Mathematical Structuralism

A good metaphysical account of logical form, must make clear why logical form is logical. However, this task has proved to be very elusive. Here, I analyze different attempts to meet this challenge and defend an inferential externalism where logical form is grounded on external logical relations as our most promising option.

A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.

Numbers are not quantities. They are discrete coherence states produced by threshold transitions in a continuous field. This paper develops a structural ontology of number grounded in Snap Geometry — the domain‑level expression of the Foundational Architecture. Under this view, numbers arise from Snap events: non‑linear threshold transitions that reorganize a continuous coherence field into stable, countable basins. -/- Arithmetic, algebra, calculus, and topology emerge as transformations of Snap sequences and the gradients between them. Numbers become indices of coherence rather (...) than objects; operations become reorganizations of boundaries rather than symbolic manipulations. -/- This framework reframes the foundations of mathematics through the lens of coherence physics, revealing number as a structural phenomenon — a geometry of coherence emerging from threshold transitions. (shrink)

This study is about the Quality. Here I have dealt with the quality that differs significantly from the common understanding of quality /as determined quality/ that arise from the law of dialectics. This new quality is the quality of the quantity /quality of the quantitative changes/, noticed in philosophy by Plato as “quality of numbers”, and later developed by Hegel as “qualitative quantity. The difference between the known determined quality and qualitative quantity is evident in the exhibit form of these (...) two qualities. The exhibit form of the known determined quality from the law of dialectics /or it transformation/ is related with discreteness and abrupt changes. The exhibit form of the qualitative quantity /and it transformation/ is related with the continuity and gradual transition from one condition, to a different condition, without any abrupt changes. In my paper “Quality of the quantity”, I have argued that one of the most ancient implementation of quality of the number can be found in the dimensional mathematical model of point – line – surface – figure - introduced by Plato. The most whole presentation of the idea of quality of number in Plato is embeded in his teaching about the "eidical number". The quality of the quantity emerges as criteria for recognizing the difference between the eidical numbers and natural arithmetical number. The thesis concerning Plato is based on the The Unwritten Doctrine of Plato and one of the most original works in the history of philosophy written in the 20th century - “Arete bei Platon und Aristoteles” – “Arete in Plato and Aristotle” /Heidelberg 1959/ written by Hans Joachim Krämer. The new quality as the quality of the quantity /quality of the quantitative changes/, first noticed in philosophy by Plato as “quality of numbers” was developed in Hegel as “qualitative quantity”. Hegel proclaimed the Qualitative quantity, or Measure in the both of his Logics -The Science of Logic / the Greater Logic/ and The Lesser Logic/ Part One of the Encyclopedia of Philosophical Sciences: The Logic. In my paper I have offered the arguments that the concept of quality of the quantity should be enhanced with the adopted methodological approach of analogy with an implementation in the field of the Topology - Analysis Situs, developed by the Jules Henri Poincare. In the topology we could see homeomorphism as exhibit form of Quality on the Quantity. The explicit form of the quality of the quantity transformation is the continuous deformation – typically known in topology as homeomorphism. The concept of qualitative quantity is linked with the concept “structural stability” and nonequilibrum phase transition. The concept of structural stability is related with the topological homeomorphism. In his book “Synergetics: Introduction and Advanced Topics” /Springer, ISBN 3-540-40824/, in the Chapter 1.13. Qualitative Changes: General approach, p. 434-435, Hermann Haken explores and illustrate the structural stability with an example /figure 1.13, p.434/ given by of D'Arcy W. Thompson, the Scottish biologist, mathematician and classics scholar and pioneering mathematical biologist, Nobel Laureate in Medicine /1960/, the author of the book, On Growth and Form, /1917/. The quality of the quantity could be seen in the Herman Haken’s citation on the D'Arcy W. Thompson. My thesis is that the topological homeomorphism is the explicit form of the quality of the quantity transformation. The qualitative quantity change which becomes phenomenon, according to Émile Boutroux, is the subject of study in Cultural Phenomenology of Qualitative quantity. Our approach to this subject is Poincaré Model of the Subconscious Mind in Mathematics, which is the most suitable tool to unfold the arhetype of qualitative quantity. (shrink)

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...) one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (shrink)

From the general history of culture, with a particular attention turned towards the personal and intellectual relationships between Hermann Weyl and Edmund Husserl, it will be possible to identify certain historical-critical moments from which a philosophical reflection concerning aspects of the ontology of mathematics may be carried out. In particular, a notable epistemological relevance of group theory methods will stand out.

In this paper I try to fortify the nominalistic objectology (cf. Meinong's 'Gegenstandstheorie') with essentialist means. This also is intended as a preparation for introducing Information Monism.

We propose a structuralist reconstruction of the real continuum, arguing that it is not a primitive ontological stage for physics but an emergent completion of the prime-exponent lattice M. By identifying the reals R as the Dedekind completion of M, we demonstrate that the natural logarithm and exponential functions are not merely analytic tools but inevitable algebraic consequences of identifying the multiplicative structure of number theory with the additive structure of the continuum. Furthermore, we show that the natural base e (...) is the unique structural invariant required to normalize the gauge freedom inherent in this identification. Finally, we refine the Erdős–Kac theorem to argue that the "Triple-Log" scale (ln ln ln n) emerges as a deterministic background limit of arithmetic, providing an intrinsic argument for a natural ultraviolet cutoff in physical theories without external imposition. In the language of Ontic Structural Realism (OSR), this framework suggests that the 'physical continuum' is not an object-like substance, but a structural instantiation of the relation between the additive and multiplicative sectors of arithmetic. (shrink)

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that seeks to (...) precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

Structures are ubiquitous in mathematics. But how should they be understood? Modelists claim they are model-theoretic structures. This thesis can be read in two ways: as a claim about what structures refer to, or about how we conceptualize them. Objects-modelism, developed by Button and Walsh, pursues the first; the second leads to concepts-modelism, which remains underexplored. In this paper we develop and defend a version of concepts-modelism, cognitive modelism, drawing on Carey’s theory of conceptual development, and we show how it (...) addresses the challenges Button and Walsh pose for a conceptual account of mathematical structures. (shrink)

Advocates of the explanatory indispensability argument for platonism say two things. First, we should believe in the parts of our best scientific theories that are explanatory. Second, mathematical objects play an explanatory role within those theories. I give a two-part response. I start by using a Bayesian framework to argue that the standards many have proposed must be met to show that mathematical objects are dispensable are too demanding. In particular, nominalistic theories may be more probable than platonistic ones even (...) if they are extremely complicated by comparison. This is true even if there are genuine cases of mathematical explanation in science. The point made here is a matter of principle, holding regardless of how one assesses nominalistic theories already on offer. I then examine my recent nominalization of second-order impure set theory in light of the correct, laxer standards. I make a tentative case that my nominalistic theory meets those standards, which would undermine the explanatory indispensability argument. While this case is provisional, I aim to bring attention to my nominalization and others in light of the revised standards for demonstrating dispensability. (shrink)

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

According to structuralism in the philosophy of mathematics, the natural numbers are individuated purely by their structural interrelations. A related metasemantic view, which I call axiomism, holds that the meanings of our arithmetical terms are determined just by our acceptance of categorical axioms for arithmetic. Against both structuralism and axiomism, I present the case of the Dyadians. These speakers accept principles identical to our Peano axioms. Nevertheless, it seems clear that they use terms like “13” and “natural number” with different (...) meanings from us. English speakers and Dyadians refer to numerical structures that are isomorphic but distinct, a possibility that axiomists and structuralists cannot adequately account for. This basic problem has been noted before, going back all the way to Russell’s Principles of Mathematics (1903), but it has often been brushed off. My primary aim in this paper is to clarify the costs of doing so. In the final section of the paper, I suggest a remedy. Contra structuralism and axiomism, we should acknowledge that the natural numbers, and the meanings of our basic arithmetical terms, are essentially tied to counting. This is also reminiscent of proposals from Russell, Frege, and the neo-Fregeans. However, I identify two senses of the transitive verb “to count” that are often conflated in this context. In only one of the two senses is counting essential to the natural numbers or our concepts of them. (shrink)

This paper presents the mathematical layer within the first system of Kasei-Theory as a non-modal readability maintainability architecture. -/- The paper does not develop a philosophy of mathematics, a foundational formalism, or a representational theory of symbolic systems. Instead, it fixes non-relational fixation, non-metric differentiation, constraint notation, topological differentiation, and non-dynamical configuration as constrained structural positions within local readability maintainability. -/- Mathematical notation does not establish representation. Structural fixation does not establish formal grounding. Differentiation does not require measurable distance, and (...) topology does not establish spatial structure. Constraint notation does not establish proof as foundational grounding, explanatory closure, or operational formalization. -/- The mathematical layer is not introduced as a complete mathematical system. Formal structures are treated as constrained readability stabilizations rather than transcendental explanatory foundations. -/- The propositions developed here remain structural fixations rather than formal or metaphysical foundations. -/- No subject is presupposed. No final philosophical closure is established. -/- This paper is part of Kasei-Theory. (shrink)

This paper presents placement within the first system of Kasei-Theory as a non-modal readability maintainability architecture. -/- The paper does not develop a geometry of placement, a theory of spatial order, or a metaphysics of positional distribution. Instead, it fixes placement, distributed configuration, locality, configurational drift, and constrained maintainability as distributed structural positions within constrained local readability maintainability. -/- Placement does not establish position, localization, or coordinate structure. Distribution does not establish extension, spatial multiplicity, or universal arrangement. Locality does not (...) establish region, containment, or spatial adjacency. -/- Configurational differentiation remains locally maintainable without transition into geometrical coherence, dimensional ordering, or centralized structural stabilization. -/- Drift does not establish displacement. Placement does not establish hierarchy, enclosure, or terminal arrangement. -/- The propositions developed here remain structural fixations rather than geometrical or spatial foundations. -/- No subject is presupposed. No spatial grounding is secured. No final philosophical closure is established. -/- This paper is part of Kasei-Theory. (shrink)

This paper establishes the Isometric Extension of the Mathematical Unified Theory of Cognitional Mechanics (CM-MUT), deriving the exact quantitative correspondence between the geometric modal functor and the algebraic modal functor over the historical category Hist. The central result is that for all admissible operational histories H, the κ-scale L¹ norm and the Frobenius norm are related by the Casimir invariant K=√3 of M₃(ℂ): ‖M_A(H)‖_κ = K·‖M_G(H)‖_F. -/- This isometric relationship is derived from two implementation axioms, A3' (Modal Norm Selection) and (...) A5 (Modal Distance Preservation), which formalize assumptions implicit throughout the CM corpus. The proof proceeds via the automorphism structure Aut(M₃(ℂ))≅PGL(3,ℂ), Axiom 4 (Redundancy Exclusion), and a formal lemma establishing that minimal descriptive complexity in the Cartan subalgebra uniquely forces uniform eigenvalue distribution Var(κ)=0 for all admissible histories. -/- Four corollaries follow from a single algebraic source: the inverse fine structure constant α⁻¹=137.03599…, the Iron Stability Attractor κ(Fe)=6/13, the chemical projection constant Π_chem=145/468, and the ABC quality bound q(a,b,c)<13/7. -/- Prior CM results on the Riemann Hypothesis, P≠NP, and the Halting Problem are unified under this framework. The Navier-Stokes regularity problem, the Hodge Conjecture, the Yang-Mills mass gap, the Poincaré Conjecture, and the Birch-Swinnerton-Dyer Conjecture are shown to be Tier-3 projections of Tier-1 structural necessities. -/- The apparent difficulty of the Millennium Problems is an artifact of their Tier-3 formulation, not of mathematical reality. All derivations proceed with zero free parameters from the axiomatic structure of M₃(ℂ). -/- Published on April 4, 2026 DOI: 10.5281/zenodo.19411822 . (shrink)

This paper is Version 3 of the e-as-finite-number programme, establishing e as a Category B Operational Invariant (Japanese: Sousa Hensuu 操作遍数) of the unique minimal operational closure C^(3)_Πd, realised as M₃(ℂ) structure, within the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, the successor framework to Cognitional Mechanics. -/- The central advance is the Grounding Uniqueness Lemma: among all values exp(t) for t∈ℝ, the unique element generating an independent Operational Invariant is exp(1)=e. This follows from exhaustive case (...) analysis — rational t are A4-redundant relative to e; nonstructural irrational t admit no M₃(ℂ)-grounding; structural irrational t (e.g. e^δ, e^π, e^γE) yield only Category E composites of existing S-elements. -/- The grounding source (G2b) is the unique unit operation of the Lie algebra exponential map of M₃(ℂ) — a Tier-2 operational concept involving no numerical parameter. Its Tier-3 representation is t=1, yielding e=exp(1) as the M3-value. e is thus established as an element of S\Fext of the general Operational Invariant solution (DOI: 10.5281/zenodo.20647095). -/- Hermite's transcendence result (1873) is repositioned as the final step in the logical chain: G2b ⟹ unit operation ⟹ Tier-3 projection ⟹ t=1, exp(1)=e ⟹ Hermite confirms e∉Q̄. It confirms, rather than grounds, the uniqueness established by G2b. The structural invariance de/dt=0 follows as a theorem (Corollary 3.6). -/- This extraction is logically necessary in the same sense as that of π (DOI: 10.5281/zenodo.20602756): the absence of a Tier-3 closure condition for the open curve exp(tH) — confirmed by Hermite — uniquely necessitates characteristic-polynomial confirmation, symmetric in logical structure with the closure condition U(θ)=I for π. -/- For those interested, the following code illustrates the Tier-3 confirmation of the exponential unit operation at 50-digit precision: starting from H=diag(1,1,-2) with integer eigenvalues only, no numerical approximation of any constant is manually supplied at any stage, and the constant appears only in the printed output. The algebraic justification is given in Section 3 of the paper. -/- ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ -/- from mpmath import mp, mpf, mpc, matrix, findroot, expm, exp, sqrt ◆ mp.dps = 50 ◆ h = [mpf(1), mpf(1), mpf(-2)] ◆ H = matrix([[h[0],0,0],[0,h[1],0],[0,0,h[2]]]) ◆ M = expm(H) ◆ def tr(A): return sum(A[i,i] for i in range(3)) ◆ def det3(A): return (A[0,0]×(A[1,1]×A[2,2] - A[1,2]×A[2,1]) - A[0,1]×(A[1,0]×A[2,2] - A[1,2]×A[2,0]) + A[0,2]×(A[1,0]×A[2,1] - A[1,1]×A[2,0])) ◆ p1 = tr(M) ◆ p2 = (p1^2 - tr(M × M)) / 2 ◆ p3 = det3(M) ◆ def char_poly(lam): return lam^3 - p1×lam^2 + p2×lam - p3 ◆ r_small = findroot(char_poly, mpf('0.5')) ◆ b = r_small - p1 ◆ c = p3 / r_small ◆ C1 = ((-b + sqrt(b^2 - 4×c)) / 2).real ◆ C2 = mp.e ◆ print(f"Tier-2: unit operation (G2b); Tier-3: parameter value t=1, char_poly(expm(H))") ◆ print(f"Logical order: G2b -> unit operation (T2) -> Tier-3 projection -> parameter value t=1 -> exp(1)=e -> Hermite confirms value not in Q-bar") ◆ print(f"confirmed eigenvalue > 1 = {C1}") ◆ print(f"library reference value = {C2}") ◆ print(f"difference = {abs(C1 - C2)}") -/- ◆◆◆ [END] ◆◆◆ -/- Published on June 14, 2026 DOI 10.5281/zenodo.20685098. (shrink)

This paper establishes that category theory is not a foundational layer of any operational system but a representational artifact: the minimal morphism-based formal language encoding the operational obstruction structure [ℐ/∼] derived from Operatiology. The Unbounded Index Obstruction criterion classifies core categorical notions individually. Arbitrary categories, functors, and natural transformations require certification over non-finitely-exhaustible index families. Limits and colimits belong to the power-set type of [ℐ/∼], the categorical analogue of the Power Set axiom in ZFC. Adjunctions belong to the unrestricted type, (...) parallel to the Axiom of Replacement. The Yoneda lemma occupies the extremal position within the power-set type: the entire relational complexity of Nat(h_A, F) collapses to a single evaluation, constituting maximum compression efficiency within the morphism-based language. The minimal categorical structures retaining executive correlates in C⁽³⁾_Πd are finite acyclic free categories and the single-object category BM₃(ℂ). A fidelity ordering among categorical, set-theoretic, and type-theoretic foundations is established as a structural taxonomy. Category theory and ZFC are dual [ℐ/∼] compression languages differing only in compression axis: membership versus morphism. Their foundational rivalry is a competition internal to the projective layer. The classification criterion (operational necessity, grounded in Axiom 2) and the classification function (Unbounded Index Obstruction) are both derived from the executive layer; no external foundation is structurally required or presupposed. -/- Published on May 31, 2026 DOI: 10.5281/zenodo.20470946. (shrink)

This paper derives two structural results from the unique minimal non-commutative algebra M₃(ℂ) of Cognitional Mechanics (CM), with zero free parameters. -/- First, the complete spectral closure terminus Z = 118 is established as an algebraic necessity of Axioms A1–A4: the slot formula N_ℓ = 2 + 4(ℓ−1) terminates at ℓ_max = Φ₆ = 7 by the A3 cyclotomic structure, and ℓ = 8 closure is forbidden by Axiom A4 (Redundancy Exclusion) via the κ = 1/2 conflict with the established (...) f-band half-closure node. Elements with Z > 118 are not algebraically excluded; Z = 118 is the terminus of complete spectral closure. -/- Second, all seven classical nuclear magic numbers 2, 8, 20, 28, 50, 82, 126 are derived as a zero-parameter consequence of the phase difference between the Cartan action (electron spectral closure, dimension n−1 = 2) and the automorphism action (nucleon spectral closure, dimension n²−1 = 8) on M₃(ℂ). The phase difference series D_k ∈ {0, ±Φ₁(3), ±2Φ₂(3)} is uniquely determined by S1–S2 antisymmetry, the conservation law Σ D_k = −2Φ₂(n) = −8 fixed by the bidirectional eigenvalue structure of the Cartan action, and a monotonicity bound. -/- New magic numbers in neutron-rich nuclei (6, 14, 16, 32, 34) are derived as Type II-modified closure nodes. The value 14 = 2Φ₆ was confirmed experimentally in 2025. The isospin-symmetric Island of Inversion at Z = N = 42 (⁸⁴Mo) is identified as an over-resonance node κ = 7/6 > 1, with the 8p-8h excitation count n_ph = 2Φ₂(n) = 8 following algebraically from A1 alone. (shrink)

This paper establishes that the standard number systems ℕ, ℤ, ℚ, ℝ, and ℂ are not pre-existing mathematical objects but structures uniquely selected by the operational constraints of the rank-3 minimal operational closure C⁽³⁾_Πd as formalised in Operatiology. The selection order is constrained by the axioms: Axiom 1 contains a geometric-persistence clause expressed using the distance function d, which is defined only in Axiom 2. Therefore ℝ, the number system selected by Axiom 2, must be established before ℂ, the number (...) system selected by the orbital structure of Axiom 1. -/- Axiom 2 selects ℝ as the unique totally ordered, complete Archimedean field. Axioms 1 and 4 identify ℕ, ℤ, and ℚ as the minimal discrete substructures of ℝ required for distinguishability, reversibility, and minimal description. Axiom 1 also generates bilateral inaccessibility between non-commutative orbit pairs, producing a canonical ℤ/2ℤ symmetry. By Galois theory, the unique minimal realisation of this symmetry over ℝ is ℂ, making ℂ the number system structurally demanded by non-commutativity. -/- Working Axiom 3 is not invoked; all results follow from A1, A2, and A4. Numbers and the algebra M₃(ℂ) are co-derived from the same axiomatic source via the chain C⁽³⁾_Πd → ℳ₁ → M₃(ℂ), showing that mathematics functions as the projective layer generated by, and subordinate to, the executive structure of Operatiology. Numbers, in this framework, are not discovered or invented; they are what operational closure structurally requires. -/- This version supersedes the prior Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20108822). The transition from Cognitional Mechanics to Operatiology replaces the earlier grounding in "intelligence" as a technical term with the structurally deeper foundation of C⁽³⁾_Πd, and reframes the mathematical projection hierarchy in terms of the five-layer distance ordering established in the companion paper on mathematics as the unique top-down projection of operational structure. Published on May 24, 2026 DOI: 10.5281/zenodo.20366742 . (shrink)

This paper establishes that geometric structure — distance, metric, curvature, and the analytic machinery built upon them — is not operationally necessary in any operational system but a representational artifact: a formal construct encoding the algebraic structure of the rank-3 minimal operational closure C⁽³⁾_Πd into an extended descriptive language. The argument proceeds from the axiomatic foundation of Operatiology, in which C⁽³⁾_Πd is derived from three axioms governing non-commutativity, Πd-saturation with finite generator rank, and redundancy exclusion, and from the companion result (...) that algebra is the unique top-down projection of the operational substrate. The Metric Reduction Theorem establishes that every admissible distance structure is a projection of the trace inner product Tr(A†B) of M₃(ℂ), the unique minimal algebraic realisation of C⁽³⁾_Πd, and is therefore not an independent operational primitive. The Geometric Unbounded Index Obstruction lemma classifies continuous curves, infinite metric spaces, topological neighbourhoods, and curvature as elements of the termination-failure equivalence class [ℐ/∼], with distinct obstruction types. Euclidean and non-Euclidean geometries are classified by the same mechanism, differentiated only by the curvature parameter of their metric projection. The Euclidean–non-Euclidean dispute is dissolved as a competition between alternative representational languages. The general relativistic metric formulation is classified as a representational artifact without affecting its empirical validity. Published on May 30, 2026 DOI: 10.5281/zenodo.20465726. (shrink)

PEMDAS is the dependency chain of arithmetic operations read in the evaluation direction. Counting, addition, multiplication, and exponentiation form a strict dependency chain in which each operation requires the previous one to exist, because multiplication is repeated addition compressed into one quantity and exponentiation is multiplication composed with itself. That there are exactly these four operations is shown by removal: remove any one and the operation above it loses its operand, and no fifth operation is required, because the output of (...) exponentiation is already a count the chain takes up again. Evaluation reverses the chain because evaluation decomposes what construction composed, with the most composed operation integrated first into its single output and the least composed last. Parentheses come before any operation because the boundary of the unit to evaluate has to be identified before the chain can operate on the interior. The convention story is incomplete because a different evaluation order would not be a different valid arithmetic but an arithmetic in which composed expressions cannot evaluate to the values they were built to represent. The four-operation count is derived here on arithmetic's own operations; the same count is reached independently at the substrate scale (Stewart, 2026g) and in algebraic form (Stewart, 2026e), by derivations that share none of arithmetic's operations. The classroom inheritance is that a teacher derives PEMDAS with the student from the dependency rather than asking the student to memorize a mnemonic, and the student arrives at the order of operations as a conclusion of arithmetic's own structure. -/- **Keywords:** order of operations, PEMDAS, BODMAS, arithmetic, exponentiation, inverse operations, mathematics education, conceptual understanding. (shrink)

This paper proposes that any mathematical construction requires four operations to be performed in a fixed structural order. The four operations are distinction, placement, identity, and composition. Logic formalizes distinction, sets formalize placement, types formalize identity, and categories formalize composition. Each foundation formalizes one operation as primary and employs the remaining three as apparatus. The four operations proceed in the order distinction, placement, identity, composition, and that order is forced by operational dependency. -/- The bilateral correspondences established by Curry and (...) Feys (1958) and Howard (1969/1980), extended by Lambek (1972), and connected to sets through topos theory by Lawvere (1970) and Tierney (1972) are cited as the published evidence that each of the four foundations has the capacity to express all four operations as a single formal system. In this sense each foundation is four-in-one, and the four foundations are structurally translatable into one another. The contribution of this paper is the recognition that the four operations, contained as one in each foundation, proceed in a fixed dependency order, with distinction preceding placement, placement preceding identity, and identity preceding composition. The bilateral correspondences are symmetric. We propose the dependencies among the four operations are not symmetric, and the asymmetry of the dependencies fixes the direction in which the correspondences are used when a construction is built up. The correspondences themselves remain equivalences. -/- We distinguish the operations any mathematics must perform from the axiomatic apparatus chosen as primary. The existing foundational debates (ZFC versus Martin-Löf type theory versus ETCS) concern which apparatus is primary and do not address the question we ask. The claim is falsifiable. A piece of mathematics that genuinely lacks one of the four operations would refute it. We predict no such piece exists, because the four operations are features of any mathematical construction. -/- **Keywords:** foundations of mathematics, philosophy of mathematics, Curry-Howard correspondence, Lambek correspondence, topos theory, propositions-as-types, logic, set theory, type theory, category theory, ETCS, mathematical structuralism. (shrink)

Uno dei problemi essenziali della ricerca matematica, e degli studi scientifici in generale, è rappresentato dalla capacità, e dunque dall’analisi predittiva di questi campi, cioè il processo di utilizzo dei dati osservati per prevedere risultati futuri. Questo è in particolare il caso di eventi casuali (cioè non necessariamente causali o deterministici, ma probabilistici o stocastici). All’interno della teoria della probabilità, un processo stocastico è un insieme ordinato di funzioni reali di un parametro specifico, che gode di determinate proprietà statistiche. Esso (...) rappresenta cioè un oggetto matematico solitamente definito come una sequenza di variabili casuali in uno spazio di probabilità, dove il parametro suddetto è spesso il tempo; quindi, rappresenta una famiglia di variabili aleatorie {X(t), t ∈ T ⊆ R +} dipendenti dal tempo t, dove X(t) è il parametro di variabili casuali reali, cioè trasformazioni dallo stato iniziale allo stato successivo, ossia dopo un certo tempo. Queste variabili sono definite su uno spazio campione Ω, assumenti valori in un insieme definito spazio degli stati del processo. (shrink)

The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a weaker version of (...) the insubstantiality thesis: mathematical objects have no non-structural essential properties. Next, I show how this rendition of structuralism alleviates a Fregean worry against insubstantiality, which is directed at the explanation of the applicability of mathematics from the structuralist perspective. (shrink)

We show that structuralism has the very serious defect of having no satisfactory notion of identity which can be associated with its central notion: structure. We also refute the structural thesis about the nature of the natural numbers by showing that there are at least two completely different structures that are entitled to be taken as ‘the structure of the natural numbers’, and any choice between them would arbitrarily favor one of them over the equally legitimate other. Finally, we argue (...) for the high plausibility of the identification of the natural numbers with the finite von Neumann ordinals. (shrink)

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all (...) adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps in other sciences, not to mention in philosophy. (shrink)

In the first part of the paper, I clarify what is at stake in the debate between accounts of the iterative conception based on the notion of metaphysical dependence and the minimalist account I have defended in previous work (Incurvati 2012; 2020). I argue that the debate concerns how to understand and motivate the central tenet of the iterative conception that every set occurs at some level of the cumulative hierarchy. This debate, I contend, should be distinguished from the debate (...) between actualist and potentialist accounts of the cumulative hierarchy. In the second part of the paper, I use the distinction drawn in the first part of the paper to assess an objection leveled by Mark Gasser (2015) against ante rem structuralism. In particular, this distinction makes it clear that there are two different objections in Gasser’s article. The first objection is that the iterative conception conflicts with dependence claims made by structuralists. As I have suggested in previous work, ante rem structuralists can address this objection by endorsing a minimalist account of the iterative conception. The second objection is that the indefinite extensibility of the set concept conflicts with the idea that the cumulative hierarchy is exhausted by the ZFC axioms. I chart various possible ways of addressing the second objection and show that by disentangling ante rem structuralism from the idea that mathematical structures ought to be given by implicit definitions opens up a novel way of addressing the second objection. I conclude by contending that although my arguments show that ante rem structuralism is compatible with the iterative conception, there are still reasons to favour a more deflationary understanding of structuralism, advocated by John Burgess, Joel Hamkins and Gasser himself. (shrink)

Uno degli scopi essenziali delle scienze matematiche è la loro applicabilità in senso pratico, cioè per risolvere problemi tecnico-scientifici. In questo senso si interpreta lo scopo della matematica applicata, contenente il sottocampo dell’Analisi Numerica. Nello specifico, l’Analisi Numerica riesce a risolvere i modelli prodotti dall'analisi matematica alle scomposizioni finite normalmente praticabili, tramite il concetto di approssimazione. Essa utilizza il calcolo integrale e le equazioni differenziali, i metodi multipasso e a passo singolo, i sistemi lineari e i metodi avanzati, come i (...) metodi diretti, l’eliminazione gaussiana, la rotazione secondo il perno (il pivoting) e la fattorizzazione, le tecniche iterative, i metodi di Jacobi e di Gauss-Siedel, il metodo (di) Monte Carlo, il Successivo Sovrarilassamento (SSR), la convergenza, la procedure di miglioramento iterativo, e metodi alternativi, per esempio di trasformazione e del rigetto, e le distribuzioni. D’interesse particolare sono gli algoritmi, essenziali tecniche caratterizzabili secondo la velocità di convergenza, la computabilità e la stabilità numerica. (shrink)

If you have enjoyed any of the 7 (seven) other books I have published over 20 years, including literally thousands of pages of mathematical and topological concepts, Python programs and conceptually expanding papers, please consider buying this book for $20.00 on google play books. -/- Introduction: -/- Though the following pages provide extensive exposition and dedicated descriptions of the phenomenological velocity formulas, theory and mystery, I thought it appropriate to write this introduction as a partial explanation for what phenomenal velocity (...) is, and describe, briefly its theory and applications. Phenomenological Velocity is a method for solving for something that ought cancel out with itself, but there are specific implicit forms for this thing that, “ought cancel out with itself,” namely the Lorentz coefficient ought cancel out with itself when applied to the height of a cone derived from the difference between the circumferences of two circles applied to the Pythagorean Theorem, or, more generally, the height implied by application of the Pythagorean theorem to the difference between two arc lengths’ equaling a third arc length. These difference equations are essential to conceptualizing differentiation, and in these further chapters, I demonstrate that the phenomenological velocity is, indeed the conditional derivative in the chapter, “Conditional Integral of Phenomenological Velocity.” The phenomenological velocity algebraic solution to the velocity within the Lorentz coefficient when applied to the height function in such a way that it ought cancel out with itself is both constructive mathematics and it employs the concept of, “bracketing,” - first introduced by Edmund Husserl in his writings on the phenomenological reduction. Phenomenological Velocity’s algebraic solution from the difference between two arc lengths applied to the Pythagorean Theorem to solve for a theoretical height (which is a projected distance in space), employs bracketing, because we, “set aside,” the existence of an undefined solution, namely due to the presence of necessitated complex analytical forms by the architecture of the equation, or the “mathetecture,” of the algebraic form. -/- With respect to theology, the phenomenological velocity is somehow symbolic of the creation itself; symbolic of creation due to the fact that we find the canceling out of the Lorentz coefficient as, “impotent,” non-existent or non-effecting to the mathetecture of the height function. However, via the modus-ponens work around to phenomenological velocity, which in itself does not require the complex field, but embeds implied complex field solutions to the equation while maintaining logical consistency, we find existence from non-existence. This is directly linguistically applicable to the concept of the big-bang, the resurrection of Yeshua the Messiah, and opens analogies for us to draw relationships between the, “fall,” of Adam and Eve as the generation of error, or the introduction of paradox, as we see the phallus representing paradox topologically. -/- The phenomenological velocity is a gestalt concept, relevant to cosmology, because we find that it is the perfect language-form for discussing dark matter. It does, however, require the reader to re-conceive or re-frame rather, some of the fundamental aspects of assumed physical reality like time, experience, solidity of dark matter, etc. We find the hidden dimension of phenomenological velocity to have been an overlooked aspect of mathematical physics by the researchers of Bell’s theorem and undoubtedly a host of other theorems. Thus, raising awareness about the real existence and necessitated reality of phenomenological velocity is in no way an endeavor deserving further procrastination by the scientific community, for doing so would be intellectually dishonest and further the propagation of incomplete or misleading theories on reality. -/- This work details how the Lorentz coefficient, when applied to the height of a cone in such a way as to cancel out with itself, permits the velocity variable to have a solution to it anyway, even though it ought cancel out with itself. This mathe-tecture, so to speak has consequences for complex analysis, and pave the way for, "transcendental relativity," building an adaptive framework for consciousness and physical reality. (shrink)

Chaos topology delves into bizarre symbol configurations of the world of quasi-quantification and pseudo-quantifiability. With quantum complexes, we are eventually able to find game play that blends wormholes with quantum computers and transcendental numbers. A conclusion that will leave you breathless, Chaos (Emmerson, 2023),.

This Element discusses the philosophical roles of definitions in the attainment of mathematical knowledge. It first focuses on the role of definitions in foundational programs, and then examines their major varieties, both as regards their origins, their potential epistemic roles, and their formal constraints. It examines explicit definitions, implicit definitions, and implicit definitions of primitive terms, these latter being further divided into axiomatic and abstractive. After discussing elucidations and explications, various ways in which definitions can yield mathematical knowledge are surveyed.

Developments in truthmaker semantics for the most part stay clear of the metaphysical issue of what sort of entities serve as the truthmakers and falsitymakers for sentences. It is assumed that perhaps facts or states of affairs (Fine 2017a; Jago 2020), with these taken sometimes as concrete particulars (Hawke 2018) could serve for the job, but nonetheless that some such entities would do. In this paper I take a closer look at the issue of what entities could or could not (...) play the role of truthmakers and falsitymakers in standard truthmaker semantics (Fine 2016, 2017a, b; Fine and Jago 2019), based on desiderata imposed by metaphysical and semantic considerations. (shrink)

These involved theorems on sweeping nets, saddle maps and complex analysis are a thorough examination of the method an its fundamental mechanics. The basic foundation of this analytical method is useful to any artificer of mechanical programs or development of software applications that involve computer vision or graphics. These methods will have application to further theories and methods in string theory and cosmology or even approximation of environmental factors for machine learning. -/- .

The bridges of Königsberg case has been widely cited in recent philosophical discussions on scientific explanation as a potential example of a structural explanation of a physical phenomenon. However, when discussing this case, different authors have focused on two different versions, depending on what they take the explanandum to be. In one version, the explanandum is the _failure_ of a given individual in performing an Eulerian walk over the bridge system. In the other version, the explanandum is the _impossibility_ of (...) performing an Eulerian walk over the bridges. The goal of this paper is to show that only the latter version amounts to a real case of a structural explanation. I will also suggest how to fix the first version, and show how my remarks apply to other purported cases of structural explanations. (shrink)

Recent years have featured the existence of a variety of structuralisms, with an important partition between methodological versus philosophical structuralism. Inside philosophical structuralism, many trends can be identified, corresponding to various ontological stances. We argue here that another main partition has contributed to organize structuralism in the twentieth century, rooted in different technical and theoretical interests. This partition is largely transversal to the ones classically identified. Concretely, the paper will focus on possible differences between an arithmetical and logical notion of (...) structure that can be traced back to the writings of Bertrand Russell and Rudolf Carnap, and a mathematical notion of structure, exemplified in the works by Bourbaki. This coexistence gives rise to a fundamental ambiguity that affects contemporary structuralism. Philosophically, in one case the attention is rather centered on a foundational and reductionist perspective, as featured by the Whitehead-Russell Principia and the Carnapian project of the Aufbau: the scientific construction of the world around the idea of structure. In the other, the focus is on epistemological and dynamical issues, as exemplified by two key issues in Bourbaki’s treatise: understanding the architecture of mathematics, offering a tool-kit to mathematicians. These two distinct meanings still coexist inside contemporary scientific practices and lead to different theoretical interests, as we will show thanks to various recent examples. (shrink)

In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages (...) over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics. (shrink)

Relationism holds that objects entirely depend on relations or that they must be eliminated in favour of the latter. In this article, I raise a problem for relationism. I argue that relationism cannot account for the order in which non-symmetrical relations apply to their relata. In Section 1, I introduce some concepts in the ontology of relations and define relationism. In Section 2, I present the Problem of Order for non-symmetrical relations, after distinguishing it from the Problem of Differential Application. (...) I also examine four main existing strategies to solve it. In Section 3, I develop my argument. The first step consists in arguing that—among those strategies—relationism can only accept directionalism. The second step consists in arguing that directionalism is affected by a serious problem: the Problem of Converses. I also show that relationists who embrace directionalism cannot solve this problem. In Section 4, I introduce and rebut several strategies on behalf of relationists to cope with my argument. In Section 5, I briefly draw some conclusions. (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)

Systems of differential equations are used to describe, model, explain, and predict states of physical systems. Experimental and theoretical branches of physics including general relativity, climate science, and particle physics have differential equations at their center. Direct solutions to differential equations are not available in many domains, which spurs on the use of creative mathematics and simulated solutions.

In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends (...) her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences. (shrink)

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language. (shrink)

In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, (...) and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”. (shrink)

Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

Structuralism is one of the most popular contemporary accounts of mathematics. Despite its popularity, it has been challenged on the grounds of consistency. In this paper, I show that existing arguments purporting to establish an inconsistency miss the mark. I then proceed to develop a new argument against realist structuralism, to show that the commitment to mathematical pluralism and the structural identity criterion embraced by the realist structuralist jointly entail a contradiction.