one can construct “relabellings” of the naturals such that the probability of drawing an even number remains 1/2, while the probability for drawing an odd number becomes 1/4. I argue that, ultimately, it is the coarseness of Cantorian cardinality that generates the probabilistic paradoxes. I then propose that finer-grained measures of infinite sets from mathematical logic and number theory can help to dissolve the paradoxes in question. Thus, in probability theory, we find that particular kinds of infinitary techniques effectively systematize our theory, while others lead to paradox. (shrink)

We propose a characterization of mathematical experiments in terms of a setup, a process with an outcome, and an interpretation. Using a broad notion of process, this allows us to consider arithmetic calculations and geometric constructions as components of mathematical experiments. Moreover, we argue that mathematical experiments should be considered within a broader context of an experimental research project. Finally, we present a particular case study of the genesis of a geometric construction to illustrate the experimental use of hand drawings (...) and computers in mathematical practice. (shrink)

Informally rigorous mathematical reasoning is relevant. So too should be the premises to the conclusions of formal proofs that regiment it. The rule Ex Falso Quodlibet induces spectacular irrelevance. We therefore drop it. The resulting systems of Core Logic $ \mathbb{C}$ and Classical Core Logic $ \mathbb{C}^{+}$ can formalize all the informally rigorous reasoning in constructive and classical mathematics respectively. We effect a revised match-up between deducibility in Classical Core Logic and a new notion of relevant logical consequence. It matches (...) better the deducibility relation of Classical Core Logic than does the Tarskian notion of consequence. It is implosive, not explosive. (shrink)

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory power. The (...) examples developed in this paper constitute compelling evidence supporting these claims. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...) broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...) is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) we can’t (...) know ourselves to have any such freedom. (shrink)

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...) model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by (...) its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice.´´aa. (shrink)

Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge - unlike the former - can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics - queer incomaths - as a liberatory activity meant to undermine said naturalization.

Financial modelling is an essential tool for studying the possibility of financial transactions. This paper argues that financial models are conventional tools widely used in formulating and establishing possibility claims about a prospective investment transaction, from a set of governing possibility assumptions. What is distinctive about financial models is that they articulate how a transaction possibly could occur in a non-actual investment scenario given a limited base of possibility conditions assumed in the model. For this reason, it is argued that (...) the epistemic contribution of financial models is that of enabling the model users to envision exactly how a prospective investment could be achieved in various ways through a detailed understanding of the available transaction mechanisms. Thus, financial models provide information about the possibility of an investment scenario by showing how a specific transaction mechanism could result from a small set of initial possibility conditions assumed in the model. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed (...) light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our access to mathematical truths requires nothing beyond (...) our ability to make logical deductions. Sections 1 and 2 define and motivate deductivism in more detail. Section 3 covers four authors (Russell, Hilbert, Pasch, Curry) who have endorsed deductivism at some point. Section 4 aims to clarify what semantic claim deductivists make. Sections 5–9 review objections to deductivism and possible replies. (shrink)

Arguments for the effectiveness, and even the indispensability, of mathematics in scientific explanation rely on the claim that mathematics is an effective or even a necessary component in successful scientific predictions and explanations. Well-known accounts of successful mathematical explanation in physical science appeals to scientists’ ability to solve equations directly in key domains. But there are spectacular physical theories, including general relativity and fluid dynamics, in which the equations of the theory cannot be solved directly in target domains, and yet (...) scientists and mathematicians do effective work there (McLarty 2023, Elder 2023). Building on extant accounts of structural scientific explanation (Bokulich 2011, Leng 2021), I argue that philosophical accounts of the role of equations in scientific explanation need not rely on scientists’ ability to solve equations independently of their understanding of the empirical or experimental context. For instance, the process of formulating solutions to equations can involve significant appeal to information about experimental contexts (Curiel 2010) or of physically similar systems (Sterrett 2023). Working from a close analysis of work in fluid mechanics by Martin Bazant and Keith Moffatt (2005), I propose an account of heuristic structural explanation in mathematics (Einstein 1921, Pincock 2021), which explains how physical explanations can be constructed even in domains where basic equations cannot be solved directly. (shrink)

In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.

In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive (...) predicativity of inductive definitions. (shrink)

Volume II deals with philosophy of mathematics and general philosophy of science. In discussing theoretical entities, the notion of antirealism formulated in Volume I is further elaborated: Contrary to what is usually attributed to antirealism or idealism, the author does not claim that theoretical entities do not really exist, but rather that their existence is not independent of the possibility to know about them.

The article attempts to clarify Weyl’s metaphorical description of Emmy Noether’s algebra as the Eldorado of axiomatics. It discusses Weyl’s early view on axiomatics, which is part of his criticism of Dedekind and Hilbert, as motivated by Weyl’s acquiescence to a phenomenological epistemology of correctness. The article then describes Noether’s work in algebra, emphasizing in particular its ancestral relation to Dedekind’s and Hilbert’s works, as well as her mathematical methods, characterized by nonelementary reasoning—that is, reasoning detached from mathematical objects. The (...) article then turns to Weyl’s remarks on Noether’s work and argues against assimilating her use of the axiomatic method in algebra to his late view on axiomatics, on the ground of the latter’s resistance to Noether’s principle of detachment. (shrink)

The paper discusses Peano's argument for preserving familiar notations. The argument reinforces the principle of permanence, articulated in the early 19th century by Peacock, then adjusted by Hankel and adopted by many others. Typically regarded as a principle of theoretical rationality, permanence was understood by Peano, following Mach, and against Schubert, as a principle of practical rationality. The paper considers how permanence, thus understood, was used in justifying Burali-Forti and Marcolongo's notation for vectorial calculus, and in rejecting Frege's logical notation, (...) and closes by considering Hahn's revival of Peano's argument against Pringsheim's reading of permanence as a logically necessary principle. (shrink)

Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).

This essay concerns Dedekind’s “mathematical structuralism,”by which we mean methodological features characteristic for the approach to mathematics in his mature writings. The discussion starts with some background on forerunners, especially Gauss, Dirichlet, and Riemann, whose “conceptual” style of work influenced him strongly. But Dedekind went further than them, by making methodological choices that are more distinctly and fully “structuralist”. This includes his resolute acceptance of actually infinite systems, understood within a “logical” framework, and studied not just axiomatically, but also in (...) terms of isomorphisms and related notions (since 1871). As an illustration, the essay discusses his early adoption and strikingly modern transformation of Galois theory, together with his contributions to algebraic number theory. After that, the essayturns to Dedekind’s more “foundational” contributions, i.e., his writings on the real numbers, the natural numbers, and set theory. It shows that the same methodological choices inform them. Yet his approach kept evolving in subtle ways too, especially in terms of the centrality of functions (_Abbildungen_, mappings). (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...) means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)

In this article, I tackle a heretofore unnoticed difficulty with the application of Pyrrhonian scepticism to science. Sceptics can suspend belief regarding a dogmatic proposition only by setting up opposing arguments for and against that proposition. Since Sextus provides arguments exclusively against particular geometrical definitions in Adversus Mathematicos III, commentators have argued that Sextus’ method is not scepticism, but negative dogmatism. However, commentators have overlooked the fact that arguments in favour of particular geometrical definitions were absent in ancient geometry, and (...) hence unavailable to Sextus. While this might explain why they are also absent from Sextus’ text, I survey and evaluate various strategies to supply arguments in support of particular geometrical definitions. (shrink)

In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis.

Frege’s conception of axioms is an old-fashioned one. According to it, each axiom is a determinate non-linguistic proposition, one with a fixed subject-matter, and with respect to which the notion of a ‘model’ or an ‘interpretation’ makes no sense. As contrasted with the fruitful modern conception of mathematical axioms as collectively providing implicit definitions of structure-types, a conception on which the range of models of a set of axioms is of the essence of those axioms’ significance, Frege’s view is a (...) dinosaur. This essay investigates some of the philosophically-important aspects of that dinosaur, in order to shed light on Frege’s understanding of the foundational role of axioms, and on some of the ways in which our current conception of such axiomatic virtues as independence and categoricity have (and in some cases have not) been informed by a move away from Frege’s understanding of the foundational role of axioms. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)