1. Jump up^ Givon, Talmy; Bertram F. Malle (2002). The Evolution of Language Out of Pre-language. John Benjamins. ISBN 1-58811-237-3.
2. Jump up^ Deacon, Terrence (1997). The Symbolic Species: The Co-evolution of Language and the Brain. W.W. Norton. ISBN 0-571-17396-9.
3. Jump up^ MacNeilage, Peter, 2008. The Origin of Speech. Oxford: Oxford University Press.
4. Jump up^ Botha, R. and C. Knight (eds) 2009. The Cradle of Language. Oxford: Oxford University Press.
5. Jump up^ Dor, D., C. Knight & J. Lewis (eds), 2014. The Social Origins of Language. Oxford: Oxford University Press.
6. ^ Jump up to:^{a b} Lewis, M. Paul, ed. (2009). Ethnologue: Languages of the World (16 ed.). Dallas: SIL International. ISBN 1-55671-216-2.
7. Jump up^ “Simulated Evolution of Language: a Review of the Field”, Journal of Artificial Societies and Social Simulation vol. 5, no. 2
8. Jump up^ Robert Spence, “A Functional Approach to Translation Studies. New systemic linguistic challenges in empirically informed didactics”, 2004, ISBN 3-89825-777-0, thesis. A pdf file

Unsolved Problems in Mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still elude solution.^[1]

An unsolved problem in mathematics does not refer to the kind of problem found as an exercise in a textbook, but rather to the answer to a major question or a general method that provides a solution to an entire class of problems. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems receive considerable attention. This article reiterates the list of Millennium Prize Problems of unsolved problems in mathematics (includes problems ofphysics and computer science) as of August 2015, and lists further unsolved problems in algebra, additive and algebraic number theories, analysis, combinatorics,algebraic, discrete and Euclidean geometries, dynamical systems, partial differential equations, and graph, group, model, number, set and Ramsey theories, as well as miscellaneous unsolved problems. A list of problems solved since 1995 also appears, alongside some sources, general and specific, for the stated problems.

Lists of unsolved problems in mathematics

Over the course of time, several lists of unsolved mathematical problems have appeared. The following is a listing of those lists.

List	Number of problems	Proposed by	Proposed in
Hilbert’s problems	23	David Hilbert	1900
Landau’s problems	4	Edmund Landau	1912
Taniyama’s problems[2]	36	Yutaka Taniyama	1955
Thurston’s 24 questions[3][4]	24	William Thurston	1982
Smale’s problems	18	Stephen Smale	1998
Millennium Prize problems	7	Clay Mathematics Institute	2000
Unsolved Problems in Mathematics for the 21st Century[5]	22	Jair Minoro Abe, Shotaro Tanaka	2001
DARPA’s math challenges[6][7]	23	DARPA	2007

Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of 2016:^[8]

• P versus NP
• Hodge conjecture
• Riemann hypothesis
• Yang–Mills existence and mass gap
• Navier–Stokes existence and smoothness
• Birch and Swinnerton-Dyer conjecture

The seventh problem, the Poincaré conjecture, has been solved.^[9] The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.^[10]

Unsolved problems

Additive number theory

• Beal’s conjecture
• Fermat–Catalan conjecture
• Goldbach’s conjecture
• The values of g(k) and G(k) in Waring’s problem
• Collatz conjecture (3n + 1 conjecture)
• Lander, Parkin, and Selfridge conjecture
• Diophantine quintuples
• Gilbreath’s conjecture
• Erdős conjecture on arithmetic progressions
• Erdős–Turán conjecture on additive bases
• Pollock octahedral numbers conjecture
• Skolem problem
• Determine growth rate of r_k(N) (see Szemerédi’s theorem)
• Minimum overlap problem

Algebra

• Homological conjectures in commutative algebra
• Hilbert’s sixteenth problem
• Hilbert’s fifteenth problem
• Hadamard conjecture
• Jacobson’s conjecture
• Existence of perfect cuboids and associated Cuboid conjectures
• Zauner’s conjecture: existence of SIC-POVMs in all dimensions
• Wild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
• Köthe conjecture
• Birch–Tate conjecture
• Serre’s conjecture II
• Bombieri–Lang conjecture
• Farrell–Jones conjecture
• Bost conjecture
• Uniformity conjecture
• Kaplansky’s conjecture
• Kummer–Vandiver conjecture
• Serre’s multiplicity conjectures
• Pierce–Birkhoff conjecture
• Eilenberg–Ganea conjecture
• Green’s conjecture
• Grothendieck–Katz p-curvature conjecture
• Sendov’s conjecture

Algebraic geometry

• André–Oort conjecture
• Bass conjecture
• Deligne conjecture
• Fröberg conjecture
• Fujita conjecture
• Hartshorne conjectures
• Manin conjecture
• Nakai conjecture
• Resolution of singularities in characteristic p
• Standard conjectures on algebraic cycles
• Section conjecture
• Tate conjecture
• Virasoro conjecture
• Whitehead conjecture
• Zariski multiplicity conjecture

Algebraic number theory

• Are there infinitely many real quadratic number fields with unique factorization (Class number problem)?
• Characterize all algebraic number fields that have some power basis.
• Stark conjectures (including Brumer–Stark conjecture)

Analysis

• The Jacobian conjecture
• Schanuel’s conjecture and four exponentials conjecture
• Lehmer’s conjecture
• Pompeiu problem
• Are (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, π^e, π^√2, π^π, e^π2, ln π, 2^e, e^e, Catalan’s constant or Khinchin’s constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?^[11][12][13]
• Khabibullin’s conjecture on integral inequalities
• Hilbert’s thirteenth problem
• Vitushkin’s conjecture

Combinatorics

• Number of magic squares (sequence A006052 in OEIS)
• Number of magic tori (sequence A270876 in OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group
• Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
• The lonely runner conjecture: if runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be at least a distance from each other runner) at some time?
• Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?
• The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
• unicity conjecture for Markov numbers
• balance puzzle ^[14]

Discrete geometry

• Solving the happy ending problem for arbitrary
• Finding matching upper and lower bounds for k-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies
• The Kobon triangle problem on triangles in line arrangements
• The McMullen problem on projectively transforming sets of points into convex position
• Ulam’s packing conjecture about the identity of the worst-packing convex solid
• Filling area conjecture
• Hopf conjecture
• Kakeya conjecture

Euclidean geometry

• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^[15]
• Inscribed square problem – does every Jordan curve have an inscribed square?^[16]
• Moser’s worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?^[17]
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?^[18]
• Shephard’s problem (a.k.a. Dürer’s conjecture) – does every convex polyhedron have a net?^[19]
• The Thomson problem – what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?
• Pentagonal tiling – 15 types of convex pentagons are known to monohedrally tile the plane, and it is not known whether this list is complete.^[20]
• Falconer’s conjecture
• g-conjecture
• Circle packing in an equilateral triangle
• Circle packing in an isosceles right triangle

Dynamical systems

• Furstenberg conjecture – Is every invariant and ergodic measure for the action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture – Is the Mandelbrot set locally connected ?
• Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?
• Is every reversible cellular automaton in three or more dimensions locally reversible?^[21]

Graph theory

• Barnette’s conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
• The Erdős–Hajnal conjecture on finding large homogeneous sets in graphs with a forbidden induced subgraph
• The Hadwiger conjecture relating coloring to clique minors
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques
• Harborth’s conjecture that every planar graph can be drawn with integer edge lengths
• The total coloring conjecture
• Hadwiger conjecture (graph theory)
• The list coloring conjecture
• The Ringel–Kotzig conjecture on graceful labeling of trees
• How many unit distances can be determined by a set of n points? (see Counting unit distances)
• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
• Lovász conjecture
• Deriving a closed-form expression for the percolation threshold values, especially (square site)
• Tutte’s conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
• Petersen coloring conjecture
• The reconstruction conjecture and new digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
• The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice.
• Does a Moore graph with girth 5 and degree 57 exist?
• Conway’s thrackle conjecture
• Negami’s conjecture on the characterization of graphs with planar covers
• The Blankenship–Oporowski conjecture on the book thickness of subdivisions
• Hedetniemi’s conjecture
• Vizing’s conjecture

Group theory

• Is every finitely presented periodic group finite?
• The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
• For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
• Is every group surjunctive?
• Andrews–Curtis conjecture
• Herzog–Schönheim conjecture
• Does generalized moonshine exist?

Model theory

• Vaught’s conjecture
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in is a simple algebraic group over an algebraically closed field.
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.^[22]
• Determine the structure of Keisler’s order^[23][24]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?^[25]
• The Stable Forking Conjecture for simple theories^[26]
• For which number fields does Hilbert’s tenth problem hold?
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality does it have a model of cardinality continuum?^[27]
• Shelah’s eventual Categority conjecture: For every cardinal \lambda there exists a cardinal \mu(\lambda) such that If an AEC K with LS(K)<= \lambda is categorical in a cardinal above \mu(\lambda) then it is categorical in all cardinals above \mu(\lambda).^[22][28]
• Shelah’s categoricity conjecture for L_{\omega_1,\omega}: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.^[22]
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[29]
• If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?^[30][31]
• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
• Kueker’s conjecture^[32]
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Lachlan’s decision problem
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?^[33]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^[34]

Number theory (general)

• Grand Riemann hypothesis
  • Generalized Riemann hypothesis
    • Riemann hypothesis
• n conjecture
  • abc conjecture (Proof claimed in 2012, currently under review.)
• Hilbert’s ninth problem
• Hilbert’s eleventh problem
• Hilbert’s twelfth problem
• Carmichael’s totient function conjecture
• Erdős–Straus conjecture
• Pillai’s conjecture
• Hall’s conjecture
• Lindelöf hypothesis
• Montgomery’s pair correlation conjecture
• Hilbert–Pólya conjecture
• Grimm’s conjecture
• Leopoldt’s conjecture
• Do any odd perfect numbers exist?
• Are there infinitely many perfect numbers?
• Do quasiperfect numbers exist?
• Do any odd weird numbers exist?
• Do any Lychrel numbers exist?
• Is 10 a solitary number?
• Catalan–Dickson conjecture on aliquot sequences
• Do any Taxicab(5, 2, n) exist for n>1?
• Brocard’s problem: existence of integers, (n,m), such that n!+1 = m² other than n=4, 5, 7
• Beilinson conjecture
• Littlewood conjecture
• Szpiro’s conjecture
• Vojta’s conjecture
• Goormaghtigh conjecture
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell’s theorem)
• Lehmer’s totient problem: if φ(n) divides n − 1, must n be prime?
• Are there infinitely many amicable numbers?
• Are there any pairs of amicable numbers which have opposite parity?
• Are there any pairs of relatively prime amicable numbers?
• Are there infinitely many betrothed numbers?
• Are there any pairs of betrothed numbers which have same parity?
• The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
• Is π a normal number (its digits are “random”)?^[35]
• Casas-Alvero conjecture
• Sato–Tate conjecture
• Find value of De Bruijn–Newman constant

Number theory (prime numbers)

• Catalan’s Mersenne conjecture
• Agoh–Giuga conjecture
• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
• New Mersenne conjecture
• Erdős–Mollin–Walsh conjecture
• Are there infinitely many prime quadruplets?
• Are there infinitely many cousin primes?
• Are there infinitely many sexy primes?
• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
• Are there infinitely many Wagstaff primes?
• Are there infinitely many Sophie Germain primes?
• Are there infinitely many Pierpont primes?
• Are there infinitely many regular primes, and if so is their relative density ?
• Are there infinitely many repunit primes to every base except perfect power and numbers of the form -4k⁴?
• Are there infinitely many Cullen primes?
• Are there infinitely many Woodall primes?
• Are there infinitely many palindromic primes to every base?
• Are there infinitely many Fibonacci primes?
• Are there infinitely many Lucas primes?
• Are there infinitely many Pell primes?
• Are there infinitely many Newman–Shanks–Williams primes?
• Are all Mersenne numbers of prime index square-free?
• Are there infinitely many Wieferich primes?
• Are there any Wieferich primes in base 47?
• Are there any composite c satisfying 2^{c – 1} ≡ 1 (mod c²)?
• For any given integer a > 0, are there infinitely many primes p such that a^{p – 1} ≡ 1 (mod p²)?^[36]
• Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) simultaneously?^[37]
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there any Wall–Sun–Sun primes?
• Is every Fermat number 2²ⁿ + 1 composite for ?
• Are all Fermat numbers square-free?
• For any given integer a which is not a square and does not equal to -1, are there infinitely many primes with a as a primitive root?
• Artin’s conjecture on primitive roots
• Is 78,557 the lowest Sierpiński number (so-called Selfridge’s conjecture)?
• Is 509,203 the lowest Riesel number?
• Fortune’s conjecture (that no Fortunate number is composite)
• Landau’s problems
• Does every prime number appear in the Euclid–Mullin sequence?
• Does the converse of Wolstenholme’s theorem hold for all natural numbers?
• Elliott–Halberstam conjecture
• Problems associated to Linnik’s theorem

Partial differential equations

• Regularity of solutions of Vlasov–Maxwell equations
• Regularity of solutions of Euler equations

Ramsey theory

• The values of the Ramsey numbers, particularly
• The values of the Van der Waerden numbers
• Erdős–Burr conjecture

Set theory

• The problem of finding the ultimate core model, one that contains all large cardinals.
• If ℵ_ω is a strong limit cardinal, then 2^ℵ_ω < ℵ_ω1 (see Singular cardinals hypothesis). The best bound, ℵ_ω4, was obtained by Shelah using his pcf theory.
• Woodin’s Ω-hypothesis.
• Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
• (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
• Does there exist a Jónsson algebra on ℵ_ω?
• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
• Does the Generalized Continuum Hypothesis entail for every singular cardinal ?
• Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?

Other

• Problems in loop theory and quasigroup theory
• Problems in Latin squares
• List of unsolved problems in statistics
• List of unsolved problems in computer science
• Invariant subspace problem
• Kaplansky’s conjectures on groups rings
• Painlevé conjecture
• Dixmier conjecture
• Baum–Connes conjecture
• Novikov conjecture
• Prove Turing completeness for all unique elementary cellular automaton
• Generalized star height problem
• Assorted sphere packing problems, e.g. the densest irregular hypersphere packings
• Closed curve problem: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.^[38]
• Keating–Snaith conjecture
• Kung-Traub conjecture
• Atiyah conjecture on configurations
• Toeplitz’ conjecture (open since 1911)
• Carathéodory conjecture
• Church–Turing thesis
• Weight-monodromy conjecture
• Berry–Tabor conjecture
• Birkhoff conjecture
• Guralnick–Thompson conjecture
• Hilbert–Smith conjecture
• MNOP conjecture
• Mazur’s conjectures
• Rendezvous problem
• Scholz conjecture
• Nirenberg–Treves conjecture
• Quantum unique ergodicity conjecture
• Density hypothesis
• Zhou conjecture
• Borel conjecture

Problems solved since 1995

• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^[39]
• Main conjecture in Vinogradov’s mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^[40]
• Erdős discrepancy problem (Terence Tao, 2015)^[41]
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^[42]
• Anderson conjecture (Cheeger, Naber, 2014)^[43]
• Goldbach’s weak conjecture (Harald Helfgott, 2013)^[44][45][46]
• Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^[47][48] (and the Feichtinger’s conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, Bourgain-Tzafriri conjecture and -conjecture)
• Virtual Haken conjecture (Agol, Groves, Manning, 2012)^[49] (and by work of Wise also virtually fibered conjecture)
• Hsiang–Lawson’s conjecture (Brendle, 2012)^[50]
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^[51]
• Ehrenpreis conjecture (Kahn, Markovic, 2011)^[52]
• Hanna Neumann conjecture (Mineyev, 2011)^[53]
• Bloch–Kato conjecture (Voevodsky, 2011)^[54] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^[55][56][57])
• Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^[58]
• Density theorem (Namazi, Souto, 2010)^[59]
• Hirsch conjecture (Francisco Santos Leal, 2010)^[60][61]
• Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^[62]
• Atiyah conjecture (Austin, 2009)^[63]
• Kauffman–Harary conjecture (Matmann, Solis, 2009)^[64]
• Surface subgroup conjecture (Kahn, Markovic, 2009)^[65]
• Scheinerman’s conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^[66]
• Geometrization conjecture (proof was completed by Morgan and Tian in 2008^[67] and it is based mostly on work of Grigori Perelman, 2002)^[68]
• Serre’s modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)^[69][70][71]
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^[72]
• Erdős–Menger conjecture (Aharoni, Berger 2007)^[73]
• Road coloring conjecture (Avraham Trahtman, 2007)^[74]
• The angel problem (Various independent proofs, 2006)^{[75][76][77][78]}
• Lax conjecture (Lewis, Parrilo, Ramana, 2005)^[79]
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^[80]
• Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)^[81]
• Robertson–Seymour theorem (Robertson, Seymour, 2004)^[82]
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)^[83] (and also Alon–Friedgut conjecture)
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^[84]
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^[85]
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)^[86][87]
• Milnor conjecture (Vladimir Voevodsky, 2003)^[88]
• Kemnitz’s conjecture (Reiher, 2003, di Fiore, 2003)^[89]
• Nagata’s conjecture (Shestakov, Umirbaev, 2003)^[90]
• Kirillov’s conjecture (Baruch, 2003)^[91]
• Poincaré conjecture (Grigori Perelman, 2002)^[68]
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^[92]
• Kouchnirenko’s conjecture (Haas, 2002)^[93]
• Vaught conjecture (Knight, 2002)^[94]
• Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^[95]
• Catalan’s conjecture (Preda Mihăilescu, 2002)^[96]
• n! conjecture (Haiman, 2001)^[97] (and also Macdonald positivity conjecture)
• Kato’s conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^[98]
• Deligne’s conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)^[99]
• Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^[100]
• Erdős–Stewart conjecture (Florian Luca, 2001)^[101]
• Berry–Robbins problem (Atiyah, 2000)^[102]
• Erdős–Graham problem (Croot, 2000)^[103]
• Honeycomb conjecture (Thomas Hales, 1999)^[104]
• Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^[105]
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)^[106][107]
• Lafforgue’s theorem (Laurent Lafforgue, 1998)^[108]
• Kepler conjecture (Ferguson, Hales, 1998)^[109]
• Dodecahedral conjecture (Hales, McLaughlin, 1998)^[110]
• Ganea conjecture (Iwase, 1997)^[111]
• Torsion conjecture (Merel, 1996)^[112]
• Harary’s conjecture (Chen, 1996)^[113]
• Fermat’s Last Theorem (Andrew Wiles and Richard Taylor, 1995)^[114][115]

References

Further reading

Books discussing recently solved problems^{[dated info]}

• Singh, Simon (2002). Fermat’s Last Theorem. Fourth Estate. ISBN 1-84115-791-0.
• O’Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• Szpiro, George G. (2003). Kepler’s Conjecture. Wiley. ISBN 0-471-08601-0.
• Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6.

Books discussing unsolved problems

• Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
• Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
• Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
• Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8.
• Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
• Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.
• Lizhen Ji, [various]; Yat-Sun Poon, Shing-Tung Yau (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 1-571-46278-3.
• Waldschmidt, Michel (2004). “Open Diophantine Problems” (PDF). Moscow Mathematical Journal 4 (1): 245–305. ISSN 1609-3321. Zbl 1066.11030.
• Mazurov, V. D.; Khukhro, E. I. (1 Jun 2015). “Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)”. arXiv:1401.0300v6.
• Derbyshire, John (2003). Prime Obsession. The Joseph Henry Press. ISBN 0-309-08549-7.

External links

• 24 Unsolved Problems and Rewards for them
• List of links to unsolved problems in mathematics, prizes and research
• Open Problem Garden The collection of open problems in mathematics build on the principle of user editable (“wiki”) site
• AIM Problem Lists
• Unsolved Problem of the Week Archive. MathPro Press.
• Aizenman, Michael. “Open Problems in Mathematical Physics”.
• Ball, John M. “Some Open Problems in Elasticity” (PDF).
• Constantin, Peter. “Some open problems and research directions in the mathematical study of fluid dynamics” (PDF).
• Serre, Denis. “Five Open Problems in Compressible Mathematical Fluid Dynamics” (PDF).
• Unsolved Problems in Number Theory, Logic and Cryptography
• 200 open problems in graph theory
• The Open Problems Project (TOPP), discrete and computational geometry problems
• Kirby’s list of unsolved problems in low-dimensional topology
• Erdös’ Problems on Graphs
• A List of Approachable Open Problems in Knot Theory
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
• Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications
• List of open problems in inner model theory
• Open Problems in Mathematical Physics
• 15 Problems in Mathematical Physics

Unsolved Problems in Medicine

This article lists currently unsolved problems in medicine.

Problems or unknown causes for diseases

• Alzheimers Disease
• Acrocyanosis
• Bell’s Palsy
• Benign Fasciculation Syndrome
• Benign Paroxysmal Vertigo of Childhood
• Calciphylaxis
• Cardiac Syndrome X
• Cluster Headache
• Complex regional pain syndrome
• Copenhagen Disease
• Cronkhite-Canada Syndrome
• Dermatomyositis
• Cyclic vomiting syndrome
• Danubian endemic familial nephropathy
• Dialysis disequilibrium syndrome
• Exploding head syndrome
• Fibromyalgia
• Fields’ disease
• Floating-Harbor Syndrome
• Geographic tongue
• Granuloma annulare
• Idiopathic intracranial hypertension
• Irritable bowel syndrome
• Interstitial cystitis
• Kawasaki disease
• Mesoamerican nephropathy
• Mortimer’s disease
• Myofascial pain syndrome
• Nodding disease
• Sweating sickness
• Pigmented villonodular synovitis
• Pityriasis rosea
• Prurigo nodularis
• SAPHO syndrome
• Sarcoidosis
• Sjögren’s syndrome
• Spontaneous cerebrospinal fluid leak
• Stiff person syndrome
• Sudden infant death syndrome
• Synovial osteochondromatosis
• TEMPI syndrome
• Trigger finger
• Tropical sprue
• Villitis of unknown etiology
• Lymphedema

(Note: Current diseases being researched for cures are not on this list e.g. Cancer, due to usually using available knowledge of the causes and epidemiology to manage symptoms, having treatments already available for that ailment or find a cure through their based epidemiological vectors.)

Unsolved Problems in Neuroscience

There are yet unsolved problems in neuroscience, although some of these problems have evidence supporting a hypothesized solution, and the field is rapidly evolving. These problems include:

• Consciousness: What is the neural basis of subjective experience, cognition, wakefulness, alertness, arousal, and attention? Is there a “hard problem of consciousness“? If so, how is it solved? What, if any, is the function of consciousness?^[1][2]
• Perception: How does the brain transfer sensory information into coherent, private percepts? What are the rules by which perception is organized? What are the features/objects that constitute our perceptual experience of internal and external events? How are the senses integrated? What is the relationship between subjective experience and the physical world?
• Learning and memory: Where do our memories get stored and how are they retrieved again? How can learning be improved? What is the difference betweenexplicit and implicit memories? What molecule is responsible for synaptic tagging?
• Neuroplasticity: How plastic is the mature brain?
• Development and evolution: How and why did the brain evolve? What are the molecular determinants of individual brain development?
• Free will, particularly the neuroscience of free will
• Sleep: What is the biological function of sleep? Why do we dream? What are the underlying brain mechanisms? What is its relation to anesthesia?
• Cognition and decisions: How and where does the brain evaluate reward value and effort (cost) to modulate behavior? How does previous experience alter perception and behavior? What are the genetic and environmental contributions to brain function?
• Language: How is it implemented neurally? What is the basis of semantic meaning?
• Diseases: What are the neural bases (causes) of mental diseases like psychotic disorders (e.g. mania, schizophrenia), Parkinson’s disease, Alzheimer’s disease, or addiction? Is it possible to recover loss of sensory or motor function?
• Movement: How can we move so controllably, even though the motor nerve impulses seem haphazard and unpredictable?^[3]
• Computational theory of mind: What are the limits of understanding thinking as a form of computing?
• Computational neuroscience: How important is the precise timing of action potentials for information processing in the neocortex? Is there a canonical computation performed by cortical columns? How is information in the brain processed by the collective dynamics of large neuronal circuits? What level of simplification is suitable for a description of information processing in the brain? What is the neural code?
• How does general anesthetic work?
• Neural computation: What are all the different types of neuron and what do they do in the human brain?^[4]

References

External links

• The Human Brain Project Homepage
• David Eagleman (August 2007). “10 Unsolved Mysteries of the Brain”. Discover Magazine.
• Dennett D (April 2001). “Are we explaining consciousness yet?”. Cognition 79 (1-2): 221–37. doi:10.1016/S0010-0277(00)00130-X. PMID 11164029.

Unsolved Problems in Philosophy

This is a list of some of the major unsolved problems in philosophy. Clearly, unsolved philosophical problems exist in the lay sense (e.g. “What is the meaning of life?“, “Where did we come from?“, “What is reality?“, etc.). However, professional philosophers generally accord serious philosophical problems specific names or questions, which indicate a particular method of attack or line of reasoning. As a result, broad and untenable topics become manageable. It would therefore be beyond the scope of this article to categorize “life” (and similar vague categories) as an unsolved philosophical problem.

Aesthetics

Essentialism

Main article: Essentialism

In art, essentialism is the idea that each medium has its own particular strengths and weaknesses, contingent on its mode of communication. A chase scene, for example, may be appropriate for motion pictures, but poorly realized in poetry, because the essential components of the poetic medium are ill suited to convey the information of a chase scene. This idea may be further refined, and it may be said that the haiku is a poor vehicle for describing a lover’s affection, as opposed to the sonnet. Essentialism is attractive to artists, because it not only delineates the role of art and media, but also prescribes a method for evaluating art (quality correlates to the degree of organic form). However, considerable criticism has been leveled at essentialism, which has been unable to formally define organic form or for that matter, medium. What, after all, is the medium of poetry? If it is language, how is this distinct from the medium of prose fiction? Is the distinction really a distinction in medium or genre? Questions about organic form, its definition, and its role in art remain controversial. Generally, working artists accept some form of the concept of organic form, whereas philosophers have tended to regard it as vague and irrelevant.

Art objects

This problem originally arose from the practice rather than theory of art. Marcel Duchamp, in the 20th century, challenged conventional notions of what “art” is, placing ordinary objects in galleries to prove that the context rather than content of an art piece determines what art is. In music, John Cage followed up on Duchamp’s ideas, asserting that the term “music” applied simply to the sounds heard within a fixed interval of time.

While it is easy to dismiss these assertions, further investigation^[who?] shows that Duchamp and Cage are not so easily disproved. For example, if a pianist plays aChopin etude, but his finger slips missing one note, is it still the Chopin etude or a new piece of music entirely? Most people would agree that it is still a Chopin etude (albeit with a missing note), which brings into play the Sorites paradox, mentioned below. If one accepts that this is not a fundamentally changed work of music, however, is one implicitly agreeing with Cage that it is merely the duration and context of musical performance, rather than the precise content, which determines what music is? Hence, the question is what the criteria for art objects are and whether these criteria are entirely context-dependent.

Epistemology

Epistemological problems are concerned with the nature, scope and limitations of knowledge. Epistemology may also be described as the study of knowledge.

Gettier problem

Plato suggests, in his Theaetetus (210a) and Meno (97a–98b), that “knowledge” may be defined as justified true belief. For over two millennia, this definition of knowledge has been reinforced and accepted by subsequent philosophers. An item of information’s justifiability, truth, and belief have been seen as the necessary and sufficient conditions for knowledge.

In 1963, however, Edmund Gettier published an article in the periodical Analysis entitled “Is Justified True Belief Knowledge?”, offering instances of justified true belief that do not conform to the generally understood meaning of “knowledge.” Gettier’s examples hinged on instances of epistemic luck: cases where a person appears to have sound evidence for a proposition, and that proposition is in fact true, but the apparent evidence is not causally related to the proposition’s truth.

In response to Gettier’s article, numerous philosophers^[who?] have offered modified criteria for “knowledge.” There is no general consensus to adopt any of the modified definitions yet proposed.

Problem of the criterion

Overlooking for a moment the complications posed by Gettier problems, philosophy has essentially continued to operate on the principle that knowledge is justified true belief. The obvious question that this definition entails is how one can know whether one’s justification is sound. One must therefore provide a justification for the justification. That justification itself requires justification, and the questioning continues interminably.

The conclusion is that no one can truly have knowledge of anything, since it is, due to this infinite regression, impossible to satisfy the justification element. In practice, this has caused little concern to philosophers, since the demarcation between a reasonably exhaustive investigation and superfluous investigation is usually clear.

Others argue for forms of coherentist systems, e.g. Susan Haack. Recent work by Peter D. Klein^[1] views knowledge as essentially defeasible. Therefore, an infinite regress is unproblematic, since any known fact may be overthrown on sufficiently in depth investigation.

Molyneux problem

The Molyneux problem dates back to the following question posed by William Molyneux to John Locke in the 17th century: if a man born blind, and able to distinguish by touch between a cube and a globe, were made to see, could he now tell by sight which was the cube and which the globe, before he touched them? The problem raises fundamental issues in epistemology and the philosophy of mind, and was widely discussed after Locke included it in the second edition of hisEssay Concerning Human Understanding.^[2]

A similar problem was also addressed earlier in the 12th century by Ibn Tufail (Abubacer), in his philosophical novel, Hayy ibn Yaqdhan (Philosophus Autodidactus). His version of the problem, however, dealt mainly with colors rather than shapes.^[3][4]

Modern science may now have the tools necessary to test this problem in controlled environments. The resolution of this problem is in some sense provided by the study of human subjects who gain vision after extended congenital blindness. One such subject took approximately a year to recognize most household objects purely by sight.^{[citation needed]} This indicates that this may no longer be an unsolved problem in philosophy.

Münchhausen trilemma

The Münchhausen trilemma, also called Agrippa‘s trilemma, purports that it is impossible to prove any certain truth even in fields such as logic and mathematics. According to this argument, the proof of any theory rests either on circular reasoning, infinite regress, or unproven axioms.

Qualia

See also: Distinguishing blue from green in language

The question hinges on whether color is a product of the mind or an inherent property of objects. While most philosophers will agree that color assignment corresponds to spectra of light frequencies, it is not at all clear whether the particular psychological phenomena of color are imposed on these visual signals by the mind, or whether such qualia are somehow naturally associated with their noumena. Another way to look at this question is to assume two people (“Fred” and “George” for the sake of convenience) see colors differently. That is, when Fred sees the sky, his mind interprets this light signal as blue. He calls the sky “blue.” However, when George sees the sky, his mind assigns green to that light frequency. If Fred were able to step into George’s mind, he would be amazed that George saw green skies. However, George has learned to associate the word “blue” with what his mind sees as green, and so he calls the sky “blue”, because for him the color green has the name “blue.” The question is whether blue must be blue for all people, or whether the perception of that particular color is assigned by the mind.

This extends to all areas of the physical reality, where the outside world we perceive is merely a representation of what is impressed upon the senses. The objects we see are in truth wave-emitting (or reflecting) objects which the brain shows to the conscious self in various forms and colors. Whether the colors and forms experienced perfectly match between person to person, may never be known. That people can communicate accurately shows that the order and proportionality in which experience is interpreted is generally reliable. Thus one’s reality is, at least, compatible to another person’s in terms of structure and ratio.

Ethics

Moral luck

The problem of moral luck is that some people are born into, live within, and experience circumstances that seem to change their moral culpability when all other factors remain the same.

For instance, a case of circumstantial moral luck: a poor person is born into a poor family, and has no other way to feed himself so he steals his food. Another person, born into a very wealthy family, does very little but has ample food and does not need to steal to get it. Should the poor person be more morally blameworthy than the rich person? After all, it is not his fault that he was born into such circumstances, but a matter of “luck”.

A related case is resultant moral luck. For instance, two persons behave in a morally culpable way, such as driving carelessly, but end up producing unequal amounts of harm: one strikes a pedestrian and kills him, while the other does not. That one driver caused a death and the other did not is no part of the drivers’ intentional actions; yet most observers would likely ascribe greater blame to the driver who killed (compare consequentialism and choice).

The fundamental question of moral luck is how our moral responsibility is changed by factors over which we have no control.

Moral knowledge

Are moral facts possible, what do they consist in, and how do we come to know them? Rightness and wrongness seem strange kinds of entities, and different from the usual properties of things in the world, such as wetness, being red, or solidity. Richmond Campbell^[5] has outlined these kinds of issues in his encyclopedia article Moral Epistemology.

In particular, he considers three alternative explanations of moral facts as: theological, (supernatural, the commands of God); non-natural (based on intuitions); or simply natural properties (such as leading to pleasure or to happiness). There are cogent arguments against each of these alternative accounts, he claims, and there has not been any fourth alternative proposed. So the existence of moral knowledge and moral facts remains dubious and in need of further investigation. But moral knowledge supposedly already plays an important part in our everyday thinking, in our legal systems and criminal investigations.

Philosophy of mathematics

Mathematical objects

What are numbers, sets, groups, points, etc.? Are they real objects or are they simply relationships that necessarily exist in all structures? Although many disparate views exist regarding what a mathematical object is, the discussion may be roughly partitioned into two opposing schools of thought: platonism, which asserts that mathematical objects are real, and formalism, which asserts that mathematical objects are merely formal constructions. This dispute may be better understood when considering specific examples, such as the “continuum hypothesis“. The continuum hypothesis has been proven independent of the ZF axioms of set theory, so according to that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question. A platonist, however, would assert that there either does or does not exist a transfinite set with acardinality less than the continuum but greater than any countable set.^{[citation needed]} So, regardless of whether it has been proven unprovable, the platonist would argue that an answer nonetheless does exist.

Metaphysics

Sorites paradox

Otherwise known as the “paradox of the heap”, the question regards how one defines a “thing.” Is a bale of hay still a bale of hay if you remove one straw? If so, is it still a bale of hay if you remove another straw? If you continue this way, you will eventually deplete the entire bale of hay, and the question is: at what point is it no longer a bale of hay? While this may initially seem like a superficial problem, it penetrates to fundamental issues regarding how we define objects. This is similar toTheseus’ paradox and the Continuum fallacy.

Counterfactuals

A counterfactual is a statement that follows this form: “If Joseph Swan had not invented the modern incandescent light bulb, then someone else would have invented it anyway.” People use counterfactuals every day; however, its analysis is not so clear. Swan, after all, did invent the modern incandescent light bulb, so how can the statement be true, if it is impossible to examine its correspondence to reality? (See correspondence theory of truth.) Similar statements have the form, “If you don’t eat your meat, then you can’t have any pudding.” This is another clear if-then statement, which is not verifiable (assuming the addressee did eat his/her meat). Two proposed analyses have resulted from this question. First, some philosophers assert that background information is assumed when stating and interpreting counterfactual conditionals. In the case of the Swan statement, certain trends in the history of technology, the utility of artificial light, and the discovery of electricity may all provide evidence for a logically sound argument. However, other philosophers assert that a modal “possible world” theory offers a more accurate description of counterfactual conditionals. According to this analysis, in the Swan example one would consider the closest possible world to the real world in whichSwan did not create the modern incandescent light bulb. When a counterfactual is used as an argument to justify an illegal act, it is known as the dirty hands argument. For example, “if I didn’t sell him drugs then someone else would have, and those drugs might not have been cut or more harmful.”

Material implication

People have a pretty clear idea what if-then means. In formal logic however, material implication defines if-then, which is not consistent with the common understanding of conditionals. In formal logic, the statement “If today is Saturday, then 1+1=2” is true. However, ‘1+1=2’ is true regardless of the content of the antecedent; a causal or meaningful relation is not required. The statement as a whole must be true, because 1+1=2 cannot be false. (If it could, then on a given Saturday, so could the statement). Formal logic has shown itself extremely useful in formalizing argumentation, philosophical reasoning, and mathematics. The discrepancy between material implication and the general conception of conditionals however is a topic of intense investigation: whether it is an inadequacy in formal logic, an ambiguity of ordinary language, or as championed by H.P. Grice, that no discrepancy exists.

Philosophy of mind

Mind–body problem

The mind–body problem is the problem of determining the relationship between the human body and the human mind. Philosophical positions on this question are generally predicated on either a reduction of one to the other, or a belief in the discrete coexistence of both. This problem is usually exemplified by Descartes, who championed a dualistic picture. The problem therein is to establish how the mind and body communicate in a dualistic framework. Neurobiology and emergencehave further complicated the problem by allowing the material functions of the mind to be a representation of some further aspect emerging from the mechanistic properties of the brain. The brain essentially stops generating conscious thought during deep sleep; the ability to restore such a pattern remains a mystery to science and is a subject of current research (see also neurophilosophy).

Cognition and AI

This problem actually defines a field, however its pursuits are specific and easily stated. Firstly, what are the criteria for intelligence? What are the necessary components for defining consciousness? Secondly, how can an outside observer test for these criteria? The “Turing Test” is often cited as a prototypical test of consciousness, although it is almost universally regarded as insufficient. It involves a conversation between a sentient being and a machine, and if the being can’t tell he is talking to a machine, it is considered intelligent. A well trained machine, however, could theoretically “parrot” its way through the test. This raises the corollary question of whether it is possible to artificially create consciousness (usually in the context of computers or machines), and of how to tell a well trained mimic from a sentient entity.

Important thought in this area includes most notably: John Searle‘s Chinese Room, Hubert Dreyfus‘ non-cognitivist critique, as well as Hilary Putnam‘s work onFunctionalism.

A related field is the ethics of artificial intelligence, which addresses such problems as the existence of moral personhood of AIs, the possibility of moral obligationsto AIs (for instance, the right of a possibly sentient computer system to not be turned off), and the question of making AIs that behave ethically towards humans and others.

Hard problem of consciousness

The hard problem of consciousness is the question of what consciousness is and why we have consciousness as opposed to being philosophical zombies. The adjective “hard” is to contrast with the “easy” consciousness problems, which seek to explain the mechanisms of consciousness (“why” versus “how,” or final causeversus efficient cause). The hard problem of consciousness is questioning whether all beings undergo an experience of consciousness rather than questioning the neurological makeup of beings.

Philosophy of science

Problem of induction

Intuitively, it seems to be the case that we know certain things with absolute, complete, utter, unshakable certainty. For example, if you travel to the Arctic and touch an iceberg, you know that it would feel cold. These things that we know from experience are known through induction. The problem of induction in short; (1) any inductive statement (like the sun will rise tomorrow) can only be deductively shown if one assumes that nature is uniform. (2) the only way to show that nature is uniform is by using induction. Thus induction cannot be justified deductively.

Demarcation problem

‘The problem of demarcation’ is an expression introduced by Karl Popper to refer to ‘the problem of finding a criterion which would enable us to distinguish between the empirical sciences on the one hand, and mathematics and logic as well as “metaphysical” systems on the other’. Popper attributes this problem to Kant. Although Popper mentions mathematics and logic, other writers focus on distinguishing science from metaphysics and pseudo-science.

Some, including Popper, raise the problem because of an intellectual desire to clarify this distinction.^{[citation needed]} Logical positivists had, in addition, a social and intellectual agenda to discredit non-scientific disciplines.^{[citation needed]}

Realism

Does a world independent of human beliefs and representations exist? Is such a world empirically accessible, or would such a world be forever beyond the bounds of human sense and hence unknowable? Can human activity and agency change the objective structure of the world? These questions continue to receive much attention in the philosophy of science. A clear “yes” to the first question is a hallmark of the scientific realism perspective. Philosophers such as Bas van Fraassenhave important and interesting answers to the second question. In addition to the realism vs. empiricism axis of debate, there is a realism vs. social constructivismaxis which heats many academic passions. With respect to the third question, Paul Boghossian‘s “Fear of Knowledge: Against Relativism and Constructivism”. Oxford University Press. 2006. is a powerful critique of social constructivism, for instance. Ian Hacking’s The Social Construction of What? (Harvard UP, 2000) constitutes a more moderate critique of constructivism, which usefully disambiguates confusing polysemy of the term “constructivism.”

See also

• List of years in philosophy
• Thought experiment

References