Numbers

Why does quantum theory need the complex numbers? With a view toward answering this question, this paper argues that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. This paper then describes the ‘indivisible interpretation’ of quantum theory, according to which a quantum system can be regarded as an ‘indivisible’ stochastic process unfolding in an old-fashioned configuration space, with wave functions and other exotic Hilbert-space ingredients demoted from having an ontological status. The complex numbers (...) end up being necessary to ensure that the Hilbert-space formalism is indeed a Markovian embedding. (shrink)

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)

Gottlob Frege and Edmund Husserl are two seemingly different philosophers in their methodology. Both have significantly influenced Western philosophy in that their contributions established fields within philosophy that are of intensive study today. Still, their differences in methodology have, in certain instances, yielded similar or distinct results. Their results ranged from the distinction of sense and reference, objectivity, and the theory of mathematics: specifically, their definition of number. Frege and Husserl have such striking similarities in their theory of sense and (...) reference and related notions that from their apparent correspondence, Frege seems to have acknowledged the coincidences in their theories (Frege & Kluge, 1972) (Hill & Rosado Haddock Intro, 4, & 30-33). That is not to say there are exact parallels between Frege and Husserl and their results, as I will acknowledge their similarities and differences within scale. So, I intend to demonstrate their respective theories and results descriptively to show their likenesses while still recognizing their divergences in scope and methodology. Indirectly, I would also hope to illustrate the ties, in terms of subject matter, of these two philosophers who are historically considered at odds with one another; be it personally, in their schools of philosophy, or methodologically (Hartimo 47-53). (shrink)

The multiplicative group ℝ₊×, equipped with its Haar measure d×r = dr/r, carries a scale-invariant spectral structure whose unitary irreducible representations are precisely the characters χγ(r) = r^(iγ) for γ ∈ ℝ, parametrised by a single real frequency. When the standard Riemann variable s = σ + iγ is introduced via the Haar-normalised coordinate s = 1/2 + iγ, this principal-series condition becomes the statement Re(s) = 1/2. The critical line is therefore the unitarity locus of the natural spectral theory (...) on ℝ₊×, not a numerical accident. Tate’s thesis establishes that the Riemann zeta function arises as the Mellin transform of the idele class group scaling action, connecting the zeros of ζ(s) to the spectral theory of the scaling generator A = −i d/d(log|a|) on L²(A×/ℚ×, d×a). Meyer’s unconditional spectral realisation then identifies those zeros as the atoms of the trace spectral measure μ_A via the Weil explicit formula. The product decomposition A×/ℚ× ≅ K × ℝ₊×, with K compact, reduces the eigenvalue equation for A to a first-order ordinary differential equation on ℝ whose solution space is one-dimensional, confining each ordinate to a single spectral atom of weight one. The functional equation ξ(s) = ξ(1 − s), combined with the Schwarz reflection principle, forces every zero at ordinate γ lying off the critical line to generate a second distinct zero of ζ(s) at the same ordinate, violating the multiplicity bound. All non-trivial zeros of ζ(s) therefore satisfy Re(s) = 1/2, and every such zero is simple. The argument extends without modification to every primitive Dirichlet L-function, yielding the Generalised Riemann Hypothesis. (shrink)

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

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

The Feynman prescription weights every loop energy equally and diverges. We replace the flat spectral weight with the Plancherel measure of SL(2, ℂ), P(λ) = πλ /sinh (πλ), the unique weight forced by locality, unitarity, and dimensional reduction in the shadow spectral representation of celestial holography. Stripped of kinematics, the bare L-loop measure obeys the unconditional bound M_L ≤ (1/8)^L, so the loop expansion is ultraviolet finite at every order with no regularization and no counterterms. The measure separates cleanly from (...) the kinematics: all transcendental content of an amplitude resides in the conformal block K_L(s, t), never in M_L. At one and two loops the measure is exactly rational, M_1 = 1/8 and M_2 = 1/90, the Bernoulli fractions of ζ(2) and ζ(4); we prove that this happens precisely because the ladder’s path graph coincides with the complete graph only at L ≤ 2, and we prove by an elementary wall-count invariant that the coincidence cannot recur at any higher order, with the certified 44-digit value of M_3 failing every integer-relation test for rationality. Loop integrands arise as shadow discontinuities of tree amplitudes, extending the construction derived in earlier work, and the one-loop box is verified to fifteen significant figures with a 19.6-fold speedup over the Feynman parametrization. -/- Keywords: self-dual Haar measure, renormalization, quantum gravity, celestial holography, shadow discontinuity, Plancherel spectral weight, all-loop finiteness theorem, Bernoulli numbers. (shrink)

The Euler product formula zeta(s) = prod_p (1 - p^{-s})^{-1} is the exact trace Tr_F(N^{-s}) of the number operator N on the bosonic Fock space F built from one-particle states labelled by primes, in which integers are Fock states, primes are elementary quanta with single-particle energies E_p = log p, and zeta(s) is the partition function. We prove that all non-trivial zeros of zeta(s) lie on the critical line Re(s) = 1/2. The proof identifies F with L^2(A^x/Q^x, d^x a) via (...) the Fundamental Theorem of Arithmetic and the adelic product formula. The scaling generator A = -i d/d(log|a|) is self-adjoint by Stone's theorem. Meyer's unconditional spectral realization (2005) identifies the non-trivial zeros as atoms of the trace spectral measure of A via the Weil explicit formula. The product decomposition A^x/Q^x = K x R_+^x, where K is compact, reduces the eigenvalue equation to a first-order ODE on R_+^x whose solution space is one-dimensional: each ordinate gamma admits exactly one eigenfunction r^{i*gamma}. The Spectral-Weil identity equates the atom weight at each ordinate with the total analytic multiplicity, which therefore equals 1. The functional equation xi(s) = xi(1-s) pairs any zero rho = sigma + it with a companion (1-sigma) + it at the same ordinate; when sigma != 1/2 these are distinct, forcing multiplicity at least 2 — contradicting the spectral bound. All zeros therefore satisfy Re(s) = 1/2, and as an immediate corollary, all zeros are simple. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

How is it that numbers can magically map the world around us? Are numbers really Real? Was Pythagoras completely crazy when he seemed to regard numbers as divine entities capable of revealing the truth about things around us? With Michael Moses, Ihran Izmirlih, and Michael Stein.

If one replaces the standard (Gödel) mathematics with Hilbert arithmetic/ mathematics thus able to merge ontomathematically reality and mathematics (in the former case, being prevented by the Gödel objection), "creatio ex nihilo " can be rigorously inferred only from the unlimited function successor, furthermore under the axiom of induction providing universal finiteness. It is caused in the final analysis by the closeness of the universe following from its definition to "be all" and thus single one, in particular excluding: the Big (...) Bang; other universes ("worlds", e.g. after Everett-Wheeler's interpretation); multiverse; etc. That deduction is called the "fundamental theorem of the universe". It is rather a new paradigm implying a class of theories belonging to "ontomathematical Postmodernity", rather than a new theory, belonging together with the Big Bang one (as causing the universe externally) to Modernity, thus needing energy conservation to be kept at any cost, and adding ad hoc external causes such as the Big Bing itself. On the contrary, the alternative paradigm, only in the framework of which the fundamental theorem of the universe can be proved, excludes any causes external to the universe by allowing violations of energy conservation, in turn equivalent to external causes in the former paradigm. Anyway, those violations obey a more general conservation of quantum information. Under the newly introduced "quantum invariance", a further conservative generalization of Einstein's principle of the physical law invariance to smooth transformations between noninertial reference frames in order to include discrete ones, all external causes are equivalent to internal ones. The metaphor of two "ship models", the one of which is built "within in a bottle" (as a figurative illustration of Modernity's restrictions) since both models are quite similar, but that within a bottle needs virtuous skills to be constructed: so both paradigms are formally equivalent, but the ontomathematical one generates much simpler explanations and implies the huge extension of human cognition in the future. (shrink)

Learning in the New Dispersed Prime Time -/- A study of concept of social time through the case of normalization of algorithmic temporality in instruction and learning. Aesthetic representations of DeepMind's development of AlphaGo is analyzed in terms of compensatory narrative strategies in the face historical discontinuities and mathematical indeterminacies. In this context, prime numbers are given a political and philosophical role as figures and techniques of mediating irreducibility, inexhaustibility and intensification of computation. Cultural philosophies of Walter Benjamin, Gilles Deleuze (...) and Felix Guattari are mobilized to conceptualize the problematics of historical time and political rhythm. (shrink)

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 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 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)

According to a familiar, simple argument, numbers exist because sentences like ‘Two is an even number’ are true. Whereas realists accept the argument as sound, anti-realists either reject that number words function referentially in such sentences (non-referentialism) or else that such sentences are true (fictionalism). We argue that this dialectic, though familiar, drastically underestimates the extent to which natural language supports realism. Indeed, if dominant accounts of number and measurement-related expressions within linguistic semantics are correct, then far more than just (...) our overt talk about numbers as objects would not be true if numbers do not exist. The purpose of this paper is to sketch this strengthened argument for realism and to survey its consequences for extant forms of anti-realism. (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 books offers a novel account of the nature of numbers firmly grounded in results from numerical cognition and the philosophy of mathematics. Drawing on empirical data on the human experience of what we call “numbers,” the author shows that numbers do not exist as abstract objects, but that the idea that they do is a useful cognitive tool. Contrary to the platonist view, according to which arithmetic is true of a realm of abstract entities, the nominalistic account presented in (...) this book shows arithmetic to be true of descriptions of structural properties of techniques such as counting and calculating procedures. This book is of interest to both philosophers and cognitive scientists who want to have a deeper understanding of what numbers are. (shrink)

The paper applies the newly introduced “KS fundamental randomness” to the nonstandardly generalized primes in Hilbert arithmetic to prove that the latter satisfies the necessary condition and separately the sufficient condition of the former. When the two conditions can be identified is also investigated. A review of other available generalizations of primes demonstrates that none of them is suitable for approaching the problem. The design aims to suggest a universal method for resolving number theory puzzles such as Goldbach’s conjecture. The (...) scheme is applicable only within Hilbert arithmetic (but not in the standard mathematics) due to the fact that primes are not KS fundamental random in the latter, but only in the former. “Mathematics enters reality” (though only “fetching” a premise for the proof) is the relevant philosophical reflection furthermore reasoned by the opposition of Hilbert space of dimensions d=1,2 versus d3 after both Gleason and Kochen - Specker theorems. (shrink)

Goldbach's conjecture is simply proved in Hilbert arithmetic. However, that proof is either invalid ("incomplete") or false ("contradictory") in the standard mathematics obeying Gödel's objections about the relation of arithmetic to set theory. The proof uses the "apophatic" (holistic) reformulation of the Kochen-Specker theorem and the fundamental randomness of primes in Hilbert arithmetic: both confirmed to be true in previous papers. A few other conjectures, about twin primes, k-twin primes, k-tuple primes (a part of the Hardy-Littlewood conjecture) including about infinite (...) tuples are deduced from Goldbach's conjecture as corollaries in Hilbert arithmetic. The hypothesis that the asymptotic behaviour predicted by Hardy and Littlewood is false in Hilbert arithmetic is reasoned. A few philosophical observations about "crossing Rubicon to reality" after interpreting the Gleason and Kochen-Specker theorems to the meta-space of human experience to be Hilbert space and considering the opposition of two-versus three-dimensionality meant by them where the former is associated with Modernity and the latter, with Postmodernity: thus, both defined rigorously and mathematically unlike today's vague humanitarian (cultural and relativistic) research about them and that historical development led to it. Another and quite simple proof of the theorem about the asymptotic limit of the number of primes until any natural number is suggested. The link between Goldbach's conjecture and Riemann's hypothesis is demonstrated to be inherent and almost obvious in Hilbert arithmetic. The conclusion explains why the proofs of centuries-old puzzles about primes are so "scandalously" brief and simple in Hilbert arithmetic. (shrink)

The proper mathematical proof of Riemann's hypothesis (RH) in Hilbert mathematics is suggested. It follows the methodological and philosophical considerations in Part I of the paper. Riemann's zeta function is continued "physically" at its single and simple pole conventionally to be square integrable there (though not being analytical only there) and thus everywhere on the complex plane in order to be interpreted as a wave function (though with a singularity at the pole, and thus generalizing the Hilbert-Polya conjecture's viewpoint). Therefore, (...) is interpreted physically corresponding to ζ(????) some quantum state described by it This allows for applying the Noether (1918) first theorem after its reformulation by means of the newly introduced "nonstandard bijection" meaning the mapping of any mathematical structure and its nonstandard model after the Löwenheim-Skolem theorem. Then, the additive semigroup of its trivial zeros implies for all nontrivial zeros to share the critical line (i.e. proving RH) furthermore reasoning the reciprocity of the former generator "2" and the constant "½" of "Re(s) = ½". That approach for proving RH implies as "byproducts" also: the fundamentally random (GUE) distribution of all nontrivial zeros due to "nonstandard bijection"; the generalization of the axiom of induction to a conservation low as for the set-theoretical "continuation" of arithmetic (thus from the axiom of induction the axiom of infinity); the generalization of the Noether theorem in a few consecutive levels reaching even to set theory and the foundations of mathematics (after Hilbert arithmetic and ontomathematics). An idea for proving Goldbach's conjecture in Hilbert arithmetic (in a forthcoming paper) is reasoned as a corollary from the same approach for proving RH. (shrink)

What should be the "physical interpretation" of Riemann's hypothesis? Can its eventual physical interpretation pioneer a pathway for the proper mathematical proof? Answers to both questions are researched in the framework of ontomathematics inherently involving the unity of physics, mathematics, and philosophy. After that viewpoint, a philosophical method for reinterpreting most fundamental mathematical problems (in particular, the seven "Millennium Problems" of CMI) is suggested. Loosely speaking, it consists in determining the ontomathematical "forest" in which the "tree" of a certain very (...) essential mathematical problem is situated, after which the shortest silogism eventually needing a relevant "Gestalt change" appears to be natural and almost obvious, furthermore rather elementarily provable. One even notices that many (if not all) most fundamental problems of contemporary mathematics appeal to the same "Gestalt change" needing the "Cartesian glasses" to be "put off" and Modernity and its episteme to be abandoned. As for mathematics itself, one can conjecture that many or all of the most fundamental problems are (or at least, are linkable to) Gödel's insoluble statements. Ontomathematics suggests a general framework (also) for resolving many essential mathematical problems by breaking the Cartesian prejudice established by Modernity: then, Riemann's hypothesis can be reformulated in terms of the qubit Hilbert space so that the "zeta function" belongs to it. If one manages to demonstrate this, Riemann's hypothesis is rather easily provable since it refers to the fundamental reducibility of any qubit to a single bit "after measurement". From the newly introduced viewpoint, the zeta function is "physically" continued also at its single pole therefore tracing its interpretation by means of the Noether (1918) first theorem. Its application for proving Riemann's hypothesis is sketched in order to be elaborated in the next, second part of the study. (shrink)

This article reexamines the classical diagonal argument underlying the claim of the existence of non-computable functions, based on binary encodings of functions N → {0, 1}. We clarify why the diagonal construction does not yield a new non-computable function in the finite case. The argument is then reconsidered within the recently introduced grossone-based computational paradigm, which allows numerical computations with different infinite and infinitesimal quantities. From this perspective, the function constructed by Turing can be interpreted not as non-computable, but as (...) belonging to a class of binary functions that remain inaccessible to classical approaches while becoming observable and computable within the grossone-based framework. (shrink)

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique (...) abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$. (shrink)

(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer a question (...) posed by Bacon and Dorr (2024). (shrink)

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of ‘ordinary’ mathematics? (2) How should Frege’s theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won’t give rise to different numbers at different worlds? (3) Can one reconstruct Frege’s theory of numbers in a non-modal setting without mathematical primitives such (...) as “the number of Fs” ( $$\#F$$ ) or mathematical axioms such as Hume’s Principle? Our answer to question (1) is ‘None’. Our answer to question (2) begins by defining ‘x numbers G’ as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege’s derivation of the Dedekind-Peano axioms in a non-modal setting. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...) along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

This paper explores a unified approach to linguistic reference and the nature of objects, addressing both abstract and concrete entities. We propose a method of redefining ultra-thin objects through a modified abstraction principle, which involves two distinct computations: subsemantic computation processes direct physical input, while semantic computation derives the semantic values of a sentence from the meanings of its constituents. These computations take different inputs—one physical and one semantic—but yield identical outputs. Among these, the subsemantic computation is more accessible. This (...) approach facilitates a consistent treatment across various types of objects, including mathematical, concrete, social, and mental entities, thereby eliminating the need for domain-specific justifications. We advocate for this innovative perspective and address potential objections related to idealism and the utility of introducing objects. Our proposal advances the discourse of the nature of objects and linguistic reference, providing a comprehensive framework for understanding the existence and reference of objects across diverse discourse domains. (shrink)

Conservation of a Circle explains the Singularity called ‘Nature’ and the Metaverse called ‘Mind.’ How Nature Operates: A Motionless Computer and A Frameless Frame of Reference, Conservation of the Circle is the only dynamic in Nature. Producing The Financial Statement that Explains Everything in Nature.

This is a book about infinity, specifically the infinity of numbers and sequences. Amazing properties arise, for instance, some kinds of infinity are argued to be greater than others. Along the way the author will demonstrate how infinity can be made to create beautiful 'art', guided by the development of underlying mathematics. This book will provide a fascinating read for anyone interested in number theory, infinity, math art, and/or generative art, and could be used a valuable supplement to any course (...) on these topics. (shrink)

According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop this (...) sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

Ordinality and cardinality, in the finite domain, are ordinarily considered as mere aspects of the very same objects, the natural numbers. Yet Steiner (Mathematics – application and applicability. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford University Press, 2005) draws attention to the intricate interplay between them, which is made implicit by this conception of them. In this chapter, I present a fitting cognitive framework and use it to account for how this situation comes (...) to be. The framework concerns basic principles of object representation in the cognitive system, and particularly when and how different “aspects” of objects come, developmentally, to be integrated into a deeper, combined representation. Our coming to master the natural numbers is then presented as the process of such integration of ordinals and cardinals, based on their well-behaved interaction in the finite domain. This regular connection in the finite domain between ordinality and cardinality, however, is grasped automatically by the cognitive system as a statistical pattern rather than being stated explicitly, mathematically. Philosophically, this case study thus serves as a warning against ignoring cognition’s inner workings. The upshot is a novel, cognitively motivated account of the natural numbers. (shrink)

Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.

Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...) account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...) or it has the cardinality of the continuum, the ①-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in ①-based numbers. (shrink)

Symbolic representation demonstrates, and proves, the conservation of a circle (is the basis for, and, thus, controls, language) (all disciplines).

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.

This paper considers hybrid systems — dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only “touches” zero is considered as the zero-crossing, as well). Determination of zero-crossings plays a crucial role in the correct simulation of the system in this case. However, for models of many real-life hybrid (...) systems, such interactions may lead to the so-called Zeno executions, i.e., situations where the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. In this case, standard numerical methods of simulating the systems may fail, since the time between two transitions can decrease significantly leading to ill-conditioning of the simulation. Correct determination of zero-crossings for a complex real-life system can require a lot of computational resources and, as a consequence, slow down the simulation significantly. This paper presents a new way to execute the simulation generating time observations of the hybrid system dynamically using numerical infinitesimals introduced recently, allowing thus to determine zero-crossings more accurately. The proposed method allows to automatically detect zero-crossings with predefined accuracy and to analyze better the behavior of the system around the zero-crossings generating observations more densely, where it is necessary. Moreover, the search for zero-crossings is performed efficiently without re-evaluation of the whole system at each observation. To show the validity of the proposed algorithm, the well-known Bouncing Ball hybrid system has been studied and the obtained simulation results were compared with the standard method. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

Number and Count Sentences like ‘The number of Martian moons is two’ and ‘Mars has two moons’ give rise to a puzzle. How can they be equivalent if only the truth of Number but not that of Count Sentences requires the existence of numbers? Proponents of Linguistic Deflationism seek to resolve this puzzle by arguing that on their correct linguistic analysis the truth of Number Sentences does not require the existence of numbers. In this paper, I argue that Katharina Felka’s (...) recent attempt to vindicate this strategy by analysing Number Sentences as so-called specificational sentences is philosophically dissatisfying and sketch a promising alternative. (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception (...) of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)