Abstract
In this preface of the Special Issue on Future Directions of Further Developments in Mathematics, we discuss about mathematics, how it should be, what it was and how we hope it will develop.
When we were asked to be editors of a special issue celebrating 20 years of the journal Open Mathematics, https://www.degruyter.com/math, we wondered what subject could be capable of including the very numerous lines of research in current mathematics. After a long reflection, we agreed that the topic that could unify the many souls of mathematics was to discuss the perspectives of our discipline.
Designing a volume on the future of mathematics ultimately means asking what it is for and what it is about. Many great mathematicians have tried to give an answer, among others Courant and Robbins [1], Hardy [2], and Penrose [3]; see also Bartocci and Oddifreddi [4]. Even one of the authors of this article recently attempted to tackle this fascinating but extremely difficult subject. We cite his contribution [5] because it extensively deals with all the historical ideas mentioned in this very brief note and, for this reason, it can be taken as a reference by the readers.
To tackle this difficult task, we decided to brainstorm in a fabulous fish restaurant in Acireale (Sicily). We let a little wine, while keeping us lucid, help give us the courage to undertake increasingly visionary reasoning. This sweet philosophizing in front of the sea and in the moonlight reminded us of the poems of the great mathematician Khayyam [6], who drew inspiration from similar environments for profound reflections on human nature. Furthermore, as a friend and teacher of ours, Professor E. DiBenedetto, used to say (cf. [7]), mathematics is, together with philosophy, the convivial discipline par excellence. The deepest ideas are often born by discussing informally, a bit like what happened in ancient Greece between Socrates and his disciples [8].
We began by agreeing that mathematics reaches its apex when it is poised between applications and pure abstraction. According to Hardy, the beauty of mathematics has the same nature as that of painting and poetry. Hardy went so far as to list some characteristics that make a mathematical theory “beautiful” and these are “unpredictability, inevitability and economy.” Simplicity and elegance are characteristics closely linked to the least possible number of assumptions. This makes mathematics very similar to physics. Newton, in the third volume of the Principia [9], as the first of the rules of philosophizing establishes that “Of natural things, more numerous causes must not be admitted than those which are true and sufficient to explain the phenomena.” But mathematics is not just aesthetics. According to Dirac, theories when they are too “beautiful” cannot be discarded and do not need experimental confirmation. In fact, if a theory is sufficiently universal and elegant, it will end up describing phenomena for which it was not originally conceived and directed.
But where does this ability of mathematics to anticipate the description of nature come from? From the fact that, as Galilei said [10], “Philosophy is written in this huge book which is continually open before our eyes (I mean the universe), but it cannot be understood unless one first learns to understand the language and know the characters in which it is written. It is written in mathematical language, and the characters are triangles, circles, and other geometric figures, without which means it is impossible to humanly understand a word of it; without these it is a vain wandering through a dark labyrinth.” The idea that the universe is written in a simple and clear language is the postulate, almost a truth of faith, on which modern science is based: the same vision of the greatest modern physicist Hawking [11], who was perpetually in search of the theory of everything. The whole universe can be perfectly described by simple and elegant principles that appear complex only because of their interactions and the enormous number of physical particles/bodies involved. Why this is true is a mystery. As De Giorgi used to say, “At the beginning and at the end we have the mystery. We could say that we have God’s plan. Mathematics brings us closer to this mystery, without penetrating it.”
However, mathematics differs from physics in one essential respect: mathematics, like theology, is eternal. Mathematical language is the only language where one can distinguish the true from the false. As written as an epitaph on Vitali’s tomb “Empires die, but Euclid’s theorems retain eternal youth.” This is not true for the experimental sciences where knowledge is negative: we know if a theory is false, but we can never say that a theory is true. At most, we can say that a theory is consistent with experimental observations. This “timelessness” of mathematics makes ridiculous the applied-to-measure metrics currently used to measure the “productivity” of mathematicians. A theorem should not be measured by its immediate impact, but by how it remains alive even decades after its publication. Galileo compared human beings to flasks full of wine [12]; mathematical theorems are also subjected to the same comparison: the really good ones age well and increase their flavor and strength over the years.
This publish or perish mentality has been harmful for mathematics (cf. also the Code of Practice of the European Mathematical Society Ethics Committee), which may be fascinating, aesthetically beautiful, and useful in its apparent uselessness (as everyone says), but it is experiencing a difficult moment. It was the protagonist of the third scientific revolution (Von Neumann, Turing and Shannon were all mathematicians) although it was unable to become the protagonist of the technological revolution currently taking place. In fact, it is not convenient to get involved to study new and significant problems that will probably be mentioned in the decades to come, but it is essential for one’s career to throw oneself on fashion topics where even a marginal contribution will certainly be mentioned. The number of citations is more important than the depth of thought. So relatively few mathematicians have the courage to do what mathematicians have always done: starting from the phenomena that surround us, through abstraction define, through induction, a model, study its properties and, finally, deduce the properties of the starting phenomenon.
At the end of the dinner, we had the courage to invite mathematicians who had the same idea of mathematics as us and who were our friends. Indeed, as DiBenedetto said (cf. [7]), mathematics is a frustrating activity. Theorems, the real ones, are difficult to demonstrate, their proof often eludes us for years and years. Without the comfort of friends, it is difficult to manage this wait that seems to never end. The mathematician often experiences the sensations described by Beckett [13] or Buzzati [14].
Our friends have not betrayed us and have gifted us seven papers [15,16,17,18,19,20,21] of excellent and profound mathematics which may become ground for reflection and further discussion about the future directions of developments in the discipline.
References
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© 2023 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Special Issue on Future Directions of Further Developments in Mathematics
- What will the mathematics of tomorrow look like?
- On H 2-solutions for a Camassa-Holm type equation
- Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
- Control of multi-agent systems: Results, open problems, and applications
- Logical perspectives on the foundations of probability
- Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
- A non-smooth Brezis-Oswald uniqueness result
- Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
- Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
- Positive solution for a nonlocal problem with strong singular nonlinearity
- Analysis of solutions for the fractional differential equation with Hadamard-type
- Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
- A comprehensive review on fractional-order optimal control problem and its solution
- The θ-derivative as unifying framework of a class of derivatives
- Review Articles
- On the use of L-functionals in regression models
- Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
- Regular Articles
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
- An extension of the Hermite-Hadamard inequality for a power of a convex function
- Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
- Relay fusion frames in Banach spaces
- Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
- On the uniqueness of limit cycles for generalized Liénard systems
- A derivative-Hilbert operator acting on Dirichlet spaces
- Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
- Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
- A symbolic approach to multiple Hurwitz zeta values at non-positive integers
- Some results on the value distribution of differential polynomials
- Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
- Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
- Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
- Homotopy cartesian squares in extriangulated categories
- A unified perspective on some autocorrelation measures in different fields: A note
- Total Roman domination on the digraphs
- Well-posedness for bilevel vector equilibrium problems with variable domination structures
- Binet's second formula, Hermite's generalization, and two related identities
- Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
- Multidimensional sampling-Kantorovich operators in BV-spaces
- A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
- Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
- Relating the super domination and 2-domination numbers in cactus graphs
- Compatibility of the method of brackets with classical integration rules
- On the inverse Collatz-Sinogowitz irregularity problem
- Positive solutions for boundary value problems of a class of second-order differential equation system
- Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
- Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
- On 2r-ideals in commutative rings with zero-divisors
- A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
- The construction of nuclei for normal constituents of Bπ-characters
- Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
- Mean square exponential stability of stochastic function differential equations in the G-framework
- Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
- Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
- The dual index and dual core generalized inverse
- Study on Birkhoff orthogonality and symmetry of matrix operators
- Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
- Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
- On semigroups of transformations that preserve a double direction equivalence
- Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
- A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
- Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
- Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
- General Randić indices of a graph and its line graph
- On functional reproducing kernels
- On the Waring-Goldbach problem for two squares and four cubes
- Singular moduli of rth Roots of modular functions
- Classification of self-adjoint domains of odd-order differential operators with matrix theory
- On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
- Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
- Remarks on hyponormal Toeplitz operators with nonharmonic symbols
- Complete decomposition of the generalized quaternion groups
- Injective and coherent endomorphism rings relative to some matrices
- Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
- Continued fractions related to a group of linear fractional transformations
- Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
- Approximate controllability for a stochastic elastic system with structural damping and infinite delay
- On extremal cacti with respect to the first degree-based entropy
- Compression with wildcards: All exact or all minimal hitting sets
- Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
- Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
- Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
- On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
- Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
- On the quadratic residues and their distribution properties
- On average theta functions of certain quadratic forms as sums of Eisenstein series
- Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
- Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
- Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
- On some spaces via topological ideals
- Linear maps preserving equivalence or asymptotic equivalence on Banach space
- Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
- On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
- Entire solutions of two certain Fermat-type ordinary differential equations
- Generalized Lie n-derivations on arbitrary triangular algebras
- Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
- Notes on pseudodifferential operators commutators and Lipschitz functions
- On Graham partitions twisted by the Legendre symbol
- Strong limit of processes constructed from a renewal process
- Construction of analytical solutions to systems of two stochastic differential equations
- Two-distance vertex-distinguishing index of sparse graphs
- Regularity and abundance on semigroups of partial transformations with invariant set
- Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
- Spin(8,C)-Higgs pairs over a compact Riemann surface
- Properties of locally semi-compact Ir-topological groups
- Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
- Ordering stability of Nash equilibria for a class of differential games
- A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
- About a dubious proof of a correct result about closed Newton Cotes error formulas
- Ricci ϕ-invariance on almost cosymplectic three-manifolds
- Schur-power convexity of integral mean for convex functions on the coordinates
- A characterization of a ∼ admissible congruence on a weakly type B semigroup
- On Bohr's inequality for special subclasses of stable starlike harmonic mappings
- Properties of meromorphic solutions of first-order differential-difference equations
- A double-phase eigenvalue problem with large exponents
- On the number of perfect matchings in random polygonal chains
- Evolutoids and pedaloids of frontals on timelike surfaces
- A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
- The 𝔪-WG° inverse in the Minkowski space
- Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
- Approximate solvability method for nonlocal impulsive evolution equation
- Construction of a functional by a given second-order Ito stochastic equation
- Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
- On pomonoid of partial transformations of a poset
- New fractional integral inequalities via Euler's beta function
- An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
- Eigenfunctions in Finsler Gaussian solitons
- On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
- Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
- A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
- A digital Jordan surface theorem with respect to a graph connectedness
- A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
- The structure fault tolerance of burnt pancake networks
- Average value of the divisor class numbers of real cubic function fields
- Uniqueness of exponential polynomials
- An application of Hayashi's inequality in numerical integration
Artikel in diesem Heft
- Special Issue on Future Directions of Further Developments in Mathematics
- What will the mathematics of tomorrow look like?
- On H 2-solutions for a Camassa-Holm type equation
- Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
- Control of multi-agent systems: Results, open problems, and applications
- Logical perspectives on the foundations of probability
- Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
- A non-smooth Brezis-Oswald uniqueness result
- Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
- Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
- Positive solution for a nonlocal problem with strong singular nonlinearity
- Analysis of solutions for the fractional differential equation with Hadamard-type
- Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
- A comprehensive review on fractional-order optimal control problem and its solution
- The θ-derivative as unifying framework of a class of derivatives
- Review Articles
- On the use of L-functionals in regression models
- Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
- Regular Articles
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
- An extension of the Hermite-Hadamard inequality for a power of a convex function
- Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
- Relay fusion frames in Banach spaces
- Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
- On the uniqueness of limit cycles for generalized Liénard systems
- A derivative-Hilbert operator acting on Dirichlet spaces
- Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
- Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
- A symbolic approach to multiple Hurwitz zeta values at non-positive integers
- Some results on the value distribution of differential polynomials
- Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
- Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
- Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
- Homotopy cartesian squares in extriangulated categories
- A unified perspective on some autocorrelation measures in different fields: A note
- Total Roman domination on the digraphs
- Well-posedness for bilevel vector equilibrium problems with variable domination structures
- Binet's second formula, Hermite's generalization, and two related identities
- Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
- Multidimensional sampling-Kantorovich operators in BV-spaces
- A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
- Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
- Relating the super domination and 2-domination numbers in cactus graphs
- Compatibility of the method of brackets with classical integration rules
- On the inverse Collatz-Sinogowitz irregularity problem
- Positive solutions for boundary value problems of a class of second-order differential equation system
- Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
- Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
- On 2r-ideals in commutative rings with zero-divisors
- A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
- The construction of nuclei for normal constituents of Bπ-characters
- Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
- Mean square exponential stability of stochastic function differential equations in the G-framework
- Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
- Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
- The dual index and dual core generalized inverse
- Study on Birkhoff orthogonality and symmetry of matrix operators
- Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
- Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
- On semigroups of transformations that preserve a double direction equivalence
- Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
- A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
- Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
- Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
- General Randić indices of a graph and its line graph
- On functional reproducing kernels
- On the Waring-Goldbach problem for two squares and four cubes
- Singular moduli of rth Roots of modular functions
- Classification of self-adjoint domains of odd-order differential operators with matrix theory
- On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
- Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
- Remarks on hyponormal Toeplitz operators with nonharmonic symbols
- Complete decomposition of the generalized quaternion groups
- Injective and coherent endomorphism rings relative to some matrices
- Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
- Continued fractions related to a group of linear fractional transformations
- Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
- Approximate controllability for a stochastic elastic system with structural damping and infinite delay
- On extremal cacti with respect to the first degree-based entropy
- Compression with wildcards: All exact or all minimal hitting sets
- Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
- Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
- Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
- On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
- Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
- On the quadratic residues and their distribution properties
- On average theta functions of certain quadratic forms as sums of Eisenstein series
- Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
- Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
- Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
- On some spaces via topological ideals
- Linear maps preserving equivalence or asymptotic equivalence on Banach space
- Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
- On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
- Entire solutions of two certain Fermat-type ordinary differential equations
- Generalized Lie n-derivations on arbitrary triangular algebras
- Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
- Notes on pseudodifferential operators commutators and Lipschitz functions
- On Graham partitions twisted by the Legendre symbol
- Strong limit of processes constructed from a renewal process
- Construction of analytical solutions to systems of two stochastic differential equations
- Two-distance vertex-distinguishing index of sparse graphs
- Regularity and abundance on semigroups of partial transformations with invariant set
- Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
- Spin(8,C)-Higgs pairs over a compact Riemann surface
- Properties of locally semi-compact Ir-topological groups
- Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
- Ordering stability of Nash equilibria for a class of differential games
- A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
- About a dubious proof of a correct result about closed Newton Cotes error formulas
- Ricci ϕ-invariance on almost cosymplectic three-manifolds
- Schur-power convexity of integral mean for convex functions on the coordinates
- A characterization of a ∼ admissible congruence on a weakly type B semigroup
- On Bohr's inequality for special subclasses of stable starlike harmonic mappings
- Properties of meromorphic solutions of first-order differential-difference equations
- A double-phase eigenvalue problem with large exponents
- On the number of perfect matchings in random polygonal chains
- Evolutoids and pedaloids of frontals on timelike surfaces
- A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
- The 𝔪-WG° inverse in the Minkowski space
- Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
- Approximate solvability method for nonlocal impulsive evolution equation
- Construction of a functional by a given second-order Ito stochastic equation
- Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
- On pomonoid of partial transformations of a poset
- New fractional integral inequalities via Euler's beta function
- An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
- Eigenfunctions in Finsler Gaussian solitons
- On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
- Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
- A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
- A digital Jordan surface theorem with respect to a graph connectedness
- A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
- The structure fault tolerance of burnt pancake networks
- Average value of the divisor class numbers of real cubic function fields
- Uniqueness of exponential polynomials
- An application of Hayashi's inequality in numerical integration