In analytic philosophy, anti-realism is the position that the truth of a statement rests on its demonstrability through internal logic mechanisms, such as the context principle or intuitionistic logic, in direct opposition to the realist notion that the truth of a statement rests on its correspondence to an external, independent reality. In anti-realism, this external reality is hypothetical and is not assumed.
There are many varieties of anti-realism, such as metaphysical,
mathematical, semantic, scientific, moral and epistemic. The term was
first articulated by British philosopherMichael Dummett in an argument against a form of realism Dummett saw as 'colorless reductionism'.
Anti-realism in its most general sense can be understood as being in contrast to a generic realism,
which holds that distinctive objects of a subject-matter exist and have
properties independent of one's beliefs and conceptual schemes. The ways in which anti-realism rejects these type of claims can vary
dramatically. Because this encompasses statements containing abstract ideal objects (i.e. mathematical objects), anti-realism may apply to a wide range of philosophical topics, from material objects to the theoretical entities of science, mathematical statements, mental states, events and processes, the past and the future.
One kind of metaphysical anti-realism maintains a skepticism
about the physical world, arguing either: 1) that nothing exists
outside the mind, or 2) that we would have no access to a
mind-independent reality, even if it exists. The latter case often takes the form of a denial of the idea that we can have 'unconceptualised' experiences (see Myth of the Given). Conversely, most realists (specifically, indirect realists) hold that perceptions or sense data
are caused by mind-independent objects. But this introduces the
possibility of another kind of skepticism: since our understanding of causality is that the same effect can be produced by multiple causes, there is a lack of determinacy about what one is really perceiving, as in the brain in a vat scenario. The main alternative to this sort of metaphysical anti-realism is metaphysical realism.
On a more abstract level, model-theoretic anti-realist arguments hold that a given set of symbols in a theory
can be mapped onto any number of sets of real-world objects—each set
being a "model" of the theory—provided the relationship between the
objects is the same (compare with symbol grounding.)
In the philosophy of mathematics, realism is the claim that mathematical entities such as 'number' have an observer-independent existence. Empiricism, which associates numbers with concrete physical objects, and Platonism, in which numbers are abstract, non-physical entities, are the preeminent forms of mathematical realism.
The "epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field. Platonism posits that mathematical objects are abstract entities. By general agreement, abstract entities cannot interact causally
with physical entities ("the truth-values of our mathematical
assertions depend on facts involving platonic entities that reside in a
realm outside of space-time"). Whilst our knowledge of physical objects is based on our ability to perceive
them, and therefore to causally interact with them, there is no
parallel account of how mathematicians come to have knowledge of
abstract objects.
Field developed his views into fictionalism. Benacerraf also developed the philosophy of mathematical structuralism,
according to which there are no mathematical objects. Nonetheless, some
versions of structuralism are compatible with some versions of realism.
Counterarguments
Anti-realist arguments hinge on the idea that a satisfactory, naturalistic
account of thought processes can be given for mathematical reasoning.
One line of defense is to maintain that this is false, so that
mathematical reasoning uses some special intuition that involves contact with the Platonic realm, as in the argument given by Sir Roger Penrose.
Another line of defense is to maintain that abstract objects are
relevant to mathematical reasoning in a way that is non causal, and not
analogous to perception. This argument is developed by Jerrold Katz in his 2000 book Realistic Rationalism. In this book, he put forward a position called realistic rationalism, which combines metaphysical realism and rationalism.
A more radical defense is to deny the separation of physical world and the platonic world, i.e. the mathematical universe hypothesis (a variety of mathematicism). In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.
According to intuitionists (anti-realists with respect to mathematical objects), the truth
of a mathematical statement consists in our ability to prove it.
According to Platonic realists, the truth of a statement is proven in
its correspondence to objective reality. Thus, intuitionists are ready to accept a statement of the form "P or Q" as true only if we can prove P or if we can prove Q. In particular, we cannot in general claim that "P or not P" is true (the law of excluded middle), since in some cases we may not be able to prove the statement "P" nor prove the statement "not P". Similarly, intuitionists object to the existence property for classical logic, where one can prove , without being able to produce any term of which holds.
Dummett argues that this notion of truth lies at the bottom of
various classical forms of anti-realism, and uses it to re-interpret phenomenalism, claiming that it need not take the form of reductionism.
In philosophy of science, anti-realism applies chiefly to claims about the non-reality of "unobservable" entities such as electrons or genes, which are not detectable with human senses.
One prominent variety of scientific anti-realism is instrumentalism,
which takes a purely agnostic view towards the existence of
unobservable entities, in which the unobservable entity X serves as an
instrument to aid in the success of theory Y and does not require proof
for the existence or non-existence of X.
Anti-representationalism
Anti-representationalism
rejects the idea that thought and language function by mirroring or
representing an independent reality. Instead, it adopts a deflationary
view of truth and reference, treating them as pragmatic tools within
discourse rather than robust semantic relations. Anti-representationalists like Richard Rorty and Huw Price
argue that all ontological commitments are framework-dependent, denying
any privileged "external" perspective to judge which claims (including
scientific ones) correspond to reality.
Moral anti-realism
In the philosophy of ethics, moral anti-realism (or moral irrealism) is a meta-ethical doctrine that there are no objective moral values or normative facts. It is usually defined in opposition to moral realism,
which holds that there are objective moral values, such that a moral
claim may be either true or false. Specifically the moral anti-realist
is committed to denying at least one of the following three statements:
The Semantic Thesis: Moral statements have meaning, they express propositions, or are the kind of things that can be true or false.
The Alethic Thesis: Some moral propositions are true.
The Metaphysical Thesis: The metaphysical status of moral facts is robust and ordinary, not importantly different from other facts about the world.
Different version of moral anti-realism deny different statements: specifically, non-cognitivism denies the first claim, arguing that moral statements have no meaning or truth content, error theory denies the second claim, arguing that all moral statements are false, and ethical subjectivism denies the third claim, arguing that the truth of moral statements is mind dependent.
There is a debate as to whether moral relativism
is actually an anti-realist position. While many versions deny the
metaphysical thesis, some do not, as one could imagine a system of
morality which requires you to obey the written laws in your country. Such a system would be a version of moral relativism, as different
individuals would be required to follow different laws, but the moral
facts are physical facts about the world, not mental facts, so they are
metaphysically ordinary. Thus, different versions of moral relativism
might be considered anti-realist or realist.
Just as moral anti-realism asserts the nonexistence of normative
facts, epistemic anti-realism asserts the nonexistence of facts in the
domain of epistemology. Thus, the two are now sometimes grouped together as "metanormative anti-realism". Prominent defenders of epistemic anti-realism include Hartry Field, Simon Blackburn, Matthew Chrisman, and Allan Gibbard, among others.
Realism in the arts is generally the attempt to represent subject-matter truthfully, without artificiality, exaggeration, or speculative or supernatural elements. The term is often used interchangeably with naturalism,
although these terms are not necessarily synonymous. Naturalism, as an
idea relating to visual representation in Western art, seeks to depict
objects with the least possible amount of distortion and is tied to the
development of linear perspective and illusionism in Renaissance Europe. Realism, while predicated upon naturalistic representation and a departure from the idealization of earlier academic art, often refers to a specific art historical movement that originated in France in the aftermath of the French Revolution of 1848. With artists like Gustave Courbet capitalizing on the mundane, ugly or sordid, realism was motivated by the renewed interest in the commoner and the rise of leftist politics. The realist painters rejected Romanticism, which had come to dominate French literature and art, with roots in the late 18th century.
In 19th-century Europe, "Naturalism" or the "Naturalist school"
was somewhat artificially erected as a term representing a breakaway
sub-movement of realism, that attempted (not wholly successfully) to
distinguish itself from its parent by its avoidance of politics and
social issues, and liked to proclaim a quasi-scientific basis, playing
on the sense of "naturalist" as a student of natural history, as the biological sciences were then generally known.
When
used as an adjective, "realistic" (usually related to visual
appearance) distinguishes itself from "realist" art that concerns
subject matter. Similarly, the term "illusionistic" might be used when
referring to the accurate rendering of visual appearances in a
composition. In painting, naturalism is the precise, detailed and accurate
representation in art of the appearance of scenes and objects. It is
also called mimesis or illusionism and became especially marked in European painting in the Early Netherlandish painting of Robert Campin, Jan van Eyck and other artists in the 15th century. In the 19th century, Realism art movement painters such as Gustave Courbet
were not especially noted for fully precise and careful depiction of
visual appearances; in Courbet's time that was more often a
characteristic of academic painting, which very often depicted with great skill and care scenes that were contrived and artificial, or imagined historical scenes.
Realism, or naturalism as a style depicting the unidealized version
of the subject, can be used in depicting any type of subject without
commitment to treating the typical or every day. Despite the general
idealism of classical art, this too had classical precedents, which came
in useful when defending such treatments in the Renaissance and Baroque. Demetrius of Alopece was a 4th-century BCE sculptor whose work (all now lost) was said to prefer realism over ideal beauty, and during the Ancient Roman Republic, politicians preferred a truthful depiction in portraits, though the early emperors favored Greek idealism. Goya's portraits of the Spanish royal family represent a sort of honest, unflattering portrayal of important people.
A recurring trend in Christian art was "realism" that emphasized the humanity of religious figures, above all Christ and his physical sufferings in his Passion. Following trends in devotional literature, this developed in the Late Middle Ages,
where some painted wooden sculptures in particular strayed into the
grotesque in portraying Christ covered in wounds and blood, with the
intention of stimulating the viewer to meditate on the suffering that
Christ had undergone on their behalf. These were especially found in
Germany and Central Europe. After abating in the Renaissance, similar
works re-appeared in the Baroque, especially in Spanish sculpture.
Renaissance theorists opened a debate, which was to last several
centuries, as to the correct balance between drawing art from the
observation of nature and from idealized forms, typically those found in
classical models, or the work of other artists generally. Some admitted
the importance of the natural, but many believed it should be idealized
to various degrees to include only the beautiful. Leonardo da Vinci
was one who championed the pure study of nature and wished to depict
the whole range of individual varieties of forms in the human figure and
other things. Leon Battista Alberti was an early idealizer, stressing the typical, with others such as Michelangelo supporting the selection of the most beautiful – he refused to make portraits for that reason.
In the 17th century, the debate continued. In Italy, it usually
centered on the contrast between the relative "classical-idealism" of the Carracci and the "naturalist" style of the Caravaggisti, or followers of Caravaggio,
who painted religious scenes as though set in the back streets of
contemporary Italian cities and used "naturalist" as a self-description.
Bellori,
writing some decades after Caravaggio's early death and no supporter of
his style, refers to "Those who glory in the name of naturalists" (naturalisti).
During the 19th century, naturalism developed as a broadly
defined movement in European art, though it lacked the political
underpinnings that motivated realist artists. The originator of the term
was the French art critic Jules-Antoine Castagnary,
who in 1863 announced that: "The naturalist school declares that art is
the expression of life under all phases and on all levels, and that its
sole aim is to reproduce nature by carrying it to its maximum power and
intensity: it is truth balanced with science". Émile Zola
adopted the term with a similar scientific emphasis for his aims in the
novel. Many Naturalist paintings covered a similar range of subject
matter as that of Impressionism, but using tighter, more traditional brushwork styles.
The term "continued to be used indiscriminately for various kinds
of realism" for several decades, often as a catch-all term for art that
was outside Impressionism and later movements of Modernism and also was not academic art. The later periods of the French Barbizon School and the Düsseldorf School of painting, with its students from many countries, and 20th-century American Regionalism
are movements that are often also described as "naturalist", although
the term is rarely used in British painting. Some recent art historians
claimed either Courbet or the Impressionists for the label.
Lord Leighton's Cimabue's Celebrated Madonna
of 1853–55 is at the end of a long tradition of illusionism in
painting, but is not Realist in the sense of Courbet's work of the same
period.
The development of increasingly accurate representations of the
visual appearances of things has a long history in art. It includes
elements such as the accurate depiction of the anatomy of humans and
animals, the perspective and effects of distance, and the detailed effects of light and color. The art of the Upper Paleolithic in Europe achieved remarkably lifelike depictions of animals. Ancient Egyptian art developed conventions involving both stylization and idealization. Ancient Greek art
is commonly recognized as having made great progress in the
representation of anatomy. No original works on panels or walls by the
great Greek painters survive, but from literary accounts and the
surviving corpus of derivative works (mostly Graeco-Roman works in mosaic), illusionism seems to be highly valued in painting. Pliny the Elder's famous story of birds pecking at grapes painted by Zeuxis in the 5th century BC may well be a legend.
As well as accuracy in shape, light, and color, Roman paintings
show an unscientific but effective knowledge of representing distant
objects smaller than closer ones and representing regular geometric
forms such as the roof and walls of a room with perspective. This
progress in illusionistic effects in no way meant a rejection of
idealism; statues of Greek gods and heroes attempt to represent with
accuracy idealized and beautiful forms, though other works, such as
heads of the famously ugly Socrates, were allowed to fall below these ideal standards of beauty. Roman portraiture, when not under too much Greek influence, shows a greater commitment to a truthful depiction of its subjects, called verism.
The art of Late Antiquity
famously rejected illusionism for expressive force, a change already
well underway by the time Christianity began to affect the art of the
elite. In the West, classical standards of illusionism did not begin to
be reached again until the Late medieval and Early Renaissance
periods and were helped first in the Netherlands in the early 15th
century, and around the 1470s in Italy by the development of new
techniques of oil painting
which allowed very subtle and precise effects of light to be painted
using several layers of paint and glaze. Scientific methods of
representing perspective were developed in Italy in the early 15th
century and gradually spread across Europe, with accuracy in anatomy
rediscovered under the influence of classical art. As in classical
times, idealism remained the norm.
The accurate depiction of landscape in painting
had also been developing in Early Netherlandish/Early Northern
Renaissance and Italian Renaissance painting and was then brought to a
very high level in 17th-century Dutch Golden Age painting,
with very subtle techniques for depicting a range of weather conditions
and degrees of natural light. After being another development of Early
Netherlandish painting, 1600 European portraiture subjects were often
idealized by smoothing features or giving them an artificial pose. Still life
paintings and still life elements in other works played a considerable
role in developing illusionistic painting, though in the Netherlandish
tradition of flower painting they long lacked "realism", in that flowers
from all seasons were typically used, either from the habit of
assembling compositions from individual drawings or as a deliberate
convention; the large displays of bouquets in vases were atypical of 17th-century habits; the flowers were displayed one at a time.
The depiction of ordinary, everyday subjects in art also has a long
history, though it was often squeezed into the edges of compositions or
shown at a smaller scale. This was partly because art was expensive and
usually commissioned for specific religious, political or personal
reasons, which allowed only a relatively small amount of space or effort
to be devoted to such scenes. Drolleries in the margins of medieval illuminated manuscripts
sometimes contain small scenes of everyday life, and the development of
perspective created large background areas in many scenes set outdoors.
Medieval and Early Renaissance art usually showed non-sacred figures in
contemporary dress by convention.
Early Netherlandish painting brought the painting of portraits as low down the social scale as the prosperous merchants of Flanders, and some of these, notably the Arnolfini Portrait by Jan van Eyck (1434) and more often in religious scenes such as the Merode Altarpiece by Robert Campin
and his workshop (circa 1427), include very detailed depictions of
middle-class interiors full of lovingly depicted objects. However, these
objects are at least largely there because they carry layers of complex
significance and symbolism that undercut any commitment to realism for
its own sake. Cycles of the Labours of the Months in late medieval art, of which many examples survive from books of hours,
concentrate on peasants laboring on different tasks through the
seasons, often in a rich landscape background, and were significant both
in developing landscape art and the depiction of everyday working-class people.
In the 16th century, there was a fashion for the depiction in large
paintings of scenes of people working, especially in food markets and
kitchens; in many, the food is given as much prominence as the workers.
Artists included Pieter Aertsen and his nephew Joachim Beuckelaer in the Netherlands, working in an essentially Mannerist style, and in Italy the young Annibale Carracci in the 1580s, using an unpolished style, with Bartolomeo Passerotti somewhere between the two. Pieter Bruegel the Elder pioneered large panoramic scenes of peasant life. Such scenes acted as a prelude for the popularity of scenes of work in genre painting in the 17th century, which appeared all over Europe, with Dutch Golden Age painting sprouting several different subgenres of such scenes, the Bamboccianti (though mostly from the Low Countries) in Italy, and in Spain the genre of bodegones, and the introduction of unidealized peasants into history paintings by Jusepe de Ribera and Velázquez. The Le Nain brothers in France and many Flemish artists including Adriaen Brouwer and David Teniers the Elder and Younger
painted peasants, but rarely townsfolk. In the 18th century, small
paintings of working people remained popular, mostly drawing on the
Dutch tradition and featuring women.
Much art depicting ordinary people, especially in the form of prints,
was comic and moralistic, but the mere poverty of the subjects seems
relatively rarely to have been part of the moral message. From the
mid-19th century onwards, the difficulties of life for the poor were
emphasized. Despite this trend coinciding with large-scale migration
from the countryside to cities in most of Europe, painters still tended
to paint poor rural people. Crowded city street scenes were popular with
the Impressionists and related painters, especially ones showing Paris.
Medieval manuscript illuminators were often asked to illustrate
technology, but after the Renaissance, such images continued in book
illustrations and prints, with the exception of marine painting which largely disappeared in fine art until the early Industrial Revolution, scenes from which were painted by a few painters such as Joseph Wright of Derby and Philip James de Loutherbourg.
Such subjects probably failed to sell very well, and there is a
noticeable absence of industry, other than a few railway scenes, in
painting until the later 19th century, when works began to be
commissioned, typically by industrialists or for institutions in
industrial cities, often on a large scale, and sometimes given a
quasi-heroic treatment.
American realism, a movement of the early 20th century, is one of many modern movements to use realism in this sense.
The Realist movement began in the mid-19th century as a reaction to Romanticism and History painting.
In favor of depictions of 'real' life, the Realist painters used common
laborers, and ordinary people in ordinary surroundings engaged in real
activities as subjects for their works. Its chief exponents were Gustave Courbet, Jean-François Millet, Honoré Daumier and Jean-Baptiste-Camille Corot. According to Ross Finocchio, formerly of the Department of European Paintings at the Metropolitan Museum of Art,
Realists used unprettified detail depicting the existence of ordinary
contemporary life, coinciding with the contemporaneous naturalist
literature of Émile Zola, Honoré de Balzac and Gustave Flaubert.
The French Realist movement had equivalents in all other Western countries, developing somewhat later. In particular the Peredvizhniki or Wanderers group in Russia who formed in the 1860s and organized exhibitions from 1871 included many realists such as Ilya Repin, Vasily Perov and Ivan Shishkin, and had a great influence on Russian art. In Britain, artists such as Hubert von Herkomer and Luke Fildes had great success with realist paintings dealing with social issues.
Broadly defined as "the faithful representation of reality", Realism as a literary movement is based on "objective reality."
It focuses on showing everyday activities and life, primarily among the
middle- or lower-class society, without romantic idealization or
dramatization. According to Kornelije Kvas, "the realistic figuration and
re-figuration of reality form logical constructs that are similar to our
usual notion of reality, without violating the principle of three types
of laws – those of natural sciences, psychological and social ones". It may be regarded as a general attempt to depict subjects as they are
considered to exist in third-person objective reality without
embellishment or interpretation and "in accordance with secular, empirical rules." As such, the approach inherently implies a belief that such reality is ontologically
independent of humankind's conceptual schemes, linguistic practices and
beliefs and thus can be known to the artist, who can in turn represent
this 'reality' faithfully. As Ian Watt
states, modern realism "begins from the position that truth can be
discovered by the individual through the senses" and as such, "it has
its origins in Descartes and Locke, and received its first full formulation by Thomas Reid in the middle of the eighteenth century."
While the preceding Romantic era was also a reaction against the values of the Industrial Revolution,
realism was in its turn a reaction to Romanticism, and for this reason
it is also commonly derogatorily referred as "traditional bourgeois
realism". Some writers of Victorian literature produced works of realism. The rigidities, conventions, and other limitations of "bourgeois realism" prompted in their turn the revolt later labeled as modernism;
starting around 1900, the driving motive of modernist literature was
the criticism of the 19th-century bourgeois social order and world view,
which was countered with an anti-rationalist, anti-realist and
anti-bourgeois program.
A photograph taken during the 1922 performance of 'Uncle Vanya' at the Moscow Art Theatre
Theatrical realism is said to have first emerged in European drama in the 19th century as an offshoot of the Industrial Revolution and the age of science. Some also specifically cited the invention of photography as the basis of the realist theater while others view that the association between realism and drama is far
older as demonstrated by the principles of dramatic forms such as the
presentation of the physical world that closely matches reality.
The achievement of realism in the theatre was to direct attention
to the social and psychological problems of ordinary life. In its
dramas, people emerge as victims of forces larger than themselves, as
individuals confronted with a rapidly accelerating world. These pioneering playwrights present their characters as ordinary,
impotent, and unable to arrive at answers to their predicaments. This
type of art represents what we see with our human eyes. Anton Chekov, for instance, used camera works to reproduce an uninflected slice of life. Scholars such as Thomas Postlewait noted that throughout the nineteenth
and twentieth centuries, there were numerous joining of melodramatic
and realistic forms and functions, which could be demonstrated in the
way melodramatic elements existed in realistic forms and vice versa.
In the United States, realism in drama preceded fictional realism
by about two decades as theater historians identified the first impetus
toward realism during the late 1870s and early 1880s. Its development is also attributed to William Dean Howells and Henry James who served as the spokesmen for realism as well as articulator of its aesthetic principles.
The realistic approach to theater collapsed into nihilism and the absurd after World War II.
Italian Neorealism was a cinematic movement incorporating elements of realism that developed in post-WWII Italy. Notable Neorealists included Vittorio De Sica, Luchino Visconti, and Roberto Rossellini. Realist films generally focus on social issues. There are two types of realism in film: seamless realism and aesthetic
realism. Seamless realism tries to use narrative structures and film
techniques to create a "reality effect" to maintain its authenticity. Aesthetic realism, which was first called for by French filmmakers in the 1930s and promoted by Andre Bazin
in the 1950s, acknowledges that a "film cannot be fixed to mean what it
shows", as there are multiple realisms; as such, these filmmakers use
location shooting, natural light and non-professional actors to ensure
the viewer can make up her/his own choice based on the film, rather than
being manipulated into a "preferred reading". Siegfried Kracauer is also notable for arguing that realism is the most important function of cinema.
Aesthetically realist filmmakers use long shots, deep focus and eye-level 90-degree shots to reduce manipulation of what the viewer sees. Italian neorealism filmmakers from after WWII took the existing realist
film approaches from France and Italy that emerged in the 1960s and
used them to create a politically oriented cinema. French filmmakers
made some politically oriented realist films in the 1960s, such as the cinéma vérité and documentary films of Jean Rouch while in the 1950s and 1960s, British, French and German new waves of filmmaking produced "slice-of-life" films (e.g., kitchen sink dramas in the UK).
Verismo was a post-Romantic operatic tradition associated with Italian composers such as Pietro Mascagni, Ruggero Leoncavallo, Umberto Giordano, Francesco Cilea and Giacomo Puccini. They sought to bring the naturalism of influential late 19th-century writers such as Émile Zola, Gustave Flaubert and Henrik Ibsen into opera. This new style presented true-to-life drama that featured gritty and flawed lower-class protagonists while some described it as a heightened portrayal of a realistic event. Although an account considered Giuseppe Verdi's Luisa Miller and La traviata as the first stirrings of the verismo, some claimed that it began in 1890 with the first performance of Mascagni's Cavalleria rusticana, peaked in the early 1900s. It was followed by Leoncavallo's Pagliacci, which dealt with the themes of infidelity, revenge, and violence.
Verismo also reached Britain where pioneers included the Victorian-era theatrical partnership of the dramatist W. S. Gilbert and the composer Arthur Sullivan (1842–1900). Specifically, their play Iolanthe is considered a realistic representation of the nobility although it included fantastical elements.
Wigner
argues that mathematical concepts have applicability far beyond the
context in which they were originally developed. He writes: "It is
important to point out that the mathematical formulation of the
physicist's often crude experience leads in an uncanny number of cases
to an amazingly accurate description of a large class of phenomena." He adds that the observation "the laws of nature are written in the language of mathematics," properly made by Galileo three hundred years ago, "is now truer than ever before."
Wigner's first example is the law of gravitation formulated by Isaac Newton.
Originally used to model freely falling bodies on the surface of the
Earth, this law was extended based on what Wigner terms "very scanty
observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations." Wigner says that "Newton...
noted that the parabola of the thrown rock's path on the earth and the
circle of the moon's path in the sky are particular cases of the same
mathematical object of an ellipse, and postulated the universal law of
gravitation on the basis of a single, and at that time very approximate,
numerical coincidence."
Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan,
and Heisenberg then proposed to replace by matrices the position and
momentum variables of the equations of classical mechanics. They applied
the rules of matrix mechanics to a few highly idealized problems and
the results were quite satisfactory. However, there was, at that time,
no rational evidence that their matrix mechanics would prove correct
under more realistic conditions." But Wolfgang Pauli found their work accurately described the hydrogen atom: "This application gave results in agreement with experience." The helium atom,
with two electrons, is more complex, but "nevertheless, the calculation
of the lowest energy level of helium, as carried out a few months ago
by Kinoshita
at Cornell and by Bazley at the Bureau of Standards, agrees with the
experimental data within the accuracy of the observations, which is one
part in ten million. Surely in this case we 'got something out' of the
equations that we did not put in." The same is true of the atomic spectra of heavier elements.
Wigner's last example comes from quantum electrodynamics:
"Whereas Newton's theory of gravitation still had obvious connections
with experience, experience entered the formulation of matrix mechanics
only in the refined or sublimated form of Heisenberg's prescriptions.
The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger,
is a purely mathematical theory and the only direct contribution of
experiment was to show the existence of a measurable effect. The
agreement with calculation is better than one part in a thousand."
There are examples beyond the ones mentioned by Wigner. Another often cited example is Maxwell's equations,
derived to model the elementary electrical and magnetic phenomena known
in the mid-19th century. The equations also describe radio waves,
discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death.
Responses
The responses the thesis received include:
Richard Hamming in computer science, "The Unreasonable Effectiveness of Mathematics".
Arthur Lesk in molecular biology, "The Unreasonable Effectiveness of Mathematics in Molecular Biology".
Peter Norvig in artificial intelligence, "The Unreasonable Effectiveness of Data"
Max Tegmark in physics, "The Mathematical Universe".
Ivor Grattan-Guinness
in mathematics, "Solving Wigner's mystery: The reasonable (though
perhaps limited) effectiveness of mathematics in the natural sciences".
Vela Velupillai in economics, "The Unreasonable Ineffectiveness of Mathematics in Economics".
Terrence Joseph Sejnowski in Artificial Intelligence: The Unreasonable Effectiveness of Deep Learning in Artificial Intelligence".
Richard Hamming
Mathematician and Turing Award laureate Richard Hamming reflected on and extended Wigner's Unreasonable Effectiveness in 1980, discussing four "partial explanations" for it, and concluding that they were unsatisfactory. They were:
1. Humans see what they look for. The belief that science
is experimentally grounded is only partially true. Hamming gives four
examples of nontrivial physical phenomena he believes arose from the
mathematical tools employed and not from the intrinsic properties of
physical reality.
Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Hamming imagines Galileo as having engaged in the following thought experiment (the experiment, which Hamming calls "scholastic reasoning", is described in Galileo's book On Motion.):
Suppose that a falling body broke into two
pieces. Of course, the two pieces would immediately slow down to their
appropriate speeds. But suppose further that one piece happened to touch
the other one. Would they now be one piece and both speed up? Suppose I
tie the two pieces together. How tightly must I do it to make them one
piece? A light string? A rope? Glue? When are two pieces one?
There is simply no way a falling body can "answer" such
hypothetical "questions." Hence Galileo would have concluded that
"falling bodies need not know anything if they all fall with the same
velocity, unless interfered with by another force." After coming up with
this argument, Hamming found a related discussion in Pólya (1963:
83-85). Hamming's account does not reveal an awareness of the 20th-century scholarly debate over just what Galileo did.
Hamming argues that Albert Einstein's pioneering work on special relativity
was largely "scholastic" in its approach. He knew from the outset what
the theory should look like (although he only knew this because of the Michelson–Morley experiment),
and explored candidate theories with mathematical tools, not actual
experiments. Hamming alleges that Einstein was so confident that his
relativity theories were correct that the outcomes of observations
designed to test them did not much interest him. If the observations
were inconsistent with his theories, it would be the observations that
were at fault.
2. Humans create and select the mathematics that fit a situation. The mathematics at hand does not always work. For example, when mere scalars proved awkward for understanding forces, first vectors, then tensors, were invented.
3. Mathematics addresses only a part of human experience. Much of human experience does not fall under science or mathematics but under the philosophy of value, including ethics, aesthetics, and political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith.
4. Evolution has primed humans to think mathematically. The earliest lifeforms must have contained the seeds of the human ability to create and follow long chains of close reasoning.
Max Tegmark
Physicist Max Tegmark
argued that the effectiveness of mathematics in describing external
physical reality is because the physical world is an abstract
mathematical structure. This theory, referred to as the mathematical universe hypothesis, mirrors ideas previously advanced by Peter Atkins. However, Tegmark explicitly states that "the true mathematical
structure isomorphic to our world, if it exists, has not yet been
found." Rather, mathematical theories in physics are successful because
they approximate more complex and predictive mathematics. According to
Tegmark, "Our successful theories are not mathematics approximating
physics, but simple mathematics approximating more complex mathematics."
Ivor Grattan-Guinness
Ivor Grattan-Guinness
found the effectiveness in question eminently reasonable and explicable
in terms of concepts such as analogy, generalization, and metaphor. He
emphasizes that Wigner largely ignores "the effectiveness of the natural
sciences in mathematics, in that much mathematics has been motivated by
interpretations in the sciences".
German scholar Moritz Drobisch
was known to have revered the "mathematical fundament" of most
sciences, as it put students in the position to "awe at the teleological
coherence" and "recognise a superhuman, ordering wisdom whose purposes
[...] [they] will gradually understand". Relevantly, he stated of astronomy:
[T]he
harmonious order, in which the celestial bodies describe their orbits,
the eternally consistent regularity, touches a deep sounding string
within us and elevates us – far from just letting the dead mechanism of
chance unwind before us – to the notion of the supreme wise being.
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